HACETTEPE UNIVERSITY

FACULTY OF SCIENCE

TURKEY

HACETTEPE JOURNAL OF

MATHEMATICS AND

STATISTICS

A Bimonthly Publication

Volume 46 Issue 1

2017

Dedicated to the memory ofLawrence Michael Brown

ISSN 1303 5010

HACETTEPE JOURNAL OF

MATHEMATICS AND

STATISTICS

Volume 46 Issue 1

February 2017

Dedicated to the memory ofLawrence Michael Brown

A Peer Reviewed Journal

Published Bimonthly by the

Faculty of Science of Hacettepe University

Abstracted/Indexed in

SCI-EXP, Journal Citation Reports, Mathematical Reviews,

Zentralblatt MATH, Current Index to Statistics,

Statistical Theory & Method Abstracts,

SCOPUS, Tübitak-Ulakbim.

ISSN 1303 5010

This Journal is typeset using LATEX.

Hacettepe Journal of Mathematics and Statistics

Cilt 46 Say 1 (2017)

ISSN 1303 5010

KÜNYE

YAYININ ADI:

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS

YIL : 2017 SAYI : 46 - 1 AY : ubat

YAYIN SAHBNN ADI : H.Ü. Fen Fakültesi Dekanl§ adna

Prof. Dr. A. Nurdan Özer

SORUMLU YAZI L. MD. ADI : Prof. Dr. Yücel Tra³

YAYIN DARE MERKEZ ADRES : H.Ü. Fen Fakültesi Dekanl§

YAYIN DARE MERKEZ TEL. : 0 312 297 68 50

YAYININ TÜRÜ : Yaygn

BASIMCININ ADI : Hacettepe Üniversitesi Hastaneleri Basmevi.

BASIMCININ ADRES : 06100 Shhye, ANKARA.

BASIMCININ TEL. : 0 312 305 1020

BASIM TARH - YER : - ANKARA

Hacettepe Journal of Mathematics and Statistics

A Bimonthly Publication Volume 46 Issue 1 (2017)

ISSN 1303 5010

EDITORIAL BOARD

Honorary Editor:

Lawrence Micheal Brown

Co-Editors in Chief:

Mathematics:

Yücel Tra³ (Hacettepe University - Mathematics - ytiras@hacettepe.edu.tr)

Statistics:

Cem Kadlar (Hacettepe University-Statistics - kadilar@hacettepe.edu.tr)

Associate Editor:

Durdu Karasoy (Hacettepe University-Statistics - durdu@hacettepe.edu.tr)

Managing Editor:

Ramazan Ya³ar (Hacettepe University - Mathematics - ryasar@hacettepe.edu.tr)Furkan Yldrm (Hacettepe University - Actuarial Science - furkany@hacettepe.edu.tr)

Members:

Ali Allahverdi (Operational research statistics, ali.allahverdi@ku.edu.kw)Olcay Arslan (Robust statistics, oarslan@ankara.edu.kw)N. Balakrishnan (Statistics, bala@mcmaster.ca)Gary F. Birkenmeier (Algebra, gfb1127@louisiana.edu)G.C. L. Brümmer (Topology, gcl.brummer@uct.ac.za)Okay Çelebi (Analysis, acelebi@yeditepe.edu.tr)Gülin Ercan (Algebra, ercan@metu.edu.tr)Alexander Goncharov (Analysis, goncha@fen.bilkent.edu.tr)Sat Gupta (Sampling, Time Series, sngupta@uncg.edu)Varga Kalantarov (Appl. Math., vkalantarov@ku.edu.tr)Ralph D. Kopperman (Topology, rdkcc@ccny.cuny.edu)Vladimir Levchuk (Algebra, levchuk@lan.krasu.ru)Cihan Orhan (Analysis, Cihan.Orhan@science.ankara.edu.tr)Abdullah Özbekler (App. Math., aozbekler@gmail.com)Ivan Reilly (Topology, i.reilly@auckland.ac.nz)Bülent Saraç (Algebra, bsarac@hacettepe.edu.tr )Patrick F. Smith (Algebra, pfs@maths.gla.ac.uk )Alexander P. ostak (Analysis, sostaks@latnet.lv)Derya Keskin Tütüncü (Algebra, derya.tutuncu@outlook.com)A§ack Zafer (Appl. Math., zafer@metu.edu.tr)

Published by Hacettepe University

Faculty of Science

This issue of the journal "Hacettepe Journal of Mathematics and Statistics" isdedicated to the memory of Lawrence Michael Brown. The issue contains 13Research Papers which were selected to the usual high standarts of the journal.We would like to thank the authors for their submissions.

Dr. Yücel Tra³, Dr. Rza Ertürk, Dr. Selma Özça§, Dr. enol Dost, Dr. FilizYldz and Dr. . U§ur Tiryaki.

LAWRENCE MICHAEL BROWN's LIFE STORY

Lawrence Michael Brown was born in 1938 inChestereld-England. He gratuated from Hull Univer-sity in 1960 and moved to Turkey in 1968 with his wifefrom Cyprus. From 1968 until his retirement in 2012 heworked in the Mathematics Department of HacettepeUniversity in Ankara.

Brown defended his PhD thesis in 1981 at Univer-sity of Glasgow. His Ph.D. thesis was entitled Dualcovering theory, conuence structures and the latticeof bicontinuous functions ". His main scientic inter-ests were logic, general topology, fuzzy topology andbitopological spaces. L.M. Brown supervised 12 MSc.students and 14 Ph.D. students. Brown published 45

papers in Scientic journals and 4 books which of them is a dictionary of mathe-matical terms. Additionally; as an editor he has a great deal of trouble and hasprovided signicant contributions to the improvement the international journal of"Hacettepe Journal of Mathematics and Statistics".

L. M. Brown passed his last 25 years of his academic life to the creation andimprovement of a theory called Texture. Much of Brown's and his students workswas devoted to the study of texture spaces. Below, we introduce this theory andrecall some of his results.

The notion of a texture space under the name fuzzy structure was introducedby L. M. Brown at the conference on Fuzzy Systems and Articial Intelligenceheld in Trabzon, 1992. Textures rst arose in connection with the representationof Hutton algebras and lattices of L-fuzzy sets in a point-based setting and havesubsequently proved to be a fruitful setting for the investigation of complement-free concepts in mathematics. We recall that a texturing on a set S is a subsetS of P(S), that is, a point-separating, complete, completely distributive latticecontaining S and ∅, and for which arbitrary meet

∧coincides with intersection⋂

and nite joins∨

with unions⋃. The pair (S, S) is called a texture. L. M.

Brown and R. Ertürk obtained one-to-one correspondence between fuzzy latticesand complemented simple texture, and then showed that all fuzzy textures aresimple but the textures are strictly more general than fuzzy lattices.

In classical topology the notion of open set is usually taken as primitive with thatof closed set being auxiliary. However, since the closed sets are easily obtained asthe complements of open sets, they often play an important, sometimes dominatingrole in topological arguments. A similar situation holds for topologies on latticeswhere the role of set complement is played by an order reversing involution. Itis the case, however, that there may be an order reversing involution available,or that the presence of such an involution is otherwise irrelevant to the topicunder consideration. To deal with such cases it is natural to consider a topologicalstructure considering of a priory unrelated families of open sets and of closedsets. This was the approach adapted from the beginning for topological structurescalled fuzzy structures originally introduced as a point-based representation for

fuzzy sets. Then these topological structures were called as dichotomous topology,or ditopology for short. They consist of a family τ of open sets and a generallyunrelated family κ of closed sets. Hence, both the open and the closed sets areregarded as primitive concepts for a ditopology and the open and the closed setshave the same role in the ditopology as a topological structure. A dichotomoustopology, or ditopology for short, on a texture (S, S) is a pair (τ, κ) of subsets of S,where the set of open sets τ satises: (1) S, ∅ ∈ τ , (2) G1, G2 ∈ τ ⇒ G1 ∩G2 ∈ τand (3) Gi ∈ τ , i ∈ I ⇒ ∨

iGi ∈ τ , and the set of closed sets κ satises (1′)S, ∅ ∈ κ, (2′) K1, K2 ∈ κ ⇒ K1 ∪ K2 ∈ κ and (3′) Ki ∈ κ, i ∈ I ⇒

⋂Ki ∈ κ.

Topological and bitopological spaces may be regarded as a ditopological texturespaces on the discrete texture (X,P(X)), that is, the ditopological structure ismore general than the topological, fuzzy topological and bitopological structures.

Together with M. Diker he showed that the lattice of intuitionistic set on a setXmay be regarded as a texture space. They also introduced the notions of dicovers,paracompactness, full normality and connectedness. Then M. Diker has researchedthe interrelations between rough sets and texture spaces and obtained importantresults. Besides, M. Diker and A. Altay U§ur generalized one-point compacti-cations and Wallman-type compactications to ditopological texture spaces andinvestigated the connections between them.

Furtheremore, L. M. Brown and A. Irkad have introduced the notions sequen-tially dinormal ditopological texture spaces and dimetrizability and then intro-ducing the binary direlations on a texture, studied the space of bicontinuous realdifunctions on a ditopological texture space.

In order to dene categories whose objects are textures it is necessary to describethe morphisms between textures. For this purpose, the theory of direlations anddifunctions was developed by L. M. Brown. The concept of difunction betweentextures, which in turn is derived from a notion of direlation in much the sameway that classical (point) functions are derived from binary relations in the usualsense. This has provided the stimulus for the most of studies. Difunctions aremainly morphisms of texture spaces theory. The categories of texture spaces anddifunctions was investigated in . Dost's Ph.D. thesis. Together with R. Ertürkand . Dost, L. M. Brown developed the fundamental aspects of the theory ofditopological texture spaces in a categorical setting and presented important linkswith the theory of L− topological spaces. Furthermore, this team dened andstudied basic separation axioms in general ditopological texture spaces, relatingthese to known separation axioms for bitopological and topological spaces.

Despite the close links with fuzzy sets and topologies, the development of thetheory of ditopological texture spaces has proceed largely independently. In thisdirection, L. M. Brown and S. Özça§ laid the foundations of a theory of uniformi-ties on textures giving descriptions in terms of direlations, dicovers and dimetrics.They also investigated the relation between quasi uniformities and di-uniformities.Direlational uniformities have a base of symmetric direlations, and it might there-fore be conjectured that they would correspond to uniformities in the classicalsense. However, by restricting attention to the discrete complemented texture(X,X, πX) on a set X it is shown that in fact direlational uniformities correspond

in a one-to-one way with quasi-uniformities on X. Moreover, a direlational unifor-mity on (X,X, πX) corresponds to a uniformity if and only if it is complemented.Together with S. Özça§ and B. Krsteska, he obtained a characterization of di-uniformities on a texture (S, S) in terms of functions. In that work they obtainedthat this characterization enables quasi uniformities in the sense of B. Hutton tobe regarded as di-uniformities on the corresponding Hutton texture, which revealsdi-uniformities as a generalization of Hutton quasi-uniformities.

L. M. Brown and F. Yldz explored a new class of textures under the namenearly-plain texture", as a major subclass of textures. By virtue of this subclass,they constructed a natural counterpart of the classical realcompactness in ditopo-logical textural theory, called real dicompactness, via the suitable notion of dilteras an analogue of the classical lter. All the related information about the nearlyplain textures and the theory of real dicompact spaces have been published in F.Yldz 's Ph.D Thesis in a categorical setting, besides in related papers publishedlater. In addition to that, they brought a dierent dimension to their works aboutreal dicompactness by characterizing all the real difunctions and constructing var-ious type of categories, as well as functors, via the point functions satisfying acompatibility condition, namely w-preserving, between textures. Following that,by describing another subclass of textures, namely almost-plain texture they ob-tained so many signicant results about the real dicompactications, and a specialtype of dicompactication. Also, in view of the some relationships with the uni-form structures in texture theory, they introduced an isomorphism between thecategory of separated quasi-uniform spaces and uniformly continuous functions,and the category of separated di-uniform plain texture spaces and uniformly bi-continuous w-preserving point functions. As an important result of these, theydescribed the notion extended real dicompactness for almost-plain textures, andby applying this theory to Hutton spaces, they brought in a new approach to thetopology literature for the suitable notion of realcompactness in fuzzy topologicalspaces, in the sense of textural structures.

L. M. Brown with .Tiryaki obtained several important results in the plaintextures. First of all, they developed relations and corelations between the latticeof fuzzy subsets of a crisp set X and that of a crisp set Y based on the theoryof relations and corelations between textures, and they showed that these notionsgeneralize in a natural way the important concept of fuzzy relation from X toY . In addition, difunctions are also characterized and their relationship withknown mappings between fuzzy sets is investigated by them. After that theyintroduce new Fuzzy Sets over poset I = [0, 1] called Soft Fuzzy Sets". Thesenew type of Fuzzy sets have a richer mathematical theory than classical I-fuzzysets. In particular soft fuzzy points behave like the points of crisp set theory withrespect to join, and moreover there exists a Lowen type functor from Top to theconstruct SF-Top that preserves both separation and compactness, and nallythey used to the characterization of plain textures in terms of posets given byMustafa Demirci (M. Demirci, Textures and C-spaces, Fuzzy Sets and Systems 158(11) (2007) (12371245) to obtain new results relating to plain textures and plainditopological texture spaces. In the last paper they showed: (1) The construct ofplain textures and ω-preserving mappings is a full, isomorphism-closed concretely

reective subconstruct of the construct of textures and ω preserving mappings.(2) Every complementation on a plain texture is grounded in the sense of S. Özça§and L. M. Brown. (3) There exists an isomorphism between the construct ofweakly pairwise T0 bitopological spaces and pairwise continuous functions and theconstruct of plain T0 ditopological texture spaces and bicontinuous ω preservingmappings. Just before he passed away, L. M. Brown and . Tiryaki started newjoint work to consider Hyperspace in the setting of ditopological texture space.

As a continuation of what L. M. Brown and S. Özça§ did, proximity spacesknown in classical structure were generalized to texture spaces as diextremityspaces by R. Ertürk and G. Yldz and the connections between them were inves-tigated.

Besides these, together with A. ostak, they published a paper titled Categoriesof Fuzzy Topology in the Context of Graded Ditopologies On Textures". Followingthat, R. Ertürk and R. Ekmekçi generalized the notions dineighbourhood, dilterand diuniformity dened by L. M. Brown and colleagues in ditopological texturespaces to the graded ditopological spaces dened by A. ostak and L. M. Brown.Also, they examined the connections between them in detail.

Additionally, after his retirement, L. M. Brown gave a talk for 45 minutes aboutthe new subject hyperspaces in textures", in the International Workshop titledWorkshop on Applications of Topology in Mathematics and Computer Science-ATMC 2015" held in Hacettepe University, in September 2015, as an invitedspeaker via the invitation of F. Yldz as the organiser of ATMC 2015, as wellas the other members of Topology Group in Hacettepe University, gave severaltalks about texture theory. Unfortunately, it was his last visiting to HacettepeUniversity and Ankara, before his death.

Also, under the leadership of R. Ertürk, Topology Group of Hacettepe Univer-sity will organise a Series of Workshops in Honor of L. M. Brown in Decemberwhich is the month of death, for each year, on the new developments in texturetheory, as the theory of him. First one of this Series of Workshops has alreadybeen done in December 2016, one year after the date of death, that is December2, 2015.

He carved out a niche for himself in the eld of topology theory with his eorts,scientic activities and studies mentioned above. He will always be rememberedand will always live in inspirational works created by him.

With all our respects,Dr. Rza Ertürk, Dr. Selma Özça§, Dr. enol Dost, Dr. Filiz Yldz and Dr. .U§ur Tiryaki.

CONTENTS

Arhangel'skii A.V.

Remainders of locally ech-complete spaces and homogeneity . . . . . . . . . . . . . . . . . . 1

Ralph Kopperman and Homeira Pajoohesh

Generalizations of metrics and partial metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Alexander ostak and Aleksandrs El,kins

LM -valued equalities, LM -rough approximation operatorsand ML-graded ditopologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Natasha Demetriou and Hans-Peter A. Künzi

A study on quasi-pseudometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

Ljubi²a D.R. Ko£inac, Amani Sabah, Moiz ud Din Khan andDjamila Seba

Semi-Hurewicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

Carmen Alegre, Hacer Da§, Salvador Romaguera andPedro Tirado

Characterizations of quasi-metric completeness in terms ofKannan-type xed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Mitrofan M. Choban and Radu N. Dumbr veanu

Functional equivalence of topological spaces and topological modules . . . . . . . . . 77

Josef lapal and John L. Pfaltz

Closure operators associated with networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Taras Banakh and Bogdan Bokalo

Weakly discontinuous and resolvable functions between topological spaces . . . . 103

D. N. Georgiou, A. C.Megaritis and F. Sereti

A study of the quasi covering dimension for nite spaces through the matrixtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Sang-Eon Han

U(k)- and L(k)-homotopic properties of digitizations ofnD Hausdor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127

Mohammad H.M. Rashid and Ljubi²a D.R. Ko£inac

Ideal convergence in 2-fuzzy 2-normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Ivan Reilly and Bill Barton

A man of words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 1 8

Remainders of locally ech-complete spaces andhomogeneity

Arhangel'skii A.V.∗

Abstract

We study remainders of locally ech-complete spaces. In particular, itis established that if X is a locally ech-complete non-ech-completespace, then no remainder of X is homogeneous (Theorem 3.1). We alsoshow that if Y is a remainder of a locally ech-complete space X, andevery y ∈ Y is a Gδ-point in Y , then the cardinality of Y doesn't exceed2ω. Several other results are obtained.

Keywords: Remainder, Compactication, Gδ-point, Homogeneous,Point-countable base, Lindelöf Σ-space, Charming space, Countable type, ech-complete.

2000 AMS Classication: Primary: 54A25; Secondary: 54B05

1. Introduction

All spaces considered in this article are assumed to be Tychono. SymbolsX,Y, Z always stand for topological spaces. In terminology and notation we follow[7]. We say that a space X has a topological property P locally, if for each x ∈ Xthere exists an open neighbourhood V of x such that the closure of V in X hasthe property P.

A compactication of a space X is any compact space bX such that X is a densesubspace of bX. A remainder Y of a space X is the subspace Y = bX \ X of acompactication bX of X.

A space X is of countable type if every compact subspace P of X is contained ina compact subspace F ⊂ X with a countable base of open neighbourhoods in X[1]. Metrizable spaces, locally compact spaces, ech-complete spaces, and Moorespaces are of countable type [1]. A remarkable classical result in the theory ofcompactications is the following theorem of M. Henriksen and J. Isbell [8]:

∗Email: arhangel.alex@gmail.com

Doi : 10.15672/HJMS.2016.398

2

1.1. Theorem. A space X is of countable type if and only if the remainder of Xin any (in some) compactication of X is Lindelöf.

A Lindelöf p-space is a preimage of a separable metrizable space under a perfectmapping [1]. K. Nagami has dened the class of Σ-spaces [9]. They can becharacterized as continuous images of Lindelöf p-spaces.

It is proved below that ifX is a locally ech-complete space with a homogeneousremainder Y , then X is ech-complete (Theorem 3.1). We show that a remainderof a locally ech-complete space needn't have a dense σ-compact subspace andneedn't be a Lindelöf Σ-space (Example 2.4). We also give a characterizationof remainders of locally ech-complete spaces (Theorem 4.1) and obtain somecorollaries from it.

2. Two examples

A space X is ech-complete if it is a Gδ-subspace of some (of any) of its com-pactications. One of the simplest duality theorems involving remainders is thenext statement: a space X is ech-complete if and only if some (every) remainderof X is σ-compact. Locally compact spaces constitute an important subclass ofthe class of ech-complete spaces. In this case, we have the classical theorem ofP.S. Alexandro: a space X is locally compact non-compact if and only if someremainder of X consists of exactly one point.

However, the next natural question remained unanswered: how to characterizein intrinsic terms the remainders of locally ech-complete spaces? We answer thisquestion in this article.

2.1. Theorem. Every remainder of any locally ech-complete spaceX is Lindelöf.

Proof. By Henriksen-Isbell Theorem, it is enough to show that X is a space ofcountable type. Since every ech-complete space is a space of countable type,we see that X is locally of countable type. It remains to use the next easy toverify assertion: if a space X is locally of countable type, then X is of countabletype.

How close are remainders of locally ech-complete spaces to remainders of ech-complete spaces? This question, Theorem 2.1, - and the obvious fact that everyremainder of a ech-complete space is σ-compact, - motivate the next question:is every remainder of any locally ech-complete space a Lindelöf Σ-space? Theanswer is in the negative.

2.2. Example. Let B be the usual space of ordinal numbers not exceeding therst uncountable ordinal ω1, and Z be the subspace of B consisting of all non-isolated points of B. Furthermore, let Y0 be the subspace of Z consisting of allisolated points of Z. Finally, put p = ω1, Y = Y0 ∪ p, and X = B \ Y .

Clearly, B is a compactication bX of X, and Y is the remainder bX \ X ofX in bX. It is also easy to see that all points of Y0 are isolated in Y and p is anon-isolated P -point in Y . Observe that every open neighbourhood of p containsall but countably many points of the set Y . Hence, Y is a Lindelöf P -space, andthe space X is locally ech-complete. It follows that every compact subspace ofY is nite, and that Y is not a Lindelöf Σ-space. Therefore, Y does not have a

3

dense σ-compact subspace and hence, X is not ech-complete. In particular, it isnot true that every remainder of any locally ech-complete space has a dense σ-compact subspace. It is also easy to see that the space Y is not locally σ-compact.In this connection, see the last section.

2.3. Corollary. There exists a locally ech-complete space X with a remainderY such that no dense subspace of Y is a Lindelöf Σ-space.

Proof. Let us take X and Y constructed in the preceding example. We also usethe notation introduced there. We have seen in Example 2.4 that X is locallyech-complete. Assume that Y1 is an arbitrary dense subspace of Y . Then Y1contains all isolated points of Y , and hence, either Y1 = Y \ p, or Y1 = Y . Inthe rst case, Y1 is discrete and uncountable, and therefore, is not Lindelöf. Inthe second case, Y1 is not a Lindelöf Σ-space, since it has been shown above thatY is not a Lindelöf Σ-space.

Alexandro's Theorem on remainders of locally compact spaces leads to thenext question: is it true that every ech-complete space X has a remainder Ysuch that |Y | ≤ 2ω? The answer is "no".

2.4. Example. a) Let X be any nowhere locally compact space metrizable by acomplete metric and satisfying the condition: the weight of X is greater than 2ω.To construct such a space, we can x any cardinal number τ such that τ > 2ω

and take X to be the countable power of a discrete space of the cardinality τ . LetbX be any compactication of X. Then the remainder Y of X in bX is dense inbX, since the spacee X is nowhere locally compact. Since the Souslin number ofX is greater than 2ω, it follows that |Y | > 2ω. Notice that the space X is, clearly,ech-complete. Its additional nice feature is that it is metrizable. On the otherhand, the Souslin number of X is quite large, and we have made a good use ofthis fact in our argument above. The next example serves the same purpose butthe Souslin number of the space constructed in it is countable.

b) Let G be the countable power of the usual space R of real numbers, and Bbe a compact topological group such that w(B) > 2c, where c = 2ω. Now dene Xas the topological product G×B. Clearly, X is a ech-complete nowhere locallycompact topological group, and the Soulin number of X is countable. However,X is not metrizable. Let bX be any compactication of X. Then the remainderY of X in bX is dense in bX, since the spacee X is nowhere locally compact.Assume that |Y | ≤ c = 2ω. Then w(bX) ≤ 2c, since Y is dense in bX. Therefore,w(bX) ≤ 2c, a contradiction. It follows that |Y | > 2ω.

3. Homogeneous remainders

Recall that a space X is said to be homogeneous if for any x, y ∈ X there existsa homeomorphism h of X onto X such that h(x) = y. In fact, we will use below amuch weaker form of homogeneity. A space X will be called meekly homogeneousif for any x, y ∈ X and any open neighbourhood Ox of x there exists an openneighbourhood Oy of y such that Oy is homeomorphic to some open subspace ofOx.

The remainder Y of the locally ech-complete space X constructed in Example2.4 is easily seen to be non-homogeneous. A natural question arises: can we

4

construct a similar example, in which Y is, in addition, homogeneous, or at least,meekly homogeneous? Somewhat unexpectedly, the answer turns out to be in thenegative.

3.1. Theorem. Suppose that X is a locally ech-complete space with a meeklyhomogeneous remainder Y . Then X is ech-complete, and Y is σ-compact.

This statement will be derived from the next slightly more general statement:

3.2. Proposition. Suppose that X is a space with an open ech-complete non-locally compact subspace U . Then any remainder Y of X has a closed σ-compactsubspace P such that the interior of P in Y is non-empty.

Proof. Fix x ∈ U such that U is not locally compact at x. Clearly, we may assumethat the closure of U in X is ech-complete, - otherwise we can replace U witha non-empty open subset W of U such that x ∈ W and the closure of W in X iscontained in U . Let Y = bX \X. We denote by F the closure of U in bX. Clearly,x is in the interior of F in bX, and x is in the closure of Y . Therefore, the interiorof the set P = F ∩ Y in Y is non-empty as well.

Obviously, P is the remainder of Z in F , where Z is the closure of U in X. Itfollows that the subspace P of Y is σ-compact, since Z is ech-complete.

3.1. Lemma. Suppose that Y is a meekly homogeneous space with a closed σ-compact subspace P such that the interior of P in Y is non-empty. Then Y islocally σ-compact.

Proof. Fix x in the interior of P , and consider the interior of P as a neighbourhoodOx of x. Now take an arbitrary y ∈ Y . Since Y is meekly homogeneous, we cannd an open neighbourhood Oy of y and an open subset V of Ox such that Oyis homeomorphic to V . Since P is σ-compact, and V is an open subspace of P ,the space V is locally σ-compact. Hence, the space Oy is also locally σ-compact.Since Y is regular and Oy is open in Y , it follows that Y is locally σ-compact aty.

Proof. (of Theorem 3.1). If X is locally compact, then we have nothing to prove.If X is not locally compact, then there exists an open ech-complete non-locallycompact subspace U of X. By Proposition 3.2, there exists a closed σ-compactsubspace P of Y such that the interior V of P in Y is non-empty. Since Y is meeklyhomogeneous, it follows from Lemma 3.1 that Y is locally σ-compact. Therefore,there exists an open covering γ of the space Y such that the closure of any memberof γ in Y is σ-compact.

Notice that by Theorem 3.1 the space Y is Lindelöf. Therefore, γ has a count-able subcovering η of Y . Since the closure in Y of each member of η is σ-compact,we conclude that Y is σ-compact. It follows that X is ech-complete.

4. A characterization of remainders of locally ech-complete spaces

It is not true that a space is locally ech-complete if and only if its remaindersare locally σ-compact. We have seen this in Example 2.4. In this section, we willcharacterize locally ech-complete spaces by a somewhat unusual, but still natural

5

and easy to use property of its remainders. Some corollaries are derived from thischaracterization.

First, let us introduce a piece of terminology. Suppose that Y is a space and Fis a subspace of Y . We will say that Y has a topological property P outside of Fif every closed subspace Z of Y such that Z ∩ F = ∅ has P.

4.1. Theorem. A space X is locally ech-complete if and only if every (some)remainder Y of X is σ-compact outside of some compact subspace F of Y .

Proof. Necessity. Suppose that X is locally ech-complete. Take any compacti-cation bX of X, and let Y = bX \X. Fix also an open covering γ of X such thatthe closure X(V ) of every V ∈ γ in X is ech-complete. For V ∈ γ, we denote byU(V ) an open subset of bX such that U(V ) ∩X = V . Put E = ∪U(V ) : V ∈ γand F = bX \ E. Clearly, E is open in bX, F is compact, X ⊂ E, and F ⊂ Y .

Let A be a closed subset of Y such that A ∩ F = ∅, and B be the closure of Ain bX. Clearly, F ∩ B = ∅. Therefore, B ⊂ E = ∪U(V ) : V ∈ γ. Since B iscompact, and each U(V ) is open in bX, there exists a nite collection V1, ..., Vk ofmembers of γ such that B ⊂ ∪U(Vi) : i = 1, ..., k.

Put Xi = X(Vi) and let Bi be the closure of Vi in bX. By the denitions above,the space Xi is ech-complete. Hence, the subspace Pi = Bi \ Xi is σ-compact.Therefore, the subpace P = ∪Pi : i = 1, ..., k is also σ-compact. Clearly,A ∩Xi = ∅ for i = 1, ..., k, since A ⊂ Y . Since A ⊂ B ⊂ ∪Xi : i = 1, ..., k, itfollows that A ⊂ P .

The set Xi is closed in X, and bX = X ∪Y . Therefore, Pi ⊂ Y , so that P ⊂ Y .Now we can conclude that A is a closed subspace of P . Finally, it follows that Ais σ-compact, since P is σ-compact.

Suciency. Suppose that some remainder Y of X is σ-compact outside of somecompact subspace F of Y . Fix a compactication bX of X such that Y = bX \X.Take any x ∈ X. The set F is closed in bX, since F is compact. We also have:x /∈ F . Hence, we can nd an open neighbourhood U of x in bX such that theclosure of U in bX doesn't intersect F . Since Y is σ-compact outside of F , it followsthat the closed subspace P = Y ∩ U of Y is σ-compact. Since U is compact, itfollows that the subspace X ∩ U is ech-complete. Note that U ∩ X is an opensubspace of X ∩ U . Therefore, the set Ox = U ∩ X is an open ech-completeneighbourhood of x in X. Thus, the space X is locally ech-complete.

We present now a few applications of the last theorem. The concept of acharming space has been introduced in [2]. A space Y is charming if there exists asubspace Z of Y such that Z is a Lindelöf Σ-space and Y \U is a Lindelöf Σ-space,for every open neighbourhood U of Z in Y .

The next statement immediately follows from Theorem 4.1.

4.2. Corollary. Every remainder of a locally ech-complete space is a charmingspace.

According to Theorem 2.1, every remainder of a locally ech-complete spaceis Lindelöf. It is still unknown whether there exists in ZFC a Lindelöf space Ysuch that every y ∈ Y is a Gδ-point in Y and |Y | > 2ω. Let us show that it isimpossible to nd a space of this kind among remainders of locally ech-completespaces.

6

4.3. Theorem. Suppose that Y is a remainder of a locally ech-complete spaceX such that every y ∈ Y is a Gδ-point in Y . Then |Y | ≤ 2ω.

Proof. By Corollary 4.2, Y is a charming space. Since the cardinality of everycharming space of countable pseudocharacter does not exceed 2ω [Theorem 3.7 in[2] ], it follows that |Y | ≤ 2ω.

The following easy to prove statement tells us that non-trivial locally ech-complete spaces are never remainder-wise dual to locally σ-compact spaces.

4.4. Theorem. A space X is ech-complete if and only if X is locally ech-complete and every (some) remainder Y of X is locally σ-compact.

Proof. The necessity is clear.Suciency. The space Y is Lindelöf, since X is locally ech-complete. Since

Y is also locally σ-compact, it follows that Y is σ-compact. Hence, X is ech-complete. 4.5. Theorem. Suppose thatX is a locally ech-complete space with a remainderY in a compactication bX. Furthermore, suppose that Y has a point-countablebase. Then Y is separable, metrizable, and σ-compact, and X is ech-complete.

Proof. By Theorem 2.1, Y is Lindelöf. Theorem 4.1 implies that Y is σ-compactoutside of some compact subspace F of Y which we now x. Clearly, F is separablemetrizable, by the well-known Theorem of A.S. Mischenko [7]. Let us also x apoint-countable base B for Y . Since F is separable, the family η of members V ofB such that V ∩ F 6= ∅ is countable. Therefore, F has a countable base for openneighbourhoods in Y . Hence F is a Gδ-set in Y . Since Y is Lindelöf, it followsthat Y \F is Lindelöf. Since Y \F is, obviously, locally σ-compact, it follows thatY \F and Y are σ-compact. Hence, X is ech-complete. Using again Mischenko'sTheorem, we conclude that Y \F is separable. Therefore, the family ξ of membersV of B such that V ∩ (Y \ F ) 6= ∅ is countable. Since, clearly, B = ξ ∪ η, weconclude that the base B is countable. Hence, Y is separable metrizable.

In connection with the last theorem, recall that, under the Continuum Hypoth-esis CH, there exists a non-metrizable Lindelöf space with a point-countable base[6].

4.6. Theorem. Suppose thatX is a locally ech-complete space with a remainderY in a compactication bX. Furthermore, suppose that Y is symmetrizable. ThenY is σ-compact, has a countable network, and is submetrizable, and X is ech-complete.

Proof. By Theorem 2.1, Y is Lindelöf. Now it follows from a theorem of S.J.Nedev in [10] that Y is hereditarily Lindelöf.

By Theorem 4.1, Y is σ-compact outside of some compact subspace F of Y .Clearly, Y \ F is Lindelöf. Since Y \ F is, obviously, locally σ-compact, it followsthat Y \ F is σ-compact. Hence, Y is σ-compact as well. Therefore, X is ech-complete.

Clearly, every compact subspace of a symmetrizable space is symmetrizable. Itis well-known that each symmetrizable compact space is metrizable and hence, hasa countable base. Now we can conclude that Y has a countable network.

7

It is not dicult to see that the remainder Y , under the assumptions in thelast statement, needn't have a countable base, and that the space X needn't bemetrizable or paracompact.

4.7. Problem. Suppose that X is a locally ech-complete space with a remainderY in a compactication bX such that every y ∈ Y is a Gδ-point in Y . Does itfollow that X is ech-complete?

5. Remainders of locally σ-compact spaces

5.1. Theorem. If a space X is locally σ-compact, then for every remainder Y ofX there exists a compact subspace F of Y such that Y is ech-complete outsideof F .

Proof. Take any compactication bX of X, and put Y = bX \X. Fix also an opencovering γ of X such that the closure X(V ) in X of any V ∈ γ is σ-compact. ForV ∈ γ, we denote by U(V ) an open subset of bX such that U(V ) ∩X = V . PutE = ∪U(V ) : V ∈ γ and F = bX \ E. Clearly, E is open in bX, F is compact,X ⊂ E, and F ⊂ Y .

Let A be a closed subset of Y such that A ∩ F = ∅, and B be the closure of Ain bX. Clearly, F ∩ B = ∅. Therefore, B ⊂ E = ∪U(V ) : V ∈ γ. Since B iscompact, and each U(V ) is open in bX, there exists a nite collection V1, ..., Vk ofmembers of γ such that B ⊂ ∪U(Vi) : i = 1, ..., k.

Put Xi = X(Vi), and let Hi be the closure of Vi in bX. Clearly, Xi is σ-compactand closed in X. Hence, the subspace P = ∪Xi : i = 1, ..., k is also closed in Xand σ-compact. The set P is dense in the closure H of P in bX, and H ∩Y is theremainder of P in H. It follows that H ∩ Y is ech-complete. Clearly, B ⊂ H,so that A = B ∩ Y ⊂ H ∩ Y . Since A is closed in H ∩ Y , it follows that A isech-complete.

5.2. Theorem. Suppose that X is a locally σ-compact space with a homogeneousremainder Y . Then Y is ech-complete.

Proof. By Theorem 5.1, there exists a compact subspace F of Y such that Y isech-complete outside of F . If Y = F , then we are done.

Assume now that Y \F 6= ∅. Fix y ∈ Y \F . Clearly, Y is locally ech-completeat y. Since Y is homogeneous, it follows that Y is locally ech-complete at everypoint. Therefore, we can nd a nite collection U1, ..., Un of open subsets of Ysuch that F ⊂ U1∪ ...∪Un and the closure Hi of Ui in Y is ech-complete, for eachi = 1, ..., n. The subspace P = Y \(U1∪...∪Un) is a closed ech-complete subspaceof Y , since Y is ech-complete outside of F . Clearly, Y = (H1 ∪ ... ∪ Hn) ∪ P ,that is, Y is the union of a nite collection of closed ech-complete subspace ofY . Hence, Y is also ech-complete.

5.3. Corollary. Suppose that X is a locally σ-compact space with a homogeneousremainder Y . Then X = S∪L, where S is a closed σ-compact subspace of X, andL is an open locally compact subspace of X.

Proof. Fix a compactication bX of X such that Y = bX \X, and let bY be theclosure of Y in bX. Put S = X ∩ bY . Then S is a closed subspace of X, and Y is

8

ech-complete, by Theorem 5.2. Since S = bY \Y , it follows that S is σ-compact.The subspace L = (bX) \ bY is, clearly, locally compact and open in bX and inX. We also have S ∪ L = X. 5.4. Corollary. Suppose that X is a locally σ-compact nowhere locally compactspace with a homogeneous remainder Y . Then X is σ-compact, and Y is ech-complete.

References

[1] A.V. Arhangel'skii, On a class of spaces containing all metric and all locally compact spaces,Mat. Sb. 67(109) (1965), 55-88. English translation: Amer. Math. Soc. Transl. 92 (1970),1-39.

[2] A.V. Arhangel'skii, Remainders of metrizable spaces and a generalization of Lindelöf Σ-spaces, Fund. Mathematicae 215 (2011), 87-100.

[3] A.V. Arhangel'skii, Remainders of metrizable and close to metrizable spaces, FundamentaMathematicae 220 (2013), 7181.

[4] A.V. Arhangel'skii, A generalization of ech-complete spaces and Lindelöf Σ-spaces, Com-ment. Math. Universatis Carolinae 54:2 (2013), 121139.

[5] A.V. Arhangel'skii and M.M. Choban, Some generalizations of the concept of a p-space,Topology and Appl. 158 (2011), 1381 - 1389.

[6] E.K. van Douwen, F. Tall, and W. Weiss, Non-metrizable hereditarily Lindelöf spaces withpoint-countable bases from CH, Proc. Amer. Math. Soc. 64 (1977), 139-145.

[7] R. Engelking, General Topology, PWN, Warszawa, 1977.[8] M. Henriksen and J.R. Isbell, Some properties of compactications, Duke Math. J. 25 (1958),

83-106.[9] K. Nagami, Σ-spaces, Fund. Mathematicae 61 (1969), 169-192.[10] S.J. Nedev, o-metrizable spaces, Trudy Mosk. Matem. O-va 24 (1971), 201-236.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 9 14

Generalizations of metrics and partial metrics

Over half a century ago, Dr. L. M. Brown made an important discovery that did

not become a theorem in any paper: You can work and live happily and productively

with people of dierent cultures, and the result will be good for all. Dr. Brown

passed on this message through the example that participants saw in the topology

conferences he organized. The two authors met at one of these in summer 2001

and talked about future collaboration. We ocially began working in October 2002

along with Steve Matthews in England for a year. Since then we have worked on

developing and studying generalized metrics, among other topics. So our special

thanks go to Dr. Brown for organizing these Hacetteppe conferences. Due to him

our lives and research are much more enjoyable and productive than they would be

otherwise.

Ralph Kopperman ∗ and Homeira Pajoohesh †

Abstract

In [14] k-metric spaces were dened for certain `-group applications,by weakening the metric triangle inequality. In this article we showthat much of the theory of metric spaces, including the Banach xedpoint theorem extends to these spaces.

Keywords: partial metrics, variety of `-groups, k-metrics

2000 AMS Classication: 06F15, 54E35

∗Department of Mathematics, City College of New York, CUNYEmail : rdkcc@ccny.cuny.edu†Department of Mathematics, Medgar Evers College, CUNY

Email : hpajoohesh@mec.cuny.eduThe second author wishes to acknowledge support for this research from the City University ofNew York (PSC-CUNY grant 68354-00 46).

Doi : 10.15672/HJMS.2016.394

10

1. Introduction

Metrics have been generalized in many ways. Steve Matthews in [12] intro-duced partial metrics. His goal was to study the reality of nding closer and closerapproximations to a given number (using, say, n-place decimal approximations),and showing that contractive algorithms would serve to nd these approximations.Specically, he identied the approximations as partially known points. For ex-ample an n-place decimal approximation to a number tells us in which interval ofthe form [d, d+ 10−n) the number lies; this interval has width 10−n; this tells theremaining uncertainty of the exact value of the point, and is usefully seen as itsself-distance of this interval.

He then showed that the usual proof of the contraction xed-point theoremworked in his more general spaces and the theorem in these spaces meant thatcontractive algorithms would converge to a fully known point that is, a pointwhose distance to itself is 0. His denition is:

1.1. Denition. A partial metric is a function p : X ×X → [0,∞) satisfying thefollowing conditions for every x, y, z ∈ X:p(x, y) ≥ p(x, x),p(x, y) = p(y, x),p(x, z) + p(y, y) ≤ p(x, y) + p(y, z),x = y if p(x, y) = p(x, x) = p(y, y).

One can easily verify that a partial metric p on the set X is a metric if and onlyif p(x, x) = 0 for every x ∈ X.

An example of a partial metric is ∨ : [0,∞)×[0,∞)→ [0,∞) such that ∨(x, y) =maxx, y.

In [11] partial metrics were generalized by allowing p : X × X → V , whereV is a value quantale or a value lattice. In [10] completions of partial metricspaces were considered and it was shown that a new form of completion calledthe spherical completion is the same as the Round ideal completion" which isimportant in computer science.

In [2], a relationship between partial metrics and metrics with a base pointwas discussed. In [6] partial metrics on `-groups (lattice ordered groups) werediscussed. There it was shown that if pn∨(x, y) = n(x ∨ y) is a partial metric onan `-group G then na+ nb = nb+ na for all a, b ∈ G (this property is sometimescalled the commutativity of the nth power). So if En is the variety of `-groupsso that pn∨ is a partial metric and Ln is the variety of `-groups such that thenth power commutes then En ⊆ Ln. There it was shown that if n is prime thenEn = Ln.

Later in [4] it was shown that En = Ln if and only if n is prime. In [4] it wasshown that Ln ∩A2 ⊆ En for every n, where A2 is the set of `-groups G such thatG has an abelian convex normal `-subgroup H such that G

H is abelian. Further,

if n is prime then Ln ∩ A2 = En but whether this equality holds for all n wasunanswered for some time. Then in [3] it was shown that this equality holds ifn = pq where p and q are two positive prime numbers. Later, in [13] it was shownthat the equality holds for every n. These papers built another bridge betweencomputer science and order theory.

11

Metrics have also been generalized by allowing them to be valued in struc-tures other than IR, such as abelian `-groups. When this is done and a notionof positive" is given, many basic topology-like notions can be characterized withthese structures - for example each Tychono topology ( [8]), each proximity, andeach uniformity ( [7]) arises from a metric into such a structure with a subset ofpositives, and if symmetry is dropped then one obtains a quasimetric, and eachtopology ( [8]), each quasiproximity and each quasiuniformity ( [7]) as well as eachneighborhood space, closure space and pretopology arises from a quasimetric intosuch a structure with a subset of positives (the last three are discussed in [9]).Similarly it was shown in [11] that each T0 topology arises from a partial metricinto such a space with a subset of positives.

When it comes to `-groups, a major tool that allows this generalization is thatabsolute value for abelian `-groups, like absolute value on the real line, yields ametric. More precisely the absolute value of the zero is zero, the absolute valuesof an element and that of its inverse are the same, and the triangle law holds.The triangle law fails for non-abelian `-groups; in fact in [5] it was shown that an`-group is abelian if and only if the triangle law for the absolute value holds. Fora general `-group we only have that for each x, y, |x+ y| ≤ 2(|x|+ |y|). This ledto dening k-metrics, see [14].

1.2. Denition. Let k be a positive integer. A k-metric on a set X is a functiond : X ×X → IR such that for all x, y, z ∈ X:

(pos) d(x, y) ≥ 0(id) d(x, y) = 0⇔ x = y(sym) d(x, y) = d(y, x)(ktri) d(x, y) ≤ k(d(x, z) + d(z, y)).

A k −metric space is a set X 6= ∅ with a k-metric on it.

If we allow k-metrics to be valued in a lattice ordered group G rather thanthe reals, the absolute value of their dierence, d(x, y) = |x − y| will always be a2-metric on G. (See [1] p. 296.)

Clearly if d is a k-metric then it is t-metric for each integer t ≥ k. Certainlyevery metric space is a 1-metric space and so it is a k-metric space whenever1 ≤ k ∈ IN. But, the converse is not true; there are k-metric spaces which are notmetric:

1.3. Example. Consider d : IR× IR→ IR dened by d(x, y) = (x− y)2. Then d isa 2-metric on IR because for every two real numbers a and b, we have (a + b)2 ≤2(a2 + b2). But d(x, y) = (x − y)2 is not a metric: for example d(−1, 1) = 4 6≤d(−1, 0)+d(0, 1) = 1+1 = 2. In general for every even integer n, d(x, y) = |x−y|nis a 2n-metric because for every a, b ∈ IR we have |a+ b|n ≤ (|a|+ |b|)n ≤ (2(|a| ∨|b|))n = 2n(|a| ∨ |b|)n ≤ 2n(|a|n + |b|n). The last inequality holds since for everya, b ∈ IR, a ∨ b = a or a ∨ b = b.

So k-metrics allow us to talk about distance between points and thus aboutsequences, series and convergence in a wider context. They relax triangularity andstill the induced topology is metrizable, see [14]. Sometimes proving triangularityof metrics is challenging but proving triangularity for k-metrics can be much easier.As we see in this article many properties of metrics hold for k-metrics.

12

The central point of this paper is to show that the basic theories of topologyand uniformity are essentially the same for k-metrics as for metrics.

We recall that for X 6= ∅, a uniformity on X is a set U of subsets of X ×Xsuch that:

(Id) 1X =⋂

U where 1X = (x, x) : x ∈ X, the identity map of X,(Sym) if U ∈ U then U−1 ∈ U (where U−1 = (y, x) : (x, y) ∈ U)(Int) if U, V ∈ U then U ∩ V ∈ U, and(Com) if U ∈ U then V V ⊆ U for some V ⊆ X × X, where denotes

composition of relations.Every uniformity U induces a topology, τU, whose open sets are those T for

which x ∈ T ⇒ (∃U ∈ U)(U [x] ⊆ T ), where U [x] = y ∈ X : (x, y) ∈ U.For a k-metric d : X×X → R, if x ∈ X and r > 0, Nr(x) = y : d(x, y) ≤ r and

τd = T ⊆ X : (∀x ∈ T )(∃r > 0)(Nr(x) ⊆ T ) is called the topology induced by thek −metric. Also, if r > 0, Nr = (x, y) : x, y ∈ X & d(x, y) ≤ r and Ud = U ⊆X ×X : (∃r > 0)(Nr ⊆ U) is the uniformity induced by the k −metric.1.4. Lemma. For any k-metric space (X, d), τd is a topology, called the k −metric topology on X and Ud is a uniformity, called the k −metric uniformityon X. Further, τd is the topology induced by Ud.

Proof: That τd is a topology was shown in [14] (further, the reader can check it).Now we prove that Ud is a uniformity.

To show (Id) note that if U ∈ Ud then for some r > 0, Nr ⊆ U ; since foreach x ∈ X, d(x, x) = 0 ≤ r, each (x, x) ∈ Nr, so 1X ⊆ Nr; thus 1X ⊆ U ,and thus 1X ⊆

⋂Ud; but if (x, y) 6∈ 1X then x 6= y so d(x, y) 6= 0 so for some

r > 0, d(x, y) 6≤ r, so (x, y) 6∈ Nr, thus (x, y) 6∈ ⋂Ud, showing⋂

Ud ⊆ 1X .For (Sym), if U ∈ U then for some r > 0, Nr ⊆ U , so Nr = N−1r ⊆ U−1 thus

U−1 ∈ U.For (Int), if U, V ∈ Ud then for some r, s > 0, Nr ⊆ U and Ns ⊆ V , thus

t = minr, s > 0; also Nt ⊆ Nr ∩Ns ⊆ U ∩ V , so U ∩ V ∈ Ud, andFor (Com) if U ∈ U then nd r > 0 such that Nr ⊆ U , and note that there

is an s > 0 so that 2ks ≤ r. Let V = Ns ∈ Ud and if (x, y), (y, z) ∈ Ns thend(x, z) ≤ k(d(x, y) + d(y, z)) ≤ 2ks ≤ r, so Ns Ns ⊆ Nr ⊆ U . ut

Thus we can dene uniform continuity, Cauchy sequences, continuity and limitsfor k-metrics exactly the way they are dened for metrics. Similarly we say a k-metric is complete if every Cauchy sequence converges.

2. The contraction xed-point theorem

2.1. Denition. For a k-metric space (X, d), a function f : X → X is a Lipschitzmap with bound q if q is so that for each x, y ∈ X, d(f(x), f(y)) ≤ qd(x, y); f is acontraction if it is a Lipschitz map with bound q < 1

k2 .

2.2. Lemma. If f : X → X is a Lipschitz map with bound q on a k-metric space

(X, d), then f is continuous.

Proof: To show continuity with respect to τd, we establish that for each r > 0and x ∈ X, there is an s > 0 such that d(x, y) ≤ s implies d(f(x), f(y)) ≤ r. Lets = r

q ; then if d(x, y) < s we have d(f(x), f(y)) < qd(x, y) < qs = q( rq ) = r. ut

13

Note that the above in fact showed uniform continuity (in other words x doesn'tmatter).

2.3. Lemma. If f : X → X is a Lipschitz map with bound q on a k-metric space

(X, d), then for each x ∈ X, n ∈ IN∪ 0, d(x, fn(x)) ≤ (Σni=1k

iqi−1)d(x, f(x)).

Proof: For n = 0 and all x ∈ X, this inequality saysd(x, f0(x)) ≤ (Σ0

i=1kiqi−1)d(x, f(x)); that is, d(x, x) = 0 ≤ (Σ∅)d(x, f(x)), which

holds. Assume our inequality for n and all x ∈ X. Then by the k-metric tri-angle inequality and the inductive hypothesis, d(x, fn+1(x)) ≤ k(d(x, f(x)) +d(f(x), fn(f(x)))) ≤ kd(x, f(x)) + kd(f(x), fn(f(x))) ≤

kd(x, f(x)) + k(Σni=1k

iqi−1)d(f(x), f(f(x))) ≤ kd(x, f(x)) + k(Σni=1k

iqi−1)qd(x, f(x)) =kd(x, f(x)) + (Σn

i=1ki+1qi)d(x, f(x)) = kd(x, f(x)) + (Σn+1

i=2 kiqi−1)d(x, f(x)) =

(Σn+1i=1 k

iqi−1)d(x, f(x)).So our inequality holds for n + 1 and arbitrary x, completing our inductive

proof. ut

2.4. Theorem. Fixed point Theorem Let (X, d) be a complete k-metric space.

If f : X → X is a contraction then f has a xed point.

Proof: If k = 1 then we have a metric and the result is true. Thus let k > 1.Let x0 ∈ X be any point and inductively dene xn = f(xn−1). Then for eachn, xn = fn(x0). Now we show that the sequence xn is Cauchy: Let ε > 0; thereis an N such that 1

k2N−21

k−1d(x0, x1) < ε. Now let m ≥ n ≥ N . Since f is a Lips-

chitz map with bound q < 1k2 , by Lemma 2.3, d(xn, xm) = d(fn(x0), fm(x0)) ≤

qn(d(x0, fm−n(x0))) ≤ qn(Σm−n

i=1 kiqi−1)d(x0, f(x0)) ≤ 1k2n (Σm−n

i=11

ki−2 )d(x0, x1) ≤1

k2n (Σ∞i=11

ki−2 )d(x0, x1)

= 1k2n−2 (Σ∞i=1

1ki )d(x0, x1) = 1

k2n−21

k−1d(x0, x1) ≤ 1k2N−2

1k−1d(x0, x1) < ε.

This proves that xn is a Cauchy sequence, and since (X, d) is a completek-metric space, it converges to a point, say a. Thus by the continuity of f (shownin Lemma 2.2):f(a) = f(limn→∞ xn) = limn→∞ f(xn) = limn→∞ xn+1 = a, so a is a xed

point. ut

3. Future Work

The continuation of this research is considering k-partial metrics. We havea denition for k-partial metrics and we must verify that it follows the idea ofpartial metric and generalizes partial metrics in a way that keeps their propertiesand their relationship with other distance functions.

References

[1] Birkho, G., Lattice Theory, 3rd Edition, American Mathematical Society Colloquium Pub-lications, Volume 25, 1967, Providence, RI.

[2] Bukatin, M., Kopperman, R., Matthews, S., and Pajoohesh, H., Partial Metric spaces,American Mathematical Monthly 116, 708-718, 2009.

[3] Darnel, M. and Holland, W.C., Minimal non-metabelian varieties of `-groups that containno nonabelian o-groups, Communications in Algebra 42, 5100-5133, 2014.

14

[4] Darnel, M., Holland, W.C., Pajoohesh, H.,The relationship of partial metric varieties and

commuting powers varieties, Order, 30, no.2, 403-414, 2013.

[5] Holland, W. C., Intrinsic metrics for lattice-ordered groups, Algebra Universalis, 19, 142-150, 1984.

[6] Holland, W.C, Kopperman, R. and Pajoohesh, H., Intrinsic generalized metrics AlgebraUniversalis, 67, no.1, 1-18, 2012.

[7] Kopperman, R., Lengths on Semigroups and Groups, Semigroup Forum 25, 345-360, 1984.

[8] Kopperman, R., All Topologies Come From Generalized Metrics, Am. Math. Monthly 95,89-97, 1988.

[9] Kopperman, R., Mynard, F., and Ruse, P., Quasi-metric representations of various cate-

gories of closure spaces, Topology Proceedings, 37: 331-347, 2011.

[10] Kopperman, R., Matthews, S., and Pajoohesh, H., Completions of partial metrics into value

lattices, Topology and Applications 156, 1534-1544, 2009.

[11] Kopperman, R., Matthews, S., and Pajoohesh, H., Universal partial metrizability, AppliedGeneral Topology 5, 115-127 2004.

[12] Matthews, S. G., Partial metric topology", Proc. 8th summer conference on topology andits applications, ed S. Andima et al., New York Academy of Sciences Annals, 728, 183-197,1994.

[13] Pajoohesh, H., The relationship of partial metric varieties and commuting powers varieties

II, Algebra Universalis, 73, issue 3-4, 291-295, 2015.

[14] Pajoohesh, H., k-metric spaces, Algebra Universalis, 69, no.1, 27-43, 2013.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 15 32

LM -valued equalities, LM -rough approximationoperators and ML-graded ditopologies

Alexander ostak ∗ and Aleksandrs El,kins†

Abstract

We introduce a certain many-valued generalization of the concept ofan L-valued equality called an LM -valued equality. Properties of LM -valued equalities are studied and a construction of an LM -valued equal-ity from a pseudo-metric is presented. LM -valued equalities are ap-plied to introduce upper and lower LM -rough approximation opera-tors, which are essentially many-valued generalizations of Z. Pawlak'srough approximation operators and of their fuzzy counterparts. Westudy properties of these operators and their mutual interrelations.In its turn, LM -rough approximation operators are used to inducetopological-type structures, called here ML-graded ditopologies.

Keywords: LM -valued equalities, LM -rough approximation operators, ML-graded ditopologies.

2000 AMS Classication: Primary 54A40, 03E75, 18B99; Secondary 46A08

1. Introduction

After the inseption of the concepts of an L-valued equality and an L-valuedset by U. Höhle [19], the study of the category of L-valued sets itself, as wellas of dierent mathematical structures, specically topological and algebraic, onL-valued sets attracted interest of many researchers, see e.g. [20], [21], [22], [24],[45] just to mention a few of them. In Section 3 of this paper we introduce theconcept of an LM -valued set (Denition 3.1), where L is an iccl-monoid (Subsection2.1.1) and M is an arbitrary innitely distributive lattice. An LM -valued set is,in a certain sense, a many-valued version of the concept of an L-valued set. We

∗Department of Mathematics, University of Latvia, Riga, LV-1002 and Institute of Mathe-matics and CS, Riga, LV-1459, LATVIAEmail : aleksandrs.sostaks@lu.lv, aleksandrs.sostaks@lumii.lv†Department of Mathematics, University of Latvia, Riga, LV-1002 LATVIA

Email : aleksandrs.elkins@gmail.com

Doi : 10.15672/HJMS.2016.400

16

consider dierent special kinds of LM -valued equalities, and study their propertiesin Section 3 of this paper. Further, in Section 6, we construct an LM -valuedequality Eρ from an ordinary pseudo-metric ρ on a setX and investigate propertiesof the obtained LM -valued set (X,Eρ). We consider this construction to be animportant source for creation many examples of LM -valued sets.

Aiming to dene a precise mathematical tool which would be appropriate andeective to deal with big data, Z. Pawlak [32] introduced in 1983 the concept ofa rough set. Pawlak's work was followed by many other researches. In particular,in 1991 D. Dubois and H. Prade [12] published a paper in which fuzzy roughsets were dened. In this way Pawlak's ideas, aimed specically to deal withthe analysis of big data, were alloyed with L. Zadeh' s vision [49] to develop aprecise mathematical tool, which would be appropriate to deal with unpreciseand vague objects. This combination gave rise to a new eld of mathematicalreserach, the eld interesting and important both from theoretical and practicalpoints of view. Namely, we mean the theory of upper and lower fuzzy roughapproximation operators. In this paper, basing on the concept of an LM -valuedset, we introduce a certain many-valued generalization of this theory. It is donein Section 4 consisting of three subsection: Subsection 4.1 where we dene andstudy upper LM -rough approximation operators induced by LM -valued equalities,Subsection 4.2 dealing with lower LM -rough approximation operators induced byLM -valued equalities, and Subsection 4.3 where some additional properties of theseoperators, in particular their mutual interrelations, are considered.

Topological properties of upper and lower Pawlak's rough approximation oper-ators where rst noticed in 1988 by A.Skowron [39] and A. Wiweger [47]. J. Korte-lainen [26] was probably the rst one to discover deep connections between fuzzyupper and lower fuzzy rough approximation operators on one side and (Alexan-dro) fuzzy topologies on the other. Later the link between fuzzy rough approx-imation operators and topological L-fuzzy closure and L-fuzzy interior operatorswas in the center of interest of dierent authors, see e.g. , [13], [18], [23], [30], [33],[34], [44], [48].‡ In our paper, we use upper and lower LM -approximation operatorsin order to deneM -graded L-fuzzy topologies, orML-graded topologies for short[6], on LM -valued sets. This is done in Section 5 under an additional assumptionthat the lattice M is completely distributive.

2. Prerequisites: The context of the work

2.1. Lattices, iccl-monoids and residuated lattices. In this work the twoobjects, lattices L and M , will play the fundamental role.

2.1.1. Lattices. By L=(L,≤L,∧L,∨L) we denote a complete lattice, that is alattice in which arbitrary suprema (joins) and inma (meets) exist. In particular,the top 1L and the bottom 0L elements in L exist and 0L 6= 1L. A lattice (L,≤L

‡Although the authors of these papers speak about fuzzy topologies, in fact they are dealingwith fuzzy ditopologies [4], [5] since the families of fuzzy open and fuzzy closed sets obtained inthis way remain unrelated unless some additional assumptions are made, for example under theassumption that the range L of fuzzy sets is an MV-algebra

17

,∧L,∨L) is called innitely distributive or a frame if ∧ distributes over arbitraryjoins:

α∧L(∨

iβi

)=∨

i(α ∧L βi) ∀α ∈ L, ∀βi : i ∈ I ⊆ L.

In the sequel we usually omit the subscript L in notation of ≤,∧,∨ when thiscould not lead to misunderstanding.

2.1.2. iccl-monoids. Following e.g. [19, 20] by an integral commutative cl-monoid(iccl-monoid for short) we call a tuple (L,≤,∧,∨, ∗) where (L,≤,∧,∨) is a com-plete lattice and (L, ∗, 1L) is a monoid such that:(1cl) ∗ is commutative: α ∗ β = β ∗ α for all α, β ∈ L;(2cl) ∗ is associative: (α ∗ β) ∗ γ = α ∗ (β ∗ γ) for all α, β, γ ∈ L;(3cl) ∗ distributes over arbitrary joins: α ∗

(∨i∈I βi

)=∨i∈I(α ∗ βi) for all

α ∈ L, for all βi | i ∈ I ⊆ L,(4cl) α ∗ 1L = α for all α ∈ L.

It is known and easy to prove that α ∗ 0L = 0L for every α ∈ L and that ∗ ismonotone:

α ≤ β =⇒ α ∗ γ ≤ β ∗ γNote that an iccl-monoid can be characterized also as an integral commutative

quantale in the sense of K.I. Rosenthal [37].

2.1. Example. Among the most important examples of iccl-monois are the fol-lowing three.

• Let L = [0, 1] and ∗ = ∧. In this case iccl-monoid (L,≤,∧,∨, ∗) justreduces to the underlying lattice (L,≤,∧,∨,∧).

• Let L = [0, 1] and let α ∗ β := α · β be the product. Then we come to theso called product t-norm.

• Let L = [0, 1] and α ∗ β = max(α + β − 1, 0). Then ∗ is the well-knownukasiewicz t-norm.

The monoidal operation ∗ : [0, 1]× [0, 1] → [0, 1] in these cases is usually referredto as a left semi-continuous t-norm, the term originating from the classic paperby [29]. These and other t-norms were studied and used by many authors, see e.g.fundamental monographs [38] and [25].

2.1.3. Residuated lattices. In an iccl-monoid a further binary operation 7→, resid-uation, is dened:

α 7→ β =∨λ ∈ L | λ ∗ α ≤ β ∀α, β ∈ L.

Residuation is connected with operation ∗ by Galois connection, see [15]:

α ∗ β ≤ γ ⇐⇒ α ≤ (β 7→ γ).

An iccl-monoid (L,≤,∧,∨, ∗) extended by 7→, that is the tuple (L,≤,∧,∨, ∗, 7→),is known also as a residuated lattice [31].

In the following proposition we collect well-known properties of the residiumwhich will be used in the main text:

2.2. Proposition. see e.g. [19], [20](1) (

∨i αi) 7→ β =

∧i (αi 7→ β) for all αi | i ∈ I ⊆ L, for all β ∈ L;

18

(2) α 7→ (∧i βi) =

∧i(α 7→ βi) for all α ∈ L, for all βi | i ∈ I ⊆ L,;

(3) 1L 7→ α = α for all α ∈ L;(4) α 7→ β = 1L whenever α ≤ β(5) α ∗ (α 7→ β) ≤ β for all α, β ∈ L;(6) (α 7→ β) ∗ (β 7→ γ) ≤ α 7→ γ for all α, β, γ ∈ L;(7) α 7→ β ≤ (α ∗ γ 7→ β ∗ γ) for all α, β, γ ∈ L.(8) α ∗ β ≤ α ∧ β for any α, β ∈ L.(10) (α ∗ β) 7→ γ = α 7→ (β 7→ γ) for any α, β, γ ∈ L.

2.1.4. Lattice M. ByM we denote a complete innitely distributive lattice (M,≤M,∧M ,∨M ), whose bottom and top elements are denoted by 0M and 1M respectively.As dierent from the lattice L, we do not exclude here the trivial case, that is Mcan be the one-element lattice • and hence in this case 0M = 1M . Although in thelarger part of this work M can be an arbitrary innitely distributive lattice, whenapplying our results for constructing M -graded L-fuzzy ditopologies in Section 5,we additionally assume thatM is completely distributive. Actually we will use notthe original denition of complete distributivity, see e.g [15, Denition I-2-8], butits characterization given by G.N. Raney [36]. Namely, given a complete latticeM and β, α ∈M following [36], see also [15, Excercise IV-3-31], we introduce theso called "wedge below" relation C on M as follows:

β C α⇐⇒(if K ⊆M and α ≤

∨K then ∃γ ∈ K, α ≤ γ

).

As shown by G.N. Raney [36], a lattice M is completely distributive if and only ifrelation C has the approximation property, that is

α =∨β ∈M | β C α for every α ∈M.

Moreover, relation C has the following nice properties (see [15, 36]) used in thesequel:(C 1) β C α implies β ≤ α;(C 2) γ ≤ β C α ≤ δ implies γ C δ;(C 3) β C α implies that there exists γ ∈ L such that β C γ C α.

2.2. Fuzzy sets. [49], [17] Recall that an L-fuzzy subset of a set X, where L isa complete lattice, is a mapping A : X → L. Given a family Ai | i ∈ I its union∨iAi : X → L and intersection

∧iAi : X → L are dened respectively by

(∨iAi

)(x) = sup

i∈IAi(x),

(∧iAi

)(x) = inf

i∈IAi(x).

2.3. L-relations, L-valued equalities and L-valued sets.

Given sets X,Y and an iccl-monoid L, by an L-relation between X and Y wecall a mapping R : X × Y → L. In case X = Y , an L-relation E : X ×X → L iscalled an L-valued equality if it is

(1) reexive, that is E(x, x) = 1L for every x ∈ X;(2) symmetric, that is E(x, y) = E(y, x) for all x, y ∈ X;(3) transitive, that is E(x, y) ∗ E(y, z) ≤ E(x, z) for all x, y, z ∈ X.

The concepts called here an L-relation and L-valued equivalence under dierent names andwith dierent degrees of generality appear in many papers, see e.g. [46], [50], [1], [2], etc.

19

A pair (X,E), where E : X ×X → L is an L-valued equality on X, is called anL-valued, or a many-valued, set.

When dealing with fuzzy subsets of L-valued sets, the property of extensionalityplays an important role. This property was considered by many authors, see e.g.U. Höhle [19], [20], F. Klawon [24], etc:

A fuzzy set A in an L-valued set (X,E) is called extensional if

A(x) ∗ E(x, x′) ≤ A(x′) ∀x, x′ ∈ X.The smallest extensional fuzzy set A in (X,E) that is larger than or equal to A( A ≤ A) is called the extensional hull of A. Explicitly the extensional hull of Acan be dened by

A(x) =∨

x′∈X(E(x, x′) ∗A(x′)) ,

see e.g. [19], [20], [24].In particular, identifying an element x0 with the characteristic function χx0

of the one-element set x0, we get the extensional hull of the point x0 called afuzzy singleton:

χx0= E(x0, x).

3. LM -valued equalities and LM -valued sets

3.1. LM -fuzzy sets. Let, as it was assumed, L = (L,≤L,∧L,∨L, ∗) be an iccl-monoid and M = (M,≤M ,∧M ,∨M ) be a complete innitely distributive lattice.Then the powerset LM = ϕ | ϕ :M → L becomes an iccl-monoid by point-wiseextension of operations ≤L,∧L,∨L, ∗ from L to LM :

(ϕ ∧ ψ)(α) = ϕ(α) ∧ ψ(α); (ϕ ∨ ψ)(α) = ϕ(α) ∨ ψ(α); (ϕ ∗ ψ)(α) = ϕ(α) ∗ ψ(α)for all ϕ,ψ ∈ LM and every α ∈M.

Applying the standard denition of a fuzzy set to this situation, we say thatan LM -fuzzy subset A of a set X is just a mapping A : X → LM . However,the special form of the range set LM allows to interpret A either as a mappingassigning to each x ∈ X the mapping A(x) = ϕx : M → L, or as an L-fuzzysubset A ∈ LX×M of X ×M , that is as a mapping A : X ×M → L assigning toa pair (x, α) ∈ X ×M the element A(x, α) = A(x)(α) ∈ L. This interpretation ofan LM -fuzzy set A allows to represent it as the family Aα : α ∈ M of L-fuzzysubsets Aα ∈ LX of X ordered by the elements of M , where the L-fuzzy sets Aα

are dened by Aα(x) = A(x, α).

3.2. LM -valued equalities: Denitions and basic properties. Adjustingthe dention of an L-valued relation (see Denition 2.3) to our situation we getthe following:

3.1. Denition. Given a set X, an LM -valued equality on it is a mapping E :X ×X → LM such that

(1ELM ) E(x, x)(α) = 1L for every x ∈ X and every α ∈M ;(2ELM ) E(x, y)(α) = E(y, x)(α) for all x, y ∈ X and every α ∈M ;(3ELM ) E(x, y)(α) ∗ E(y, z)(α) ≤ E(x, z)(α) for all x, y, z ∈ X, α ∈M .

20

(4ELM ) E(x, y)(·) is not-increasing, that isα < β =⇒ E(x, y)(α) ≥ E(x, y)(β) for all x, y ∈ X, α, β ∈M .

Sometimes we will speak about some special properties of an LM -valued rela-tions collected in the next denition:

3.2. Denition. An LM -valued equality E will be called upper semi-continuousif

(5ELM ) E(x, y)(∨

i∈I αi)=∧i∈I E(x, y)(αi) for all x, y ∈ X, αi | i ∈ I ⊆M .

An LM -valued equality E will be called lower semi-continuous if(6ELM ) E(x, y)

(∧i∈I αi

)=∨i∈I E(x, y)(αi) for all x, y ∈ X, αi | i ∈ I ⊆M .

An LM -valued equality satisfying both properties (5ELM ) and (6ELM ) is calledcontinuous.An LM -valued equality E will be called global if it satises properties (7ELM ) and(8ELM ) below:

(7ELM ) E(x, y)(0M ) = 1L for all x, y ∈ X,(8ELM )

E(x, y)(1M ) =

1L if x = y0L otherwise.

Note that each one of the properties (5ELM ) and (6ELM ) implies the property(4ELM ).

3.3. Remark. Sometimes we interpret an LM -equality E : X × X → LM as amapping E : X ×X ×M → L dened by E(x, y, α) = E(x, y)(α) satisfying corre-sponding analogues of conditions (1ELM ) - (8ELM ) reformulated in an obviousway. In what follows we will use both entries E(x, y)(α) and E(x, y, α) and inter-pret E as a mapping E : X ×X → LM and as a mapping E : X ×X ×M → L,when it is more convenient. Besides we usually write just E instead of E when itcannot lead to misunderstanding.

The proof of the following proposition is straightforward:

3.4. Proposition. A mapping E : X ×X ×M → L is an LM -valued equality ona set X if and only if for every α ∈M the restriction Eα of E to X ×X ×α isan L-valued equality on X and α ≤ β =⇒ Eα ≥ Eβ . Thus an LM - valued equalityon a set X can be represented as a non-increasing family of L-valued equalities onthis set ordered by the elements of the lattice M .

3.5. Example. Let (X,E) be an L-valued set and M be an arbitrary completelattice. Then setting E(x, y, α) = E(x, y) for every α ∈M we obtain a continuousLM -valued equality E : X ×X ×M → L. In this way the L-valued set (X,E) canbe identied with the LM -valued set (X, E). In particular, in the role of M , onecan take here the one-element lattice M = •.3.6. Denition. An LM -fuzzy set B is called extensional, if B(x, α)∗E(x, x′α) ≤B(x′, α) for every x, x′ ∈ X and for every α ∈ M . By the LM -etensioanal hull ofan L-fuzzy set A ∈ LX we call the smallest extensional LM -fuzzy set B ∈ (LM )X

which is larger than or equal to A, that is A(x) ≤ B(x, α) for all x ∈ X and forall α ∈M.

21

From the denitions one can straightforward get the following

3.7. Proposition. An LM -fuzzy set B is extensional if and only for each α ∈Mthe L-fuzzy set Bα is extensional. Specically, an LM -fuzzy set B is the extensionalhull of the LM -fuzzy set A if an only for each α ∈ M Bα is the extensional hullof Aα.

4. LM -rough approximation operators on an LM -valued set

4.1. Upper LM -rough approximation operator on an LM -valued set. LetE : X × X → LM be an LM -valued equality on a set X. Given an L-fuzzy setA ∈ LX we dene an LM -fuzzy set uE(A) ∈ (LM )X as follows:

uE(A)(x)(α) =∨

x′∈X(E(x, x′)(α) ∗A(x′)) .

In such a way we obtaine an operator uE : LX →= (LM )X that, in an obviousway, can be interpreted also as an operator uE : LX → LM×X

4.1. Denition. Let (X,E) be an LM -valued set We call operator uE : LX →(LM )X the upper LM -fuzzy rough approximation operator induced on the LM -valued set (X,E).

Such operator can be represented as a family of L-fuzzy rough approximationoperators uαE : LX → LX : α ∈M dened by

uα(A)(x) = u(A)(x)(α) ∀A ∈ LX , ∀x ∈ X.This family is ordered by elements of the lattice M in such a way that

α ≤ β =⇒ uαE(A) ≥ uβE(A) ∀A ∈ LX ,see Proposition 4.2 (5u).

We dene the reduced composition uE uE : LX → (LM )X for operator uE bysetting

(uE uE)(A)(x)(α) = uE(uE(A)(x)(α))(x)(α) ∀A ∈ LX , ∀x ∈ X.The most important properties of operator uE are collected in the following

proposition:

4.2. Proposition. Let (X,E) be an LM -valued set and uE : LX → (LM )X be theinduced upper LM -fuzzy rough approximation operator. Then uE : LX → (LM )X

has the following properties:

(1u) uE(0X)(x, 0M ) = 0L for all x ∈ X;(2u) uE(A)(x, α) ≥ A(x) for every x ∈ X,α ∈M.(3u) uE(

∨iAi) =

∨i uE(Ai) ∀Ai | i ∈ I ⊆ LX in particular

(3′u) uE(A1 ∨A2) = uE(A1) ∨ u(A2)∀A1, A2 ∈ LX ;(4u) (uE uE)(A) = uE(A) ∀A ∈ LX ;(5u) α ≤ β ⇒ uE(A)(x, α) ≥ uE(A)(x, β) ∀x ∈ X;(6u) If E is upper semicontinuous, then uE(A)(x,

∧i αi) =

∨i uE(A)(x, αi) for

every set αi | i ∈ I ⊆M ;(7u) If E is global, then uE(A)(x, 0M ) =

∨x′∈X A(x

′)and uE(A)(x, 1M ) = A(x).

22

Proof Statement (1u) is obvious. Statement (2u) follows easily taking intoaccount reexivity of the LM -relation E.

We prove property (3u) as follows:

uE(∨

i

Ai)(x)(α) =∨

x′(E(x, x, α′)∗(

∨iAi(x

′))) =∨

x′(∨

iE(x, x′, α)∗Ai(x′)) =

∨i(∨

x′(E(x, x′, α) ∗Ai(x′))) =

∨i(uE(Ai)(x, α)) =

(∨i(uE(Ai)

)(x, α).

To prove property (4u) we x α ∈ M and x ∈ X and taking into accounttransitivity of the LM -relation we have:

(uE uE)(A)(x)(α) = uαE(uαE(A))(x) =

∨x′(uαE(A)(x

′) ∗ Eα(x, x′)) =∨

x′′

∨x′(A(x′′) ∗ Eα(x, x′) ∗ Eα(x′, x′′)) ≤

∨x′′A(x′′) ∗ Eα(x, x′′) =

uαE(A)(x) = uE(A)(x)(α)

Since the converse inequality follows from (2u), we get property (4u).Property (5u) is clear from the denitions taking into account that the LM -

valued equality E is non-increasing.We prove property (6u) as follows. Let αi | i ∈ I ⊆ M and let α =

∧i∈I αi.

Then for every x ∈ X we have:

uE(A)(x, α) = uE(A)(x,∧

iαi

)=∨

x′

(∨i∈I

(E(x, x′, αi) ∗A(x′)))=

∨i∈I

∨x′(E(x, x′, αi) ∗A(x′)) =

∨i∈IuE(A)(x, αi).

In case E is global, we prove property (7u) as follows:

uE(A)(x, 0M ) =∨

x′

(E(x, x′, 0M ) ∗A(x′)) =∨

x′

(1L ∗A(x′)) =∨

x′

A(x′) and

uE(A)(x, 1M ) =∨

x′

E(x, x′, 1M ) ∗A(x) = A(x).

2

4.3. Corollary. L-fuzzy set uE(A) ∈ (LM )X is the LM -extensional hull of theL-fuzzy set A ∈ LX .

The proof is straightforward from the denitions and taking into account prop-erty (2u) in Proposition 4.2.

4.2. Lower LM -rough approximation operator on an LM -valued set. LetE : X × X → LM be an LM -valued equality on a set X. Given an L-fuzzy setA ∈ LX we dene the LM -fuzzy set lE(A) ∈ (LM )X as follows:

lE(A)(x)(α) =∧

x′∈X(E(x, x′)(α) 7→ A(x′)) .

In such a way we obtain an operator lE : LX → (LM )X . In an obvious way it canbe interpreted also as an operator lE : LX → LM×X

4.4. Denition. Let (X,E) be an LM -valued set. We call lE : LX → (LM )X

by the lower LM -fuzzy rough approximation operator induced by the LM -valuedequality E.

23

Such operator can be represented as a family of lower L-fuzzy rough approxi-mation operators lαE : LX → LX : α ∈M dened by

lα(A)(x) = l(A)(x) ∀A ∈ LX , ∀x ∈ X.This family is ordered by elements of the lattice M in such a way that

α ≤ β =⇒ lαE(A) ≤ lβE(A) ∀A ∈ LX ,see Proposition 4.5 (5l).

We dene the reduced composition lE lE : LX → (LM )X for operator lE bysetting

(lE lE)(A)(x)(α) = lE(lE(A)(x)(α))(x)(α) ∀A ∈ LX , ∀x ∈ X.The most important properties of this operator are collected in the following

proposition:

4.5. Proposition. Let (X,E) be an LM -valued set.

(1l) lE(1X)(x, α) = 1L ∀α ∈M, ∀x ∈ X;(2l) A(x) ≥ lE(A)(x, α) ∀A ∈ LX , ∀α ∈M ;(3l) lE(

∧iAi) =

∧i lE(Ai) ∀Ai | i ∈ I ⊆ LX in particular

(3′l) lE(A1 ∧A2) = lE(A1) ∧ u(A2)∀A1, A2 ∈ LX ;(4l) (lE lE)(A)(x)(α) = lE(A)(x)(α);(5l) If E is non-increasing, then α ≤ β =⇒ lE(A)(x, α) ≤ lE(A)(x, β);(6l) If E is upper semicontinuous, then lE(A)(x,

∨i αi) =

∧i lE(A)(x, αi);

(7l) If E is global, then lE(A)(x, 0M ) =∧x′ A(x′) and lE(A)(x, 1M ) = A(x).

Proof Statement (1l) is obvious. Statement (2l) follows easily taking intoaccount reexivity of the LM -equivalence E. We prove property (3l) as follows:

lE(∧

iAi)(x, α) =

∧x′

(E(x, x′, α) 7→

∧iAi(x

′))=∧

x′

∧i(E(x, x′, α) 7→ Ai(x

′)) =

∧i

∧x′(E(x, x′, α) 7→ Ai(x

′)) =∧

ilE(Ai).

To prove property (4l) we take into account transitivity of the L-valued equalityEα and are reasoning as follows:

(lE lE)(A)(x)(α) = lαE(lαE(A))(x) =

∧x′(Eα(x, x′) 7→ lαE(A)(x

′)) =∧

x′(Eα(x, x′) 7→

∧x′(Eα(x′, x′′) 7→ A(x′′))) =

∧x′(∧

x′′(Eα(x, x′) ∗ Eα(x′, x′′) 7→ A(x′′))) ≥

∧x′′(Eα(x, x′′) 7→ A(x′′)) = lαE(A)(x) = lE(A)(x)(α).

Since the converse inequality follows from (2l), we get property (4l).Property (5l) is clear from the denitions taking into account that the LM -

valued equality E is non-increasing.We prove property (6l) as follows. Let x ∈ X and αi : i ∈ I ⊆M. Then

lEA(x,∨

iαi) =

∧x′

(E(x, x′,

∨iαi) 7→ A(x′)

)=

∧x′

(∧iE(x, x′, αi) 7→ A(x′)

)=∧

x′

∧i(E(x, x′, αi) 7→ A(x′)) =

24

∧i

∧x′(E(x, x′, αi) 7→ A(x′)) =

∧ilE(A)(x, αi).

To prove property (7l) we notice that in case of the global LM -valued equalitywe have

lE(A)(x, 0M ) =∧

x′

(E(x, x′, 0M ) 7→ A(x′)) =

∧

x′

(1L 7→ A(x′)) = 1 7→∧

x′

A(x′) =∧

x′

A(x′);

lE(A)(x, 1M ) =∧

x′

(E(x, x′, 1M ) 7→ A(x′)) = E(x, x, 1M ) 7→ A(x)) = A(x).

4.3. Additional properties of the LM -rough approximation operators.

In this section rst of all, we are interested in the interchange properties of theupper and lower rough approximation operators uE : LX → (LM )X and lE :LX → (LM )X . Since we need to deal with combination of operators uE and lE ,we have to specify how to "compose" them. We dene the operation of reducedcomposition uE lEX → (LM )X and lE uEX → (LM )X in the same manner asit was done in the previous two subsections:

(uE lE)(A)(x)(α) = uE(lE(A)(x)(α))(x)(α) ∀A ∈ LX , ∀x ∈ X;

(lE uE)(A)(x)(α) = lE(uE(A)(x)(α))(x)(α) ∀A ∈ LX , ∀x ∈ X.4.6. Proposition. Given an LM -valued set (X,E) we have uE lE = lE, or,explicitely,

uE(lE(A)(x)(α))(x, α) = lE(A)(x, α) for any x ∈ X and any α ∈M.

Proof From the denition of the operators uE , lE : LX → (LM )X and oper-ation we have:

(uE lE)(A)(x)(α) =∨y∈X

(E(x, y, α) ∗

∧z∈X

(E(z, y, α) 7→ A(z)))≤

∨y∈X

∧z∈X

(E(x, y, α) ∗ (E(z, y, α) 7→ A(z))) ≤∧

z∈X

∨y∈X

E(x, y, α) ∗ (E(z, y, α) 7→ A(z)) ≤∧

z∈X

∨y∈X

((E(x, y, α) 7→ E(y, z, α)) 7→ A(z)) ≤∧

z∈X(∧

y∈Y(E(x, y, α) 7→ E(y, z, α)) 7→ A(z)) ≤

∧z∈X

(E(x, z, α) 7→ A(z)) = lE(A)(x)(α).

The rst two inequalities in the above series are obvious; The third and the fourthinequalities in the above series are ensured by the easily established inequalitiesa ∗ (b 7→ c) ≤ (a ∗ b 7→ c) and

∨i(ai 7→ b) ≤ (

∧i ai 7→ b) which hold in every iccl-

monoid; the last inequality follows from the denition of an L-valued equality: thecondition E(x, y, α) ≤ E(x, z, α)∗E(z, y, α) implies that E(x, z, α) ≤ E(z, y, α) 7→E(y, x, α),∀y ∈ X.Thus we have (uE lE)(A)(x)(α) ≤ lE(A)(x)(α). We complete the proof noticingthat the inequality lE(A)(x)(α) ≤ (uE lE)(A)(x)(α) is obvious. 2

25

4.7. Remark. In case M is the one-element lattice, the corresponding result iscontained in [14] In particular, in the special case when L = [0, 1] is viewed asa Gödel algebra, that is ∗ = ∧ is the minimum t-norm and M is the one pointlattice, the statement of the above theorem is contained in Proposition 9 in [35].

4.8. Proposition. For every L-fuzzy set A in an LM -valued set (X,E) its lowerLM -rough approximation lE(A) is an extensional fuzzy set.

Proof From Proposition 4.6 we know that uE lE = lE , that is for everyα ∈ M and for every A ∈ LX the equality uαE(l

αE(A)) = lαE(A) holds. Now from

Proposition 3.7 it follows that lαE(A) is extensional for every α ∈ M . Finally,applying Proposition 4.3 we conclude that lE(A) is extensional. 2

4.9. Denition. Let (X,E) be an LM -valued set (X,E) and A ∈ LX be itsL-fuzzy subset. By the extensional kernel of A in (X,E) we call the smallestextensional LM -fuzzy set A0 ∈ (LM )X which is smaller than or equal to A.

From the denitions one can easily prove

4.10. Proposition. A0 ∈ (LM )X is the extensional kernel of A ∈ LX if andonly if for each α ∈ L the L-fuzzy set (A0)α is the extensional kernel of A in theL-valued set (X,Eα).

4.11. Proposition. Let A be an L-fuzzy subset of an LM -valued set (X,E) andlet A0 be its kernel. Then A0 ≤ lE(A)

Proof Referring to Proposition 3.7 we conclude that for every α ∈M L-fuzzyset A0,α is extensional in (X,E). Therefore we have

(A0)α(x) ∗ Eα(x, x′) ≤ (A0)α(x′) for every x, x′ ∈ X,and hence

(A0)α(x) ≤ Eα(x, x′) 7→ (A0)α(x′) ≤ Eα(x, x′) 7→ A(x′), ∀x, x′ ∈ X.It follows from here that

(A0)α(x) ≤∧

x′∈X(Eα(x, x′) 7→ A(x′)) = lαE(A)(x), ∀x ∈ X,

that is (A0)α ≤ lα(A).Referring to Proposition 3.7 again we conclude that A0 ≤ lE(A) 2

From Propositions 4.6 and 4.11 we get the following important result:

4.12. Theorem. For every L-fuzzy set A in an LM -valued set (X,E) the lowerfuzzy rough approximation operator lE assigns to A its kernel A0: That is lE(A) =A0.

From this theorem we get

4.13. Corollary. The equality lE uE = uE holds. Explicetely

(lE uE)(A)(x)(α) = uE(A)(x)(α)

for every L-fuzzy set A in an LM -valued set (X,E).

26

4.14. Remark. Let L = 0, 1 =: 2, M = • be the one-element lattice andlet E : X × X → 0, 1 be an equivalence relation. Obviously, in this case E isactually the crisp equivalence relation on X. Then the images of a set A ∈ 2X

under operators uE : 2X → 2 and lE : 2X → 2 make the pair (uE(A), lE(A))which is actually Pawlak's originally dened rough set (AH, AN) determined bythe set A. Indeed, notice rst that uE(A) in this case is just the set of all elementsx ∈ A whose classes [x]E of E-equivalence have non-empty intersections with A:[x]E ∩ A 6= ∅, and hence uE(A) = AH. On the other hand, lE(A) is the set of allelements x ∈ A, whose classes of equivalence [x]E are contained in A: [x]E ⊆ A,and hence lE(A) = AN.

5. ML-graded ditopology induced by an LM -valued equality

In this section we apply upper and lower LM -rough approximation operatorsinduced by an LM -valued equality on a set X in order to present a construction ofan ML-graded ditopology on this set. However rst we have to make commentson the terminology used here.

Generalizing the concept of an L-fuzzy topology in the sense of Chang-Goguen(see [7], [17] [16]), T. Kubiak [27] and A.ostak [40] independently introducedan alternative, in a certain sense more consequent, concept of a fuzzy topology.According to this denition the topology itself is an L-fuzzy subset (and not acrisp one as it is in the case of Chang-Goguen's denition) of the family of L-fuzzysubsets of the ground set X, see Subsection 5.1 To distinguish such approach fromthe one in the sense of Chang-Goguen, we call it here a graded topology.¶ In orderto specify the role of the iccl-monoid L and the lattice M in this case, we usea more precise term an M -graded L-fuzzy topology or an ML-graded topology forshort.

In classical topology, as well as, to a large extent, in fuzzy topology, the notionof an open set is usually taken as the primitive and that of a closed set being anauxiliary one, since closed sets are easily obtained from open by taking comple-ments. However in some cases it is reasonable to consider open and closed sets asindependent notions. This is especially crucial when dealing with L-fuzzy topolo-gies in case when the lattice L is not equipped with an order reversing involution.To handle with such and analogous more general problems, L.M. Brown with co-aurthors has developed the theory of a dichotomous topology, or just ditopology inshort [3], [4], [5], etc. Developing the idea of a ditopology, we have introduced andstudied the graded version of a ditopology in [6]. In the context of this work thetermML-ditopology on a set means just a pair of mutually independent mappingsT : LX → (LM )X and K : LX → (LM )X satisfying certain topological axioms, seeSubsections 5.1, 5.2 for the precise denitions. It is the aim of this section to elab-orate a construction ofML-ditopologies induced on LM -valued sets by LM -valuedequalities.

5.1. ML-graded topology on an LM -valued set. Let (X,E) be an LM -valuedset and let lE : LX → (LM )X be the lower LM -rough approximation operatorinduced on this set. Further, let as before, its α-levels lαE : LX → LX be dened

¶This term was already used by some authors, [8], [9].

27

by lαE(A)(x) = lE(A)(x, α). Then the properties (1l)− (4l) of lE related to lαE canbe reformulated as follows:(1lα) lα(1X) = 1L;(2lα) A ≥ lαE(A) ∀A ∈ LX ;(3lα) lαE(

∧iAi) =

∧i lE(Ai) ∀Ai | i ∈ I ⊆ LX in particular

(3'lα) lαE(A1 ∧A2) = lαE(A1) ∧ u(A2)∀A1, A2 ∈ LX .(4lα) lαE(l

αE(A)) = lαE(A) ∀A ∈ LX ;

However, this means that lαE : LX → LX can be interpreted as an L-fuzzy interioroperator on the set X. (This fact is well-known, see, e.g. [28], [41], [42]). Henceby setting Tα = A ∈ LX : lαE(A) = A, we obtain the L-fuzzy topology corre-sponding to this L-fuzzy interior operator. Moreover, the property (3l) allows toconclude that it is actually an Alexandro L-fuzzy topology (see e.g. [26], [10]),that is the intersection axiom holds also for innite families. Thus for each α thefamily Tα satises the following axioms of an Alexandro L-fuzzy topology:

(1) 1X ∈ Tα;(2) Ai : i ∈ I ⊆ Tα =⇒ ∧

iAi ∈ Tα;(3) Ai : i ∈ I ⊆ Tα =⇒ ∨

iAi ∈ TαTaking such L-fuzzy topologies for all α ∈M , we obtain the family Tα : α ∈M.Besides, since lαE ≤ lβE whenever α ≤ β, we conclude that

α ≤ β =⇒ Tα ⊃ Tβ ,that is the family Tα : α ∈ M is non-increasing. We use this family of L-fuzzytopologies to dene an (Alexandro) ML-graded topology T on the set X, bysetting

T(A) =∨α ∈M : A ∈ Tα.

5.1. Theorem. If M is completely distributive, then T is an M -graded L-fuzzytopology on the LM -valued set (X,E), that is T : LX → M satises the followingaxioms:

(1) T(1X) = 1M ;(2) T(

∧iAi) ≥

∧i T(Ai) for every family Ai : i ∈ I ⊆ LX ;

(3) T(∨iAi) ≥

∧i T(Ai) for every family Ai : i ∈ I ⊆ LX ;

Proof The rst property is obvious, since 1X ∈ Tα for all α ∈M .To prove the second property, take any family Ai : i ∈ I ⊆ LX and assume

that∧i T(Ai) = α. In case α = 0M the inequality is obvious, therefore we assume

that α > 0M . Take any β C α where C is the way below relation on the completelydistributive lattice M . From the denition of T it is clear that Ai ∈ Tβ for everyi ∈ I and hence, recalling that Tβ is an Alexandro L-fuzzy topology, we concludethat also

∧iAi ∈ Tβ . Therefore T(

∧iAi) ≥ β. Since this is true for any β C α and

lattice M is completely distributive, we conclude that T(∧iAi) ≥ α =

∧i T(Ai).

The proof of the third property is similar and we omit it.

5.2. Graded co-topology of an LM -valued set. Let (X,E) be an LM -valuedset and let uE : LX → (LM )X be the upper rough approximation operator inducedby the LM -valued equality E on the set X. Further, as before, let its α-levels uαE :LX → LX be dened by uαE(A)(x) = uE(A)(x, α). Then properties (1u)− (4u) of

28

the upper LM -rough approximation operator uE related to uαE can be reformulatedas follows:

(1uα) uα(1X) = 1L;(2uα) A ≤ uαE(A) ∀A ∈ LX ;(3uα) uαE(

∨iAi) =

∨i lE(Ai) ∀Ai | i ∈ I ⊆ LX in particular

(3'uα) uαE(A1 ∧A2) = uαE(A1) ∧ u(A2)∀A1, A2 ∈ LX .(4uα) uαE(u

αE(A)) = uαE(A) ∀A ∈ LX ;

However, this means that uαE : LX × LX can be interpreted as an L-fuzzy closureoperator on the set X (This fact is well-known, see, e.g. [28], [41], [42]). Henceby setting Kα = A ∈ LX : uαE(A) = A, we obtain the L-fuzzy co-topologycorresponding to this L-fuzzy closure operator. Moreover, the property (3u) allowsto conclude that it is actually an Alexandro L-fuzzy co-topology [10]: this meansthat the union axiom holds also for innite families. Thus, for each α the familyKα satises the following axioms of an Alexandto L-fuzzy co-topology:

(1) 1X ∈ Kα;(2) Ai : i ∈ I ⊆ Kα =⇒ ∨

iAi ∈ Kα;(3) Ai : i ∈ I ⊆ Kα =⇒ ∧

iAi ∈ Kα

Taking such L-fuzzy co-topologies for all α ∈ M , we obtain the family Kα : α ∈M. Besides, since uαE ≥ uβE whenever α ≤ β, we conclude that

α ≤ β =⇒ Kα ⊃ Kβ ,

that is the family Kα : α ∈ M is non-increasing. We use this family of L-fuzzyco-topologies to dene an (Alexandro) L-fuzzy co-topology K on the set X, bysetting

K(A) =∨α ∈M : A ∈ Kα.

5.2. Theorem. IfM is completely distributive, then K is anM -graded L-fuzzy co-topology on the LM -valued set (X,E). This means that the mapping K : LX →Msatises the following axioms:

(1) K(1X) = 1M ;(2) K(

∨iAi) ≥

∧iK(Ai) for every family Ai : i ∈ I ⊆ LX ;

(3) K(∧iAi) ≥

∧iK(Ai) for every family Ai : i ∈ I ⊆ LX ;

Proof The rst property is obvious, since 1X ∈ Kα for all α ∈M .To prove the second property, take any family Ai : i ∈ I ⊆ LX and assume

that∧iK(Ai) = α. In case α = 0M the inequality is obvious, therefore we

assume that α > 0M . Take any β C α where C is the wedge-below relation inthe completely distributive lattice. Then from the denition of K it is clear thatAi ∈ Kβ for every i ∈ I, and hence, recalling that Kβ is an Alexandro L-fuzzyco-topology, we conclude that also

∨iAi ∈ Kβ . Therefore K(

∨iAi) ≥ β. Since

this is true for any β C α and lattice M is completely distributive, we concludethat K(

∨iAi) ≥ α =

∧iK(Ai).

The proof of the third property is similar and we omit it.

29

6. Construction of an LM -valued equality from a pseudo-metric

In this section we construct an LM -valued equality Eρ from an ordinary pseudo-metric ρ on a set X. We think that this construction presents an important sourcefor creation of many examples of LM -valued sets with prescribed properties.

Let L =M = [0, 1] be the unit intervals viewed as lattices and let ∗ : L×L→ Lbe a continuous t-norm. Further, let X be a set and ρ : X×X → [0, 1] be a pseudo-metric on this set. We dene a mapping Eρ : X ×X × [0, 1]→ [0, 1] by setting

Eρ(x, y)(α) =

1−α1−α+αρ(x,y) if α 6= 1 or ρ(x, y) 6= 0

1 if α = 1 and ρ(x, y) = 0.

It is easy to see that the mapping E(x, y)(·) : [0, 1] → [0, 1] is continuous for anyx, y ∈ [0, 1]. Indeed, the statement is obvious if ρ(x, y) 6= 0 or α 6= 1, otherwiselimα→1E(x, y)(α) = limα→1

1−α1−α+αρ(x,y) = 1.

6.1. Proposition. For every pseudo-metric ρ : X × X → [0, 1] the mappingEρ : X × X × [0, 1] → [0, 1] satises conditions (1ELM ), (2ELM ), (4ELM ),(5ELM ), (6ELM ), (7ELM ) and (8ELM ). The mapping Eρ : X × X × [0, 1] →[0, 1] satises condition (3ELM ) in cases of the product t-norm ∗ = · and ofthe ukasiewicz t-norm ∗ = ∗L. If ρ is an ultra pseudo-metric, then mappingEρ : X × X × [0, 1] → [0, 1] satises condition (3ELM ) in case of the minimumt-norm ∗ = ∧.

The validity of conditions (1ELM ) and (2ELM ) follows directly from the de-nition of the mapping Eρ : X ×X × [0, 1]→ [0, 1].

To prove (3ELM ) consider separately the cases of the three t-norms:∗ = ∧ Since in this case ρ is assumed to be an ultra pseudo-metric, we have

ρ(x, y) ≤ maxρ(x, z), ρ(z, y) for all x, y, z. It is straightforward to con-clude from here that

1− α1− α+ αρ(x, y)

≥ 1− α1− α+ αρ(x, z)

∧ 1− α1− α+ αρ(z, y)

.

∗ = · The inequality1− α

1− α+ αρ(x, y)≥ 1− α

1− α+ αρ(x, z)· 1− α1− α+ αρ(z, y)

can be easily established taking into account the triangular property ρ(x, y) ≤ρ(x, z) + ρ(z, y) of the pseudo-metric ρ.

∗ = ∗L It is well known that ∗L ≤ · and hence this property follows from theanalogous property of the product t-norm establish above.

Property (4ELM ) follows directly from the denition of the LM valued equalityEρ.

To prove Property (5ELM ) let α =∨n∈N αn for some α ∈ [0, 1] and αn : n ∈

N ⊂ [0, 1]. Without loss of generality we may assume that

n ≤ n+ 1⇒ αn ≤ αn+1 for every n ∈ N.Then, referring to the continuity and already the established non-increaseness ofthe mapping Eρ(x, y) : [0, 1]→ [0, 1] we have

Eρ(x, y, α) = Eρ(x, y, limn→∞

αn) = limn→∞

Eρ(x, y, αn) =∧

n∈NEρ(x, y, αn).

30

To prove Property (6ELM ), let α =∧n∈N αn for some αn ∈ [0, 1]. Without

loss of generality we may assume that n ≤ n + 1 ⇒ αn ≥ αn+1 for every n ∈ N.Then, referring to the continuity and already established non-increaseness of themapping Eρ(x, y) : [0, 1]→ [0, 1] we have

Eρ(x, y, α) = Eρ(x, y, limn→∞

αn) = limn→∞

Eρ(x, y, αn) =∨

n∈NEρ(x, y, αn).

From the denition of Eρ it is clear that Eρ(x, y, 0) = 1 for every x, y ∈ X and

Eρ(x, y)(1M ) =

1L if x = y0L otherwise,

and hence (7ELM ) and (8ELM ) hold. 2

6.2. Corollary. In case ∗ = · and ∗ = ∗L the mapping Eρ : X×X → [0, 1]→ [0, 1]is a global continuous LM -equality for any pseudo-metric ρ : X × X → [0, 1]. Ifρ is an ultra pseudo-metric, then Eρ is a global continuous LM -valued equality incase ∗ = ∧.6.3. Remark. It is well known that for every pseudo-metric d : X ×X → (0,∞)there exist fuzzy metrics ρ : X×X → [0, 1] equivalent to the given pseudo-metric d.By saying equivalent we mean that d and ρ induce the same topology on the set X.Therefore, if we start with an arbitrary pseudo-metric d : X×X → (0,∞), then wetake the equivalent pseudo-metric ρ : X ×X → [0, 1] dened by ρ(x, y) = d(x,y)

1+d(x,y)

as its counterpart. In this case LM -valued equality Eρ can be rewritten as

Ed(x, y, α) =(1− α)(1 + d(x, y)

1− α+ d(x, y.

7. Conclusion

We have introduced the notions of an LM -valued equality and an LM -valuedset, which conceptionally generalize the concepts of an L-valued equality and anL-valued set, well-known to people working in this eld. We have studied the basicproperties of these concepts. An example of an LM -valued equality induced by abounded pseudometric was presented. We showed that LM -equalities induce in anatural way a certain kind of many-valued rough approximation operators; we callthem an upper and a lower LM -rough approximation operators. Finally we applythese operators to construct an ML-graded ditopology on the LM -valued set.

We view this work as the rst part of the reserach in this direction. Amongimportant, in our opinion, issues, which remained beyond the scope of this work,we mention here the following:

In this work we did not touch the question how special properties of LM -equalities (upper and lower semicontinuity, etc.,) are reected in the structure ofthe constructedML-graded topologies? Can we characterize the class of ditopolo-gies which are induced by an LM -equality with a certain property? In particular,how do the levels Tα and Kα of the ML-graded topology T and K are related tothe L-fuzzy topology Tα and co-topology Kα depending on the properties of theLM -valued equality E ?

Having LM -valued sets on one side and ML-graded ditopogical spaces on theother it seems important to study their relations on the categorical level, that

31

is when certain ordinary functions or fuzzy functions [11], [43], [24] are takenas morphisms in the corresponding category. A similar question was studied forordinary L-valued sets in our paper [14].

Related to the previous question: what are the connections between the opera-tions in the (prospective!) category of LM -valued sets (products, coproducts, etc)and the corresponding operations in the category of LM -graded ditopologies?

References

[1] U. Bodenhofer, Ordering of fuzzy sets based on fuzzy orderings. I: The basic approach,Mathware Soft Comput. 15 (2008) 201218.

[2] U. Bodenhofer, Ordering of fuzzy sets based on fuzzy orderings. II: Generalizations, Math-ware Soft Comput. 15 (2008) 219249.

[3] L.M. Brown, M. Diker, Ditopological texture spaces and intuitionistic sets Fuzzy Sets andSyst. 98, (1998) 217224.

[4] L.M. Brown, R. Ertürk, . Dost, Ditopological texture spaces and fuzzy topology, I. Basic

concepts, Fuzzy Sets and Syst. 147, (2004) 171199.[5] L.M. Brown, R. Ertürk, . Dost, Ditopological texture spaces and fuzzy topology, II. Topo-

logical considerations, Fuzzy Sets and Syst. 147, (2004) 201231. 18861912.[6] L.M. Brown, A. ostak, Categories of fuzzy topologies in the context of graded ditopologies,

Iranian J. Of Fuzzy Systems, Systems 11, No. 6, (2014) pp. 1-20[7] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24, 182190 (1968)[8] K.C. Chattopadhyay, R.N. Nasra, S.K. Samanta, Gradation of openness: Fuzzy topology,

Fuzzy Sets and Syst. 48 (1992), 237242.[9] K.C. Chattopadhyay, S.K. Samanta, Fuzzy closure operators, fuzzy compactness and fuzzy

connectedness, Fuzzy Sets and Syst. 54 (1992), 237242.[10] P. Chen, D. Zhang, Alexandro L-cotopological spaces, Fuzzy Sets and Systems 161 (2010)

2505 2525.[11] M. Demirci, Fuzzy functions and their fundamental properties, Fuzzy Sets Syst. 106 (1999),

239246[12] D. Dubois, H. Prade,Rough fuzzy sets and fuzzy rough sets. Internat. J. General Syst. 17,

191209 (1990)[13] A. El

,kins, A. ostak, On some categories of approximate systems generated by L-relations.

In: 3rd Rough Sets Theory Workshop, pp. 1419 Milan, Italy (2011)[14] A. El

,kins, A. ostak, I. Ul

,jane, On a category of extensional fuzzy rough approximation

L-valued spaces, In: Communication in Computer and Information Science, vol 611 (2016).16th. International Conference IPMU 2016, Eindhofen, The Netherlands, June 20-24, 2016,Proceedings, Part II, 36-47.

[15] G. Gierz, K.H. Homan, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott, ContinuousLattices and Domains, Cambridge University Press, Cambridge (2003)

[16] J.A. Goguen, The fuzzy Tychono theorem, J. Math. Anal. Appl. 43, 734742 (1973)[17] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967), 145174.[18] J. Hao, Q. Li, The relation between L-fuzzy rough sets and L-topology, Fuzzy Sets and

Systems, 178 (2011)[19] U. Höhle, M-valued sets and sheaves over integral commutative cl-monoids Chapter 2 in

Applications of Category Theory to Fuzzy Sets, 1992, Kluwer Acad. Press, S.E. Rodabaugh,E.P. Klement and U. Höhle eds, pp. 3372.

[20] U. Höhle Commutative, residuated l-monoids, in: S.E. Rodabaugh, E.P. Klement, U. Höhleeds., Non-classical logics and their applications to Fuzzy Sets. Kluwer Acad. Publ. Dodrecht,Boston 1995, pp. 53106.

[21] U. Höhle Many-valued equalities, singletons and fuzzy partitions Soft Computing 2 (1998),134140.

[22] U. Höhle, Many Valued Topology and its Application Kluwer Acad. Publ. 2001, Boston,Dodrecht, London.

32

[23] J. Järvinen, J. Kortelainen, A unied study between modal-like operators, topologies and

fuzzy sets, Fuzzy Sets and Systems 158 (2007) 12171225.[24] F. Klawonn, Fuzzy points, fuzzy relations and fuzzy functions, in: V. Novák, I. Perlieva

(Eds.), Discovering the World with Fuzzy Logic, Springer, Berlin, 2000, pp. 431453.[25] E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Acad. Publ., 2000.[26] J. Kortelainen, On relationship between modied sets, topological spaces and rough sets,

Fuzzy Sets and Systems 61 (1994) 91-95.[27] T. Kubiak, On fuzzy topologies, PhD Thesis, Adam Mickiewicz University Pozna«, Poland

(1985)[28] Liu Yingming, Luo Maokang, Fuzzy Topology Advances in Fuzzy Systems - Applications

and Topology. World Scientif. Singapore, New Jersey, London, Hong Kong, 1997.[29] K. Menger, Probabilistic geometry, Proc. N.A.S. 27 (1951), 226229.[30] J.S. Mi, B.Q. Hu Topological and lattice structure of L-fuzzy rough sets determined by upper

and lower sets, Information Sciences 218 (2013) 194204.[31] W. Morgan, and R. P. Dilworth Residuated lattices Trans. Amer. Math. Soc. 45 (1939) 335-

354. Reprinted in K.Bogart, R. Freese, and J. Kung eds. The Dilworth Theorems: SelectedPapers of R.P. Dilworth Basel, 1990 Birkhauser.

[32] Z. Pawlak, Rough sets , International J. of Computer and Information Sciences, 11 (1982)341-356.

[33] K. Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems151 (2005) 601613.

[34] K. Qin, Z. Pei, Generalized rough sets based on reexive and transitive relations, InformationSciences 178 (2008) 41384141.

[35] A.M. Radzikowska, E.E. Kerre, A comparative study of fuzzy rough sets. Fuzzy Sets andSyst. 126, 137155 (2002)

[36] G.N. Raney, A subdirect-union representation for completely distibutive complete lattices,Proc. Amer. Math. Soc. 4 (1953), 518522.

[37] K.I. Rosenthal Quantales and Their Applications, Pirman Research Notes in Mathematics234. Longman Scientic & Technical (1990)

[38] B. Schweizer, A. Sklar, Statisitcal metric spaces, Pacic J. Math. 10, 215229.[39] A. Skowron, On the topology in information systems, Bull. Polon. Acad. Sci. Math. 36

(1988), 477480.[40] A. ostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. Palermo Ser II 11

(1985), 89-103[41] A. ostak, Two decades of fuzzy topology: Basic ideas, notions and results, Russian Math.

Surveys, 44 (1989), 125186[42] A. ostak, Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), 662701.[43] A. ostak, Fuzzy functions and an extension of the category L-TOP of Chang-Goguen L-

topological spaces, Proceedings of the 9th Prague Topological Symposium (2001), 271294.[44] S.P. Tiwari, A.K. Srivastava, Fuzzy rough sets. fuzzy preoders and fuzzy topoloiges, Fuzzy

Sets and Syst., 210 (2013), 6368.[45] I. Ul

,jane, On the order type L-valued relations on L-powersets, Soft Computing 14 (2007),

183-199.[46] L. Valverde, On the structure of F -indistinguishibility operators, Fuzzy Sets and Syst. 17

(1985) 313328.[47] A. Wiweger, On topological rough sets, Bull. Polon. Acad. Sci. Math. 37 (1988), 5162.[48] H. Yu, W.R. Zhan, On the topological properties of generalized rough sets, Information

Sciences 263 (2014), 141152.[49] L. Zadeh, Fuzzy sets, Information and Control (1965)[50] L. Zadeh Similarity relations and fuzzy orderings, Inf. Sci. 3 (1971) 177200.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 33 52

A study on quasi-pseudometrics

In memory of Professor Lawrence M. Brown

Natasha Demetriou ∗ and Hans-Peter A. Künzi †

Abstract

We study some aspects of the space QPM(X) of all quasi-pseudometrics on a set X equipped with the extended T0-quasi-metricAX(f, g) = sup(x,y)∈X×X(f(x, y)−g(x, y)) whenever f, g ∈ QPM(X).We observe that this space is bicomplete and exhibit various closedsubspaces of (QPM(X), τ((AX)s)).In the second part of the paper, as a rough way to measure the asym-mety of a quasi-pseudometric f on a set X, we investigate some prop-erties of the value (AX)s(f, f−1).

Keywords: quasi-pseudometric; T0-quasi-metric; nonnegatively weightable quasi-pseudometric

2000 AMS Classication: 54E35; 54E15; 54E05; 54E55

1. Introduction

On the set QPM(X) of all quasi-pseudometrics on the set X we introduce theextended T0-quasi-metric AX dened by

AX(f, g) = sup(x,y)∈X×X

(f(x, y)−g(x, y))

whenever f, g ∈ QPM(X).‡ Let us immediately mention that obviously thespecialization order ≤AX of AX is the usual order on QPM(X), that is, for f, g ∈∗Department of Mathematics and Applied Mathematics, University of Cape Town, Ronde-

bosch 7701, South AfricaEmail : natashademetriou7@gmail.com†Department of Mathematics and Applied Mathematics, University of Cape Town, Ronde-

bosch 7701, South AfricaEmail : hans-peter.kunzi@uct.ac.zaThe authors would like to thank the South African National Research Foundation for partialnancial support under grants IFR1202200082 and CPRR14071175245.‡For a, b ∈ R we set a−b = maxa− b, 0 = (a− b) ∨ 0.

Doi : 10.15672/HJMS.2016.396

34

QPM(X) we have f ≤AX g i AX(f, g) = 0 i f(x, y) ≤ g(x, y) whenever (x, y) ∈X ×X.

1. Remark. We could also consider the bounded counterpart of AX dened byminAX , 1. In the analogous metric construction this approach was for instancechosen for the studies [23, 24]. Since however we are mainly interested in largedistance values as they are investigated for instance in the theory of coarse spaces(e.g. [22]), this is not the approach that we have chosen in this paper.

Below we establish that the space (QPM(X), AX) is bicomplete. We also showthat various natural subspaces of QPM(X) are τ((AX)s)-closed and thus bicom-plete, for instance the set of all totally bounded quasi-pseudometrics on X, the setof all ultra-quasi-pseudometrics on X and the set of all nonnegatively weightablequasi-pseudometrics on X.

In the second part of the paper we consider for any quasi-pseudometric f on Xits value of asymmetry dened by Af := (AX)s(f, f−1). The denition is obviouslymotivated by the fact that f is a pseudometric on X if and only if (AX)s(f, f−1) =0. ¶

We discuss some properties of the introduced concept and consider variousinequalities that are useful to compute it for suitable quasi-pseudometric spaces(X, f).

2. The space QPM(X) of all quasi-pseudometrics

After recalling the main denitions of the notions used in this paper, we shall es-tablish bicompleteness of the space (QPM(X), AX) and exhibit various τ((AX)s)-closed subspaces of (QPM(X), AX). For a more detailed discussion of the basicconcepts dealt with in this paper the reader may want to consult [7, 13].

1. Denition. Let X be a set and let d : X ×X → [0,∞) be a function mappinginto the set [0,∞) of the nonnegative reals. Then d is called a quasi-pseudometric

on X if(a) d(x, x) = 0 whenever x ∈ X, and(b) d(x, z) ≤ d(x, y) + d(y, z) whenever x, y, z ∈ X.We shall say that d is a T0-quasi-metric provided that d also satises the fol-

lowing condition (c): For each x, y ∈ X,d(x, y) = 0 = d(y, x) implies that x = y.The specialization order ≤d of d is dened by x ≤d y i d(x, y) = 0 whenever

x, y ∈ X.2. Remark. In some cases it is more natural to assume that a quasi-pseudometricd indeed maps into [0,∞].We shall then speak of an extended quasi-pseudometric.‖

It should also be mentioned that the terminology in the literature is fairly diverse(compare for instance [10, Chapter 6]).

For later use we note that the extended T0-quasi-metric AX can indeed be dened forarbitrary functions f, g : X ×X → [0,∞). Let us mention that we shall however not dene AX

in the case of extended functions f and g in this paper.¶We remark that in the paper [21] a measure of asymmetry is considered that is based on the

quotient ff−1 instead of the dierence f−f−1 .

‖For extended quasi-pseudometrics the triangle inequality is interpreted in the obvious way.

35

1. Example. (compare for instance [8, Example 2]) On the set R of the reals setu(x, y) = x−y whenever x, y ∈ R. Then u is the standard T0-quasi-metric on R.

3. Remark. Let d be a quasi-pseudometric on a set X. Then d−1 : X ×X → [0,∞) dened by d−1(x, y) = d(y, x) whenever x, y ∈ X is also a quasi-pseudometric on X, called the conjugate or dual quasi-pseudometric of d. As usual,a quasi-pseudometric d on X such that d = d−1 is called a pseudometric. Note thatfor any (T0-)quasi-pseudometric d, ds = supd, d−1 = d ∨ d−1 is a pseudometric(metric).

The following auxiliary result is well known. Its proof is included here for theconvenience of the reader.

1. Lemma. (see for instance [14, Lemma 8]) Let (X, d) be a quasi-pseudometricspace and a, b, x, y ∈ X. Then |d(x, y)− d(a, b)| ≤ ds(x, a) + ds(y, b).

Proof. We have that d(x, y) ≤ d(x, a) + d(a, b) + d(b, y), and therefore d(x, y)−d(a, b) ≤ d(x, a)+d(b, y). Similarly d(a, b) ≤ d(a, x)+d(x, y)+d(y, b), and therefored(a, b)− d(x, y) ≤ d(a, x) + d(y, b). Thus |d(x, y)− d(a, b)| ≤ ds(x, a) + ds(y, b). 2

As we have announced above, we equip the setQPM(X) of all quasi-pseudometricson X with the (extended) function

AX(f, g) = sup(x,y)∈X×X

(f(x, y)−g(x, y))

whenever f, g ∈ QPM(X).

1. Proposition. We have that (QPM(X), AX) is an extended T0-quasi-metricspace.

Proof. The argument is obvious and left to the reader. 2

4. Remark. Note that by denition AX(d, e) = AX(d−1, e−1) whenever d, e ∈QPM(X). In particular for any quasi-pseudometric d on a set X we have thatAX(d, d−1) = AX(d−1, d) = (AX)s(d, d−1).

5. Remark. Let X be a set, d a quasi-pseudometric on X and 0 the con-stant quasi-pseudometric equal to 0. Then AX(d, 0) is equal to the diameter

δd = sup(x,y)∈X×X d(x, y) of (X, d).

2. Lemma. Let d, e, f, g be quasi-pseudometrics on a set X.(a) Then AX(d+ e, f + g) ≤ AX(d, f) +AX(e, g), where d+ e, f + g are quasi-

pseudometrics on X.(b) Furthermore AX(αd, αf) = αAX(d, f) whenever α is a nonnegative real,

where αd and αf are quasi-pseudometrics on X.(c) If f ≥ g and h ≥ e, then AX(f, e) ≥ AX(g, h).

Proof. All these computations are straightforward. 2

In the following ∆X will denote the diagonal (x, x) : x ∈ X of the set X.

36

2. Example. Let≤ be a partial order on a setX. Set, for each x, y ∈ X, d≤(x, y) =0 if x ≤ y and d≤(x, y) = 1 otherwise. Then d≤ is a T0-quasi-metric on X, whichis called the natural T0-quasi-metric of (X,≤) (compare for instance [2, Section4]). We now consider the following specic example of this construction: Let Xbe the set of integers Z. Set

≤= ∆Z ∪ (2n, 2n+ 1) : n ∈ Z ∪ (2n, 2n− 1) : n ∈ Z.Then ≤ is a partial order on Z. Of course, ≥= (≤)−1 = ∆Z ∪ (2n + 1, 2n) :

n ∈ Z ∪ (2n − 1, 2n) : n ∈ Z. We have that d≤ ∧ (d≤)−1 = 0, since ≤ ∪(≥) = (x, y) ∈ Z× Z : |x− y| ≤ 1. Here we have (d≤)−1 = d≥ and d≤ ∧ d≥ is thelargest quasi-pseudometric which is ≤ d≤ and ≤ d≥. ∗∗

It follows that d≤ ∧ (d≤)−1 < mind≤, (d≤)−1. Obviously mind≤, (d≤)−1does not satisfy the triangle inequality.

3. Lemma. Let X be a set and functions d1, d2 : X ×X → [0,∞) be given. Setb := mind1, d2 and s := d1 ∨ d2 = maxd1, d2. ††

Then (AX)s(d1, d2) = (AX)s(s, b). (Of course, AX(b, s) = 0.)

Proof. By Lemma 2(c) we have that AX(s, b) ≥ AX(d1, d2) and analogouslyAX(s, b) ≥ AX(d2, d1). Therefore AX(s, b) ≥ (AX)s(d1, d2).

Let x, y ∈ X. By considering the various possibilities in any case we have thats(x, y) − b(x, y) ≤ (d1(x, y) − d2(x, y)) ∨ (d2(x, y) − d1(x, y)) ≤ AX(d1, d2) ∨AX(d2, d1) = (AX)s(d1, d2). Hence AX(s, b) ≤ (AX)s(d1, d2). We conclude thatAX(s, b) = (AX)s(d1, d2). 2

1. Corollary. Let X be a set and functions d1, d2 : X × X → [0,∞) be given,and s and b as dened in Lemma 3.

Then AX(s, d2) = AX(d1, d2) and AX(d1, b) = AX(d1, d2).

Proof. By Lemma 2(c) we have that AX(s, d2) ≥ AX(d1, d2).Let x, y ∈ X. By considering the various possibilities, in any case we have

s(x, y)−d2(x, y) ≤ d1(x, y)−d2(x, y) ≤ AX(d1, d2) and thusAX(s, d2) ≤ AX(d1, d2).The second part of the proof is similar: AX(d1, b) ≥ AX(d1, d2) by Lemma

2(c). Let x, y ∈ X. Then by considering the various possibilities, in any case wehave d1(x, y) − b(x, y) ≤ d1(x, y) − d2(x, y) ≤ AX(d1, d2). Therefore AX(d1, b) ≤AX(d1, d2). 2

2. Proposition. Let X be a set and functions d, e, f, g : X × X → [0,∞) begiven. Then AX(d ∨ e, f ∨ g) ≤ AX(d, f) ∨AX(e, g).

Proof. Let x, y ∈ X. Then we consider the four cases:Case 1: (d ∨ e)(x, y) = d(x, y) and (f ∨ g)(x, y) = f(x, y). Then (d ∨ e)(x, y)−

(f ∨ g)(x, y) ≤ AX(d, f).Case 2: (d ∨ e)(x, y) = d(x, y) and (f ∨ g)(x, y) = g(x, y). Then (d ∨ e)(x, y)−

(f ∨ g)(x, y) ≤ d(x, y)− f(x, y) ≤ AX(d, f), because f(x, y) ≤ g(x, y).

∗∗The general construction of the inmum of two quasi-pseudometrics will be discussed brieybelow in the last section of this paper.††Note that if d1, d2 are quasi-pseudometrics, then s is a quasi-pseudometric, while b need

not satisfy the triangle inequality, as Example 2 shows.

37

Case 3: (d ∨ e)(x, y) = e(x, y) and (f ∨ g)(x, y) = f(x, y). Then (d ∨ e)(x, y)−(f ∨ g)(x, y) ≤ e(x, y)− g(x, y) ≤ AX(e, g), because g(x, y) ≤ f(x, y).

Case 4: (d ∨ e)(x, y) = e(x, y) and (f ∨ g)(x, y) = g(x, y). Then (d ∨ e)(x, y)−(f ∨ g)(x, y) ≤ AX(e, g).

The assertion follows. 2

2. Corollary. Let X be a set and functions d, e, f, g : X ×X → [0,∞) be given.Then AX(mind, e,minf, g) ≤ AX(d, f) ∨AX(e, g).

Proof. Let x, y ∈ X. Then we consider the four cases:Case 1: (mind, e)(x, y) = d(x, y) and (minf, g)(x, y) = f(x, y).

Then (mind, e)(x, y)− (minf, g)(x, y) ≤ AX(d, f).Case 2: (mind, e)(x, y) = d(x, y) and (minf, g)(x, y) = g(x, y).

Then (mind, e)(x, y)− (minf, g)(x, y) = d(x, y)− g(x, y) ≤ AX(e, g), becausee(x, y) ≥ d(x, y).

Case 3: (mind, e)(x, y) = e(x, y) and (minf, g)(x, y) = f(x, y).Then (mind, e)(x, y)− (minf, g)(x, y) = e(x, y)− f(x, y) ≤ AX(d, f), becaused(x, y) ≥ e(x, y).

Case 4: (mind, e)(x, y) = e(x, y) and (minf, g)(x, y) = g(x, y).Then (mind, e)(x, y)− (minf, g)(x, y) ≤ AX(e, g).

The assertion follows. 2

4. Lemma. Let dn (n ∈ N) and d be quasi-pseudometrics on a set X such thatlimn→∞AX(d, dn) = 0. Then limn→∞AX(d−1, (dn)−1) = 0 and

limn→∞

AX(ds, (dn)s) = 0.

Proof. The rst statement follows from Remark 4. The second statementis a consequence of Proposition 2: Indeed we conclude that AX(ds, (dn)s) ≤AX(d, dn)∨AX(d−1, (dn)−1) whenever n ∈ N. The assertion now is a consequenceof the rst statement. 2

3. Example. Let X be a set and for each λ ∈ [0, 1] set K(f, g, λ) = λf + (1−λ)gwhere f, g ∈ QPM(X) (compare [19]).

Note that K(f, g, λ) = K(g, f, 1− λ) whenever f, g ∈ QPM(X) and λ ∈ [0, 1].Furthermore, obviously, eachK(f, g, λ) is a quasi-pseudometric onX, K(f, g, 0) =

g and K(f, g, 1) = f.Let λ, λ′ ∈ [0, 1]. Suppose that λ′ ≤ λ.Then by a straightforward computation we see that

AX(K(f, g, λ),K(f, g, λ′)) = (λ− λ′)AX(f, g)

and

AX(K(f, g, λ′),K(f, g, λ)) = (λ− λ′)AX(g, f).

In particular, since for any quasi-pseudometric d on a set X we have thatAX(d, d−1) = AX(d−1, d) by Remark 4, for any λ, λ′ ∈ [0, 1] we get that

AX(K(d, d−1, λ),K(d, d−1, λ′)) = AX(K(d, d−1, λ′),K(d, d−1, λ)) =

|λ− λ′|AX(d, d−1).

38

3. Corollary. Let X be a set and let d be a quasi-pseudometric on X. Set d+ =d+ d−1. Then d+ is a quasi-pseudometric on X.

We have AX(d, d+

2 ) = AX(K(d, d−1, 1),K(d, d−1, 12 )) = 1

2AX(d, d−1) and simi-

larly AX(d+

2 , d−1) = AX(K(d, d−1, 1

2 ),K(d, d−1, 0)) = 12AX(d, d−1).

Indeed

AX(d,d+

2) = AX(

d+

2, d−1) =

1

2AX(d, d−1) =

1

2AX(d−1, d) = AX(d−1,

d+

2) = AX(

d+

2, d).

Proof. The assertion follows from Remark 4 and Example 3. 2

3. The dab-construction

In the following we recall a modication of a T0-quasi-metric d studied in [8,Section 5]. Below we give some of the details of the proofs that were omitted in[8, 9].

3. Proposition. (compare [8, Lemma 2]) Given a T0-quasi-metric d on X anda, b ∈ X be such that d(a, b) > 0 and d(b, a) > 0, we dene dab(x, y) = mind(x, a)+d(b, y), d(x, y) whenever x, y ∈ X. Then dab is the largest T0-quasi-metric satis-fying e ≤ d on X such that e(a, b) = 0.

Proof. The statement that dab ≤ d is obvious by denition of dab. Furthermoredab(a, b) = 0, hence dab < d. It is easy to see that dab is a quasi-pseudometric: Weonly have to show that dab(x, z) ≤ dab(x, y) + dab(y, z) whenever x, y, z ∈ X.

We consider the four cases:(1) dab(x, y) = d(x, y) and dab(y, z) = d(y, z).(2) dab(x, y) = d(x, a) + d(b, y) and dab(y, z) = d(y, z).(3) dab(x, y) = d(x, y) and dab(y, z) = d(y, a) + d(b, z).(4) dab(x, y) = d(x, a) + d(b, y) and dab(y, z) = d(y, a) + d(b, z).In Case (1) we obtain dab(x, z) ≤ d(x, z) ≤ d(x, y) + d(y, z).In Case (2) we obtain dab(x, z) ≤ d(x, a) + d(b, z) ≤ d(x, a) + d(b, y) + d(y, z).In Case (3) we obtain dab(x, z) ≤ d(x, a) + d(b, z) ≤ d(x, y) + d(y, a) + d(b, z).In Case (4) we obtain dab(x, z) ≤ d(x, a) + d(b, z) ≤ d(x, a) + d(b, y) + d(y, a) +

d(b, z).Hence we are done. In the proof of [8, Lemma 2] it is argued that dab satises

the T0-condition (c), because d does so and because d(b, a) > 0.Let us now note that if e ≤ d is a quasi-pseudometric on X such that e(a, b) = 0,

then we have that for any x, y ∈ X, e(x, y) ≤ e(x, a) + e(a, b) + e(b, y) ≤ d(x, a) +d(b, y) and e(x, y) ≤ d(x, y). Therefore e ≤ dab. 2

6. Remark. Let (X, d) be a T0-quasi-metric space and let a, b ∈ X be ≤d-incomparable. Then (dab)

−1 = (d−1)ba according to [9, Remark 1]: Indeed letx, y ∈ X. Then (dab)

−1(x, y) = mind(y, a) + d(b, x), d(y, x) = mind−1(x, b) +d−1(a, y), d−1(x, y) = (d−1)ba(x, y).

4. Proposition. Let d be a T0-quasi-metric on a set X and let a, b ∈ X beincomparable with respect to the specialization order of d, that is, d(a, b) > 0 andd(b, a) > 0.

39

(a) We have that AX(dab, d) = 0.(b) Moreover the equation AX(d, dab) = d(a, b) holds.

Proof. (a) The statement immediately follows from dab ≤ d.(b) By denition AX(d, dab) = sup(x,y)∈X×X(d(x, y)−dab(x, y)). We need to

consider two possible dierences in the latter expression: d(x, y)−d(x, y) = 0 ord(x, y)−(d(x, a) + d(b, y)). But d(x, y)− d(x, a)− d(b, y) ≤ d(a, b) by the triangleinequality. Note that equality in the latter inequality holds for (x, y) = (a, b).Indeed d(a, b)−dab(a, b) = d(a, b)− 0. We conclude that AX(d, dab) = d(a, b). 2

5. Proposition. Let (X, d) be a T0-quasi-metric space and let a, b ∈ X be ≤d-incomparable. Then d(b, a) ≤ AX(dab, (dab)

−1) ≤ d(a, b) +AX(d, d−1).

Proof. The rst inequality follows from the fact that dab(b, a)− (dab)−1(b, a) =

d(b, a)− 0 = d(b, a).We then have the following chain of inequalities: By the triangle inequal-

ity, Remark 6 and Proposition 4 we see that AX(dab, (dab)−1) ≤ AX(dab, d) +

AX(d, d−1)+AX(d−1, (dab)−1) = 0+AX(d, d−1)+AX(d−1, (d−1)ba) = AX(d, d−1)+

d−1(b, a). 2

4. Corollary. Let (X,m) be a metric space and let a, b ∈ X be two distinct pointsin X. Then AX(mab, (mab)

−1) = m(a, b).

Proof. The result follows from Proposition 5, sincem is a metric andAX(m,m−1) =0. 2

4. Some bicomplete subspaces of the space of all quasi-pseudometrics

An (extended) quasi-pseudometric space (X, d) is called bicomplete if the (ex-tended) pseudometric space (X, ds) is complete, that is, each ds-Cauchy sequencein X converges with respect to the pseudometric topology τ(ds).

5. Lemma. The extended metric space (QPM(X), (AX)s) is complete, hence(QPM(X), AX) is bicomplete.

Proof. The standard proof that the set of real-valued functions on a set X withthe uniform sup-metric is complete shows that each Cauchy sequence (dn)n∈N ofquasi-pseudometrics in (QPM(X), (AX)s) has a [0,∞)-valued limit function a onX×X to which it converges uniformly. Therefore we only need to show that a is aquasi-pseudometric on X. But this follows from the observation that the pointwiselimit of a sequence of quasi-pseudometrics is a quasi-pseudometric: Indeed for eachx ∈ X we have d(x, x) = limn→∞ dn(x, x) = limn→∞ 0 = 0. Furthermore we seethat for any x, y, z ∈ X we have that dn(x, z) ≤ dn(x, y) + dn(y, z). Thereforetaking limits in the reals equipped with the usual topology, we get that d(x, z) ≤d(x, y) + d(y, z) whenever x, y, z ∈ X. 2

A quasi-pseudometric d on a set X is called bounded if there is b ∈ [0,∞) suchthat d(x, y) ≤ b whenever x, y ∈ X, that is, its diameter δd <∞. By BQPM(X)we shall denote the set of bounded quasi-pseudometrics on X.

40

6. Proposition. The set BQPM(X) of bounded quasi-pseudometrics is closedin (QPM(X), τ((AX)s)).

Proof. Suppose that (dn)n∈N is a sequence of bounded quasi-pseudometrics onX such that (AX)s(d, dn) → 0 where d ∈ QPM(X). There is n ∈ N such that(AX)s(dn, d) < 1. By assumption there is a ∈ [0,∞) such that δdn ≤ a. Then forany (x, y) ∈ X ×X we have that d(x, y) ≤ (d(x, y)− dn(x, y)) + dn(x, y) ≤ 1 + a.Therefore the quasi-pseudometric d is bounded, too. 2

6. Lemma. Given a set X with at least 2 points, the set of all T0-quasi-metricsis not closed in (QPM(X), τ((AX)s)).

Proof. For any xed T0-quasi-metric d on X, the indiscrete quasi-pseudometrici(x, y) = 0 whenever (x, y) ∈ X×X is obviously the uniform limit of the sequence( 1nd)n∈N in (QPM(X), τ((AX)s)), but i is not a T0-quasi-metric in case that Xcontains at least two points. 2

7. Proposition. LetX be a set and PM(X) the set of all pseudometrics belongingto QPM(X). Then PM(X) is closed in (QPM(X), τ((AX)s)).

Proof. Suppose that the sequence (mn)n∈N of pseudometrics on X convergesto the quasi-pseudometric d on X in the sense that (AX)s(mn, d)→ 0. Therefored(x, y) = limn→∞mn(x, y) = limn→∞mn(y, x) = d(y, x) whenever x, y ∈ X. Thestatement follows. 2

Recall that a quasi-pseudometric d on a set X is called totally bounded providedthat given any ε > 0, there is a nite subset Fε of X such that for each x ∈ Xthere is f ∈ Fε such that ds(x, f) < ε.

Of course, the standard proof shows that each totally bounded quasi-pseudometricis bounded: Indeed given a totally bounded quasi-pseudometric d on X choose anite subset F1 of X as given by the denition. Then for any x, y ∈ X we havethat d(x, y) ≤ 1 + maxf,f ′∈F1

d(f, f ′) + 1 by an obvious application of the triangleinequality.

8. Proposition. Let X be a set and let TQPM(X) be the set of all totallybounded quasi-pseudometrics on X.

Then TQPM(X) is closed in (QPM(X), τ((AX)s)).

Proof. Let (dn)n∈N be a sequence of totally bounded quasi-pseudometrics on Xconverging to a quasi-pseudometric d in (QPM(X), τ((AX)s)).

Let ε > 0. There is m ∈ N such that (AX)s(d, dm) < ε. Furthermore thereis a nite subset F of X such that for any x ∈ X there is an f ∈ F such that(dm)s(x, f) < ε. Thus for any x ∈ X there is f ∈ F such that d(x, f) ≤ (d(x, f)−dm(x, f)) + dm(x, f) ≤ (AX)s(d, dm) + ε = 2ε and similarly, d(f, x) ≤ (d(f, x) −dm(f, x)) + dm(f, x) ≤ (AX)s(d, dm) + ε = 2ε. We conclude that d is totallybounded. 2

Recall that a quasi-pseudometric d on a setX is called an ultra-quasi-pseudometric

provided that d(x, z) ≤ maxd(x, y), d(y, z) whenever x, y, z ∈ X. The latter in-equality is called the strong triangle inequality for d.

41

9. Proposition. The set of all ultra-quasi-pseudometrics on a set X is τ((AX)s)-closed in QPM(X).

Proof. Let (un)n∈N be a sequence of ultra-quasi-pseudometrics on X convergingto the quasi-pseudometric d with respect to the topology τ((AX)s).

Using (uniform) convergence, the existence of x, y, z ∈ X such that d(x, z) >maxd(x, y), d(y, z) would imply the existence of an n ∈ N such that dn(x, z) >maxdn(x, y), dn(y, z) a contradiction. The assertion follows. 2

7. Lemma. Each quasi-pseudometric space (X, d) with d having a nite range isbicomplete.

Proof. The statement obviously holds for the indiscrete quasi-pseudometric onX. So we can assume that d is not indiscrete. Suppose that (xn)n∈N is a ds-Cauchysequence in X. Then there is ε > 0 such that ε ≤ min(d(X ×X) \ 0). Hence wehave that there is Nε ∈ N such that 0 = d(xn, xm) < ε whenever n,m ∈ N withn,m ≥ Nε. We conclude that (xn)n∈N converges to xNε in (X, ds) and thus d isbicomplete. 2

Our next example shows that the subset of complete pseudometrics need not beclosed in (QPM(X), τ((AX)s)), which also shows that the subset of bicompletequasi-pseudometrics need not be closed in (QPM(X), τ((AX)s)).

4. Example. Let X = [0, 1) ⊆ R and let d(x, y) = |x− y| whenever x, y ∈ X bethe usual metric on X.

Furthermore for any x ∈ X suppose that p(x) = 0.e1e2e3 . . . en . . . is a xeddecimal representation of x with innitely many digits. Of course, d(x, y) = |p(x)−p(y)| whenever x, y ∈ X.

For each n ∈ N let pn(x) = 0.e1e2 . . . en. Of course, for each n ∈ N, dn(x, y) =|pn(x) − pn(y)| whenever x, y ∈ X is a pseudometric. Note that each dn has anite range.

Obviously limn→∞(AX)s(dn, d) = 0, since by Lemma 1

(AX)s(dn, d) = sup(x,y)∈X×X

|dn(x, y)− d(x, y)|

= sup(x,y)∈X×X

||pn(x)− pn(y)| − |p(x)− p(y)||

≤ supx∈X|p(x)− pn(x)|+ sup

y∈X|p(y)− pn(y)| ≤ 2

10n.

Furthermore (1 − 1n )n∈N is a d-Cauchy sequence that is not convergent in

(X, τ(d)) and thus d not complete. However by Lemma 7 each pseudometricdn is complete and (AX)s(dn, d)→ 0.

The following concept was introduced by Steve Matthews.

2. Denition. (see for instance [5, 18, 15]) Let (X, f) be a quasi-pseudometricspace. If there exists a function w : X → [0,∞) such that f(x, y) + w(x) =f(y, x) +w(y) whenever x, y ∈ X, then f is called nonnegatively weightable and wis said to be a nonnegative weight for (X, f).

42

7. Remark. Note that the weight of a nonnegatively weightable quasi-pseudometricis not unique; for instance for a given metric space (X,m) any nonnonegative realconstant function yields a nonnegative weight function.

That is why in the proof given below, if n ∈ N and wn is a weight functionfor a nonnegatively weightable quasi-pseudometric space (X, dn), we cannot ex-pect that the sequence (wn)n∈N converges to some nonnegative weight function oflimn→∞ dn, even if the latter limit exists. 2

10. Proposition. The set WQPM(X) of all nonnegatively weightable quasi-pseudometrics on X is τ((AX)s)-closed in QPM(X).

Proof. Suppose that (dn)n∈N is a sequence of nonnegatively weightable quasi-pseudometrics on X and (AX)s(d, dn) → 0 where d ∈ QPM(X). For each n ∈ Nand x, y ∈ X set Fn(x, y) := dn(x, y) − dn(y, x), that is, Fn is the disymmetry

function of dn in the sense of [5].Then |Fn(x, y)−Fm(x, y)| ≤ |dn(x, y)−dm(x, y)|+|dn(y, x)−dm(y, x)| whenever

x, y ∈ X and n,m ∈ N.Since (dn)n∈N is a Cauchy sequence in (QPM(X), (AX)s), we conclude that for

each (x, y) ∈ X ×X, (Fn(x, y))n∈N is a Cauchy sequence in (R, us).For each (x, y) ∈ X ×X set F (x, y) = limn→∞ Fn(x, y). By the previous argu-

ment we see that indeed limn→∞(AX)s(Fn, F ) = 0.It is known by [5, Theorem 3.5] and readily checked that, by the weightability

of dn, Fn(x, z) = Fn(x, y) + Fn(y, z) whenever n ∈ N and x, y, z ∈ X. By takinglimits we have therefore F (x, z) = F (x, y) + F (y, z) whenever x, y, z ∈ X. Wededuce that F (x, y) = d(x, y)−d(y, x) = φ(y)−φ(x) for some function φ : X → Rby Sincov's functional equation [11].

It remains to be seen that we can choose the function φ in such a way thatφ(y) ≥ 0 whenever y ∈ X.

By the argument above we can nd n ∈ N such that |Fn(x, y) − F (x, y)| < 1whenever (x, y) ∈ X ×X.

Fix x ∈ X. Since Fn stems from a nonnegatively weightable quasi-pseudometricdn with a nonnegative weight φn : X → [0,∞), we have Fn(x, y) = dn(x, y) −dn(y, x) = φn(y)− φn(x) ≥ −φn(x) whenever y ∈ X.

Hence −φn(x) ≤ Fn(x, y) whenever y ∈ X and therefore −φn(x) − F (x, y) ≤Fn(x, y) − F (x, y) < 1. Thus −φn(x) − 1 ≤ F (x, y) = φ(y) − φ(x) whenevery ∈ X. We conclude that −φn(x) + φ(x) − 1 ≤ φ(y) whenever y ∈ X. Thereforew(y) := φ(y) + φn(x)− φ(x) + 1 whenever y ∈ X is a nonnegative weight for d. 2

5. The dierence approach to the skewness of a quasi-pseudometric

In this section we are interested in measuring the asymmetry or skewness of aT0-quasi-metric f on a set X. Several methods suggest themselves.

For instance we could compare the specialization orders ≤f and ≤f−1 , or wecould compare the topologies τ(f) and τ(f−1). Observe that ≤f=≤f−1 i thespecialization order ≤f is equality, that is, f is a T1-quasi-metric. (A quasi-pseudometric d on X satisfying the condition that d(x, y) 6= 0 whenever x, y ∈X with x 6= y is called a T1-quasi-metric.) Of course, τ(f) = τ(f−1) if andonly if for any x ∈ X and sequence (xn)n∈N in X, limn→∞ f(x, xn) = 0 ilimn→∞ f−1(x, xn) = 0.

43

We could also study relationships between the induced quasi-uniformities Ufand Uf−1 , or the induced totally bounded quasi-uniformities (Uf )ω and (Uf−1)ω.

‡‡

Observe that Uf = Uf−1 i Uf is a uniformity. Similarly (Uf )ω = (Uf−1)ω i(Uf )ω is a uniformity (compare [7, Corollary 1.40]).

In the following we shall consider a metric approach to asymmetry that ismore in the spirit of paper [5] where the function F (x, y) = d(x, y) − d(y, x)(whenever x, y ∈ X) of disymmetry is considered. The following sets might be ofspecial interest for a more detailed study on asymmetry, which will be conductedelsewhere.

5. Example. Let (X, d) be a T0-quasi-metric space and let k, r ∈ [0,∞).(a) Let Sd,k = (x, y) ∈ X × X : |d(x, y) − d(y, x)| ≤ k. Then Sd,k is a

τ(ds) × τ(ds)-closed symmetric reexive relation. We can call it the set of k-symmetric pairs.

(b) Ad,k = (x, y) ∈ X ×X : |d(x, y) − d(y, x)| ≥ k is a τ(ds) × τ(ds)-closedsymmetric relation. We can call it the set of k-asymmetric pairs.

(c) Further interesting tools to measure asymmetry could be the sets of realsσd,k;r = d(x, y) : (x, y) ∈ X × X and |d(y, x) − r| ≤ k and αd,k;r = d(x, y) :(x, y) ∈ X ×X and |d(y, x)− r| ≥ k.

In particular we can speak of a symmetric pair (x, y) ∈ X×X if d(x, y) = d(y, x)and call x ∈ X a symmetric point of (X, d) provided that d(x, y) = d(y, x) whenevery ∈ X.

In the present paper we shall concentrate on investigating the following muchsimpler concept.

3. Denition. Let (X, d) be a quasi-pseudometric space. We dene Ad :=AX(d, d−1) = sup(x,y)∈X×X(d(x, y)−d(y, x)) = sup(x,y)∈X×X |d(x, y)− d(y, x)|.8. Remark. Of course if X is nite, it may be more reasonable to consider theT0-quasi-metric SX(d, e) :=

∑(x,y)∈X×X(d(x, y)−e(x, y)) for d, e ∈ QPM(X) and

then for instance to investigate the value

S⊕X(d, d−1) =1

2

∑

(x,y)∈X×X|d(x, y)− d(y, x)|

in order to make sure that all the relevant dierences can contribute to the valueof asymmetry.

But we shall restrict our study in the following to the value AX(d, d−1), whichis much easier to handle.

Let us consider some examples.

6. Example. Let X = [a, b] be the closed interval with endpoints a and b of theset R. Then Au = AX(u, u−1) ≥ u(b, a)− u−1(b, a) = b− a, where u denotes alsothe restriction of u to [a, b].

The following observation was already stated in the introduction.

‡‡Here as usual, for any quasi-uniformity U on a set X, Uω will denote the nest totallybounded quasi-uniformity coarser than U on X.

44

9. Remark. Let f be a quasi-pseudometric on a set X. Then Af = 0 if and onlyif f is a pseudometric on X.

7. Example. Let (X, d,w) be a nonnegatively weighted quasi-pseudometric space,that is, d(x, y) + w(x) = d(y, x) + w(y) whenever x, y ∈ X where w : X → [0,∞)is the weight function. Therefore Ad = sup(x,y)∈X×X |w(y)− w(x)|.

8. Example. Let X = [0,∞) and for all x, y ∈ X set d(x, y) = 0 if x ≤ y andd(x, y) = x if x 6≤ y, where ≤ is the usual order on X. We rst note that d isa T0-ultra-quasi-metric on X : Observe that if x, y ∈ X such that x < y, thends(x, y) ≥ y, which shows that the T0-condition (c) is satised by d.

We next verify that d satises the strong triangle inequality: Otherwise thereare x, y, z ∈ X such that d(x, z) 6≤ maxd(x, y), d(y, z). Then x 6≤ z and thusd(x, z) = x. Note that the case that x ≤ y and y ≤ z is impossible, since x 6≤ z.

If x 6≤ y, then d(x, y) = x and the strong triangle inequality for d is satised.On the other hand, if x ≤ y and y 6≤ z, then d(y, z) = y and the strong triangle

inequality is satised for d, because d(x, z) ≤ d(y, z). Hence d is a T0-ultra-quasi-metric.

We now conclude the following: Let x, y ∈ [0,∞). If y < x, then d(x, y)−d(y, x) =x−0 = x. If y = x, then d(x, y)−d(y, x) = 0−0 = 0. If y > x, then d(x, y)−d(y, x) =0−y = 0.

Therefore for each x ∈ X, supy∈X(d(x, y)−d(y, x)) = x and for each y ∈ X,

supx∈X(d(x, y)−d(y, x)) =∞. In particular Ad =∞. 2

8. Lemma. Let (X, d) be a quasi-pseudometric space. Then Ad ≤ δd where δddenotes the diameter of (X, d).

Proof. For any (x, y) ∈ X ×X we have that d(x, y)− d(y, x) ≤ d(x, y). 2

9. Lemma. Let d, d′ be quasi-pseudometrics on a set X and λ ∈ [0,∞). Then thefollowing inequalities hold:

(a) Aλd = λAd.(b) Ad+d′ ≤ Ad +Ad′ .(c) Ad∨d′ ≤ Ad ∨ Ad′ . Furthermore Amind,d′ ≤ Ad ∨ Ad′ (where mind, d′ in

general is not a quasi-pseudometric on X).(d) Ad = Ad−1 .

Proof. The statements follow from Lemma 2(b), Lemma 2(a), Proposition 2,Corollary 2 and Remark 4. 2

10. Remark. Given a quasi-pseudometric d on a set X, we cannot establish anynontrivial lower bounds for Ad+d−1 and Ad∨d−1 in (b) and (c) above: Note that forany quasi-pseudometric d on X we have that Ad+d−1 = 0 = Ad∨d−1 . Consideringthe space (R, u), we observe that u ∧ u−1 = minu, u−1 = 0 is the constantindiscrete quasi-pseudometric equal to 0 on R× R. Since A0 = 0, we deduce thatthere is also no nontrivial lower bound for Ad∧d−1 .

The following result shows that quasi-pseudometrics that are close to each otherhave asymmetry values that are close to each other, too.

45

10. Lemma. For any quasi-pseudometrics p and q on a setX such that (AX)s(p, q) <∞ we have that either (AX)s(p, p−1) = (AX)s(q, q−1) = ∞ or |(AX)s(p, p−1) −(AX)s(q, q−1)| ≤ 2(AX)s(p, q).

Proof. Suppose that (AX)s(p, p−1) = ∞. Then by the triangle inequality wehave that (AX)s(p, p−1) ≤ (AX)s(p, q) + (AX)s(q, q−1) + (AX)s(q−1, p−1). ByRemark 4 and our assumption we see that (AX)s(q, q−1) =∞, too. The case that(AX)s(q, q−1) =∞ implies similarly that (AX)s(p, p−1) =∞.

So assume that both (AX)s(p, p−1) and (AX)s(q, q−1) are <∞. By Remark 4we conclude analogously as in Lemma 1 that |(AX)s(p, p−1) − (AX)s(q, q−1)| ≤(AX)s(p, q) + (AX)s(p−1, q−1) = 2(AX)s(p, q). 2

According to [21, p. 131] a costfunction is an arbitrary function g : [0,→) −→[0,→) with g(0) = 0 that is concave (so g((1 − λ)s + λt) ≥ (1 − λ)g(s) + λg(t)whenever s, t ∈ [0,∞) and λ ∈ [0, 1]).∗ For instance g(x) =

√x whenever x ∈

[0,∞) denes such a costfunction.

11. Proposition. Let d be a quasi-pseudometric on a set X and let g be acostfunction on [0,∞). Then Agd ≤ g(Ad).

Proof. We rst note that g d is a quasi-pseudometric on X (compare [21,Theorem 5, Lemma 3 (2) and (3)]). Now we are going to establish the statedinequality.

Case 1: Let x, y ∈ X. If g(d(x, y))− g(d(y, x)) ≤ 0, then obviously g(d(x, y))−g(d(y, x)) ≤ 0 = g(0) ≤ g(Ad), because g is nondecreasing [21, Lemma 3 (3)] and0 ≤ Ad.

Case 2: Suppose now that g(d(x, y)) − g(d(y, x)) > 0. Thus g(d(x, y)) >g(d(y, x)). Then d(x, y) ≤ d(y, x) is impossible, since g is nondecreasing [21,Lemma 3 (3)]. Thus necessarily d(x, y) > d(y, x). Therefore g(d(x, y)) = g(d(x, y)−d(y, x) + d(y, x)) ≤ g(d(x, y) − d(y, x)) + g(d(y, x)) using [21, Lemma 3 (2)]. Itfollows that g(d(x, y)) − g(d(y, x)) ≤ g(d(x, y) − d(y, x)) ≤ g(Ad), since g is non-decreasing and d(x, y)− d(y, x) ≤ Ad. We conclude that Agd ≤ g(Ad). 2

9. Example. Let (X, d) be a quasi-pseudometric space. It is well known that d1+d

is a bounded quasi-pseudometric on X. See for instance [21, Example 1]: Indeedit suces to note that s 7→ s

1+s is a costfunction. By Proposition 11 we then have

that A d1+d≤ Ad

Ad+1 if Ad <∞, and A d1+d≤ 1 if Ad =∞.

6. Asymmetrically normed real vector spaces

We next recall the concept of an asymmetric norm (see for instance [6]; compare[21, Section 2.5] or [20, p. 183]), which leads to many interesting examples of quasi-pseudometrics.

4. Denition. Let X be a real vector space and let ‖ · | → [0,∞) be a map suchthat

(1) ‖0| = 0.(2) ‖x+ y| ≤ ‖x|+ ‖y| whenever x, y ∈ X.∗ For possible use in our two next results we also set g(∞) := supx∈[0,∞) g(x).

46

(3) ‖αx| = α‖x| whenever x ∈ X and α ≥ 0. Furthermore suppose that ‖x| =‖ − x| = 0 implies that x = 0.

The function ‖ · | is called an asymmetric norm on X. It is known that eachasymmetrically normed vector spaceX induces a T0-quasi-metric d onX by settingd(x, y) = ‖x− y| whenever x, y ∈ X.

To motivate the preceding denition we recall the concept of the asymmetricsegment.

10. Example. [1, Remark 2] Let X = [0, 1]. Find a, b ∈ [0,∞) such that a+b 6= 0.Set d[ab](x, y) = (x − y)a if x > y and d[a,b](x, y) = (y − x)b if y ≥ x. Then([0, 1], d[ab]) is a T0-quasi-metric space induced by the asymmetric norm n[ab] onR dened by n[ab](x) = xa if x > 0 and n[ab](x) = −xb if x ≤ 0.

The following related example then yields another illustration of Proposition11.

11. Example. Let X = [−1, 1] be the real interval and set for x, y ∈ X d(x, y) =|x − y| if x ≥ y and d(x, y) = 2|x − y| if x < y. Then by Example 10 d is aT0-quasi-metric on X.

Using the costfunction g(x) =√x (x ∈ [0,∞)) we compute that

Ad = sup(x,y)∈X×X |x− y| = 2 and hence√Ad =

√2,

while A√d = sup(x,y)∈X×X(√

2− 1)√|x− y| = 2−

√2, which is indeed <

√2.

11. Remark. Given a set X, it is often useful to abuse the notation and writeAX(f, g) = ‖f − g| where f, g ∈ QPM(X), although in this case obviously not allconditions of Denition 4 are satised, since the vector space structure is missing.

12. Proposition. Let X be a non-trivial real vector space, let ‖·| be an asymmet-ric norm on X and let d be the induced T0-quasi-metric as dened above. ThenAd = supx∈X |‖ − x| − ‖x||. Hence Ad =∞ if ‖ · | is not a norm. 2

Proof. The rst statement is obvious. For the second statement, without lossof generality there is x0 ∈ X such that ‖−x0| > ‖x0|. Let α > 0. Then d(0, αx0)−d(αx0, 0) = ‖0−αx0|−‖αx0−0| = α(‖−x0|−‖x0|), which can be made arbitrarilylarge by choosing α appropriately. 2

12. Remark. In [21] a multiplicative approach to an asymmetry measure σd of aT0-quasi-metric d on a set X (with at least two elements) is chosen: σd is computedas

sup(x,y)∈(X×X)\∆X

d(x, y)

d(y, x),

where the latter expression is dened to be innite in case that d(y, x) = 0 forsome (x, y) ∈ (X × X) \ ∆X . Hence this denition is mainly suitable for a T1-quasi-metric. We also note that this approach is very useful in an asymmetricallynormed space (X, ‖ · |), since in this case for an induced T1-quasi-metric d thevalue σd does not depend on the length ‖z| of the vector z ∈ X and thus can bedetermined on the unit sphere z ∈ X : ‖z| = 1 (see Proposition 12 and compare[21, Lemma 10]).

We refer the reader to [4, Section 4] for a short discussion of connections betweenadditive and multiplicative approaches to distance functions.

47

7. Some properties of Ad where d is a quasi-pseudometric

Given a quasi-pseudometric d on a set X, in this section we prove two simplefacts about the asymmetry value Ad of d.

13. Proposition. Let (X, d) be a quasi-pseudometric space such that the topologyτ(ds) is compact. Then there is (a, b) ∈ X ×X such that Ad = d(a, b) − d(b, a),that is, the supremum Ad is attained.

Proof. We sketch the standard argument. By compactness of the pseudomet-ric topology τ(ds), we see that d is bounded. Hence Ad < ∞ by Lemma 8.Therefore there is a sequence (xn, yn)n∈N in X × X such that the real sequence(F (xn, yn))n∈N, where for each n ∈ N F (xn, yn) = d(xn, yn)−d(yn, xn), convergesto the value Ad. By compactness of τ(ds) there is a subsequence (nk)k∈N of (n)n∈Nand x, y ∈ X such that (xnk)k∈N resp. (ynk)k∈N τ(ds)-converges to x resp. y inX. Since limn→∞ F (xn, yn) = Ad, we conclude that F (x, y) = Ad by continuity ofd on (X ×X, τ(ds)× τ(ds)). 2

11. Lemma. Let (X, d) be a quasi-pseudometric space and Y ⊆ X. Thensup

(x,y)∈Y×Y|d(x, y)− d(y, x)| ≤ sup

(x,y)∈X×X|d(x, y)− d(y, x)|.

Proof. The argument is obvious. 2

Our next result considers a density condition under which the inverse inequalityalso holds.

14. Proposition. Let Y be a subspace of a quasi-pseudometric space (X, d) suchthat clτ(ds)Y = X. Then AY (d|Y×Y , d−1|Y×Y ) = AX(d, d−1).

Proof. Let x, y ∈ clτ(ds)Y. Then there are sequences (xn)n∈N and (yn)n∈N in Xsuch that ds(x, xn) → 0 and ds(yn, y) → 0. Fix n ∈ N. Then |d(x, y)− d(y, x)| ≤|d(x, y) − d(xn, yn)| + |d(xn, yn) − d(yn, xn)| + |d(yn, xn) − d(y, x)| ≤ ds(x, xn) +ds(y, yn)+ |d(xn, yn)−d(yn, xn)|+ds(yn, y)+ds(xn, x) ≤ 2ds(xn, x)+2ds(yn, y)+sup(x,y)∈Y×Y |d(x, y)− d(y, x)| by Lemma 1. Therefore

sup(x,y)∈X×X

|d(x, y)− d(y, x)| ≤ sup(x,y)∈Y×Y

|d(x, y)− d(y, x)|.

Hence the stated equality is established. 2

5. Corollary. Let (X, d) be a T0-quasi-metric with bicompletion (X, d) (see [13,

Example 2.7.1]). Then AX(d, d−1) = AX(d, (d)−1).

Proof. It is known that X is τ((d)s)-dense in X. 2

8. The q-hyperconvex hull of a T0-quasi-metric space

We rst recall some basic facts about the q-hyperconvex hull of a T0-quasi-metric space. For additional information we refer the reader to [12, 17] and theliterature cited in these papers.

Let (X, d) be a T0-quasi-metric space. We consider the set QX of all functionpairs f = (f1, f2) on (X, d), where fi : X → [0,∞) (i = 1, 2), satisfying

48

f1(x) = supd(y, x)−f2(y) : y ∈ X and f2(x) = supd(x, y)−f1(y) : y ∈ Xwhenever x ∈ X.

We equip QX with the T0-quasi-metric D dened by

D(f, g) = supx∈X

(f1(x)−g1(x)) = supx∈X

(g2(x)−f2(x))

whenever f, g ∈ QX .Then the map e dened for each x ∈ X by x 7→ e(x) = fx where (fx)1(y) :=

d(x, y) and (fx)2(y) := d(y, x) whenever y ∈ X yields an isometric embedding of(X, d) into (QX , D). The T0-quasi-metric space (QX , D) is called the q-hyperconvexhull of (X, d).

Let us mention that for each f, g ∈ QX , we haveD(f, g) = sup(D(fx1

, fx2)−D(fx1

, f)−D(g, fx2)) ∨ 0 : x1, x2 ∈ X (∗)

according to [12, Remark 7].

15. Proposition. Let (X, d) be a T0-quasi-metric space and let (QX , D) be itsq-hyperconvex hull. Then δd = AD = δD.

Proof. We rst show that the diameter δD of the q-hyperconvex hull (QX , D)of a T0-quasi-metric space (X, d) is equal to the diameter δd of (X, d).

Obviously δD ≥ δd, since (X, d) embeds as an isometric subspace into (QX , D).Note that for any f, g ∈ QX we have that by the result (∗) stated above,

D(f, g) = sup(x,y)∈X×X

D(x, y)−D(x, f)−D(g, y), 0 = sup(x,y)∈X×X

D(x, y) ≤ δd.

Thus δD ≤ δd. Hence the equality of the two diameters δD and δd is established.We next consider now the case that the diameter δd <∞. Dene a function pair

⊥ by setting ⊥1(x) = 0 and ⊥2(x) = supa∈X d(x, a) whenever x ∈ X. Further-more dene a function pair > by setting >1(x) = supa∈X d(a, x) and >2(x) = 0whenever x ∈ X.

One veries that ⊥,> ∈ QX by checking the dening equations: Indeed foreach x ∈ X,

⊥1(x) = 0 = supy∈X

(d(y, x)−⊥2(y)) = supy∈X

(d(y, x)− supa∈X

d(y, a))

and similarly

⊥2(x) = supy∈X

(d(x, y)−⊥1(y)) = supy∈X

(d(x, y)−0).

Analogously for each x ∈ X,>1(x) = sup

y∈Xd(y, x) = sup

y∈X(d(y, x)−>2(y)) = sup

y∈X(d(y, x)−0)

and

>2(x) = 0 = supy∈X

(d(x, y)−>1(y)) = supy∈X

(d(x, y)− supa∈X

d(a, y)).

Hence ⊥,> ∈ QX , as asserted.Furthermore one computes

D(⊥, f) = supx∈X

(⊥1(x)−f1(x)) = supx∈X

(0−f1(x)) = 0

49

and similarly D(f,>) = supx∈X(>2(x)−f2(x)) = supx∈X(0−f2(x)) = 0 when-ever f ∈ QX . Hence ⊥ is the bottom and > the top of QX with respect to thespecialization order ≤D of D on QX .

Thus D(>,⊥)−D(⊥,>) = D(>,⊥)− 0 = supx∈X(>1(x)−⊥1(x))= supx∈X(supa∈X d(a, x)−0) = δd. We conclude that AD ≥ δd.

Hence we know by Lemma 8 that Ad ≤ δd ≤ AD ≤ δD ≤ δd and conclude thatδd = AD = δD.

Suppose now that (X, d) is an unbounded T0-quasi-metric space and let (QX , D)be the q-hyperconvex hull of (X, d).

Choose x0 ∈ X. For each n ∈ N set Xn = x ∈ X : ds(x0, x) ≤ n and denotethe restriction of d to Xn ×Xn by dn.

Note that for each n ∈ N we have that δdn ≤ 2n, thus (Xn, dn) is bounded. Wealso observe that

⋃n∈NXn = X where the sequence (Xn)n∈N of subspaces of X is

increasing.Let (QXn , Dn) denote the q-hyperconvex hull of the subspace (Xn, dn) of (X, d).

Denote by >n resp. ⊥n the top resp. bottom element of (QXn , Dn), as constructedin the rst part of the present proof.

For each n ∈ N consider an isometry τn : QXn → QX as given in [1, Proposition4].∗

For each n ∈ N set fn := τn(>n) and gn := τn(⊥n). We have that

δdn = supx∈Xn

( supa∈Xn

dn(a, x)) = Dn(>n,⊥n) = D(τn(>n), τn(⊥n)) = D(fn, gn)

and 0 = Dn(⊥n,>n) = D(τn(⊥n), τn(>n)) = D(gn, fn) whenever n ∈ N, as wehave noted above.

Thus AD ≥ D(fn, gn) −D(gn, fn) = D(fn, gn) − 0 = δdn whenever n ∈ N andtherefore AD ≥ supn∈N δdn = δd. Consequently in the unbounded case Ad ≤ δd ≤AD ≤ δD ≤ δd, too. Hence the stated equality is also established in the case thatδd =∞. 2

12. Example. Let (X,m) be a metric space and let (QX , D) be its q-hyperconvexhull. Then Am = 0, but AD = δm.

Proof. The assertion follows from the previous result and the trivial fact thatAm = 0. 2

9. The Hausdor quasi-pseudometric

In this section we consider a T0-quasi-metric space (X, d) with associated Haus-dor quasi-pseudometric space (B0(X), dH) where B0(X) denotes the set of allbounded nonempty subsets of (X, d).

Recall that for any A,B ∈ B0(X) we dene dH−(A,B) = supa∈A d(a,B) anddH+(A,B) = supb∈B d(A, b). It is known that dH− and dH+ are both quasi-pseudometrics on B0(X). Finally we set dH = dH+∨dH− . Then dH is the Hausdorquasi-pseudometric on B0(X) (compare for instance [3, 16]).

∗ The latter result states that if (Z, d) is a T0-quasi-metric space and S is a nonempty subspaceof (Z, d), then there exists an isometric embedding τ : QS → QZ such that τ(f)|S = f wheneverf ∈ QS .

50

Below we shall make use of the fact that (dH+)−1 = (d−1)H− , which can beveried by a straightforward computation with the help of the denitions of dH+

and dH−1 .

16. Proposition. Let (X, d) be a T0-quasi-metric space. Then AdH+ = δd.

Proof. By Lemma 8 we have AdH+ ≤ δdH+ . Furthermore the inequality δdH+ ≤δd holds by the denition of dH+ : Indeed in order to reach a contradiction supposethat for some A,B ∈ B0(X) we have dH+(A,B) > δd. Then there must be b ∈ Bsuch that d(A, b) > δd and so for each a ∈ A we have that d(a, b) > δd acontradiction. Hence δdH+ ≤ δd.

Let (xn, yn)n∈N be a sequence in X ×X such that (d(xn, yn))n∈N converges toδd, where δd could possibly be innite.

Set for each n ∈ N, An = xn, yn and Bn = xn. Obviously all these setsbelong to B0(X). Then dH+(Bn, An) − dH+(An, Bn) = d(xn, yn) − 0 whenevern ∈ N. We conclude that AdH+ ≥ δd.

Hence the stated equality AdH+ = δd is established. 2

6. Corollary. Let (X, d) be a T0-quasi-metric space. Then AdH− = δd.

Proof. We conclude by Proposition 16 and Lemma 9(d) thatAdH− = A((d−1)H+ )−1 = A(d−1)H+

= δd−1 = δd. 2

7. Corollary. Let (X, d) be a T0-quasi-metric space. Then AdH ≤ AdH+ ∨AdH− =δd.

Proof. The statement follows from the denition dH = dH+ ∨ dH− and Lemma9(c), Corollary 6 and Proposition 16. 2

13. Remark. Let (X,m) be a metric space. Then mH is a pseudometric, since(mH+)−1 = (m−1)H− = mH− . Thus AmH = 0.

10. The inmum-problem

We nish this paper by stating a problem. Given two quasi-pseudometrics fand g on a set X, f ∧ g denotes the largest quasi-pseudometric which is ≤ f and≤ g.

Indeed the following explicit form of f ∧ g is well known (compare [21, Lemma6]).

12. Lemma. Let X be a set and let f, g be quasi-pseudometrics on X. For anyx, y ∈ X set (f ∧ g)(x, y) = inf∑n−1

i=0 h(xi, xi+1) : x0 = x, xn = y;x1, . . . , xn−1 ∈X;n ∈ N;h ∈ f, g. Then f ∧ g is the largest quasi-pseudometric which is ≤ fand ≤ g.

Proof. The standard proof is left to the reader. 2

14. Remark. Note that for any d ∈ QPM(X), d∧ d−1 is indeed a pseudometric.

Proof. For any x, y ∈ X, by denition we clearly have that (d ∧ d−1)(x, y) =(d ∧ d−1)(y, x). 2

Of course, d1∧d2 ≤ mind1, d2 and the two functions can be distinct, as Exam-ple 2 above shows. The authors have only been able to establish the upper bound

51

for Ad1∧d2 given in Lemma 13 below. It should be mentioned that on the otherhand Plastria obtained an interesting upper bound for σd1∧d2 , the correspondingmultiplicative counterpart of Ad1∧d2 : He namely proved that σd1∧d2 ≤ σd1 ∨ σd2[21, Lemma 14.6].

13. Lemma. Let d1, d2 be quasi-pseudometrics on a set X. Then Ad1∧d2 ≤ δd1 ∧δd2 .

Proof. We have that Ad1∧d2 ≤ δd1∧d2 ≤ δdi whenever i ∈ 1, 2 by Lemma 8.2

1. Problem. Let d1 and d2 be quasi-pseudometrics on a set X. Is it possible thatAd1∧d2 > Ad1 ∨Ad2?

References

[1] C.A. Agyingi, P. Haihambo and H.-P.A. Künzi, Tight extensions of T0-quasi-metricspaces, in: V. Brattka, H. Diener, D. Spreen (Eds.), Logic, Computation, Hierarchies,Festschrift in Honour of V.L. Selivanov's 60th Birthday, Ontos Verlag, De Gruyter Berlin,Boston, 2014, pp. 922.

[2] C.A. Agyingi, P. Haihambo and H.-P.A. Künzi, Endpoints in T0-quasi-metric spaces: PartII, Abstract and Applied Analysis, Vol. 2013, Article ID 539573, 10 pages.

[3] G. Berthiaume, On quasi-uniformities in hyperspaces, Proc. Amer. Math. Soc. 66 (1977),335343.

[4] M. Bukatin, R. Kopperman and S. Matthews, Some corollaries of the correspondencebetween partial metrics and multivalued equalities, Fuzzy Sets Systems 256 (2014), 5772.

[5] M.J. Campión, E. Induráin, G. Ochoa and O. Valero, Functional equations related toweighable quasi-metrics, Hacettepe J. Mat. Stat. 44 (4) (2015), 775787.

[6] . Cobza³, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics,Springer, Basel, 2012.

[7] P. Fletcher and W.F. Lindgren, Quasi-uniform Spaces, Dekker, New York, 1982.[8] Y.U. Gaba and H.-P.A. Künzi, Splitting metrics by T0-quasi-metrics, Topology Appl. 193

(2015), 8496.[9] Y.U. Gaba and H.-P.A. Künzi, Partially ordered metric spaces produced by T0-quasi-

metrics, Topology Appl. 202 (2016), 366383.[10] J. Goubault-Larrecq, Non-Hausdor Topology and Domain Theory, Selected Topics in

Point-Set Topology, Cambridge University Press, Cambridge, 2013.[11] D. Gronau, A remark on Sincov's functional equation, Not. S. Afr. Math. Soc. 31 (2000),

1-8.[12] E. Kemajou, H.-P.A. Künzi and O.O. Otafudu, The Isbell-hull of a di-space, Topology

Appl. 159 (2012), 24632475.[13] H.-P.A. Künzi, An introduction to quasi-uniform spaces, Beyond Topology, Contemp.

Math. 486 (2009), 239304.[14] H.-P.A. Künzi and C. Makitu Kivuvu, A double completion for an arbitrary T0-quasi-

metric space, J. Logic Algebraic Programming 76 (2008), 251269.[15] H.-P.A. Künzi and S. Romaguera, Weightable quasi-uniformities, Acta Math. Hungar.

136 (1-2), (2012), 107128.[16] H.-P.A. Künzi and C. Ryser, The Bourbaki quasi-uniformity, Topology Proceedings 20

(1995), 161183.[17] H.-P.A. Künzi and M. Sanchis, The Kat¥tov construction modied for a T0-quasi-metric

space, Topology Appl. 159 (2012), 711720.[18] H.-P.A. Künzi and V. Vajner, Weighted quasi-metrics, Ann. New York Acad. Sci. 728

(1994), 6477.[19] H.-P. A. Künzi and F. Yldz, Convexity structures in T0-quasi-metric spaces, Topology

Appl. 200 (2016), 218.

52

[20] L. Nel, Continuity Theory, Springer, Switzerland, 2016.[21] F. Plastria, Asymmetric distances, semidirected networks and majority in Fermat-Weber

problems, Ann. Oper. Res. (2009) 167: 121155.[22] I.V. Protasov, Coronas of balleans, Topology Appl. 149 (2005), 149160.[23] T. alát, J. Tóth and L. Zsilinszky, Metric space of metrics dened on a given set, Real

Anal Exch., 18 no. 1 (19921993), 225231.[24] T. alát, J. Tóth and L. Zsilinszky, On the structure of the space of metrics dened on a

given set, Real Anal. Exch. 19, no. 1 (19931994), 321327.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 53 66

Semi-Hurewicz spaces

Ljubi²a D.R. Ko£inac∗, Amani Sabah †, Moiz ud Din Khan ‡ and Djamila Seba¶

Abstract

In this paper we study some covering properties in topological spacesby using semi-open covers. We introduce and investigate the propertiesof s-Hurewicz and almost s-Hurewicz spaces and their star versions.

2000 AMS Classication: Primary 54D20; Secondary 54B05, 54C08

Keywords: Semi-open set, s-Hurewicz space, star s-Hurewicz space.

1. Introduction

The properties of Menger and Hurewicz, which are the basic and oldest selectionprinciples, take their origin in papers [12] and [6]. Both of them appeared ascounterparts of σ-compactness. A topological space X has the Menger (resp.Hurewicz ) property, if for every sequence (Un : n ∈ N) of open covers of X thereexists a sequence (Vn : n ∈ N) such that every Vn is a nite subset of Un andthe family

⋃V : V ∈ Vn, n ∈ N is a cover of X (resp. each x ∈ X belongs to⋃Vn =

⋃V : V ∈ Vn for all but nitely many n). Clearly, every σ-compactspace X has the Hurewicz property and every Hurewicz space has the Mengerproperty. Every Menger space is Lindelöf. As a generalization of Hurewicz spaces,

Doi : 10.15672/HJMS.2016.405∗University of Ni² Faculty of Sciences and Mathematics, 18000 Ni², Serbia,

Email: lkocinac@gmail.com†Department of Mathematics, COMSATS Institute of Information Technology, Chak

Shahzad, Park road, Islamabad 45550, Pakistan,Email: amaniussabah@gmail.com‡Department of Mathematics, COMSATS Institute of Information Technology, Chak

Shahzad, Park road, Islamabad 45550, Pakistan,Email: moiz@comsats.edu.pk

Corresponding Author.¶Department of Mathematics, Faculty of Sciences, University, M'Hamed Bougara of

Boumerdes, 35000 Boumerdes, Algeria,Email: djamilaseba@gmail.com

Doi : 10.15672/HJMS.2016.405

54

the authors of [18] dened a topological space X to be almost Hurewicz if for eachsequence (Un : n ∈ N) of open covers of X there exists a sequence (Vn : n ∈N) such that for each n ∈ N, Vn is a nite subset of Un and for each x ∈ X,x ∈ ⋃Cl(V ) : V ∈ Vn for all but nitely many n. Clearly, the Hurewiczproperty implies the almost Hurewicz property. The authors [18] showed thatevery regular almost Hurewicz space is Hurewicz and gave an example that thereexists a Urysohn almost Hurewicz space that is not Hurewicz. On the study ofHurewicz and almost Hurewicz spaces (and other weaker versions) the readers cansee the references [6, 7, 9, 16, 17, 18].

In 1963, N. Levine [11] dened semi-open sets in topological spaces. A setA in a topological space X is semi-open if and only if there exists an open setO ⊂ X such that O ⊂ A ⊂ Cl(O), where Cl(O) denotes the closure of the set O.If A is semi-open, then its complement is called semi-closed [2]. The collectionof all semi-open subsets of X is denoted by SO(X). The union of any collectionof semi-open sets is semi-open, while the intersection of two semi-open sets neednot be semi-open. It happens if X is an extremally disconnected space [13]. Theintersection of open and semi-open set is semi-open. According to [2], the semi-closure and semi-interior were dened analogously to the closure and interior: thesemi-interior sInt(A) of a set A ⊂ X is the union of all semi-open subsets of A; thesemi-closure sCl(A) of A ⊂ X is the intersection of all semi-closed sets containingA. A set A is semi-open if and only if sInt(A) = A, and A is semi-closed if andonly if sCl(A) = A. Note that for any subset A of X

Int(A) ⊂ sInt(A) ⊂ A ⊂ sCl(A) ⊂ Cl(A).

The n-th power of a semi-open set in X is a semi-open set in Xn, whereas a semi-open set in Xn may not be written as a product of semi-open sets of X. A subsetA of a topological space X is called a semi-regular set if it is semi-open as well assemi-closed or equivalently, A = sCl(sInt(A)) or A = sInt(sCl(A)).

A mapping f : (X, τX)→ (Y, τY ) is called:

(1) semi-continuous if the preimage of every open set in Y is semi-open in X;(2) s-open [1] if the image of every semi-open set in X is open in Y ;(3) s-closed if the image of every semi-closed set in X is closed in Y ;(4) quasi-irresolute if for every semi-regular set A in Y the set f←(A) is semi-

regular in X [4].

For more details on semi-open sets and semi-continuity, we refer to [2, 3, 11].A space X is semi-regular if for each semi-closed set A and x /∈ A there exist

disjoint semi-open sets U and V such that x ∈ U and A ⊂ V [5].

1.1. Lemma. ([5]) The following are equivalent for a space X:

(i): X is a semi-regular space;

(ii): For each x ∈ X and U ∈ SO(X) such that x ∈ U , there exists V ∈SO(X) such that x ∈ V ⊂ sCl(V ) ⊂ U ;

(iii): For each x ∈ X and each U ∈ SO(X) with x ∈ U , there is a semi-regular

set V ⊂ X such that x ∈ V ⊂ U .The purpose of this paper is to investigate Hurewicz and almost Hurewicz spaces

(and their star versions) and their topological properties using semi-open covers.

55

2. Preliminaries

A semi-open cover U of a space X is called;

• an sω-cover if X does not belong to U and every nite subset of X iscontained in a member of U;

• an sγ-cover if it is innite and each x ∈ X belongs to all but nitely manyelements of U;

• s-groupable if it can be expressed as a countable union of nite, pairwisedisjoint subfamilies Un, n ∈ N, such that each x ∈ X belongs to

⋃Un for

all but nitely many n;• weakly s-groupable if it is a countable union of nite, pairwise disjoint sets

Un, n ∈ N, such that for each nite set F ⊂ X we have F ⊂ ⋃Un forsome n.

For a topological space X we denote:

• sO the family of semi-open covers of X;• sΩ the family of sω-covers of X;• sOgp the family of s-groupable covers of X;• sOwgp the family of weakly s-groupable covers of X.

For notation and terminology, we refer the reader to [10, 17].Let A be a subset of X and U be a collection of subsets of X, then St(A,U) =⋃U ∈ U : U ∩A 6= ∅. We usually write St(x,U) for St(x,U).Let A and B be the sets whose elements are covers of a space X.

2.1. Denition. ([8]) Sfin∗(A,B) denotes the selection hypothesis:

For each sequence (Un : n ∈ N) of elements of A there is a sequence (Vn : n ∈ N)such that for each n ∈ N, Vn is a nite subset of Un, and

⋃n∈NSt(V,Un) : V ∈ Vn

is an element of B.

2.2. Denition. ([8]) SSfin∗(A,B) denotes the selection hypothesis:

For each sequence (Un : n ∈ N) of elements of A there is a sequence (Kn : n ∈ N)of nite subsets of X such that St(Kn,Un) : n ∈ N is an element of B.

2.3. Denition. ([8]) S1∗(A,B) denotes the selection hypothesis:

For each sequence (Un : n ∈ N) of elements of A there is a sequence (Un : n ∈ N)such that for each n, Un ∈ Un and St(Un,Un) : n ∈ N is an element of B.

2.4. Denition. ([8]) Ufin∗(A,B) denotes the selection hypothesis:

For each sequence (Un : n ∈ N) of elements of A there is a sequence (Vn : n ∈ N)such that for every n, Vn is a nite subset of Un and St(

⋃Vn,Un) : n ∈ N ∈ B.

2.5. Denition. A space X is said to have:

• [15] star s-Menger property if it satises Sfin∗(sO, sO).

• [15] star s-Rothberger property if it satises S1∗(sO, sO).

For the denitions of star-Hurewicz and strongly star-Hurewicz spaces see [10].

3. Semi-Hurewicz and related spaces

3.1. Denition. Call a space X:

56

• semi-Hurewicz (or shortly s-Hurewicz ) if it satises: For each sequence(Un : n ∈ N) of elements of sO there is a sequence (Vn : n ∈ N) such thatfor each n ∈ N, Vn is a nite subset of Un and for each x ∈ X for all butnitely many n, x ∈ ⋃Vn;

• almost s-Hurewicz if for every sequence (Un : n ∈ N) of semi-open coversof X, there exists a sequence (Vn : n ∈ N) such that for every n ∈ N, Vnis a nite subset of Un and each x ∈ X belongs to sCl(

⋃Vn) for all but

nitely many n.

Evidently, we have

Hurewicz⇐ s−Hurewicz⇒ almost s−Hurewicz.3.2. Example. (1) Every semi-compact space is s-Hurewicz. The converse is nottrue. The real line R with the cocountable topology is a T1 semi-Hurewicz spacewhich is not semi-compact.

(2) The Sorgenfrey line S and the space of irrationals with the usual metrictopology are not semi-Hurewicz (because they are not Hurewicz, as it is wellknown).

3.3. Example. There is a Hurewicz space which is not s-Hurewicz,The real line with the usual metric topology is a Hurewicz space. On the other

hand, it is not an s-Hurewicz space, because from a sequence of covers whoseelements are sets of the form [a, b), a, b ∈ R, one cannot choose nite subfamilieswhose union covers R. It follows from the fact that the sets [a, b), a, b ∈ R, forma base for the Sorgenfrey line S which is not a Hurewicz space.

3.4. Theorem. Let X be a semi-regular space. If X is an almost s-Hurewiczspace, then X is s-Hurewicz.

Proof. Let (Un : n ∈ N) be a sequence of semi-open covers of X. Since X is asemi-regular space, by Lemma 1.1, there exists for each n a semi-open cover Vnof X such that sCl(V ) : V ∈ Vn forms a renement of Un. By assumption,applied to the sequence (Vn : n ∈ N), there exists a sequence (Wn : n ∈ N)such that for each n, Wn is a nite subset of Vn and each x ∈ X belongs to⋃sCl(W ) : W ∈ Wn. For every n ∈ N and every W ∈ Wn we can chooseUW ∈ Un such that sCl(W ) ⊂ UW . Let U′n = UW : W ∈ Wn. Then U′n is anite subset of Un, n ∈ N. It is easy to see that each x ∈ X belongs to

⋃U′n all

but nitely many n, which means that X is s-Hurewicz.

3.5. Theorem. A space X is almost s-Hurewicz if and only if for each sequence

(Un : n ∈ N) of covers of X by semi-regular sets, there exists a sequence (Vn : n ∈N) such that for every n ∈ N, Vn is a nite subset of Un and each x ∈ X belongs

to⋃

Vn for all but nitely many n ∈ N.

Proof. Let X be an almost s-Hurewicz space and let (Un : n ∈ N) be a sequenceof covers of X by semi-regular sets. Since every semi-regular set is semi-open,(Un : n ∈ N) is a sequence of semi-open covers of X. By assumption, there existsa sequence (Vn : n ∈ N) such that for every n ∈ N, Vn is a nite subset of Un andeach x ∈ X belongs to

⋃Vn for all but nitely many n.

57

Conversely, let (Un : n ∈ N) be a sequence of semi-open covers of X. Let(U′n : n ∈ N) be the sequence dened by U′n = sCl(U) : U ∈ Un. Then elementsof each U′n are semi-regular sets and thus, by assumption, there exists a sequence(Vn : n ∈ N) such that for every n ∈ N, Vn is a nite subset of U′n and each x ∈ Xbelongs to

⋃Vn for all but nitely many n. For each n ∈ N and V ∈ Vn there

exists UV ∈ Un such that V = sCl(UV ). Hence, x ∈ sCl(⋃UV ) : V ∈ Vn for all

but nitely many n. So X is an almost s-Hurewicz space. 3.6. Theorem. Every semi-regular subspace of an s-Hurewicz (almost s-Hurewicz)space is s-Hurewicz (almost s-Hurewicz).

Proof. Because the proofs for both case are similar, we consider only the almosts-Hurewicz case. Let A be a semi-regular subset of an almost s-Hurewicz spaceX and let (Un : n ∈ N) be a sequence of semi-open covers of A. Each semi-opensubset of A is semi-open in X [14], so that Vn = Un

⋃X \A is a semi-open coverfor X for every n ∈ N. Since X is almost s-Hurewicz, there exist nite subsetsWn of Vn, n ∈ N, such that each x ∈ X belongs to sCl(

⋃Wn) for all but nitely

many n ∈ N. By semi-regularity of A, sCl(X \ A) = X \ A and thus each a ∈ A,belongs to sCl(

⋃(Wn \ (X \ A))) for all but nitely many n, i.e. the sequence

(Wn \ (X \A) : n ∈ N) witnesses for (Un : n ∈ N) that A is almost s-Hurewicz. Now we consider preservation (in the image or preimage direction) of the prop-

erties we consider under some kinds of mappings.The proof of the next theorem is easy, obtained by applying denitions, and

thus is omitted.

3.7. Theorem. Let f : X → Y be a semi-continuous surjection. If X is an

s-Hurewicz space, then Y is a Hurewicz space.

3.8. Corollary. Let f : X → Y be a continuous surjection. If X is an s-Hurewiczspace, then Y is a Hurewicz space.

We dene now the notion of (strong) θ-semi-continuity which is an importantslight generalization of semi-continuity.

A mapping f : X → Y is θ-semi-continuous (resp. strongly θ-semi-continuous)if for each x ∈ X and each semi-open set V ⊂ Y containing f(x) there is a semi-open set U ⊂ X containing x such that f(sCl(U)) ⊂ sCl(V ) (resp. f(sCl(U)) ⊂V ).

Evidently, each strongly θ-semi-continuous mapping is θ-semi-continuous.Call a space X an almost semi-γ-set if for each sequence (Un : n ∈ N) of s-ω-

covers of X there is a sequence (Un : n ∈ N) such that Un ∈ Un for each n ∈ Nand Un : n ∈ N is an s-γ-cover of X.

3.9. Theorem. A θ-semi-continuous image of an almost semi-γ-set is an almost

semi-Hurewicz space.

Proof. Let f : X → Y be a θ-semi-continuous mapping of a semi-γ-setX to a spaceY . Let (Vn : n ∈ N) be a sequence of semi-open covers of Y and x ∈ X. For eachn ∈ N there is a set Vx,n ∈ Vn containing f(x). Since f is θ-semi-continuous thereis a semi-open set Ux,n ⊂ X containing x such that f(sCl(Ux,n)) ⊂ sCl(Vx,n). Foreach n let Un be the set of all nite unions of sets Ux,n, x ∈ X. Evidently, each Un

58

is an s-ω-cover of X. As X is an almost semi-γ-set there is a sequence (Un : n ∈ N)such that for each n, Un ∈ Un and for each x ∈ X the set n ∈ N : x /∈ sCl(Un)is nite.

Let Un = Ux1,n

⋃Ux2,n

⋃. . .⋃Uxin ,n

. To each Uxj ,n, j ≤ in, assign a setVxj ,n ∈ Vn with f(sCl(Uxj ,n)) ⊂ sCl(Vxj ,n). Let y = f(x) ∈ Y . Then from x ∈sCl(Un) for all n ≥ n0 for some n0 ∈ N, we get x ∈ sCl(Uxp,n) for some 1 ≤ p ≤ inwhich implies y ∈ f(sCl(Uxp,n)) ⊂ sCl(Vxp,n). If we put Wn =

⋃Vxj ,n : j =1, 2, . . . , in, we obtain the sequence (Wn : n ∈ N) of nite subsets of Vn, n ∈ N,such that each y ∈ Y belongs to all but nitely many sets

⋃sCl(W ) : W ∈Wn.This just means that Y is an almost semi-Hurewicz space.

3.10. Theorem. A strongly θ-semi-continuous image Y of an almost semi-Hurewicz

space X is a semi-Hurewicz space.

Proof. Let (Vn : n ∈ N) be a sequence of semi-open covers of Y . Let x ∈ X.For each n ∈ N there is a set Vx,n ∈ Vn containing f(x). Since f is stronglyθ-semi-continuous there is a semi-open set Ux,n ⊂ X containing x such thatf(sCl(Ux,n)) ⊂ Vx,n. Therefore, for each n the set Un := Ux,n : x ∈ Xis a semi-open cover of X. As X is almost semi-Hurewicz, there is a sequence(Fn : n ∈ N) of nite sets such that for each n, Fn ⊂ Un and each x ∈ X belongsto sCl (

⋃Fn) for all but nitely many n. To each F ∈ Fn assign a set WF ∈ Vn

with f(sCl(F )) ⊂ WF and put Wn = WF : F ∈ F. We obtain the sequence(Wn : n ∈ N) of nite subsets of Vn, n ∈ N, which witnesses for (Vn : n ∈ N) thatY is a semi-Hurewicz space, as it is easily checked.

A mapping f : X → Y is called contra-semi-continuous if the preimage of eachsemi-open set in Y is semi-closed in X . f is said to be pre-semi-continuous iff←(V ) ⊂ sInt(sCl(f←(V ))) whenever V is semi-open in Y .

3.11. Theorem. A contra-semi-continuous, pre-semi-continuous image Y of an

almost semi-Hurewicz space X is a semi-Hurewicz space.

Proof. Let (Vn : n ∈ N) be a sequence of semi-open covers of Y . Since f iscontra-semi-continuous, for each n ∈ N and each V ∈ Vn the set f←(V ) is semi-closed in X. On the other hand, because f is pre-semi-continuous f←(V ) ⊂sInt(sCl(f←(V ))), so that f←(V ) ⊂ sInt(f←(V )), i.e. f←(V ) = sInt(f←(V )).Therefore, for each n, the set Un = f←(V ) : V ∈ Vn is a cover of X by semi-open sets. As X is an almost semi-Hurewicz space there is a sequence (Gn : n ∈ N)such that for each n, Gn is a nite subset of Un and each x ∈ X belongs to⋃sCl(G) : G ∈ Gn. Then Wn = f(G) : G ∈ Gn is a nite subset of Vn foreach n ∈ N and each z ∈ Y belongs to sCl(

⋃Wn) for all but nitely many n. This

means that Y is a semi-Hurewicz space.

A mapping f : X → Y is called weakly semi-continuous if for each x ∈ X andeach semi-open neighbourhood V of f(x) there is a semi-open neighbourhood Uof x such that f(U) ⊂ sCl(V ).

3.12. Theorem. A weakly semi-continuous image Y of a semi-Hurewicz space Xis an almost semi-Hurewicz space.

59

Proof. Let (Vn : n ∈ N) be a sequence of open covers of Y . Let x ∈ X. Thenfor each n ∈ N there is a V ∈ Vn such that f(x) ∈ V . Since f is weakly semi-continuous there is a semi-open set Ux,n in X such that x ∈ Ux,n and f(Ux,n) ⊂sCl(V ). The set Un := Ux,n : x ∈ X is a semi-open cover of X. Apply thefact that X is a semi-Hurewicz space to the sequence (Un : n ∈ N) and nd asequence (Fn : n ∈ N) of nite sets such that for each n, Fn ⊂ Un and each x ∈ Xbelongs to

⋃Fn for all but nitely many n. To each n and each U ∈ Fn assign

a set VU ∈ Vn such that f(U) ⊂ sCl(VU ) and put Wn = VU : U ∈ Fn. Theneach z ∈ Y belongs to sCl(

⋃Wn) for all but nitely many n, i.e. Y is an almost

semi-Hurewicz space.

4. Star semi-Hurewicz property

4.1. Denition. Call a space X:

• star s-Hurewicz (shortly denoted SsH) if it satises: For each sequence(Un : n ∈ N) of elements of sO there is a sequence (Vn : n ∈ N) such thatfor each n ∈ N, Vn is a nite subset of Un, and each x ∈ X belongs toSt(⋃Vn,Un) for all but nitely many n;

• strongly star s-Hurewicz (denoted SSsH) if it satises: For each sequence(Un : n ∈ N) of elements of sO there is a sequence (An : n ∈ N) of nitesubsets of X, and each x ∈ X belongs to St(An,Un) for all but nitelymany n.

Recall that a space X is star semi-compact, denoted SsC, (star semi-Lindelöf,denoted SsL) if for each semi-open cover U of X there is a nite (countable)V ⊂ U such that St(

⋃V,U) = X. X is strongly star semi-compact, shortly SSsC,

(strongly star semi-Lindelöf, SSsL) if for each semi-open cover U of X there is anite (countable) A ⊂ X such that St(A,X) = X.

Evidently, we have the following diagram:

SSsC ⇒ SSsH ⇒ SSsL

⇓ ⇓ ⇓SsC ⇒ SsH ⇒ SsL

Call a space X σ-strongly star semi-compact if it is union of countably manystrongly star semi-compact spaces.

4.2. Theorem. Every σ-strongly star semi-compact space is strongly star s-Hurewicz.

Proof. Let X be σ-strongly star semi-compact space. Let X =⋃n∈NXn, where

each Xn is strongly star semi-compact. Suppose that X1 ⊃ X2 ⊃ . . . ⊃ Xn ⊃ . . .,because the union of nitely many strongly star semi-compact spaces is stronglystar semi-compact. Let (Un : n ∈ N) be a sequence of covers of X by semi-opensets. For each n let An be a nite subset of Xn such that St(An,Un) ⊃ Xn. Itfollows that each point of X belongs to all but nitely many sets St(An,Un). Thatis, the sequence (An : n ∈ N) shows that X is strongly star s-Hurewicz space.

4.3. Theorem. Let X be an extremally disconnected space, X is star s-Hurewiczspace if and only if X satises Ufin

∗(sO, sOgp).

60

Proof. Let (Un : n ∈ N) be a sequence of covers of X by semi-open sets. Since Xis star s-Hurewicz space, there exists a sequence (Vn : n ∈ N) such that for eachn ∈ N, Vn is a nite subset of Un, and each x ∈ X belongs to St(

⋃Vn,Un) for all

but nitely many n. This implies that St(⋃

Vn,Un) : n ∈ N is an sγ-cover ofX. Since each countable sγ-cover is s-groupable, St(

⋃Vn,Un) : n ∈ N ∈ sOgp.

Conversely, let (Un : n ∈ N) be a sequence of covers of X by semi-open sets.Let

Hn =∧

i≤nUi.

Then (Hn : n ∈ N) is a sequence of semi-open covers of X since X is extremallydisconnected. Use now Ufin

∗(sO,sOgp) property of X. For each Hn and for eachn ∈ N select a nite set Vn ⊂ Hn such that the set St(

⋃Vn,Hn) : n ∈ N

is an s-groupable cover of X. Let n1 < n2 < ... < nk < . . . be a sequence ofnatural numbers which witnesses this fact, i.e. for each x ∈ X, x belongs to⋃St(

⋃Vi,Hi) : nk < i ≤ nk+1 for all but nitely many k. Put

Wn =⋃

i<n

Vi, for n < n1;

Wn =⋃

nk<i≤nk+1

Vi, for nk ≤ n < nk+1.

Then the sequence (Wn : n ∈ N) shows that X satises star s-Hurewicz propertybecause each x ∈ X belongs to all but nitely many St(

⋃Wn,Un).

4.4. Denition. A space X is said to satisfy SsH≤n if for each sequence (Un : n ∈N) of elements of sO there is a sequence (Vn : n ∈ N) such that for each n ∈ N,Vn ∈ [Un]≤n, and the set St(

⋃Vn,Un) : n ∈ N is an sγ-cover.

4.5. Theorem. Let X be an extremally disconnected space satisfying SsH≤n. Then

X satises S1∗(sO, sOgp).

Proof. Let (Un : n ∈ N) be a sequence of semi-open covers of X. For each n dene

Vn =∧Ui : (n− 1)n/2 < i ≤ n(n+ 1)/2.

As X is extremally disconnected, each Vn is a semi-open cover of X. By applyingSsH≤n to the sequence (Vn : n ∈ N), we can nd a sequence (Wn : n ∈ N) such thatfor each n, Wn is a subset of Vn having ≤ n elements, and St(

⋃Wn,Vn) : n ∈ N

is an sγ-cover of X. Write Wn = Wi : (n − 1)n/2 < i ≤ n(n + 1)/2, and eachWi ∈Wn as the intersection of some elements from Uj , (n−1)n/2 < j ≤ n(n+1)/2.For each Wi take also the set Uj ∈ Uj which is a term in the above representationof Wi. The set St(Un,Un) : n ∈ N is a semi-open groupable cover of X. For,consider the sequence n1 < n2 < . . . < nk < . . . in N, where nk = k(k − 1)/2.Then, as it is easily checked, for each x ∈ X the fact x ∈ ⋃nk<i≤nk+1

St(Wi,Ui) for

all but nitely many k, implies that the cover St(Un,Un) : n ∈ N is s-groupable,i.e. that X satises S1

∗(sO, sOgp).

In a similar way we prove the following theorem.

61

4.6. Theorem. Let an extremally disconnected space X satises the condition

SSsH≤n: For each sequence (Un : n ∈ N) of semi-open covers of X there is

a sequence (Sn : n ∈ N) of subsets of X such that for each n |Sn| ≤ n and

St(Sn,Un) : n ∈ N is an s-γ cover of X. Then X satises SS1∗(sO, sOgp).

Proof. Let (Un : n ∈ N) be a sequence of semi-open covers of X. For each n, as inthe proof of the previous theorem, let

Vn =∧

(n−1)n2 <i≤n(n+1)

2

Ut.

Apply now SSsH≤n to the sequence (Vn : n ∈ N), and nd a sequence (Fn : n ∈ N)of subsets of X such that for each n, |Fn| ≤ n and St(Fn,Vn) : n ∈ N isan sγ cover of X. For every x ∈ X there exists positive integer n0 such that

x ∈ St(Fn,Vn) for all n > n0. Write for each n, Fn = xj : (n−1)n2 < j ≤ n(n+1)

2 .The sequence n1 < n2 < . . . < nk < . . . of natural numbers dened by nk = k(k−1)

2 ,witnesses for (Un : n ∈ N) that X satises SS1

∗(sO, sOgp). Indeed, it is evidentthat each x ∈ X belongs to

⋃nk<j≤nk+1

St(xj ,Uj) for all but nitely many k.

4.7. Theorem. If a space X satises Ufin∗(sO, sOwgp), then any open, semi-closed

subset of X satises Ufin∗(sO, sΩ).

Proof. Let A be an open and semi-closed subset of X and let (Hn : n ∈ N)be a sequence of semi-open covers of A. Because A is open, hence semi-open,for each n and each H ∈ Hn the set H is semi-open in X. Therefore, settingSn = Hn

⋃(X \ A), we get a sequence (Sn : n ∈ N) of semi-open covers of X.

Applying Ufin∗(sO, sOwgp) for X we nd a sequence (Wn : n ∈ N) such that for

each n, Wn is a nite subset of Sn and St(⋃

Wn, Sn) : n ∈ N is an s-weaklygroupable cover of X, i.e. there is a sequence n1 < n2 < · · · < nk < · · · of naturalnumbers such that for each nite set F in X one has

F ⊂⋃St(∪Wi, Si) : nk < i ≤ nk+1

for some k. For each n ∈ N put Kn = Wn \ X \ A. Then each Kn is a nitesubset of Hn. Dene now

Gn =⋃

i<n

Ki, for n < n1,

Gn =⋃

nk<i≤nk+1

Ki, for nk < n ≤ nk+1.

Each Gn is a nite subset ofHn and for each nite E ⊂ A we have E ⊂ St(∪Gi,Ui).Hence, A satises Ufin

∗(sO, sΩ).

4.8. Theorem. For an extremally disconnected space X the following are equiv-

alent:

(1) X has the strongly star-s-Hurewicz property;(2) X satises SSfin

∗(sO, sOgp).

Proof. (1) ⇒ (2): It is obvious because countable sγ-covers are s-groupable andSSfin

∗ is monotone in the second variable.

62

(2) ⇒ (1): Let (Un : n ∈ N) be a sequence of covers of X by semi-open sets.Let for each n,

Wn =∧

i≤nUi.

Each Wn is a semi-open cover of X. Apply SSfin∗(sO, sOgp) to the sequence (Wn :

n ∈ N). We nd a sequence (Bn : n ∈ N) of nite subsets of X such thatSt(Bn,Wn) : n ∈ N is an s-groupable cover of X. Let n1 < n2 < · · · < nk < · · ·be sequence of natural numbers such that for every y in X, we have

y ∈⋃

nk≤n<nk+1

St(Bi,Wi)

for all but nitely many k ∈ N. For each n, let

Sn =⋃

i<n1

Bi, for n < n1;

Sn =⋃

nk≤i≤nk+1

Bi, for nk ≤ n < nk+1.

Each Sn is a nite subset of X. We claim that the set St(Sn,Wn) : n ∈ N is ansγ-cover of X.

Let x ∈ X. There exist t ∈ N such that x ∈ ⋃nk≤n<nk+1Bi for all k > t. Since

St(Bi,Wi) ⊂ St(Si,Ui) for all i with nk ≤ i < nk+1, we have that for each k > t,x ∈ St(Sk,Uk), that is St(Sn,Un) : n ∈ N is an sγ-cover of X.

Another characterization of strongly star s-Hurewicz spaces is given in the nexttheorem.

4.9. Theorem. A space X is a strongly star s-Hurewicz space if and only if

for every sequence (Un : n ∈ N) of semi-open covers of X there is a sequence

(Sn : n ∈ N) of nite subsets of X such that for every x ∈ X, St(x,Un) ∩ Sn 6= ∅for all but nitely many n.

Proof. Let (Un : n ∈ N) be a sequence of covers of X by semi-open sets. Thereexists a sequence (Fn : n ∈ N) of nite subsets of X such that each x ∈ X belongsto St(Fn,Un) for all but nitely many n. In other words, for each x ∈ X thereexists n0(x) ∈ N such that x ∈ St(Fn,Un) for all n > n0. St(Fn,Un) is the unionof those elements of Un which intersect Fn. St(x,Un) is the union of thoseelements of Un which contains x. This implies St(x,Un)∩Fn 6= ∅ for all n > n0.

Conversely, let (Un : n ∈ N) be a sequence of covers of X by semi-open sets.Then, by assumption, there is a sequence (An : n ∈ N) of nite subsets of X suchthat for every x ∈ X there exists n0 ∈ N such that St(x,Un) ∩ An 6= ∅ for alln > n0 . This implies x ∈ St(An,Un) for all but nitely many n. Therefore,x ∈ St(An,Un) for all but nitely many n, i.e. X is strongly star s-Hurewicz.

Now we consider preservation of (stronly) star s-Hurewicz property under usualtopological operations.

4.10. Theorem. A semi-open Fσ-subset of a strongly star s-Hurewicz space is

strongly star s-Hurewicz.

63

Proof. Let X be a strongly star s-Hurewicz space and let A =⋃Mn : n ∈ N

be a semi-open Fσ-subset of X, where each Mn is closed in X for each n ∈ N.Without loss of generality, we can assume that Mn ⊂Mn+1 for each n ∈ N. Nowwe show that A is strongly star s-Hurewicz space. Let (Un : n ∈ N) be a sequenceof semi-open covers of A. We need to nd a sequence (Fn : n ∈ N) of nite subsetsof A such that for each a ∈ A, a ∈ St(Fn,Un) for all but nitely many n. For eachn ∈ N, let Vn = Un

⋃X \Mn. Then (Vn : n ∈ N) is a sequence of semi-opencovers of X. There exists a sequence (F ′n : n ∈ N) of nite subsets of X such thatfor each x ∈ X, x ∈ St(F ′n,Vn) for all but nitely many n, since X is a stronglystar s-Hurewicz space. For each n ∈ N, let Fn = F ′n ∩ A. Thus (Fn : n ∈ N) isa sequence of nite subsets of A. For every a ∈ A, there exists k ∈ N such thata ∈ Fn and a ∈ St(F ′n,Vn) for each n > k. Hence a ∈ St(Fn,Un) for n > k, whichshows that A is strongly star s-Hurewicz.

4.11. Theorem. If each nite power of a space X is star s-Hurewicz, then Xsatises Ufin

∗(sO, sΩ).

Proof. Let (Un : n ∈ N) be a sequence of covers of X by semi-open sets. Let N =N1

⋃N2

⋃ · · · be a partition of N into innitely many innite pairwise disjoint sets.For every k ∈ N and every t ∈ Nk let Wt = U1×U2×· · ·×Uk : U1, . . . Uk ∈ Ut =Ukt . Then (Wt : t ∈ Nk) is a sequence of semi-open covers of Xk, and since Xk is astar s-Hurewicz space, we can choose a sequence (Ht : t ∈ Nk) such that for each t,Ht is a nite subset of Wt and

⋃t∈NkSt(H,Wt) : H ∈ Ht is a semi-open cover of

Xk. For every t ∈ Nk and everyH ∈ Ht we haveH = U1(H)×U2(H)×· · ·×Uk(H),where Ui(H) ∈ Ut for every i ≤ k. Set Vt = Ui(H) : i ≤ k,H ∈ Ht. Then foreach t ∈ Nk, Vt is a nite subset of Ut.

We claim that St(⋃Vn,Un) : n ∈ N is an sω-cover of X. Let F = x1, ..., xpbe a nite subset of X. Then y = (x1, ..., xp) ∈ Xp so that there is an n ∈ Np suchthat y ∈ St(H,Wn) for some H ∈ Hn . But H = U1(H)× U2(H)× · · · × Up(H),where U1(H), U2(H), . . . , Up(H) ∈ Vn. The point y belongs to some W ∈ Wn ofthe form V1×V2×· · ·×Vp, Vi ∈ Un for each i ≤ p, which meets U1(H)×U2(H)×· · · × Up(H). This implies that for each i ≤ p, we have xi ∈ St(Ui(H),Un) ⊂St(⋃Vn,Un), that is, F ⊂ St(

⋃Vn,Un). Hence, X satises Ufin

∗(sO, sΩ).

In a similar way one proves the following theorem.

4.12. Theorem. If all nite powers of a space X are strongly star s-Hurewicz,then X satises SSfin

∗(sO, sΩ).

Proof. Let (Un : n ∈ N) be a sequence of covers of X by semi-open sets. LetN = N1

⋃N2

⋃... be a partition of N into innite pairwise disjoint sets. For every

k ∈ N and every t ∈ Nk let Wt = Ukt . Then (Wt : t ∈ Nk) is a sequence of semiopen covers of Xk. Applying strongly star s-Hurewicz property of Xk we can geta sequence (Vt : t ∈ Nk) of nite subsets of Xk such that each x ∈ Xk belongsto St(Vt,Wt) : t ∈ Nk, for all but nitely many t. For each t consider At a nitesubset of X such that Vt ⊂ Akt .

We show that St(An,Un) : n ∈ N is an sω-cover of X. Let F = x1, ..., xpbe a nite subset of X. Then (x1, ..., xp) ∈ Xp such that there is n ∈ Np and(x1, ..., xp) ∈ St(Vn,Wn) ⊂ St(Apn,Wn). Consequently, F ⊂ St(An,Un).

64

The following two theorems give relations between strongly star s-Hurewiczspaces and s-Hurewicz and s-Lindelöf spaces.

A space X is called meta semi-compact if every semi-open cover U of X has apoint-nite semi-open renement V (that is, every point of X belongs to at mostnitely many members of V).

4.13. Theorem. Every strongly star s-Hurewicz meta semi-compact space is s-Hurewicz space.

Proof. Let X be a strongly star s-Hurewicz meta semi-compact space. Let (Un :n ∈ N) be a sequence of semi-open covers of X. For each n ∈ N, let Vn be a point-nite semi-open renement of Un. Since X is strongly star s-Hurewicz, there isa sequence (Fn : n ∈ N) of nite subsets of X such that each x ∈ X belongs toSt(Fn,Vn) for all but nitely many n.

Since Vn is a point-nite renement and each Fn is nite, elements of eachFn belong to nitely many members of Vn say Vn1 , Vn2 , Vn3 , . . . , Vnk

. Let V′n =Vn1 , Vn2 , Vn3 , . . . , Vnk

. Then St(Fn,Vn) =⋃V′n for each n ∈ N. We have that

each x ∈ X belongs to⋃V′n for all but nitely many n. For every V ∈ V′n choose

UV ∈ Un such that V ⊂ UV . Then, for every n, UV : V ∈ V′n = Wn is a nitesubset of Un and each x ∈ X belongs to

⋃Wn for all but nitely many n, that is

X is an s-Hurewicz space.

4.14. Denition. ([15]) A space X is said to be meta semi-Lindelöf if everysemi-open cover U of X has a point-countable semi-open renement V.

4.15. Theorem. Every strongly star s-Hurewicz meta semi-Lindelöf space is a

semi-Lindelöf space.

Proof. Let X be a strongly star s-Hurewicz meta semi-Lindelof space. Let U be asemi-open cover of X then there exists V, a point-countable semi-open renementof U. Let Vn = V for each n ∈ N. Since X is strongly star s-Hurewicz, thereexists a sequence (Fn : n ∈ N) of nite subsets of X such that for each x ∈ X,x ∈ St(Fn,Vn) for all but nitely many n.

For every n ∈ N denote by Wn the collection of all members of V which intersectFn. Since V is point-countable and Fn is nite, Wn is countable. So, the setW =

⋃n∈N Wn is a countable subset of V and is a cover of X. For every W ∈ W

pick a member UW ∈ U such that W ⊂ UW . Then UW : W ∈W is a countablesubcover of U. Hence, X is a semi-Lindelöf space.

We end this section with few observations on almost star s-Hurewicz spaces.

4.16. Denition. Call a space X almost star s-Hurewicz if for each sequence(Un : n ∈ N) of semi-open covers of X there exists a sequence (Vn : n ∈ N) suchthat for every n ∈ N, Vn is a nite subset of Un and each x ∈ X belongs tosCl(St(

⋃Vn,Un)) for all but nitely many n.

4.17. Theorem. A space X is an almost star s-Hurewicz space if and only if

for each sequence (Un : n ∈ N) of covers of X by semi-regular sets there exists a

sequence (Vn : n ∈ N) such that for every n ∈ N, Vn is a nite subset of Un and

each x ∈ X belongs to sCl(St(⋃

Vn,Un)) for all but nitely many n ∈ N.

65

Proof. (=⇒) Since every semi-regular set is semi-open, it is obvious.

(⇐=) Conversely, let (Un : n ∈ N) be a sequence of semi-open covers of X.Let U′n = sCl(U) : U ∈ Un. Then U′n is a cover of X by semi-regular sets.By assumption there exists a sequence (Vn : n ∈ N) such that for every n ∈ N,Vn is a nite subset of U′n and each x ∈ X there is n0(x) ∈ N such that x ∈sCl(St(

⋃Vn,U

′n)) for all n ≥ n0(x).

For each V ∈ Vn we can nd UV ∈ Un such that V = sCl(UV ). Let V′n =UV : V ∈ Vn. It is easy to see now that x belongs to sCl(

⋃V′n,Un) for all

n ≥ n0(x).

4.18. Theorem. A quasi-irresolute image of an almost star s-Hurewicz space is

an almost star s-Hurewicz space.

Proof. Let X be an almost star s-Hurewicz space and Y be a topological space.Let f : X → Y be a quasi-irresolute surjection and let (Un : n ∈ N) be a sequenceof covers of Y by semi-regular sets. Let U′n = f←(U) : U ∈ Un. Then each U′n isa cover of X by semi-regular sets since f is quasi-irresolute. Since X is an almoststar s-Hurewicz space, there exists a sequence (V′n : n ∈ N) such that for everyn ∈ N, V′n is a nite subset of U′n and each x ∈ X belongs to sCl(St(

⋃V′n,U

′n))

for all but nitely many n.It is not hard to verify that setting Vn = f(V ) : V ∈ V′n, each y ∈ Y belongs

to all but nitely many sets sCl(St(⋃

Vn,Un)) which means that Y is an almosts-Hurewicz space.

References

[1] Cameron, D.E., Woods, G. s-continuous and s-open mappings, Preprint, 1987.[2] Crossley, S.G., Hildebrand, S.K. Semi-closure, Texas J. Sei. 22, 99112, 1971.[3] Crossley, S.G., Hildebrand, S.K. Semi-topological properties, Fund. Math. 74 , 233254,

1972.[4] Di Maio, G., Noiri, T. Weak and strong forms of irresolute functions, Suppl. Rend. Circ.

Math, Ser. II 18(3), 255273, 1988.[5] Dorsett, C. Semi-regular spaces, Soochow J. Math. 8 , 4553, 1982.

[6] Hurewicz, W. Über die Verallgemeinerung des Borelschen Theorems, Math. Z. 24, 401-425,1925.

[7] Hurewicz, W. Über Folgen stetiger Functionen, Fund. Math. 9, 193204, 1927.[8] Ko£inac, Lj.D.R. Star-Menger and related spaces, Publ. Math. Debrecen 55, 421431, 1999.[9] Ko£inac, Lj.D.R. The Pixley-Roy topology and selection principles, Questions Answers Gen.

Topology. 19, 219225, 2001.[10] Ko£inac, Lj.D.R. Star selection principles: A survey, Khayyam J. Math. 1(1), 82106, 2015.[11] Levine, N., Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly

70(1), 3641, 1963.

[12] Menger, K., Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte Abt. 2a,Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie, Wien)133, 421444, 1924.

[13] Njåstad, O. On some classes of nearly open sets, Pacic J. Math. 15 , 961970, 1965.[14] Noiri, T. On semi-continuous mappings, Atti Accad. Naz. Lincei, Ser. VIII 54(2), 210214,

1973.[15] Sabah, A., Moiz ud Din Khan, Ko£inac, Lj.D.R. Covering properties dened by semi-open

sets, J. Nonlinear Sci. Appl. 9(6), 43884398, 2016.

66

[16] Sakai, M. The weak Hurewicz property of Pixley-Roy hyperspaces, Topology Appl. 160,25312537, 2013.

[17] Scheepers, M. Combinatorics of open covers I: Ramsey theory, Topology Appl. 69, 3162,1996.

[18] Song, Y.K., Li, R. The almost Hurewicz spaces, Questions Answers Gen. Topology. 31,131136, 2013.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 67 76

Characterizations of quasi-metric completeness interms of Kannan-type xed point theorems

Dedicated to the memory of Professor Lawrence M. Brown

Carmen Alegre∗, Hacer Da§†, Salvador Romaguera‡ and Pedro Tirado

Abstract

We obtain quasi-metric versions of Kannan's xed point theorem forself-mappings and multivalued mappings, respectively, which are usedto deduce characterizations of d-sequentially complete and of left K-sequentially complete quasi-metric spaces, respectively.

Keywords: Quasi-metric space, complete, Kannan mapping, xed point.

2000 AMS Classication: 54H25, 54E50, 47H10.

1. Introduction and preliminaries

Since Hu proved in [10] that a metric space (X, d) is complete if and only iffor any closed subspace C of (X, d), every Banach contraction on C has xedpoint, several authors have investigated the problem of characterizing the metriccompleteness with the help of xed point theorems (see e.g. [13, 18, 25, 26, 27, 28]).Next we recall those characterizations which will be related with our approach.

∗Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València,46022 Valencia, SpainEmail : calegre@mat.upv.es†Departamento de Matemática Aplicada, Universitat Politècnica de València, 46022 Valencia,

SpainEmail : hada@doctor.upv.es‡Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València,

46022 Valencia, SpainDepartamento de Matemática Aplicada, Universitat Politècnica de València, 46022 Valencia,SpainEmail : sromague@mat.upv.es, Corresponding author.

Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València,46022 Valencia, SpainEmail : pedtipe@mat.upv.es

Doi : 10.15672/HJMS.2016.395

68

Caristi proved in [6] the following important generalization of the Banach con-traction principle.

1.1. Theorem (see [6]). Let (X, d) be a complete metric space. If T is a self-

mapping of X such that there is a lower semicontinuous function ϕ : X → [0,∞)satisfying

(1.1) d(x, Tx) ≤ ϕ(x)− ϕ(Tx),for all x ∈ X, then T has a xed point.

A self-mapping T on a metric space (X, d) for which there is a lower semicon-tinuous function ϕ : X → [0,∞) satisfying condition (1.1) for all x ∈ X, is calleda Caristi mapping on (X, d).

Kirk proved in [13] that Caristi's xed point theorem allows to characterize themetric completeness as follows.

1.2. Theorem (see [13]). A metric space (X, d) is complete if and only if every

Caristi mapping on (X, d) has a xed point.

Almost simultaneously, Subrahmanyam [26] showed that the well-known Kan-nan xed point theorem (see Theorem 1.3 below) also allows to characterize themetric completeness.

1.3. Theorem (see [11]). Let (X, d) be a complete metric space. If T is a

self-mapping of X such that there is a constant c ∈ [0, 1/2) satisfying

(1.2) d(Tx, Ty) ≤ c(d(x, Tx) + d(y, Ty)),

for all x, y ∈ X, then T has a unique xed point.

The above result suggests the following well-established notion: A self-mappingT of a metric space (X, d) is said to be a Kannan mapping on (X, d) if there existsa constant c ∈ [0, 1/2) for which condition (1.2) is satised for all x, y ∈ X.

Then, Subrahmanyam proved the following.

1.4. Theorem (see [26]). A metric space (X, d) is complete if and only if every

Kannan mapping on (X, d) has a xed point.

On the other hand, and motivated in part by the fact that quasi-metric spacesprovide suitable frameworks in several areas of asymmetric functional analysis,domain theory, and complexity analysis of algorithms dened by recurrence equa-tions (see [8] and its bibliography, [4, 20, 21, 23, 24] etc.), the development ofthe xed point theory for theses spaces is receiving a signicant boost (see e.g.[1, 2, 3, 5, 7, 9, 12, 15, 16, 17]). In this setting, the problem of characterizingquasi-metric completeness via xed point theorems arises in a natural way. Thisproblem has an extra appeal due to the existence of several dierent notions ofquasi-metric completeness in the literature, so it seems reasonable to expect the ex-istence of interesting dierences with respect to the classical metric setting. In this

69

paper we show that this is the case. Indeed, Romaguera and Tirado [22] extendedKirk's characterization (Theorem 1.2) to the realm of Smtyh complete quasi-metricspaces, while here we discuss the problem of characterizing the quasi-metric com-pleteness by using appropriate versions of Kannan's xed point theorem. In thisfashion, we shall obtain characterizations of d -sequentially complete and of leftK-sequentially complete quasi-metric spaces, respectively.

We conclude this section by recalling some pertinent notions and properties onquasi-metric spaces which will be useful later on. (By N we will denote the set ofall positive integer numbers.)

Following the modern terminology (see [8]), a quasi-metric on a set X is afunction d : X ×X → [0,∞) such that for all x, y, z ∈ X :

(i) x = y ⇔ d(x, y) = d(y, x) = 0, and

(ii) d(x, z) ≤ d(x, y) + d(y, z).

A quasi-metric space is a pair (X, d) such that X is a set and d is a quasi-metricon X.

Given a quasi-metric d on a setX the function ds dened onX×X by ds(x, y) =maxd(x, y), d(y, x) for all x, y ∈ X, is a metric on X.

Each quasi-metric d on X induces a T0 topology τd onX which has as a basethe family of open balls Bd(x, r) : x ∈ X, ε > 0, where Bd(x, ε) = y ∈ X :d(x, y) < ε for all x ∈ X and ε > 0.

If τd is a T1 topology on X, we say that d is a T1 quasi-metric on X.

A sequence (xn)n∈N in a quasi-metric space (X, d) is called left K-Cauchy [19] iffor each ε > 0 there exists nε ∈ N such that d(xn, xm) < ε whenever nε ≤ n ≤ m.

A quasi-metric space (X, d) is called left K-sequentially complete (resp. d -sequentially complete) [8, 19] if every left K-Cauchy sequence in (X, d) (resp.every Cauchy sequence in the metric space (X, ds)) converges for the topologyτd, and it is called Smyth complete (see e.g. [14, 22, 23]) if every left K-Cauchysequence in (X, d) converges for the topology τds .

The following implications are obvious for a quasi-metric space (X, d):

Smyth complete ⇒ left K-sequentially complete ⇒ d -sequentially complete.

The converse implications do not hold in general. The following known exam-ples illustrate this fact.

1.5. Example. Let X = N∪0 and let d be the T1 quasi-metric on X givenby d(x, x) = 0 for all x ∈ X, d(0, x) = 1/x for all n ∈ N, and d(x, y) = 1 other-wise. Then (X, d) is clearly left K-sequentially complete (note that τd is a compacttopology on X), but it is not Smyth complete because the sequence (n)n∈N is leftK-Cauchy sequence but does not converge for τds .

1.6. Example. Let R be the set of all real numbers and let d be the T1 quasi-metric on R given by d(x, y) = y−x if x ≤ y, and d(x, y) = 1 if x > y. Then (R, d) isd -sequentially complete because the Cauchy sequences in the metric space (R, ds)are eventually constant. However, it is not left K-sequentially complete becausethe sequence (−1/n)n∈N is left K-Cauchy but does not converge for τds . Observethat τd is the well-known Sorgenfrey topology on R.

70

2. The results

In [22], Smyth complete quasi-metric spaces were characterized by means of anappropriate quasi-metric version of Caristi's xed point theorem.

According to [22], a self-mapping T of a quasi-metric space (X, d) is said to bea ds-Caristi mapping on (X, d) if there exists a function ϕ : X → [0,∞) which islower semicontinuous for τds and satises d(x, Tx) ≤ ϕ(x)− ϕ(Tx), for all x ∈ X.

Then it was proved the following.

2.1. Theorem (see [22]). A quasi-metric space (X, d) is Smyth complete if and

only if every ds-Caristi mapping on (X, d) has a xed point.

In the sequel we shall prove that, however, quasi-metric versions of Kannan'sxed point theorem for self-mappings and multivalued mappings characterize d -sequential completeness and left K-sequential completeness, respectively.

2.2. Denition. Let (X, d) be a quasi-metric space. By a d-Kannan mapping on(X, d) we mean a self-mapping T of X such that there exists a constant c ∈ [0, 1/2)satisfying

(2.1) d(Tx, Ty) ≤ c(d(x, Tx) + d(y, Ty)),

for all x, y ∈ X.

2.3. Lemma. Let T be a d-Kannan mapping on a quasi-metric space (X, d) withconstant c ∈ [0, 1/2). Then:

(a) ds(Tx, Ty) ≤ c(d(x, Tx) + d(y, Ty)), for all x, y ∈ X.(b) T is a Kannan mapping on the metric space (X, ds).

(c) For any x0 ∈ X, the sequence (Tnx0)n∈N is a Cauchy sequence in the metric

space (X, ds).

Proof. (a) Given x, y ∈ X we have

d(Tx, Ty) ≤ c(d(x, Tx)+d(y, Ty)) and d(Ty, Tx) ≤ c(d(y, Ty)+d(x, Tx)),so

ds(Tx, Ty) ≤ c(d(x, Tx) + d(y, Ty)) ≤ c(d(x, Tx) + d(y, Ty)).

(b) Since d(x, Tx) ≤ ds(x, Tx) and d(y, Ty) ≤ ds(y, Ty) for all x, y ∈ X, itfollows from assertion (a) that T is a Kannan mapping on (X, ds), with constantc.

(c) Since, by (b), T is a Kannan mapping for the metric space (X, ds), theclassical proof of Kannan's xed point theorem [11] shows that for any x0 ∈ X,(Tnx0)n∈N is a Cauchy sequence in the metric space (X, ds).

Related to Lemma 2.3 (b) we give an example of a self-mapping of a quasi-metric space (X, d) which is a Kannan mapping on (X, ds) but not a d -Kannan

71

mapping.

2.4. Example. Let X = [0,∞) and let d be the quasi-metric on X given byd(x, y) = maxy−x, 0 for all x, y ∈ X. It is well known that (X, d) is Smyth com-plete. Now dene T : X → X as Tx = 0 if x ∈ [0, 1] and Tx = x/4 if x ∈ (1,∞).If x > y > 1 we have d(Tx, Ty) = (x− y)/4 but d(x, Tx) = d(y, Ty) = 0, so thatT is not d -Kannan on (X, d). However, it is easy to check that T is a Kannanmapping on (X, ds) for c = 1/3 (note that ds is the Euclidean metric on X).

2.5. Theorem. Let (X, d) be a d-sequentially complete quasi-metric space. Then,

every d-Kannan mapping on (X, d) has a unique xed point.

Proof. Let T be a d -Kannan mapping on (X, d). Then, there exists c ∈ [0, 1/2)such that the contraction condition (2.1) follows for all x, y ∈ X. Fix x0 ∈ X.From Lemma 2.3 (c), (Tnx0)n∈N is a Cauchy sequence in the metric space (X, ds).Since (X, d) is d-sequentially complete, there exists z ∈ X such that (Tnx0)n∈Nconverges to z for τd, i.e., d(z, T

nx0)→ 0 as n→∞.Next we show that Tz is the unique xed point of T. To this end, we rst show

that d(z, Tz) = 0. Indeed, we have

d(z, Tz) ≤ d(z, Tnx0) + d(Tnx0, T z)

≤ d(z, Tnx0) + c(d(Tn−1x0, Tnx0) + d(z, Tz)),

for all n ∈ N. Since d(z, Tnx0) → 0 and (Tnx0)n∈N is a Cauchy sequence inthe metric space (X, ds), we deduce that d(z, Tz) ≤ cd(z, Tz). Consequently,d(z, Tz) = 0.

Since by Lemma 2.3 (a),

ds(Tz, T 2z) ≤ c(d(z, Tz) + d(Tz, T 2)),

we deduce that ds(Tz, T 2z) ≤ cd(Tz, T 2z), so ds(Tz, T 2z) = 0, i.e., Tz is a xedpoint of T.

Finally, if Tu = u, it follows from Lemma 2.3 (a) that

ds(u, Tz) = ds(Tu, T 2z) ≤ c(d(u, Tu) + d(Tz, T 2z)).

Since d(u, Tu) = d(Tz, T 2z) = 0, we deduce that ds(u, Tz) = 0, i.e., u = Tz. Thisconcludes the proof.

The following examples illustrate Theorem 2.5.

2.6. Example. Let X = [0,∞) and let d be the quasi-metric on X given byd(x, y) = maxx − y, 0 for all x, y ∈ X. Since ds is the Euclidean metric on X,(X, d) is d -sequentially complete (in fact, it is left K-sequentially complete becauseevery sequence in X converges to 0 for τd). Dene T : X → X as in Example 2.4.Let x, y ∈ X, and assume, without loss of generality, that x ≤ y. If x, y ∈ [0, 1],then ds(Tx, Ty) = 0. If x ∈ [0, 1] and y ∈ (1,∞) we obtain

ds(Tx, Ty) =y

4≤ 1

3(x+

3y

4) =

1

3(d(x, Tx) + d(y, Ty)).

72

Finally, if x, y ∈ (1,∞) we obtain

ds(Tx, Ty) =y − x4

<1

3(3x

4+

3y

4) =

1

3(d(x, Tx) + d(y, Ty)).

Therefore T is a d -Kannan mapping on (X, d) for c = 1/3. Thus, all conditions ofTheorem 2.5 are satised. In fact z = 0 is the unique xed point of T.

2.7. Example. Let X = [0, 1] ∪ 2 and let d be the quasi-metric on X givenby d(2, x) = 0 for all x ∈ X, nd d(x, y) = |x− y| otherwise. Clearly (X, d) isd-sequentially complete. Dene T : X → X as T2 = 0 and Tx = x/4 if x ∈ [0, 1].It is easy to check that T is a d-Kannan mapping on (X, d) for c = 1/3. Thus,all condition of Theorem 2.5 are satised. It is interesting to observe that for anyx0 ∈ X the sequence (Tnx0)n∈N converges to 2 for τd but 2 is not the xed pointof T. This situation illustrates the proof of Theorem 2.5 which shows that T2 isthe unique xed point of T ; in fact (Tnx0)n∈N converges to T2 for τds .

2.8. Theorem. A quasi-metric space (X, d) is d-sequentially complete if and

only if every d-Kannan mapping on (X, d) has a xed point.

Proof. Suppose that (X, d) is d-sequentially complete. Then, every d -Kannanmapping on (X, d) has a (unique) xed point by Theorem 2.5.

For the converse suppose that (X, d) is not d-sequentially complete. Then thereexists a Cauchy sequence (xn)n∈N in (X, ds) that does not converge for τd. Then,for each x ∈ X there exists nx ∈ N such that d(x, xn) > 0, for all n ≥ nx (indeed,otherwise there is x ∈ X such that for each n ∈ N we can nd mn ≥ n forwhich d(x, xmn

) = 0; since (xn)n∈N is a Cauchy sequence in (X, ds) it follows that(xn)n∈N converges to x for τd, a contradiction).

Now, for each x ∈ X put Cx = xn : n ≥ nx. Clearly d(x,Cx) > 0 (indeed,if d(x,Cx) = 0, for some x ∈ X, reasoning as in the parenthetical part of thepreceding paragraph, we deduce that that sequence (xn)n∈N converges to x for τd,a contradiction).

Since (xn)n∈N is a Cauchy sequence in (X, ds), for each x ∈ X there existsn(x) ≥ nx such that

ds(xn, xm) <1

4d(x,Cx),

for all m,n ≥ n(x).Dene T : X → X as Tx = xn(x) for all x ∈ X.Since n(x) ≥ nx, we have that d(x, xn(x)) > 0, and hence T has not xed point.

We shall show that, nevertheless, T is a d -Kannan mapping on (X, d) for c =1/4. Indeed, let x, y ∈ X and suppose, without loss of generality, that n(x) ≤ n(y).Then

ds(Tx, Ty) = ds(xn(x), xn(y)) <1

4d(x,Cx)

≤ 1

4d(x, xn(x)) =

1

4d(x, Tx).

73

Since d(Tx, Ty) ≤ ds(Tx, Ty) and d(Ty.Tx) ≤ ds(Tx, Ty), we conclude that Tis a d -Kannan mapping on (X, d) for c = 1/4. This contradiction nishes theproof.

Let (X, d) be a quasi-metric space. The closure for τd of a subset A of X willbe denoted by A, and the set of all non-empty closed subsets of the topological(X, τd) by Cld(X).

2.9. Denition. Let (X, d) be a quasi-metric space. By a left-Kannan mul-tivalued mapping on (X, d) we mean a multivalued mapping T : X → Cld(X)such that there exists a constant c ∈ [0, 1/2) for which the following condition issatised:

For each x, y ∈ X and each u ∈ Tx there exists v ∈ Ty such that

(2.2) d(u, v) ≤ c(d(x, u) + d(y, v)).

2.10. Theorem. Let (X, d) be a left K-sequentially complete quasi-metric space.

Then, every left-Kannan multivalued mapping on (X, d) has a xed point, i.e.,

there is z ∈ X such that z ∈ Tz.

Proof. Let T be a left-Kannan multivalued mapping on (X, d). Then, thereexists c ∈ [0, 1/2) such that the contraction condition (2.2) in Denition 2.9 followsfor all x, y ∈ X.

Fix x0 ∈ X. Choose x1 ∈ Tx0. Then, there exists x2 ∈ Tx1 such that

d(x1, x2) ≤ c(d(x0, x1) + d(x1, x2)).

Therefore

d(x1, x2) ≤c

1− cd(x0, x1).

Following this process we construct a sequence (xn)n∈N where xn ∈ Txn−1 and

d(xn, xn+1) ≤c

1− cd(xn, xn−1),

for all n ∈ N. Hence

d(xn, xn+1) ≤(

c

1− c

)n

d(x0, x1),

for all n ∈ N. Consequently (xn)n∈N is a left K-Cauchy sequence in (X, d) [8,Proposition 1.2.6]

Since (X, d) is left K-sequentially complete there exists z ∈ X such that d(z, xn)→0 as n→∞.We shall show that z ∈ Tz. Indeed, for each n ∈ N there exists zn ∈ Tzsuch that

(2.3) d(xn+1, zn) ≤ c(d(xn, xn+1) + d(z, zn)).

From the triangle inequality and (5) it follows that

d(z, zn) ≤ d(z, xn+1) + c(d(xn, xn+1) + d(z, zn)),

74

for all n ∈ N. Since d(z, xn+1) → 0 and d(xn, xn+1) → 0 as n → ∞, we deducethat d(z, zn)→ 0 as n→∞, so z ∈ Tz because Tz is closed for τd. This concludesthe proof.

2.11. Lemma (see [8, Proposition 1.2.4]). Let (X, d) be a quasi-metric space.

If a left K-Cauchy sequence in (X, d) has a subsequence that converges for τd to

some x ∈ X, then the sequence converges to x ∈ X for τd.

2.12. Theorem. A quasi-metric space (X, d) is left K-sequentially complete if

and only if every left-Kannan multivalued mapping on (X, d) has a xed point.

Proof. Suppose that (X, d) is left K-sequentially complete. Then, every left-Kannan multivalued mapping on (X, d) has a xed point by Theorem 2.10.

For the converse suppose that (X, d) is not left K-sequantially complete. Thenthere exists a left K-Cauchy sequence (xn)n∈N in (X, d) that does not converge forτd. Similarly to the proof of Theorem 2.8, and using Lemma 2.11, we deduce thatfor each x ∈ X there exists nx ∈ N such that d(x, xn) > 0, for all n ≥ nx.

Now, for each x ∈ X put Cx = xn : n ≥ nx. Then x /∈ Cx and thusd(x,Cx) > 0, where, as usual, d(x,Cx) := infd(x, y) : y ∈ Cx.

Since (xn)n∈N is a left K-Cauchy sequence in (X, d), for each x ∈ X there existsn(x) ≥ nx such that

d(xn, xm) <1

4d(x,Cx),

whenever m ≥ n ≥ n(x).For each x ∈ X put Dx = xn : n ≥ n(x). Then Dx ⊆ Cx, so Dx ⊆ Cx.

Dene T : X → Cld(X) as Tx = Dx for all x ∈ X.Since, for each x ∈ X, x /∈ Cx it follows that x /∈ Tx, and thus T has no xed

points.

We shall show that, nevertheless, T is a left-Kannan multivalued mapping on(X, d) for c = 1/3. Indeed, let x, y ∈ X and suppose, without loss of generalitythat n(x) ≤ n(y). Then Dy ⊆ Dx, so Ty ⊆ Tx, and hence for each u ∈ Ty we cantake v = u ∈ Tx, and thus d(u, v) = 0. On the other hand, given u ∈ Tx thereexists v ∈ Ty such that d(u, v) < d(x,Cx)/12 + d(u, Ty). Since for each ε > 0there exists nε ≥ n(x) such that d(u, xnε

) < ε we deduce (recall that xn(y) ∈ Tyand Tx ⊆ Cx):

d(u, v) <1

12d(x,Cx) + d(u, Ty) ≤ 1

12d(x,Cx) + d(u, xnε

) + d(xnε, Ty)

<1

12d(x,Cx) + ε+ d(xnε

, xn(y)) <1

12d(x,Cx) + ε+

1

4d(x,Cx)

≤ ε+1

3d(x, Tx) ≤ ε+

1

3d(x, u).

Since ε is arbitrary we deduce that

d(u, v) ≤ 1

3d(x, u).

75

We have shown that T is a left-Kannan multivalued mapping on (X, d) for c = 1/3.This nishes the proof.

2.13. Remark. Let (R, d) be the quasi-metric space of Example 1.6. By Theorem2.5, every d -Kannan mapping on (R, d) has a unique xed point. However thereexists a left-Kannan multivalued mapping on it without xed points, by Theorem2.12. Finally, if (X, d) is the quasi-metric space of Example 2.4 or the quasi-metricspace of Example 2.6, then every left-Kannan multivalued mapping on (X, d) hasa xed point by Theorem 2.12.

Acknowledgement. Carmen Alegre, Salvador Romaguera and Pedro Tirado aresupported under grant MTM2015-64373-P (MINECO/FEDER, UE).

References

[1] C. Alegre and J. Marín, Modied w -distances on quasi-metric spaces and a xed pointtheorem on complete quasi-metric spaces, Top. Appl. 203 (2016), 32-41.

[2] C. Alegre, J. Marín and S. Romaguera, A xed point theorem for generalized contractionsinvolving w -distances on complete quasi-metric spaces, Fixed Point Theory Appl. 2014,2014:40.

[3] S. Al-Homidan, Q.H. Ansari and J.C. Yao, Some generalizations of Ekeland-type variationalprinciple with applications to equilibrium problems and xed point theory, Nonlinear Anal.TM&A 69 (2008), 126-139.

[4] M. Ali-Akbari, B. Honari, M. Pourmahdian and M.M. Rezaii, The space of formal balls andmodels of quasi-metric spaces, Math. Struct. Comput. Sci. 19 (2009), 337-355.

[5] I. Altun, N. Al Ari, M. Jleli, A. Lashin and B. Samet, A new concept of (α, Fd)-contractionon quasi metric space, J. Nonlinear Sci. Appl. 9 (2016), 3354-3361.

[6] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer.Math. Soc. 215 (1976), 241-251.

[7] S. Cobza³, Completeness in quasi-metric spaces and Ekeland Variational Principle, Top.Appl. 158 (2011), 1073-1084.

[8] S. Cobza³, Functional Analysis in Asymmetric Normed Spaces, Birkhäuser, Springer Basel,2013.

[9] H. Da§, G. Minak and I. Altun, Some xed point results for multivalued F -contractions onquasi metric spaces, RACSAM, DOI: 10.1007/s13398-016-0285-3, to appear.

[10] T.K. Hu, On a xed point theorem for metric spaces, Amer. Math. Monthly 74 (1967),436-437.

[11] R. Kannan, Some results on xed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.[12] E. Karapinar and S. Romaguera, On the weak form of Ekeland's Variational Principle in

quasi-metric spaces, Top. Appl. 184 (2015), 54-60.[13] A.W. Kirk, Caristi's xed point theorem and metric convexity, Colloq. Math. 36 (1976),

81-86.[14] H.P.A. Künzi, Nonsymmetric distances and their associated topologies: About the origins

of basic ideas in the area of asymmetric topology, in: C.E. Aull, R. Lowen (Eds.), Handbookof the History of General Topology, vol. 3, Kluwer, Dordrecht, 2001, pp. 853-968.

[15] A. Latif, and S.A. Al-Mezel, Fixed point results in quasimetric spaces, Fixed Point TheoryAppl. 2011 (2011), Article ID 178306, 8 pages.

[16] J. Marín, S. Romaguera and P. Tirado, Q-functions on quasi-metric spaces and xed pointsfor multivalued maps, Fixed Point Theory Appl. 2011 (2011), Article ID 603861, 10 pages.

[17] J. Marín, S. Romaguera and P. Tirado, Generalized contractive set-valued maps on completepreordered quasi-metric spaces, J. Funct. Spaces Appl. 2013 (2013), Article ID 269246, 6pages.

[18] S. Park, Characterizations of metric completeness, Colloquium Mathematicum 49 (1984),21-26.

76

[19] I.L. Reilly, P.V. Subrahmanyam and M.K. Vamanamurthy, Cauchy sequences in quasi-pseudo-metric spaces, Mh. Math. 93 (1982), 127-140.

[20] S. Romaguera, M.P. Schellekens and O. Valero, Complexity spaces as quantitative domainsof computation, Top. Appl. 158 (2011), 853-860.

[21] S. Romaguera and P. Tirado, The complexity probabilistic quasi-metric space, J. Math.Anal. Appl. 376 (2011), 732-740.

[22] S. Romaguera and P Tirado, A characterization of Smyth complete quasi-metric spaces viaCaristi's xed point theorem, Fixed Point Theory Appl. 2015, 2015:183.

[23] S. Romaguera and O. Valero, Domain theoretic characterisations of quasi-metric complete-ness in terms of formal balls, Math. Struct. Comput. Sci. 20 (2010), 453-472.

[24] M.P. Schellekens, A characterization of partial metrizability: domains are quantiable,Theor. Comput. Sci. 305 (2003), 409-432.

[25] N. Shioji, T. Suzuki, W. Takahashi, Contractive mappings, Kannan mappings and metriccompleteness, Proc. Amer. Math. Soc. 126 (1998), 3117-3124.

[26] P.V. Subrahmanyam, Completeness and xed-points, Mh. Math. 80 (1975), 325-330.[27] T. Suzuki, W. Takahashi, Fixed point theorems and characterizations of metric complete-

ness, Top. Methods Nonlinear Anal. 8 (1996), 371-382.[28] T. Suzuki, A generalized Banach contraction principle that characterizes metric complete-

ness, Proc. Amer. Math. Soc., 136 (2008), 1861-1869.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 77 90

Functional equivalence of topological spaces andtopological modules

Mitrofan M. Choban ∗ and Radu N. Dumbr veanu†

Abstract

Let R be a topological ring and E, F be unitary topological R-modules.Denote by Cp(X,E) the class of all continuous mappings of X into Ein the topology of pointwise convergence. The spaces X and Y arecalled lp(E,F )-equivalent if the topological R-modules Cp(X,E) andCp(Y, F ) are topological isomorphic. Some conditions under which thetopological property P is preserved by the lp(E,F )-equivalence (Theo-rems 6.3, 6.4, 7.3 and 8.1) are given.

Keywords: Function space, topology of pointwise convergence, support, linearhomeomorphism, perfect properties, open nite-to-one properties

2000 AMS Classication: Primary 54C35, 54C10, 54C60; Secondary 13F99,54C40, 54H13

1. Preliminaries

Throughout this paper, by a space we will mean a Tychono space [9].A topological semiring is a topological space R equipped with two continuous

binary operations +, ·, called addition and multiplication, such that (see [10, 11,15]):

1. (R,+) is a topological commutative monoid with identity element 0 andproprieties: (a + b) + c = a + (b + c), 0 + a = a + 0 = a, a + b = b + a for alla, b, c ∈ R.

2. (R, ·) is a topological monoid with identity element 1 6= 0 and proprieties:(a · b) · c = a · (·c), 1 · a = a · 1 = a, a · b = b · a, a · (b+ c) = (a · b) + (a · c), 0 · a =0 for all a, b, c ∈ R.

∗Department of Mathematics Tiraspol State University, MD-2069, Chi³in u, Moldova,Email : mmchoban@gmail.com†Department of Mathematics, B lµl State University, MD-3121, B lµi, Moldova

Email : rndumbraveanu@gmail.com

Doi : 10.15672/HJMS.2016.402

78

Let G be a topological Abelian group under addition operation and R be atopological semiring. We call G a topological R-module if on it is dened thecontinuous operation of multiplication · : R × G −→ G between an element of Rand an element of G, say ra ∈ G, where r ∈ R and a ∈ G, with the followingproperties: 1 ·a = a, 0 ·a = 0, r(a+ b) = ra+rb, (r+s)a = ra+sa, r(sa) = (rs)a,for any r, s ∈ R and a, b ∈ G.

Let R be a topological semiring and E, F be topological R-modules. Themapping ϕ : E → F is a linear mapping if it satises the conditions: ϕ(x + y) =ϕ(x) + ϕ(y) and ϕ(αx) = αϕ(x) for any x, y ∈ E and α ∈ R.

Fix a space X, a topological semiring R, and topological R-modules E and F .By C(X,E) we will denote the family of all E-valued continuous functions with

the domain X and by Cp(X,E) we will denote the space C(X,E) endowed withthe topology of pointwise convergence. Recall that the family of sets of the formW (x1, x2, ..., xn, U1, U2, ..., Un) = f : C(X,E) : f(xi) ∈ Ui for any i ≤ n, wherex1, x2, ..., xn ∈ X, U1, U2, ..., Un are open sets of E and n ∈ N, is a base of thespace Cp(X,E).

By Hp(E,F ) we denote the space of all linear mappings of E into F as asubspace of the space Cp(E,F ).

The spaces X and Y are called lp(E,F )-equivalent if the spaces Cp(X,E) and

Cp(Y, F ) are linearly homeomorphic and we denote XE,F∼ Y .

A space X is zero-dimensional if indX = 0 (small inductive dimension is zero),i.e., X has a base of clopen (open and closed) subsets.

The following two assertions are evidently.

1.1. Proposition. Fix a topological R-module E. Then Cp(X,E) is a topologicalR-module and E is embedded in a natural way in Cp(X,E) as a closed submoduleof Cp(X,E).

1.2. Proposition. If E is a zero-dimensional topologicalR-module, then Cp(X,E)is a zero-dimensional topological R-module too.

2. The evaluation mapping

Let X be a space, R be a topological semiring and E be a non-trivial topologicalR-module. Fix x ∈ X. Then the mapping ξx : Cp(X,E)→ E dened by ξx(f) =f(x) is called the evaluation mapping at x (see, by instance, [1]).

We now dene the canonical evaluation mapping eX : X → Cp(Cp(X,E), E),where eX(x) = ξx for any x ∈ X.

The proofs of the following two assertions are standard (see [8]).

2.1. Proposition. The evaluation mapping ξx : Cp(X,E)→ E is continuous andlinear for every point x ∈ X.

2.2. Proposition. The canonical evaluation mapping eX : X → Cp(Cp(X,E), E)is continuous. Moreover, the set eX(X) is closed in the space Cp(Cp(X,E), E).

Let X and Y be spaces, Φ be a family of functions f : X → Y . We say that Φseparates points of X (or simply is separating [1]) if for any x, y ∈ X, x 6= y, thereexists f ∈ Φ such that f(x) 6= f(y). We also say that Φ separates points from

closed sets (or is regular [1]) if for any non-empty closed subset B of X, any point

79

x ∈ X \B and any two points a, b ∈ Y there exists f ∈ Φ such that f(x) = a andf(B) = b.

2.3. Proposition. If Cp(X,E) is a regular family, then the canonical evaluationmapping eX : X → Cp(Cp(X,E), E) is a homeomorphism from X to the closedsubspace eX(X) of Cp(Cp(X,E), E).

A space X is called R-Tychono if for any closed subset B of X, any pointa ∈ X \ F there exists g ∈ C(X,R) such that g(a) = 1 and B ⊆ g−1(0).

The product of R-Tychono spaces is an R-Tychono space. The subspace ofan R-Tychono is an R-Tychono space.

Remark. Let X be an R-Tychono space and E be a non-trivial topological R-module. Then X is a Tychono space, and for each closed set B of X, any pointa ∈ X \ B and any points b, c ∈ E there exists f ∈ C(X,E) such that f(a) = band f(B) = c.

The proofs of the following two assertion is elementary.

2.4. Proposition. If indX = 0, then the space X is R-Tychono.

Let R be a topological semiring. A topological R-module E is called:(i) simple if it does not contain a non-trivial submodule over R.(ii) locally simple if E is not trivial and there exists an open subset U of E such

that 0 ∈ U and U do not contains non-trivial R-submodules of E.

2.5. Example. If R is a eld, then R is a simple topological R-module. Let Rbe the eld of real numbers and K be the eld of complex numbers. Then K islocally simple and not simple R-module.

We mention the following obvious fact.

2.1. Lemma. Let R be a topological semiring and E be an R-module. Then Rais an R-submodule for any a ∈ E.

Fix a space X and two topological R-modules E and F . We deneMp(X,E, F )= Hp(Cp(X,E), F ) the subspace of all linear mappings from Cp(X,E) into F . LetMp(X,E) = Mp(X,E,E). Now we dene Lp(X,E) ⊆ Cp(Cp(X,E), E) as followsLp(X,E) = α1x1 + α2x2 + ...+ αnxn : αi ∈ R, xi ∈ eX(X), i ≤ n ∈ N.

By construction, we have Lp(X,E) ⊆Mp(X,E). As a rule Lp(X,E) 6= Mp(X,E)(see [8]).

2.6. Proposition. Let R be a topological semiring, E be a topological R-moduleand X be a space. Then for any g ∈ C(X,E) there exists a unique linear mappingg ∈ Hp(Lp(X,E), E) such that g = g eX , where eX : X → Lp(X,E) is theevaluation mapping.

Proof. Let Ef = E for any f ∈ Cp(X,E). By denition, eX(X) ⊆ Lp(X,E) ⊆EC(X,E) = ΠEf : f ∈ C(X,E). We have eX(x) = ξx for any x ∈ X andLp(X,E) is the submodule of EC(X,E) generated by the set eX(X). We considerthe projection πf : EC(X,E) −→ Ef = E. Let f = πf |Lp(X,E) : Lp(X,E) −→ Eis the desired linear mapping.

80

2.7. Theorem. Let R be a semiring, E be a topological R-module and X bea space. Consider the space eX(X), where eX : X → Lp(X,E) is the evalu-ation mapping. Then the topological R-modules Cp(X,E), Cp(eX(X), E) andHp(Lp(X,E), E) are linearly homeomorphic.

Proof. Let Ef for any f ∈ Cp(X,E). By denition, eX(X) ⊆ Lp(X,E) ⊆ECp(X,E) = ΠEf : f ∈ C(X,E). We consider the projection πf : EC(X,E) −→Ef = E. Let f = πf |Lp(X,E) : Lp(X,E) −→ E. Then f and πf are continuouslinear mappings.

If g : eX(X) → E is a continuous mapping, then g eX = f for a uniquef ∈ C(X,E). Therefore, g = πf |eX(X) and the correspondence f → πf |eX(X) isa linear homeomorphism of Cp(X,E) onto Cp(eX(X), E).

Hence, without loss of generality, we can assume that X = eX(X) ⊆ Lp(X,E).By virtue of Proposition 2.6, the correspondence

ψ : Cp(X,E) −→ Hp(Lp(X,E), E), where ψ(f) = f , is a one-to-one linear map-ping of C(X,E) onto Hp(Lp(X,E), E).

For each y ∈ Lp(X,E) there exist the minimal n = n(y) ∈ N, the uniquepoints x1(y), ..., xn(y) ∈ X and the unique points α1(y), ..., αn(y) ∈ R such thaty = α1(y)x1(y) + ...+αn(y)xn(y). Hence, the correspondence ψ is continuous andlinear. Since ψ(f)|X = f , the mapping ψ−1 is continuous.

2.8. Corollary. Let X, Y be spaces and R be a locally simple R-module. Thespaces Cp(X,R) and Cp(Y,R) are linearly homeomorphic if and only if the spacesLp(X,R) and Lp(Y,R) are linearly homeomorphic.

2.2. Lemma. Let X be an R-Tychono space, Z be a closed subspace of X, E be

a topological R-module and g : X −→ E be a continuous mapping. For any nite

subset B of X \Z and any function f : B −→ E there exists a continuous function

ϕ : X −→ E such that f = ϕ|B and ϕ|Z = g|Z.Proof. Fix a family Ux : x ∈ B of open subsets of X such that x ∈ Ux ⊆ X \ Zfor each x ∈ B and Ux ∩ Uy = ∅ for each distinct points x, y ∈ B. For eachx ∈ B x a continuous function fx : X −→ E such that fx(x) = f(x)− g(x) andfx(X \Ux) = 0. Let fB(y) =

∑fx(y) : x ∈ B. By construction, the function fBis continuous, fB(Z) = 0 and fB(x) = f(x) − g(x) for each x ∈ B. Obviously, ϕ= fB + g is the desired function.

For any subspace Y of a space X we put Cp(Y |X,E) = f |Y : f ∈ C(X,E).A subspace Y of X is E-full if C(Y |X,E) = C(Y,E).

A space X is called compactly E-full if C(Y |X,E) = C(Y,E) for any compactsubspace Y of X.

The following assertion is well-known (see [8]).

2.3. Lemma. Let X be a zero-dimensional space and E be a metrizable space.

Then X is a compactly E-full space. Moreover, for any compact subset Y of X and

any f ∈ C(Y,E) there exists g ∈ C(X,E) such that g(X) ⊆ f(Y ) and f = g|Y .

3. The support mapping

Fix a topological semiring R and non-trivial topological R-modules E and F .

81

Consider a spaceX and a functional µ ∈Mp(X,E, F ). We put S(µ) = B ⊆ X :if B ⊆ f−1(0), then µ(f) = 0. Obviously, X ∈ S(µ). Thus the set S(µ) is non-empty.

The set suppX(µ) is the family of all points x ∈ X such that for each neigh-bourhood U of x in X there exists f ∈ Cp(X,E) such that f(X \ U) = 0 andµ(f) 6= 0 (see [2, 12], for E = R = R, [3, 14] for R = R, [8] when R is a topologicalring).

If f ∈ Cp(X,E) and U is an open neighbourhood of 0 in E, then we putA(f, L, U) = g ∈ Cp(X,E) : f(x) − g(x) ∈ U for any x ∈ L. The familyA(f, L, U) : f ∈ Cp(X,E), L is nite subset of X,U is open neighbourhood of 0in E is an open base of the space Cp(X,E).

3.1. Theorem. Let X be a R-Tychono space, E and F be non-trivial topologicalR-modules, µ ∈ Mp(X,E, F ) and µ 6= 0. If F is a locally simple topological R-module, then:

1. There exists a nite set K ∈ S(µ) such that suppX(µ) ⊆ K.2. suppX(µ) ∈ S(µ) and suppX(µ) is a nite non-empty subset of X.3. suppX(µ) = ∩S(µ).

Proof. Fix an open subset U0 of Cp(X,E) such that 0 ∈ U0 and an open subsetW0 of F such that 0 ∈W0, W0 do not contains non-trivial R-submodules of F andµ(U0) ⊆W0.

There exist a nite subset K of X and an open subset V0 of E such that 0 ∈ V0and 0 ∈ A(0,K, V0) ⊆ U0. Hence µ(f) ∈W0 for each f ∈ A(0,K, V0).

Let f ∈ Cp(X,E) and f(K) = 0. Then αf ∈ A(0,K, V0) for each α ∈ R.Hence µ(αf) ∈W0 for each α ∈ R. Thus R · µ(f) ⊆ V0 and R · µ(f) is the trivialR-submodule. Thus µ(f) = 0 and K ∈ S(µ). In this case suppX(µ) ⊆ K. HencesuppX(µ) is a nite set and K is a nite set from S(µ).

Let L ∈ S(µ) be a nite set and x0 ∈ L\suppX(µ). Then L1 = L\x0 ∈ S(µ).Really, since x0 /∈ suppX(µ), there exists an open subset H of X such that x0 ∈ Hand µ(f) = 0 provided f(X \ H) = 0. We can assume that H ∩ L = x0. Letf ∈ Cp(X,E) and f(L1) = 0. There exists h ∈ C(X,E) such that h(x0) = f(x0)and h(X \H) = 0. We put g(x) = f(x)−h(x) for any x ∈ X. Since h(X \H) = 0,we have µ(h) = 0. By construction, g(L) = 0 and µ(g) = 0. Hence f = g + hand µ(f) = µ(g + h) = µ(g) + µ(h) = 0. Hence L1 ∈ S(µ). Since K ∈ S(µ) andK \ suppX(µ) is a nite set, we have suppX(µ) ∈ S(µ). In particular, we havesuppX(µ) = ∩S(µ).

The following assertions are obviously:

3.2. Proposition. Let n ≥ 1, x1, x2, ..., xn are distinct points ofX, α1, α2, ..., αn ∈R and µ(f) = Σαif(xi) : i ≤ n for each for each f ∈ Cp(X,E), then:

1. µ ∈ Lp(X,E) and suppX(µ) ⊆ x1, x2, ..., xn.2. If for each i ≤ n the set αiE is a non-trivial R-submodule of E, then

suppX(µ) = x1, x2, ..., xn.

4. Topological properties of the mapping suppX

Fix a topological semiring R. Let X be a space, E and F be two non-trivialtopological R-modules.

82

Recall that a set-valued mapping f : X → 2Y is lower semicontinuous (l.s.c) iffor every open subset U of Y the inverse image of U , f−1(U) = x ∈ X : f(x)∩U 6=∅ is open in X.

The correspondence suppX is a set-valued mapping of the space Mp(X,E, F )into X. For H ⊆Mp(X,E, F ) we put suppX(H) = ∪suppX(µ) : µ ∈ H.4.1. Proposition. If F is a locale simple R-module, then the set-valued mappingsuppX : Mp(X,E, F )→ X is l.s.c.

Proof. We follow very closely the proof of [3, Property 4.2] and [12, Lemma 6.8.2(4)].

Let U be an open subset of X, and put V = supp−1X (U), i.e., V = µ ∈Mp(X,E, F ) : suppX(µ) ∩ U 6= ∅. Let µ ∈ V , and take x0 ∈ suppX(µ) ∩ U .Fix an open subset W of X such that x0 ∈ W ⊆ clXW ⊆ U . Then there existsf ∈ C(X,E) such that f(X\W ) = 0 and µ(f) 6= 0. LetH = η ∈Mp(X,E, F ) :η(f) 6= 0. Since the set 0 is closed in F , H is the basic open set W (f, F \ 0)= η ∈Mp(X,E, F ) : η(f) ∈ F \ 0 and µ ∈W (f, F \ 0).

We arm that H ⊆ V . By contradiction, suppose that η ∈ H \V , i.e. η(f) 6= 0and suppX(η) ∩ U = ∅. Then X \ clXW is an open neighbourhood of suppX(η)and, since f(X \ clXW ) = 0, applying Theorem 3.1, we get that η(f) = 0. Acontradiction, hence V is open in Mp(X,E, F ).

A subset L of a space X is bounded if any continuous real-valued functionf : X −→ R is bounded on L.

A subset L of a topological R-module E is called:(i) precompact or totally a-bounded if for any neighbourhood U of 0 in E there

exists a nite subset A of E such that L ⊆ A+ U = U +A;(ii) a-bounded if for any neighbourhood U of the 0 in E there exists n ∈ N such

that L ⊆ nU .Any bounded set is precompact. In a topological vector space over eld of reals

any precompact set is a-bounded.A topological R-module E is called locally bounded if there exists an a-bounded

neighbourhood U of 0 in E such that E = ∪nU : n ∈ N and for any a ∈ E,a 6= 0, and any n ∈ N there exists t ∈ R such that ta /∈ nU . In this case the set Udoes not contain R-submodules of E and E is a locally simple R-module.

4.2. Example. Let E be a normed vector space over reals R. Then E is a locallybounded R-module.

4.3. Example. Let E be a topological vector space over reals R and there existsa number q > 0 and a functional ||.|| : E −→ R such that:

1. 0 < q ≤ 1.2. ||x|| ≥ 0 for any x ∈ E.3. If ||x|| = 0, then x = 0.4. ||x+ y|| ≤ ||x||+ ||y|| for all x, y ∈ E.5. ||λx|| ≤ |λ|q||x|| for all x ∈ E and λ ∈ R.6. If x 6= 0 then limλ→+∞||λx|| = +∞.The functional ||.|| is called a q-norm, if the family V (0, r) = x : ||x|| < r :

r > 0 is a base of E at 0. Any q-normed space is locally bounded.

83

4.4. Theorem. Let F be a locally bounded topological R-module, B be a sub-module of F and X be an R-Tychono space with the following properties:

(b) : for any non-bounded subset L of X there exists f ∈ C(X,B) such that theset f(L) is not a-bounded in F ;

(r) : B is topological isomorphic to some R-submodule of E.Then:(i) The set suppX(H) is bounded inX for any a-bounded subsetH ofMp(X,E, F ).(ii) The set suppX(H) is bounded in X for any totally a-bounded subset H of

Mp(X,E, F ).(iii) The set suppX(H) is bounded inX for any bounded subsetH ofMp(X,E, F ).

Proof. We can assume that B ⊆ E too. Since B is a non a-bounded subset of Fthere exists an open subset W0 of F such that 0 ∈W0 and B \ nW0 6= ∅ for eachn ∈ N. Moreover, If H ⊆ B is a non a-bounded of F then H is a non a-boundedof B too.

Since F is locally bounded we can x an open neighbourhoodW1 of 0 in E suchthat the set W1 is a-bounded, F =

⋃nW1 : n ∈ N and for any a ∈ F , a 6= 0,and for any n ∈ N there exists t ∈ R such that ta /∈ nW1.

Now x two open neighbourhoodsW2 andW3 of 0 in F such thatW2 = −W2 ⊂3W2 = W2 +W2 +W2 ⊆W3 = −W3 ⊆W1 ∩W0.

By construction, W1 ⊆ kW2 for some k ∈ N.Hence the sets W2 and W3 have the following properties:- W2 and W3 are a-bounded subsets of E;- F =

⋃nW2 : n ∈ N =⋃nW3 : n ∈ N;

- if L is a bounded or a precompact subset of F , then L ⊆ nW2 for some n ∈ N;- if a ∈ F , a 6= 0, then for any n ∈ N there exists t ∈ R such that ta /∈ nW3.Since B is a non a-bounded subset of F and W3 is an a-bounded of F , we have

B \ nW3 6= ∅ for each n ∈ N.If µ ∈ Mp(X,E, F ) and µ 6= 0, then suppX(µ) is a nite non-empty subset of

X.We can assume that C(X,B) ⊆ C(X,E) and C(X,B) ⊆ C(X,F ).Suppose that the set H is a-bounded or precompact inMp(X,E, F ) and the set

suppX(H) is not bounded in X. Fix f ∈ C(X,B) such that the set f(suppX(H))is not a-bounded in F .

By induction, we shell construct a sequence µn : n ∈ N ⊆ H, a sequenceUk : k ∈ N of open subsets of X, a sequence xn ∈ suppX(µn) : n ∈ N and asequence hk ∈ C(X,B) : n ∈ N with properties:

1. xi ∈ Ui, hi(X \ Ui) = 0 for any i ∈ N;2. Un : n ∈ N is a discrete family of subsets of X;3. µn(hn) /∈ nW ;4. suppXµ1, µ2, ..., µn ∩ clXUn+1 = ∅;5. f(Un) ⊆ f(xn) +W0 and f(xn+1) /∈ ⋃f(xi) +W : i ≤ n for each n ∈ N;Fix µ1 ∈ H and x1 ∈ suppX(µ1). There exists an open subset U1 of X and

g1 ∈ C(X,B) such that f(U1) ⊆ W0 + f(x1), g1(X \ U1) = 0 and µ1(g1) 6= 0.There exists α1 ∈ R such that α1µ1(g) /∈W3. We put h1 = α1g1.

Assume that n ≥ 1 and the objects hi, xi, Ui, µi : i ≤ n are constructed.We put Mn =

⋃suppX(µi) : i ≤ n. The set Mn is nite. Hence the set

84

f(suppX(H))\f(Mn) is not a-bounded in F . For some mn ∈ N we have f(Mn) ⊆mnW0.

Fix µn+1 ∈ H and xn+1 ∈ suppX(H) such that f(xn+1) ∈ B \mnW . Thereexists an open subset Un+1 of X and gn+1 ∈ C(X,B) such that xn+1 ∈ Un+1,f(Un+1) ⊆ f(xn+1)+W0, gn+1(X\Un+1) = 0, clXUn+1∩Mn = ∅ andMn+1(gn+1) 6=0. There exists αn+1 ∈ R such that αn+1µn+1(gn+1) /∈ (n + 1)W . We puthn+1 = αn+1gn+1. That completes the inductive construction. The objectsxm, µn, hn, Un are constructed for all n ∈ N. Let h = Σhn : n ∈ N. SinceUn : n ∈ N is a discrete family and hn(X \ Un) = 0 for any n ∈ N, wehave h ∈ C(X,B). By construction, µn(h) = µn(hn) /∈ nW0 for any n. Thenµn(h) : n ∈ N is a not a-bounded subset of E. Since the set H is a-bounded, theset µ(h) : µ ∈ H is a-bounded too, a contradiction. The proof is complete.

Remark. Any normed space is a locally bounded R-module. If E is a non-trivialnormed space, then for any non-bounded subset L of the space X there existsf ∈ C(X,E) such that the set f(L) is not bounded in E. For a normed space ETheorem 4.4 was proved by V. Valov in [14]. For a ring R and E = F Theorem4.4 was proved in [8].

A space X is µ-complete if any closed bounded subset of X is compact.A space X is Dieudonné complete if the maximal uniformity on X is complete.

Any Dieudonné complete space is µ-complete.Denote by PX the space X with the Gδ-topology generated by the Gδ-subsets

of X. The set δ − clXH = clPXH is called the Gδ-closure of the set H in X. Ifδ − clXH = H, then we say the set H is Gδ-closed.

If the space X is µ-complete, then any Gδ-closed subspace of X is µ-complete.A tightness of a space X is the minimal cardinal number τ for which for any

subset L ⊆ X and any point x ∈ clXL there exists a subset L1 ⊆ L such that|L1| ≤ τ and x ∈ clXL1.

We denote by t(X) and l(X) the tightness and the Lindelöf numbers respectivelyof a space X.

The following four propositions were proved in [8] (see [1] for E = R).

4.5. Proposition. Assume that E is a metrizable and l(Xn) ≤ τ for any n ∈ N.Then t(Cp(X,E)) ≤ τ .4.6. Proposition. Let X and E be spaces and t(X) ≤ ℵ0. Then Cp(X,E) is aGδ-closed subspace of the space EX . Moreover, if E is µ-complete then the spaceCp(X,E) is µ-complete too.

4.7. Proposition. Let F and E be topological R-modules and Hp(F,E) be thespace of all linear continuous mappings of F into E. Then Hp(F,E) is a closedsubspace of the space Cp(F,E).

4.8. Corollary. Let E and F be topological R-modules and t(F ) ≤ ℵ0. ThenHp(F,E) is a Gδ-closed subset of EF . In particular, if E is µ-complete, then spaceHp(F,E) is µ-complete too.

4.9. Proposition. Let Y be a subspace of the space X, E be a non-trivialtopological R-module, X be an R-Tychono space and pY (f) = f |Y for each

85

f ∈ Cp(X,E). Then the mapping pY : Cp(X,E) −→ Cp(Y |X,E) has the follow-ing properties:

(i) pY is a continuous mapping.(ii) If the set Y is closed in X, then the mapping pY is open.(iii) If Y is dense in X, then pY is a one-to-one correspondence.(iv) The subspace Cp(Y |X,E) is dense in the Cp(Y,E).

4.10. Theorem. Let E be a metrizable R-module, F be a locally bounded metriz-able R-module, B be a closed submodule of F and X be an R-Tychono spacewith the following properties:

(b) : for any non-bounded subset L of X there exists f ∈ C(X,B) such that theset f(L) is not a-bounded in F ;

(r) : B is topological isomorphic to some R-submodule of E;(c) : X be an R-Tychono compactly E-full space.Then the space X is µ-complete if and only if the space Mp(X,E, F ) is µ-

complete.

Proof. By virtue of Proposition 2.3, we can assume that X = eX(X) is a subspaceof the space Mp(X,E,B). From Proposition 2.2 it follows that the subspace X isclosed in Mp(X,E,B). Obviously, Mp(X,E,B) is a closed subspace of the spaceMp(X,E, F ).

Let Mp(X,E, F ) be a µ-complete space. Since X is a closed subspaces ofMp(X,E,B) and Mp(X,E, F ), the space X is µ-complete too.

Assume that X is a µ-complete space. Let Φ be a closed bounded subset ofMp(X,E, F ). Then the closure Y of the set ∪suppX(µ) : µ ∈ Φ is a compactsubset of X.

The restriction mapping pY : Cp(X,E) −→ Cp(Y,E) is an open continuouslinear mapping of the R-module Cp(X,E) onto the R-module Cp(Y,E).Claim 1. The dual mapping ϕ : FC(Y,E) −→ FC(X,E) is a linear embedding

and the set ϕ(FC(Y,E)) is closed in FC(X,E).The proof of this fact is similar with the prof of Proposition 0.4.6 from [1].By construction, we have Φ ⊆ ϕ(Mp(Y,E, F )) ⊆Mp(X,E, F ).Claim 2. ϕ(Mp(Y,E, F )) is a closed subset of the subspaces Mp(X,E, F ) and

Cp(Cp(X,E), E) of the space EC(X,E).Follows from Claim 1 and Proposition 4.7.Claim 3. ϕ(Cp(Cp(Y,E), F )) ⊆ Cp(Cp(X,E), F ).Follows from the continuity of the mapping pY .Claim 4. The sets ϕ(Mp(X,E, F )) and ϕ(Cp(Cp(Y,E), F )) are Gδ-closed in

FC(X,E).Since Y is compact, from Proposition 4.5 it follows that t(Cp(Y,E)) = ℵ0.

Then, from Proposition 4.6 it follows that Cp(Cp(Y,E), F ) is a Gδ-closed subsetof the space FC(Y,E). From Claim 1 it follows that ϕ(Cp(Cp(Y,E), F )) is Gδ-closedin FC(X,E). Corollary 4.8 completes the proof of the claim.

Let G be the Gδ-closure of the set Cp(Cp(X,E), E)) in EC(X,E). We haveMp(X,E, F ) ⊆ G. Hence Φ is a bounded subset of the space G.Claim 5. The sets ϕ(Mp(X,E, F )) and ϕ(Cp(Cp(Y,E), F )) are closed in G.Follows from Claim 4.

86

Since F is a metrizable space, F is a µ-complete space. Thus Φ is a closedbounded subset of the µ-complete space G. Therefore the set Φ is compact. Theproof is complete.

5. Relations between linear equivalent spaces

Let R be a topological semiring and E, F be non-trivial locally bounded topo-logical R-modules. The R-module E × F is locally bounded. We identify E withthe R-submodule E×0 of E×F and F with the R-submodule 0×F of E×F .

Fix two non-empty R-Tychono spaces X and Y with the properties:- for any non-bounded subset L of X there exists f ∈ C(X,E) such that the

set f(L) is not a-bounded in E;- for any non-bounded subset L of Y there exists f ∈ C(Y, F ) such that the set

f(L) is not a-bounded in F .Fix now a continuous linear homeomorphism u : Cp(X,E) −→ Cp(Y, F ). Then

the mapping v : Mp(Y, F,E × F ) −→ Mp(X,E,E × F ), where v(η) = η u foreach η ∈Mp(Y, F,E × F ), is a linear homeomorphism.

For each x ∈ X and each f ∈ Cp(X,E) we put εx(f) = (ξx(f), 0) = (f(x), 0) ∈E ⊆ E × F . For each y ∈ Y and each g ∈ Cp(Y, F ) we put δy(g) = (0, ξy(g))= (0, g(y)) ∈ F ⊆ E × F . Realy, we can assume that εx = ξx and δy = ξy.Obviously, v−1(εx) = εx u−1 ∈ Mp(Y, F,E × F ) \ 0 and v(δy) = δy u ∈Mp(X,E,E × F ) \ 0. Hence, for each x ∈ X and each y ∈ Y ) we can putϕ(x) = suppY (v−1(εx)) and ψ(y) = suppX(v(δy)).Property 7.1. ϕ : X → Y and ψ : Y → X are l.s.c. set-valued mappings and

ϕ(x), ψ(y) are nite non-empty sets for all points x ∈ X and y ∈ Y .

Proof. Follows from Proposition 4.1 and Theorem 5.1.

Property 7.2. Let y0 ∈ Y , f ∈ C(X,E) and f(ψ(y0)) = 0. Then u(f)(y0) = 0.

Proof. For any η ∈ Mp(Y, F,E × F ) and g ∈ C(X,E) we have v(η)(g) = η(u(g))(η u)(g). Since f(suppX(v(δy0))) = f(ψ(y0)) = 0, we have (δy0 u)(f) = 0 andu(f)(y0) = δy0(u(f)) = (δy0 u)(f) = 0. The proof is complete.

5.1. Corollary. If f, g ∈ C(X,E) and f |ψ(y) = g|ψ(y), then u(f)(y) = u(g)(y).

Property 7.3. x ∈ ψ(ϕ(x)) for every point x ∈ X and y ∈ ϕ(ψ(y)) for every

point y ∈ Y .

Proof. For every x ∈ X the sets ϕ(x) and ψ(ϕ(x)) are nite and closed. Assumethat x0 ∈ X and x0 /∈ ψ(ϕ(x0)) = H. Fix f ∈ C(X,E) such that f(x0) = b 6= 0and f(H) = f(ψ(ϕ(x0))) = 0. Since ψ(y) ⊆ H and f(H) = 0 for any y ∈ ϕ(x0) byvirtue of Property 7.2, we have u(f)(y) = 0 for each y ∈ ϕ(x0). Since u(f)(y) = 0for each y ∈ ϕ(x0), by virtue of Property 7.2, we have f(x0) = u−1(u(f))(x0) = 0.By construction, we have f(x0) 6= 0, a contradiction.

Property 7.4. If H is dense subset of Y , then ψ(H) is a dense subset of Xprovided u is an injection.

87

Proof. Assume that x0 /∈ clXψ(H). Then there exists f ∈ C(X,E) such thatf(x0) 6= 0 and f(ψ(H)) = 0. Since f(ψ(H)) = 0 for any y ∈ Y , by virtue ofProperty 7.2, we have u(f)(y) = 0 for any y ∈ Y . Thus u(f) = 0. Hence f = 0, acontradiction.

From the above properties follows

5.2. Corollary. The space X is separable if and only if the space Y is separable.In general, d(X) = d(Y ).

Property 7.5. ϕ(H) is a bounded set of Y for each bounded set H of X.

Proof. LetH be a bounded subset ofX. ThenH is a bounded subset ofMp(X,E,E×F ) and respectively v−1(H) is a bounded subset of Mp(Y, F,E × F ). By Theo-rem 4.4 the set suppY (v−1(H)) is a bounded subset of Y . The proof is com-plete.

Property 7.6. Let E and F be metrizable spaces, X be a compactly E-full space

and Y be a compactly F -full space. Then the space X is µ-complete if and only if

the space Y is µ-complete.

Proof. Let X be a µ-complete space. ThenMp(X,E,E×F ) andMp(Y, F,E×F ),by virtue of Theorem 4.10, are µ-complete spaces. By Theorem 4.10 the space Yis µ-complete too. The proof is complete.

As in [3] we say that the pair of set-valued mappings θ : X −→ Y and π : Y −→X is called lower-reective if it has the following conditions:

1l. θ and π are l.s.c.2l. θ(x) and π(x) are nite sets for all points x ∈ X and y ∈ Y .3l. x ∈ π(θ(x)) and y ∈ θ(π(y)) for all points x ∈ X and y ∈ Y .Also, as in [3] we say that the pair of set-valued mappings θ : X −→ Y and

π : Y −→ X is called upper-reective if it has the following conditions:1u. θ(F ) is a bounded subset of Y for each bounded subset F of X.2u. π(Φ) is a bounded subset of X for each bounded subset Φ of Y .3u. x ∈ clXπ(θ(x)) and y ∈ clY θ(π(y)) for all points x ∈ X and y ∈ Y .General conclusion: The set valued mappings ϕ : X −→ Y and ψ : Y −→ X

forms an equivalence of X and Y in sense of article [3]. Thus the general theoremsfrom [3] can be extended for the mappings in topological R-modules. In thefollowing sections we formulate the general theorems for the R-modules, where Ris a topological semiring.

6. Application to perfect properties

We say that the property P is a perfect property if for any continuous perfectmapping f : X −→ Y of X onto Y we have X ∈ P if and only if Y ∈ P. We saythat the property P is a strongly perfect property if it is perfect and any spacewith property P is µ-complete.

6.1. Example. From the Example 6.2 [3] the following properties are perfect: tobe a compact space; to be a paracompact p-space; to be a paracompact space; tobe a metacompact space; to be a k-scattered space; to be a monotonically p-space;

88

to be a monotonically ech complete space; to be a ech complete space; to be aLindelöf space; to be a Lindelöf Σ-space; to be a subparacompact space; to be alocally compact space.

6.2. Example. The following properties are strongly perfect: to be a compactspace; to be a paracompact p-space; to be a paracompact space; to be a µ-completemetacompact space; to be a k-scattered µ-complete space; to be a µ-completemonotonically p-space; to be a µ-complete monotonically ech complete space; tobe a µ-complete ech complete space; to be a Lindelöf space; to be a LindelöfΣ-space; to be a µ-complete subparacompact space; to be a µ-complete locallycompact space.

A space X is called a wq-space if for any point x ∈ X there exists a sequenceUn : n ∈ N of open subsets of X such that x ∈ ∩Un : n ∈ N and each setxn ∈ Un : n ∈ N is bounded in X.

A space X is pseudocompact if the set X is bounded in the space X. Anypseudocompact space is a wq-space.

6.3. Theorem. Let R be a topological semiring and E and F be non-trivial locallybounded topological R-modules. Fix two non-empty R-Tychono spaces X andY with the properties:

- for any non-bounded subset L of X there exists f ∈ C(X,E) such that theset f(L) is not a-bounded in E;

- for any non-bounded subset L of Y there exists f ∈ C(Y, F ) such that the setf(L) is not a-bounded in F .

Assume that u : Cp(X,E) −→ Cp(Y, F ) is a linear homeomorphism. Then:1. X is a pseudocompact space if and only if Y is a pseudocompact space.2. If P is a perfect property and X, Y are µ-complete wq-spaces, then X ∈ P

if and only if Y ∈ P.

Proof. Consider the set-valued mappings ϕ : X −→ Y and ψ : Y −→ X con-structed in the Section 7.

Let X be a pseudocompact space. Then X is a bounded subset of the spaceX. Hence Y = ϕ(X) is a bounded subset of Y and Y is a pseudocompact space.Assertion 1 is proved.

Assume that P is a perfect property and X, Y are µ-complete wq-spaces. Sup-pose that X ∈ P. By virtue of Theorem 2.5 from [3], there exist a space Z andtwo perfect single-valued mappings f : Z −→ X and g : Z −→ Y onto X and Y ,respectively. Hence, Y, Z ∈ P. Assertion 2 is proved. The proof is complete.

6.4. Theorem. Let R be a topological semiring and E and F be non-trivialmetrizable locally bounded topological R-modules. Fix two non-empty spaces Xand Y with the properties:

- X is an R-Tychono compactly E-full space and for any non-bounded subsetL of X there exists f ∈ C(X,E) such that the set f(L) is not a-bounded in E;

- Y is an R-Tychono compactly E-full space and for any non-bounded subsetL of Y there exists f ∈ C(Y, F ) such that the set f(L) is not a-bounded in F .

Assume that u : Cp(X,E) −→ Cp(Y, F ) is a linear homeomorphism. Then:1. The space X is µ-complete if and only if the space Y is µ-complete.

89

2. X is a compact space if and only if Y is a compact space.3. If P is a strongly perfect property and X, Y are wq-spaces, then X ∈ P if

and only if Y ∈ P.

Proof. Consider the set-valued mappings ϕ : X −→ Y and ψ : Y −→ X con-structed in the Section 7. Assertion 1 follows from Property 7.7.

Assume that P is a strongly perfect property and X, Y are wq-spaces. Supposethat X ∈ P. By denition of a strongly perfect property, X is a µ-completespace. From assertion 1 it follows that Y is a µ-complete space too. By virtue ofTheorem 2.5 from [3], there exist a space Z and two perfect single-valued mappingsf : Z −→ X and g : Z −→ Y onto X and Y , respectively. Hence, Y, Z ∈ P.Assertion 3 is proved.

Let X be a compact space. By virtue of Theorem 6.3, Y is a pseudocompactspace. Hence X and Y are wq-spaces. Assertion 3 completes proof of Assertion 2.The proof is complete.

7. Application to open properties

We say that the property P is an of -property (open nite property) if for anycontinuous open nite-to-one mapping f : X −→ Y and any subspace Z of X wehave Z ∈ P if and only if f(Z) ∈ P (see [3]).

7.1. Example. From the results from [3] and [5] the following properties are of -properties: to be hereditarily Lindelöf; to be σ-space; to be hereditarily separable;to be σ-metrizable; to be σ-scattered; to be σ-discrete space.

7.2. Example. Let τ be an innite cardinal. Consider the properties: X ∈ e(τ)if and only if e(X) ≤ τ ; X ∈ d(τ) if and only if d(X) ≤ τ ; X ∈ hd(τ) if and onlyif hd(X) ≤ τ ; X ∈ hl(τ) if and only if hl(X) ≤ τ .

Then e(τ), d(τ), hd(τ), hl(τ) are of -properties.

7.3. Theorem. Let R be a topological semiring and E, F be non-trivial locallybounded topological R-modules. Fix two non-empty R-Tychono spaces X andY with the properties:

- for any non-bounded subset L of X there exists f ∈ C(X,E) such that theset f(L) is not a-bounded in E;

- for any non-bounded subset L of Y there exists f ∈ C(Y, F ) such that the setf(L) is not a-bounded in F .

Assume that u : Cp(X,E) −→ Cp(Y, F ) is a linear homeomorphism. If P is anof -property, then X ∈ P if and only if Y ∈ P.

Proof. Consider the set-valued mappings ϕ : X −→ Y and ψ : Y −→ X con-structed in the Section 7. As in [3] (see Theorem 2.1 from [3]) we put Z =∪x×ϕ(x) : x ∈ X and S = ∪ψ(y)×y : y ∈ Y as subspaces of the spacesX×Y , f(x, y) = x and g(x, y) = y for any point (x, y) ∈ X×Y . Then f : Z −→ Xand g : S −→ Y are continuous open nite-to-one mappings. If D = Z ∩ S, thenfrom Property 7.4 it follows that f(D) = X and g(D) = Y . Hence X ∈ P if andonly if Y ∈ P. The proof is complete.

90

8. lp(E,F )-equivalence and metrizability

8.1. Theorem. LetR be a topological semiring and E, F be non-trivial metrizablelocally bounded topological R-modules. Fix two non-empty spaces X and Y withthe properties:

- X is an R-Tychono compactly E-full space and for any non-bounded subsetL of X there exists f ∈ C(X,E) such that the set f(L) is not a-bounded in E;

- Y is an R-Tychono compactly E-full space and for any non-bounded subsetL of Y there exists f ∈ C(Y, F ) such that the set f(L) is not a-bounded in F .

Let X and Y be lp(E)-equivalent spaces. Then:1. X is a compact metrizable space if and only if Y is a compact metrizable

space.2. If X is a metrizable space, then the space Y is metrizable if and only if Y is

a wq-space.

Proof. Any metrizable space is a wq-space.Let X be a metrizable space and Y be a wq-space. Since X is metrizable, by

virtue of Theorem 6.3, Y is a paracompact p-space. From Theorem 7.3 it followsthat Y is a σ-space. If a paracompact space Y is a σ-space and a p-space, then Yis metrizable [13]. Assertion 2 is proved.

Assertion 1 follows from the Assertion 2 and Theorem 6.3. The proof is com-plete.

References

[1] A. V. Arhangel'skii, Topological Function Spaces, Mathematics and its Applications (SovietSeries), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992.

[2] A. V. Arhangel'skii, On linear homomorphisms of function spaces, Doklady Acad. NaukSSSR 264 (1982), no. 6, 1289-1292. English translation: Soviet Math. Dokl. 25 (1982),852-855.

[3] M.M. Choban, General theorems on functional equivalence of topological spaces, TopologyAppl. 89 (1998), 223-239.

[4] M.M. Choban, Algebraical equivalence of topological spaces, Buletinul Acad. de tiinµe aRepublicii Moldova, Matematica, 1 (2001), 12-36.

[5] M. Choban, Open nite-to-one mappings, Soviet Math. Dokl. 8 (1967), 603-603.[6] M.M. Choban, On the theory of topological algebraic systems, Trudy Moskovskogo Matem.

Obshchestva 48 (1985), 106-149. English translation: Trans. Moscow Math. Soc. 48, 1986,115-159.

[7] M.M. Choban, Some topics in topological algebra, Topology Appl. 54 (1993), 183-202.[8] M.M. Choban, R. N. Dumbr veanu, lp(R)-equivalence of topological spaces and topological

modules, Buletinul Academiei de Stiinte a Rep. Moldova, Matematica 1 (2015), 20-47.[9] R. Engelking, General Topology, PWN, Warsawa 1977.[10] J. S. Golan, Semirings and their applications, Springer, 1999.[11] V. P. Maslov, Idempotent analysis, American Mathematical Society, 1992.[12] J. van Mill, The innite-dimensional topology of function spaces, North-Holland Mathemat-

ical Library, Amsterdam, vol. 64, 2001.[13] A. Okuyama, A survey of the theory of σ-spaces, General Topology Appl. 1 (1971), 57-63.[14] V. Valov, Function spaces, Topology Appl. 81 (1997), no. 1, 1-22.[15] S. Warner, Topological rings, North-Holland mathematics studies. Elsevier, v. 178, 1993.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 91 101

Closure operators associated with networks

Josef lapal ∗† and John L. Pfaltz‡

Abstract

We study network (i.e., undirected simple graph) structures by investi-gating associated closure operators and the corresponding closed sets.To describe the dynamic behavior of networks, we employ continu-ous transformations and neighborhood homomorphisms between them.These transformations and homomorphisms are then studied. In par-ticular, the problem of preserving generators by continuous transforma-tions and that of preserving minimal dominating sets by neighborhoodhomomorphisms are dealt with.

Keywords: Network, Closure operator, Transformation.

2000 AMS Classication: 05C82, 54A05, 54C10, 90B18.

1. Introduction

Networks are ubiquitous in science and engineering, cf. [12] for a bibliography ofmore than 400 network applications. Network structures are invariably describedin combinatorial terms, that is, numerically. Scientists typically count and mea-sure. This paper, instead, uses the closed sets of an associated closure operator todene the structure of a network. Like open sets in continuous manifolds, closedsets can be a powerful tool for analyzing the structure of discrete systems.

In economics, closure is associated with rational choice operators [8, 11, 10]. TheGalois closure can be used to extract rules from data sets for subsequent usage inarticial intelligence reasoning systems [16, 17]. If a system can be partially (ortotally) ordered, then the closed sets are usually intervals, ideals or lters [7, 9]. Inthis paper, we apply closed sets to the structure of undirected graphs representingnetworks.

∗IT4Innovations Centre of Excellence, Brno University of Technology,Email: slapal@fme.vutbr.cz†Corresponding Author.‡Dept. of Computer Science, University of Virginia,

Email: jlp@virginia.edu

Doi : 10.15672/HJMS.2016.397

92

Since networks change over time [2, 18], this paper also seeks a mechanism fordescribing such changes. We examine the properties of smooth, or continuous,transformations of networks [14]. The question is, what changes to the structureof the network, as revealed by its closure properties, must ensue. We show, amongothers, that continuous transformations preserve generating sets. Another kindof network transformations discussed are neighborhood homomorphisms, whichare shown to preserve dominating sets. Their special cases, the strong homomor-phisms, even preserve the minimal dominating sets under certain conditions.

Throughout the paper, we will use the sux denotation for mappings to avoidmultiple parentheses and emphasize the importance of the domain elements. Thus,given a mapping f : X → Y (X,Y sets) and a point x ∈ X, we write x.f todenote the f -image of x. Similarly, if g : Y → Z is another mapping, we writef.g to denote the composition of f and g. Hence, x.f.g denotes the f.g-image ofx. We will also often simplify the denotation of sets given by enumerating theirelements - we will omit the commas between the elements and, in many cases,also the curly brackets. Thus, for example, abcd or abcd will be used for shortrather than a, b, c, d (we will delimit a set with the curly brackets only if wewant to emphasize its set nature). In particular, we will usually not distinguishnotationally between elements and singleton subsets of a set.

2. Networks and neighborhood closure operators

Let N = (S, ρ) be a set S of points, elements or nodes together with an ad-jacency relation, i.e., a reexive and symmetric binary relation ρ on S. (Notethat, although in the literature, adjacency relations are usually dened to beirreexive and symmetric, our denition of adjacency relations causes no con-fusion because there is a bijection between the reexive and irreexive binaryrelations on a set.) We then call N a network. For any subset Y ⊆ S, we putY.ρ = x| ∃y ∈ Y : (x, y) ∈ ρ. The set Y.ρ is called the neighborhood of Y .Clearly, ∅.ρ = ∅ and x ∈ y.ρ ⇔ y ∈ x.ρ whenever x, y ∈ S. Next, we haveX ⊆ X.ρ =

⋃x∈X x.ρ for every X ⊆ S. It follows that X ⊆ Y ⇒ X.ρ ⊆ Y.ρ and

(X ∪ Y ).ρ = X.ρ ∪ Y.ρ for all X,Y ⊆ S. Clearly, we also have S.ρ = S.Given a network N = (S, ρ) and subsets X,Y ⊆ S, X is said to dominate Y in

N if Y ⊆ X.ρ.We can represent a network (S, ρ) as an undirected simple graph with the vertex

set S and each edge being a two-element subsets x, y ⊆ S with (x, y) ∈ ρ (or,equivalently, (y, x) ∈ ρ). The neighborhood of any point is then the set of thosevertices that are adjacent to the point in the graph. For example, in the network(S, ρ) with S = abcdefgh representd by the undirected graph in Figure 1, wehave a.ρ = a, b, c or, more simply, a.ρ = abc. Clearly, a dominates abcand ch dominates S. There is a large literature on dominating sets in undirectednetworks, c.f. [5, 6].

When studying the structure of a network (S, ρ), we found it to be advantageousto consider a convenient associated operator on S. In this note, to be able toapply topological methods in the study of network structure, we employ a closureoperator. Let us recall rst the denition of a closure operator.

93

f

a

c

bd

e

g

h

Figure 1. An undirected graph.

Let S be a set. A mapping u : 2S → 2S (where 2S denotes the power set of S)is said to be a closure operator on S (cf. [15]) if, for all Y,Z ⊆ S,

(C1) Y ⊆ Y.u (extensivity),(C2) Y ⊆ Z ⇒ Y.u ⊆ Z.u (monotonicity), and,(C3) Y.u.u = Y.u (idempotency).

A subset Y ⊆ S is closed if Y = Y.u. A subset X of a closed set Y ⊆ S is said togenerate Y if X.u = Y .

It is well known that the intersection of closed sets must be closed so thata closure operator u on S is uniquely determined by the system of closed sets.Indeed, for every subset Y ⊆ S, we have Y.u =

⋂Z| Z closed and Y ⊆ Z.By (C1), the set S must be closed. The empty set may or may not be closed.

If it is, then u is said to be grounded.By a closure system we mean a pair (S, u) with S a set (of points or elements)

and u a closure operator on S.Let N = (S, ρ) be a network and Y ⊆ S a subset. Then, we dene the neigh-

borhood closure of Y to be the subset Y.uρ ⊆ S given as follows:

Y.uρ = x ∈ S| x.ρ ⊆ Y.ρ.Clearly, for all Y ⊆ S, Y.uρ ⊆ Y.ρ and Y.uρ.ρ = Y.ρ.Assigning to every subset of S its neighborhood closure, we get a mapping

uρ : 2S → 2S .

2.1. Proposition. uρ is a grounded closure operator on S.

Proof. Clearly, uρ is grounded and extensive by denition.Let X,Y ⊆ S be subsets, X ⊆ Y , and let z ∈ X.uρ. Then, z.ρ ⊆ X.ρ ⊆ Y.ρ,hence, z ∈ Y.uρ. Therefore, uρ is monotone.Let X ⊆ S be a subset and let z ∈ Y.uρ.uρ. Then, z.ρ ⊆ Y.uρ.ρ =

⋃x∈Y.uρ x.ρ ⊆⋃

x∈Y x.ρ = Y.ρ, hence, z ∈ Y.uρ. Thus, uρ is idempotent and the proof iscomplete.

2.2. Denition. The closure operator uρ and the closure system (S, uρ) aresaid to be the neighborhood closure operator and the neighborhood closure system,respectively, associated with (S, ρ).

2.3. Remark. If, in the denition of a grounded closure operator v on a setS, we replace the axiom (C3) of idempotency by the axiom of additivity, i.e.,v(X ∪ Y ) = vX ∪ vY whenever X,Y ⊆ S, we get a so-called pretopology v on S(and a pretopological space (S, v)) - cf. [20]. A neighborhood of a subset X ⊆ S in

94

a pretopological space (S, v) is any subset Y ⊆ S such that X ∩ (S − Y ).v = ∅.There is a natural way of associating a pretopology vρ on a set S with any reexivebinary relation ρ on S: we put X.vρ = x ∈ S| ∃y ∈ X : (x, y) ∈ ρ for everyX ⊆ S - see [21]. Let X ⊆ S. If ρ is a reexive and symmetric binary relationon S, then X.vρ = X.ρ. But then, X.ρ is also the smallest neighborhood (withrespect to set inclusion) of X in the closure space (S, vρ). This fact justies ourcalling X.ρ the neighborhood of X.

2.4. Proposition. For every pair of sets X,Y ⊆ S, X.uρ ⊆ Y.uρ if and only ifX.ρ ⊆ Y.ρ.Proof. Let X,Y ⊆ S be subsets with X.uρ ⊆ Y.uρ. Let x ∈ X.uρ implyingx ∈ y.uρ for some y ∈ X ⊆ X.uρ ⊆ Y.uρ. Then, y.ρ ⊆ Y.ρ so that x ∈ Y.ρ.Consequently, X.ρ ⊆ Y.ρ.To prove the converse inclusion, suppose that X.ρ ⊆ Y.ρ. Let z ∈ X.uρ, implyingz.ρ ⊆ X.ρ ⊆ Y.ρ. Then, z ∈ Y.uρ and, therefore, X.uρ ⊆ Y.uρ.

As an immediate consequence of Proposition 2.4 we get

2.5. Corollary. For every pair of sets X,Y ⊆ S, X.uρ = Y.uρ if and only ifX.ρ = Y.ρ.

One might expect that every point in a discrete space must be closed withrespect to the neighborhood closure. But this need not be true, as shown inFigure 1, where c.ρ = abcdef , a.ρ = abc ⊆ c.ρ and b.ρ = abcd ⊆ c.ρ, whiled.ρ = bcdg 6⊆ c.ρ, e.ρ = cefg 6⊆ c.ρ, and f.ρ = cefh 6⊆ c.ρ, so that c.uρ = abc.2.6. Proposition. Let X,Y ⊆ S be subsets, X ⊆ Y and Y closed in (S, uρ).Then, X generates Y in (S, uρ) if and only if X.ρ = Y.ρ.

Proof. Let X generate Y , i.e., let X.uρ = Y . Since X.uρ.ρ = X.ρ, we haveY.ρ = X.ρ.Conversely, letX.ρ = Y.ρ and let y ∈ Y . Then, y.ρ ⊆ Y.ρ = X.ρ, so that y ∈ X.uρ.Therefore, we have Y ⊆ X.uρ. As the converse inclusion is obvious, the proof iscomplete.

Let (S, u) be a closure system. Then, it is useful to deal with the structure ofclosed sets because closed sets uniquely determine the closure operator u. In thecase of a neighborhood closure operator associated with a network, the structureof closed sets may be regarded as the structure of the network. The closed setsmay be partially ordered by inclusion so that they create a complete lattice, Lu,in which inma coincide with intersections - see [15] for more details.

The neighborhood closure lattice Lu corresponding to the neighborhood closureoperator associated with the network of Figure 1 is shown in Figure 2. Thislattice has none of the regular structures one usually sees in textbook examples.Nevertheless, it carries considerable information.

3. Network transformations

Let S and S′ be sets. By a transformation Sf−→ S′ (between S and S′) we

mean a mapping f : 2S → 2S′. Given a mapping f : S → S′, its extension is the

95

abcf abdg abdhabcde

abc

gh

h

ab

a e d f g

af

dg

Ø

ah

de

ae

abcdefgh

defghaefh

efhdh

efghabd

Figure 2. Neighborhood closure lattice of Figure 1.

transformation Sf+

−→ S′ dened by Y.f+ = f(y)|y ∈ Y for every Y ⊆ S (in theliterature, f+ is often said to be lifted from f). As usual, we will write f insteadof f+ so that a mapping and its extension will be denoted by the same symbol -it will always be clear from the context whether f means a mapping f : S → S′

or its extension Sf−→ S′. Of course, there can be many transformations S

f−→ S′

other than those extended from mappings f : S → S′.

A transformation Sf−→ S′ is said to be monotone if, whenever X,Y ⊆ S,

X ⊆ Y implies X.f ⊆ Y.f . Note that a transformation that has been extendedfrom a mapping must be monotone.

Let Sf−→ S′ be a transformation and Z ⊆ S′ a subset. If the set of all subsets

Y ⊆ S with Y.f = Z is nonempty and has a greatest element (with respect to setinclusion), then the greatest element is said to be the inverse image of Z underf . Thus, if every subset of S′ has an inverse image, we get a transformationbetween S′ and S assigning to every subset of S′ its inverse image under f . This

transformation will be denoted by S′f−1

−→ S. Clearly, if Sf−→ S′ is an extension

of a mapping, then every subset of S′ has an inversion.Let (S, ρ) and (S′, ρ′) ((S, u) and (S′, u′)) be networks (closure systems) and let

Sf−→ S′ be a transformation. We then write (S, ρ)

f−→ (S′, ρ′) ((S, u)f−→ (S′, u′))

and say that f is a transformation between (S, ρ) and (S′, ρ′) ((S, u) and (S′, u′)).

3.1. Denition. A transformation (S, ρ)f−→ (S′, ρ′) between networks is said to

be neighborhood monotone if, whenever X,Y ⊆ S,X.ρ ⊆ Y.ρ⇒ X.f.ρ′ ⊆ Y.f.ρ′.

Note that a transformation that is monotone need not be neighborhood mono-tone, and vice versa.

3.2. Denition. ([13, 14, 21]) A transformation (S, u)f−→ (S′, u′)) between clo-

sure systems is said to be continuous if, whenever Y ⊆ S,Y.u.f ⊆ Y.f.u′.

96

A transformation (S, ρ)f−→ (S′, ρ′) between networks is called continuous if the

transformation (S, uρ)f−→ (S′, uρ′) between neighborhood closure systems is con-

tinuous.

3.3. Proposition. Let (S, u)f−→ (S′, u′) be a monotone transformation between

closure systems. If f is continuous, then Y.u.f = Y.f for every subset Y ⊆ S withY.f closed in (S′, u′). The converse is true provided that every subset of S′ has an

inverse image under f and the transformation S′f−1

−→ S is monotone.

Proof. Let f be continuous and let Y ⊆ S be a subset with Y.f closed in (S′, u′).Then, Y.u.f ⊆ Y.f.u′ = Y.f . The converse inclusion follows from the extensivenessof u and monotonicity of f .Conversely, let Y.u.f = Y.f for every subset Y ⊆ S with Y.f closed in (S′, u′) andlet every subset of S′ have an inverse image under f . Let X ⊆ S be an arbitrarysubset. Then, X.f.u′ is closed in (S′, u′) and X.f.u′ = X.f.u′.f−1.f . Therefore,X.f.u′.f−1.u.f = X.f.u′. Next, since X.f ⊆ X.f.u′, we have X ⊆ X.f.f−1 ⊆X.f.u′.f−1. Hence, X.u ⊆ X.f.u′.f−1.u, which yields X.u.f ⊆ X.f.u′.f−1.u.f .Consequently, X.u.f ⊆ X.f.u′ and the continuity of f is proved.

3.4. Corollary. Let (S, u)f−→ (S′, u′) be a monotone transformation between clo-

sure systems. If f is continuous, then, for every closed subset Z of (S′, u′), theinverse image of Z under f (if it exists) is closed in (S, u). The converse is trueprovided that every subset of S′ has an inverse image under f and the transfor-

mation S′f−1

−→ S is monotone.

Proof. Let f be continuous and let Z be a closed subset of (S′, u′) having an in-verse Y image under f . Since Y.u.f = Y.f by Proposition 3.3, we have Y.u ⊆ Y .Therefore, Y is closed in (S, u).Conversely, suppose that every subset of S′ has an inverse image under f and the

transformation S′f−1

−→ S is monotone. Let, for every closed subset Z of (S′, u′),the inverse image of Z under f (if it exists) is closed in (S, u) and let Y ⊆ S be anarbitrary subset with Y.f closed in (S′, u′). Then, Y.f.f−1.u = Y.f.f−1 becauseY.f.f−1 is closed in (S, u). Since Y ⊆ Y.f.f−1, we have Y.u.f ⊆ Y.f.f−1.u.f =Y.f.f−1.f = Y.f . As the converse inclusion is evident (it follows from the ex-tensiveness of u and monotonicity of f), we have Y.u.f = Y.f . Therefore, f iscontinuous by Proposition 3.3.

It is well known that, for the transformations between closure systems that areextensions of mappings, also the converse of Corollary 3.4 is true.

3.5. Proposition. If (S, ρ)f−→ (S′, ρ′) is a monotone and continuous transfor-

mation between networks, then Y.u.f.u′ = Y.f.u′.

Proof. Continuity implies Y.u.f ⊆ Y.f.u′ so that Y.u.f.u′ ⊆ Y.f.u′.u′. Since Y.f ⊆Y.u.f by monotonicity of u and f , we get Y.f.u′ ⊆ Y.u.f.u′.

3.6. Proposition. Let (S, u)f−→ (S′, u′), (S′, u′)

g−→ (S′′, u′′) be transforma-

tions and let g be monotone. If both f and g are continuous, then so is Sf.g−→ S′′.

97

Proof. We have X.u.f ⊆ X.f.u′ for any X ⊆ S and Y.u′.g ⊆ Y.g.u′′ for anyY ⊆ S′. Consequently, as g is monotone, X.u.f.g ⊆ X.f.u′.g ⊆ X.f.g.u′′. Thismeans that f.g is continuous.

In his seminal work [13], Ore considered only extended, continuous transforma-tions.

3.7. Proposition. Let (S, ρ)f−→ (S′, ρ′) be a monotone transformation between

networks. Then f is continuous if and only if f is neighborhood monotone.

Proof. Let f be continuous and let X,Y ⊆ S be subsets with X.ρ ⊆ Y.ρ. ByProposition 2.4, X ⊆ X.uρ ⊆ Y.uρ. Thus, X.f ⊆ Y.uρ.f ⊆ Y.f.uρ′ by continuity.So, X.f.uρ′ ⊆ Y.f.uρ′ , which yields X.f.ρ′ ⊆ Y.f.ρ′ by Proposition 2.4.Conversely, let f be neighborhood monotone and let Y ⊆ S be a subset. SinceY.uρ.ρ = Y.ρ, we have Y.uρ.f.ρ′ ⊆ Y.f.ρ′ by neighborhood monotinicity. Thus,by Proposition 2.4, Y.uρ.f.uρ′ ⊆ Y.f.uρ′ . Now, by the extensivity of u′, Y.u.f ⊆Y.u.f.u′ ⊆ Y.f.u′.

3.8. Proposition. Let (S, ρ)f−→ (S′, ρ′) be a monotone and continuous trans-

formation between networks. If X generates Z in (S, uρ) and Z.f is closed in(S′, uρ′), then X.f generates Z.f in (S′, uρ′).

Proof. Let X generate Z in (S, uρ). Then, X.ρ = Z.ρ by Proposition 2.6. Sincef is continuous, f is neighborhood monotone by Proposition 3.7 and we haveX.f.ρ = Z.f.ρ. Therefore, X.f generates Z.f in (S′, uρ′) by Proposition 2.6.

3.9. Denition. A transformation (S, u)f−→ (S′, u′) between closure systems is

said to be closed if Y.f is closed in (S′, u′) whenever Y is closed in (S, u).

3.10. Proposition. Let (S, u)f−→ (S′, u′) be a transformation between closure

systems. If Y.f.u′ ⊆ Y.u.f for all Y ⊆ S, then f is closed. The converse is trueprovided that f is monotone.

Proof. Let Y.f.u′ ⊆ Y.u.f for every subset Y ⊆ S. If Y is closed, then Y.f.u′ ⊆ Y.f ,so that Y.f is closed. Hence, f is closed.Conversely, let f be monotone and closed and let Y ⊆ S be a subset. Then,Y.f.u′ ⊆ Y.u.f.u′ = Y.u.f because Y.u.f is closed in (S′, u′) (as Y.u is closed in(S, u)).

3.11. Denition. A transformation (S, ρ)f−→ (S′, ρ′) between networks is said

to be a homomorphism if, for every Y ⊆ S,

Y.ρ.f ⊆ Y.f.ρ′.Homomorphisms are common in graph theory (and, more generally, theory

of binary relations - cf. [19]). If G = (S,E) and G′ = (S′,E′) are undirectedgraphs with edge sets E and E′, then a mapping f : S → S′ is said to be agraph homomorphism if x, y ∈ E implies f(x), f(y) ∈ E′ [1, 3, 4]. The abovedenition of homomorphisms is obtained by naturally extending the denition ofgraph homomorphisms.

98

Homomorphisms need not be continuous even if they are monotone. Consider

the monotone homomorphism (S, ρ)f−→ (S′, ρ′) displayed in Figure 3, which is

an extension of the mapping f : x, y1, y2, z → x′, y′, z′ given by x.f = x′,y1.f = y2.f = y′, and z.f = z′. So, x.uρ = xy1 and z.uρ = y2z while x′.uρ′ = x′

and z′.uρ′ = z′. Thus, x.uρ.f = x′y′ 6⊆ x′ = x.f.uρ′ .

x’ y’ z’x y

y z

1

2

f

Figure 3. A homomorphism that is not continuous.

Monotone and continuous transformations, too, need not be homomorphisms.

Consider the monotone, continuous transformation (S, ρ)f−→ (S′, ρ′) displayed in

Figure 4, which is an extension of the bijection f : w, x, y, z → w′, x′, y′, z′given by t.f = t′ for every t ∈ w, x, y, z.

x’ y’ z’w’x y zwf

Figure 4. A monotone, continuous map that is not a homomorphism.

Clearly, wxy = wxy.f = x.ρ.f 6⊆ x.f.ρ′ = w′x′, but it is easyto verify that f is continuous because the closed sets in (S, ρ) and (S′, ρ′) are∅, wxyz, wx, yz, w, z and ∅, w′x′y′z′, w′x′, y′z′, respectively

3.12. Proposition. Let (S, ρ)f−→ (S′, ρ′) be a monotone homomorphism between

networks. If X dominates Y in S, then X.f dominates Y.f in S′.

Proof. If Y ⊆ X.ρ, then Y.f ⊆ X.ρ.f ⊆ X.f.ρ′

3.13. Denition. A transformation (S, ρ)f−→ (S′, ρ′) between networks is called

a strong homomorphism if, for every Y ⊆ S,

Y.ρ.f = Y.f.ρ′.

Eectively, if (x′, y′) ∈ ρ′ and x ∈ S is a point with x.f = x′, then there existsy ∈ S such that y.f = x′ and x, y ∈ ρ is an edge in S. The transformation in Fig-ure 5 represents the typical conguration of a strong homomorphism. The trans-formation is an extension of the mapping f : x1, x2, y1, y2, z1, z2 → x′, y′, z′given by x1.f = x2.f = x′, y1.f = y2.f = y′, and z1.f = z2.f = z′.

3.14. Proposition. Let (S, ρ)f−→ (S′, ρ′) be a strong homomorphism between

networks that is an extension of a mapping. Then,(a) f is continuous,(b) f is closed provided that, for every X ⊆ S, x ∈ X.ρ and x.f ⊆ X.f imply

99

y

y

x’ y’ z’f

1

2x2

x1

z1

2z

Figure 5. Typical conguration for a neighborhood homomorphism.

x ∈ X, and(c) Y ′.f−1.ρ ⊆ Y ′.ρ′.f−1.

Proof. (a) Let Y ⊆ S be a subset and z′ ∈ Y.uρ.f a point. Then, there exists z ∈Y.uρ with z.f = z′. Thus, z.ρ ⊆ Y.ρ and z′.ρ′ = z.f.ρ′ = z.ρ.f ⊆ Y.ρ.f = Y.f.ρ′.This yields z′ ∈ Y.f.uρ′ . Therefore, Y.uρ.f ⊆ Y.f.uρ′ and the continuity of f isproved.(b) Let Y ⊆ S be a subset and z′ ∈ Y.f.uρ′ a point. Then, z′.ρ ⊆ Y.f.ρ. Sincez′ ∈ Y.f.uρ′ ⊆ Y.f.ρ = Y.ρ.f , there exists z ∈ Y.ρ such that z′ = z.f . Letx ∈ z.ρ ⊆ Y.ρ.ρ. Then, x.f ⊆ z.ρ.f = z.f.ρ′ = z′.ρ′ ⊆ Y.f.ρ′ = Y.ρ.f . Because ofthe extra provision, x ∈ Y.ρ. Thus, z.ρ ⊆ Y.ρ, i.e., z ∈ Y.uρ. Hence, z′ ⊆ Y.uρ.f .Therefore, Y.f.uρ′ ⊆ Y.uρ.f and the closedness of f is proved.(c) Let Y ⊆ S′ be a subset. Then, Y.f−1.ρ ⊆ Y.f−1.ρ.f.f−1 = Y.f−1.f.ρ′.f−1 =Y.ρ′.f−1. Conversely, let z ∈ Y.ρ′.f−1 be a point and put z′ = z.f =∈ Y ′.ρ′.Then, there exists y ∈ Y with z′ ∈ y.ρ′, i.e., with y ∈ z′.ρ′. Now, since f is astrong homomorphism, z.ρ.f = z.f.ρ′ = z′.ρ′. Therefore, there exists x ∈ z.ρ suchthat x.f = y ∈ Y . Thus, we have z ∈ x.ρ and x ∈ y.f−1 ⊆ Y.f−1. Therefore,z ∈ Y.f−1.ρ and the inclusion Y ′.ρ′.f−1 ⊆ Y ′.f−1.ρ is proved.

In contrast to monotone homomorphisms, which may not be continuous (seeFigure 3), monotone strong homomorphisms always are.

We have seen that the domination is preserved by monotone homomorphisms(Proposition 3.12). It would be convenient if minimal dominating sets were pre-served by (monotone) strong homomorphisms. Unfortunately, this need not betrue as illustrated by the strong homomorphism displayed in Figure 6, which is anextension of the mapping f : a, b, c, d, e, f, g, h → a′, b′, e′, g′ given by a.f = a′,b.f = c.f = d.f = b′, e.f = f.f = e′, and g.f = h.f = g′. The minimal dominatingsets of S and S′ are enumerated. Here, aeh is a minimal dominating set of S, buta′e′g′, although dominating in S′, is not minimal. Only the minimal dominatingsets aef , agh, bef , and cg are mapped onto the minimal dominating sets a′e′, a′g′,b′e′, and b′g′, respectively.

3.15. Proposition. Let (S, ρ)f−→ (S′, ρ′) be a strong homomorphism that is an

extension of a mapping. Let X be a minimal dominating set of Y in (S, ρ) suchthat, for every element x ∈ Y.ρ, x′ ∈ X.f and x′ = x.f ∈ X.f imply x ∈ X. Then,X.f is a minimal dominating set of Y.f in (S′, ρ′).

Proof. Put X ′ = X.f and Y ′ = Y.f . By Proposition 3.12, X ′ dominates Y ′,i.e., Y ′ ⊆ X ′.ρ′. If Y ′ = ∅, then Y = ∅, so that x = ∅. Hence, X ′ = ∅ and,consequently, X ′ is a minimal dominating set of Y ′ in (S′, ρ′). Let Y ′ 6= ∅ and

100

a’e’ a’g’

b’e’ b’g’

a

b

c

d

e

f

g

h

a’ b’ e’ g’

aef aeh afg agh bef

bfg bfh ceh cg deh

f

Figure 6. Neighborhood homomorphisms may not preserve minimaldominating sets.

suppose that X ′ is not a minimal dominating set of Y ′. Then, there exists a propernon-empty subset X ′1 ⊂ X ′ dominating Y ′, i.e., fullling Y ′ ⊆ X ′1.ρ

′. Let x′0 ∈X ′−X ′1 be an arbitrary element. Since X ′1 ⊆ X ′−x′0, we have Y ′ ⊆ (X ′−x′0).ρ′.Let x0 ∈ X be a point with x′0 = x0.f and put Y ′0 = Y ′ ∩ x′0.ρ′. Let y ∈ Y be anarbitrary element and put y′ = y.f . Then, y′ ∈ Y ′.If y′ /∈ Y ′0 , then y.f /∈ x′0.ρ′ = x0.f.ρ

′ = X0.ρ.f . Thus, y /∈ x0.ρ, so that Y ∩X0.ρ =∅. Since Y ⊆ X.ρ = x0.ρ∪(X−x0).ρ, we have Y ⊆ (X−x0).ρ. Therefore, X−X0

dominates Y , which is a contradiction.Suppose that y′ ∈ Y ′0 . Then, Y ′0 ⊆ Y ′ implies y′ ∈ (X ′ − x′0).ρ′, which means thatthere exists z′ ∈ X ′−x′0 such that y′ ∈ z′.ρ′. This yields z′ ∈ y′.ρ′ = y.f.ρ′ = y.ρ.f .Thus, there exists a point z ∈ y.ρ such that z′ = z.f . Therefore, y ∈ z.ρ and wehave z ∈ Y.ρ and z.f = z′ ∈ X.f . Hence, z ∈ X by the assumption of thestatement. But we also have z 6= x0 because, otherwise, z.f = x′0 6= z′, which is acontradiction. Thus, z ∈ X − x0 so that y ∈ (X − x0).ρ. Again, we have shownthat Y ⊆ (X−x0).ρ, i.e., that X−x0 dominates Y , which is a contradiction. Thisproves the statement.

In Figure 6, only the minimal dominating sets aef , agh, bef and cg satisfy thecondition of Proposition 3.15. Each of them is mapped onto one of the minimalgenerating sets of S′.

Reducing networks to simpler forms by strong homomorphisms provides analternative method of approaching some of the classic problems in dominationtheory. However, it should be emphasized that "minimal" in domination theorymeans minimal cardinality whereas we use "minimal" in the set inclusion sense.Nevertheless, only sets that are minimal in the set inclusion sense can have minimalcardinality. So, network reduction by strong homomorphisms can still yield someinsight into these classic problems.

Acknowledgement. The rst author acknowledges support by the Ministry of Ed-ucation, Youth and Sports of the Czech Republic from the National Programme ofSustainability (NPU II) project "IT4Innovations Excellence in Science - LQ1602".

101

References

[1] Agnarsson, G. and Greenlaw, R. Graph Theory: Modeling, Applications and Algorithms

(Prentice Hall, Upper Saddle River, NJ, 2007).[2] Bourqui R., Gilbert, F., Simonetto, P., Zaidi, F., Sharan, U. and Jourdan, F. Detecting

structural changes and command hierarchies in dynamic social networks, in: 2009 Advancesin Social Network Analysis and Mining, 8388 (Athens, Greece, 2009).

[3] Godsil, Ch. and Royle, G. Algebraic Graph Theory (Springer, New York, 2001).[4] Harary, F. Graph Theory (Addison-Wesley, 1969).[5] Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. (editors) Domination in Graphs, Advanced

Topics (Marcel Dekker, New York, 1998).[6] Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. Fundamentals of Domination in Graphs

(Marcel Dekker, New York, 1998).[7] Jankovic, D. and Hamlett, T.R. New topologies from old via ideals, Amer. Math. Monthly

97 (4), 295310, 1990.[8] Koshevoy, G.A. Choice functions and abstract convex geometries, Mathematical Social Sci-

ences, 38 (1), 3544, 1999.[9] McKee, T.A. and McMorris, F.R. Topics in Intersection Graph Theory, SIAM Monographs

on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics(Philadelphia, PA, 1999).

[10] Monjardet, B. Closure operators and choice operators: a survey, in: Fifth InternationalConference on Concept Lattices and their Applications (Montpellier, France, 2007).

[11] Monjardet, M and Raderinirina, V. The duality between the antiexchange closure operators

and the path independent choice operators on a nite set, Mathemetical Social Sciences41(2), 131150, 2001.

[12] Newman, M.E.J. The structure of function of complex networks, SIAM Review 45, 167256,2003.

[13] Oystein Ore. Mappings of closure relations, Annals of Math. 47 (1), 5672, 1946.[14] Pfaltz, J.L. and lapal, J. Transformations of discrete closure systems, Acta Math. Hungar.

138 (4), 386405, 2013.[15] Pfaltz, J.L. Closure lattices, Discrete Mathematics 154, 217236, 1996.[16] Pfaltz, J.L. Logical implication and causal dependency, in: Schärfe, H., Hitzler, P. and

Ohrstrom, P. (editors) Conceptual Structures: Inspiration and Application, Lecture Notesin Articial Intelligence 4068, 145157, 2006.

[17] Pfaltz, J.L. Establishing logical rules from empirical data, Intern. Journal on Articial In-telligence Tools 17 (5), 9851001, 2008.

[18] Pfaltz, J.L. Mathematical continuity in dynamic social networks, in: Datta, A., Shulman,S., Zheng, B., Lin, S., Sun, A. and Lim, E.-P (editors) Third International Conference onSocial Informatics 2011, Lecture Notes Comput. Sci. 6984, 3650, 2011.

[19] lapal, J. Direct arithmetics of relational systems, Publ. Math. Debrecen 38, 3948, 1991.[20] lapal, J. Convenient closure operators on Z2, in: Wiederhold, P. and Barneva, R.P. (edi-

tors) Combinatorial Image Analysis, Lecture Notes Comput. Sci. 5852, 425436, 2009.[21] lapal, J. A Galois correspondence for digital topology, in: Denecke, K., Erné, M. and

Wismath, S.L. (editors) Galois Connections and Applications, 413424 (Kluwer AcademicPublishers, Dordrecht, 2004).

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 103 110

Weakly discontinuous and resolvable functionsbetween topological spaces

Dedicated to the memory of Prof. Dr. L. Michael Brown

Taras Banakh ∗ and Bogdan Bokalo†

Abstract

We prove that a function f : X → Y from a rst-countable (moregenerally, Preiss-Simon) space X to a regular space Y is weakly dis-continuous (which means that every subspace A ⊂ X contains an opendense subset U ⊂ A such that f |U is continuous) if and only if f isopen-resolvable (in the sense that for every open subset U ⊂ Y thepreimage f−1(U) is a resolvable subset of X) if and only if f is resolv-able (in the sense that for every resolvable subset R ⊂ Y the preimagef−1(R) is a resolvable subset of X). For functions on metrizable spacesthis characterization was announced (without proof) by Vinokurov in1985.

Keywords: Weakly discontinuous function, resolvable function, Preiss-Simonspace

2000 AMS Classication: 54C08

1. Introduction and Main Result

In this paper we present a proof of a characterization of weakly discontinuousfunctions announced (without proof) by Vinokurov in [14].

A function f : X → Y between topological spaces is called weakly discontinu-

ous if every subspace A ⊂ X contains a dense open subset U ⊂ A such that therestriction f |U is continuous. It is well-known that for weakly discontinuous mapsf : X → Y and g : Y → Z the composition g f : X → Z is weakly discontin-uous. Weakly discontinuous functions were introduced by Vinokurov [14]. Manyproperties of functions, equivalent to the weak discontinuity were discovered in

∗Instytut Matematyki, Jan Kochanowski University in Kielce (Poland)Email : t.o.banakh@gmail.com†Department of Mathematics, Ivan Franko National University of Lviv (Ukraine)

Email : b.m.bokalo@gmail.com

Doi : 10.15672/HJMS.2016.399

104

[1, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14]. By [14, Theorem 8], a function f : X → Y from ametrizable space X to a regular space Y is weakly discontinuous if and only if forevery open set U ⊂ Y the preimage f−1(U) is a resolvable subset of X. We recall[10, I.12] that a subset A of a topological space X is resolvable if for every closed

subset F ⊂ X the set F ∩A∩F \A is nowhere dense in F . Observe that a subsetA ⊂ X is resolvable if and only if its characteristic function χA : X → 0, 1 isweakly discontinuous. It is known [10, I.12] that the family resolvable subsets ofa topological space X is closed under intersections, unions, and complements.

A function f : X → Y between topological spaces is called (open-)resolvableif for every (open) resolvable subset R ⊂ Y the preimage f−1(R) is a resolvablesubset of X. It is clear that each resolvable function is open-resolvable.

1.1. Proposition. If a function f : X → Y between topological spaces is weakly

discontinuous, then f is resolvable.

Proof. If a subset A ⊂ Y is resolvable, then its characteristic function χA : Y →0, 1 is weakly discontinuous. Since the weak discontinuity is preserved by com-positions (see, e.g., [2, 4.1]), the composition g = χA f : X → 0, 1 is weaklydiscontinuous, which implies that the set g−1(1) = f−1(A) is resolvable in X.

By [14, Theorem 8], for functions between metrizable spaces, Proposition 1.1can be reversed. However the paper [14] contains no proof of this important fact.In this paper we present a proof of this Vinokurov's characterization in a moregeneral case of functions dened on Preiss-Simon spaces.

We dene a topological space X to be Preiss-Simon at a point x ∈ X if for anysubset A ⊂ X with x ∈ A there is a sequence (Un)n∈ω of non-empty open subsetsof A that converges to x in the sense that each neighborhood of x contains all butnitely many sets Un. By PS(X) we denote the set of points x ∈ X at which X isPreiss-Simon. A topological space X is called a Preiss-Simon space if PS(X) = X(that is X is Preiss-Simon at each point x ∈ X).

It is clear that each rst-countable space is Preiss-Simon and each Preiss-Simonspace is Fréchet-Urysohn. A less trivial fact due to Preiss and Simon [12] assertsthat each Eberlein compact space is Preiss-Simon.

A base B of the topology of a space X will be called countably additive if theunion ∪C of any countable subfamily C ⊂ B belongs to B.

A function f : X → Y between topological spaces will be called base-resolvable

if there exists a countably additive base B of the topology of Y such that for everyset B ⊂ Y the preimage f−1(B) is a resolvable subset of X.

It is clear that for any function f : X → Y we have the implications:

weakly discontinuous ⇒ resolvable ⇒ open-resolvable ⇒ base-resolvable.

For functions on Preiss-Simon spaces these implications can be reversed, whichis proved in the following characterization. For functions on metrizable spaces itwas announced (without written proof) by Vinokurov in [14, Theorem 8].

1.2. Theorem. For a functions f : X → Y from a Preiss-Simon space X to a

regular space Y the following conditions are equivalent:

(1) f is weakly discontinuous;

(2) f is resolvable;

105

(3) f is open-resolvable;

(4) f is base-resolvable.

This theorem will be proved in Section 3 after some preliminary work made inSection 2.

By Theorem 1.2, any open-resolvable map f : X → Y from a Preiss-Simonspace X to a regular space Y is resolvable. We do not know if this implication stillholds for any function between regular spaces. The authors are grateful to SergeyMedvedev for turning their attention to this intriguing question.

1.3. Problem (Medvedev). Is each open-resolvable function f : X → Y betweenregular spaces resolvable?

The following example indicates that Problem 1.3 can be dicult and showsthat the countable additivity of the base B cannot be removed from the denitionof a base-resolvable function.

1.4. Example. Let RQ be the real line endowed with the metrizable topologygenerated by the countable base B =

(a, b) : a < b, a, b ∈ Q

∪q : q ∈ Q

.

The identity map id : R → RQ is not (open-)resolvable as the preimage Q =

id−1(Q) of the open set Q ⊂ RQ is not resolvable in R. Yet, for every basic set

B ∈ B the preimage id−1(B) is a resolvable set in R.

2. Five Lemmas

In this section we shall prove some auxiliary results, which will be used in the

proof of Theorem 1.2. For a subset A of a topological space by A, A, and A

we denote the closure, the interior, and the interior of the closure of A in X,respectively. A family B of non-empty open subsets of a topological space X iscalled a π-base if each non-empty open set U ⊂ X contains some set B ∈ B.

Following [2], we dene a function f : X → Y between topological spaces tobe scatteredly continuous if for every non-empty subset A ⊂ X the restriction f |Ahas a continuity point. It is easy to see that each weakly discontinuous functionis scatteredly continuous. For maps into regular spaces the converse implicationis also true (see [1], [4] or [2, 4.4]):

2.1. Lemma. A function f : X → Y from a topological space X to a regular space

Y is weakly discontinuous if and only if f is scatteredly continuous.

We recall that for a topological spaceX its tightness t(X) is the smallest cardinalκ such that for every subset A ⊂ X and point a ∈ A there exists a subset B ⊂ Aof cardinality |B| ≤ κ such that a ∈ B. The following lemma was proved in [2,2.3].

2.2. Lemma. A function f : X → Y between topological spaces is scatteredly

continuous if and only if for any non-empty subset A ⊂ X of cardinality |A| ≤ t(X)the restriction f |A has a continuity point.

2.3. Lemma. If A,B are disjoint resolvable subsets of a topological space X, then

A ∩ B is nowhere dense in X.

106

Proof. To derive a contradiction, assume that the set F = A∩ B has a non-emptyinterior U in X. Then U ∩A and U ∩B are two dense disjoint sets in U . By theresolvability of A, the dense subset A ∩ U of U has nowhere dense boundary inU . Consequently, the interior UA of the set A ∩ U is dense in U . By the samereason, the interior UB of the set B ∩ U is dense in U . Then the non-empty spaceU contains two disjoint dense open sets UA and UB , which is not possible.

A function f : X → Y between topological spaces is dened to be almost

continuous (weakly continuous) at a point x ∈ X if for any neighborhoodOy ⊂ Y ofthe point y = f(x) the (interior of the) set f−1(Oy) is dense in some neighborhoodof the point x in X. By AC(f) (resp. WC(f) ) we shall denote the set of pointof almost (resp. weak-) continuity of f .

2.4. Lemma. Let f : X → Y be a base-resolvable map from a topological space

X to a Hausdor space Y . Then

(1) AC(f) = WC(f).(2) If D is dense in X, Y is regular, and f |D has no continuity point, then

D \AC(f) also is dense in X.

(3) If X has a countable π-base, then for any countable dense set D ⊂ X there

is a point y ∈ f(D) such that for every neighborhood Oy of y the preimage

f−1(Oy) has non-empty interior in X.

(4) The family f−1(y)

: y ∈ Y is disjoint.Proof. Since f is basic-resolvable, there exists a countably additive base B of thetopology of Y such that for every U ∈ B the preimage f−1(U) is resolvable in X.

1. The inclusion WC(f) ⊂ AC(f) is trivial. To prove that AC(f) ⊂ WC(f),take any point x ∈ AC(f). To show that x ∈WC(f), take any neighborhood Oy ∈B of the point y = f(x) and consider the preimage f−1(Oy). Since x ∈ AC(f), the

closure F = f−1(Oy) is a neighborhood of x. Since the set f−1(Oy) is resolvable,

the boundary F ∩ f−1(Oy) ∩ F \ f−1(Oy) is nowhere dense in F . Consequently,the interior of the set F ∩ f−1(Oy) in F is dense in F and x ∈WC(f).

2. Assume that D ⊂ X is dense, Y is regular, and f |D has no continuity point.Given a point x ∈ D, and a neighborhood Ox ⊂ X of x we should nd a pointx′ ∈ Ox ∩D \AC(f). If x /∈ AC(f), then we can take x′ = x. So we assume thatx ∈ AC(f) and hence x ∈WC(f) by the preceding item. Since x is a discontinuitypoint of f |D, there is a neighborhood Of(x) of f(x) such that f(D ∩ Ux) 6⊂ Of(x)

for every neighborhood Ux of x. Using the regularity of Y choose a neighborhoodUf(x) ⊂ Y of f(x) with Uf(x) ⊂ Of(x). Since f is weakly continuous at x, the

closure of the interior of the preimage f−1(Uf(x)) contains some open neighborhoodWx of x. By the choice of Of(x), we can nd a point x′ ∈ D ∩ Ox ∩ Wx with

f(x′) /∈ Of(x). Consider the neighborhood Of(x′) = Y \Uf(x) of f(x′) and observe

that Wx ∩ f−1(Of(x′)) is a nowhere dense subset of Ox (because it misses the

interior of f−1(Uf(x)) which is dense in Wx). This witnesses that x′ /∈ AC(f).

3. Assume that X has countable π-base Wnn∈ω. We lose no generalityassuming that the subfamilies W2nn∈ω and W2n+1n∈ω are countable π-basesin X. Given a countable dense subset D ⊂ X, we should nd a point y ∈ f(D)such that for every neighborhood Oy ⊂ Y the preimage f−1(Oy) has non-empty

107

interior in X. Assume conversely that each point y ∈ f(D) has a neighborhoodOy ∈ B such that the preimage f−1(Oy) has empty interior in X. The resolvabilityof f−1(Oy) implies that this set is nowhere dense in X. We shall inductivelyconstruct a sequence (xn)n∈ω of points of D and a sequence (Un)n∈ω of open setsin Y such that

(a) f(xn) ∈ Un ∈ B and the set f−1(Un) is nowhere dense in X;(b) xn ∈ D ∩Wn \

⋃k<n f

−1(Un);(c) Un ∩ f(xk)k<n = ∅.

Taking into account that W2nn∈ω and W2n+1n∈ω are π-bases in X, weconclude that the disjoint sets x2nn∈ω and x2n+1n∈ω are dense in X. Thecountable additivity of the base B guarantees that the open sets Ue =

⋃n∈ω U2n

and Uo =⋃

n∈ω U2n+1 belong to B. Then their preimages f−1(Ue) ⊃ x2nn∈ωand f−1(Ue) ⊃ x2n+1n∈ω are disjoint dense resolvable sets in X. But thiscontradicts Lemma 2.3.

4. Assuming that the family f−1(y)

: y ∈ Y is not disjoint, nd two distinctpoints y, z ∈ Y such that the intersection

W = f−1(y) ∩ f−1(z)

is not empty. Observe that the sets W ∩ f−1(y) and W ∩ f−1(z) both are densein W .

By the Hausdor property of Y the points y, z have disjoint open neighborhoodsOy,Oz ∈ B. The choice of B guarantees that the sets f−1(Oy) and f−1(Oz) are

resolvable. By Lemma 2.3, the intersection f−1(Oy) ∩ f−1(Oz) is nowhere densein X, which is not possible as this intersection contains the non-empty open setW . 2.5. Lemma. Let f : X → Y be a base-resolvable map from a topological space

X to a regular space Y and D be a countable dense subset of X such that f |D has

no continuity point.

(1) For any nite subset F ⊂ Y there is a dense subset Q ⊂ D \ f−1(F ) in Xsuch that f |Q has no continuity point.

(2) If X has a countable π-base, then for any sequence (Un)∞n=1 of non-empty

open subsets of X there are an innite subset I ⊂ N and sequences (Vn)n∈Iand (Wn)n∈I of pairwise disjoint non-empty open sets in X and Y , respec-tively, such that Vn ⊂ Un ∩ f−1(Wn) for all n ∈ I.

(3) If D ⊂ PS(X), then there is a countable rst countable subspace Q ⊂ Dsuch that Q contains no nite non-empty open subsets and the restriction

f |Q is a bijective map whose image f(Q) is a discrete subspace of Y .

Proof. Fix a countably additive base B of the topology of Y such that for everyU ∈ B the preimage f−1(U) is resolvable in X.

1. The rst statement will be proved by induction on the cardinality |F | ofthe set F . If |F | = 0, then we can put Q = D and nish the proof. Assumethat for some n > 0 the rst statement is proved for all sets F ⊂ Y of cardinality|F | < n. Take any nite subset F ⊂ Y of cardinality |F | = n. Choose any pointy ∈ F . By the inductive hypothesis, for the set F \ y there exists a dense subsetE ⊂ D \ f−1(F \ y) such that the function f |E has no continuity points. We

108

claim that the set E\f−1(y) is dense in X. In the opposite case, there exists a non-empty open set U ⊂ X such that E∩U ⊂ f−1(y). It follows that E∩U ⊂ AC(f).By Lemma 2.4(2), the set E \ AC(f) ⊂ E \ U is dense in X, which is a desiredcontradiction showing that the set E \ f−1(y) is dense in X and so is the set

Q := (E \ f−1(y)) \(E ∩ f−1(y) \ E ∩ f−1(y)

) ⊂ D \ f−1(F ).It remains to check that the restriction f |Q has no continuity points. To derive

a contradiction, assume that some point x0 ∈ Q is a continuity point of therestriction f |Q. If x0 /∈ E ∩ f−1(y), then the discontinuity of the map f |E at x0implies the discontinuity of f |Q at x0. So, x0 belongs to the interior E ∩ f−1(y)

of E ∩ f−1(y).Let y0 = f(x0) and observe that y0 6= y (because x0 /∈ f−1(y)). By the

Hausdor property of Y the points y0 and y have disjoint open neighborhoodsOy0, Oy ∈ B. By the continuity of f |Q at x0, there is an open neighborhood

Ox0 ⊂ E ∩ f−1(y)of x0 such that f(Ox0 ∩ Q) ⊂ Oy0. It follows that the

preimages f−1(Oy0) and f−1(Oy) are disjoint resolvable subsets ofX. The densityof the set Q in X implies the density of the set Ox0 ∩ f−1(Oy0) ⊃ Ox0 ∩ Qin Ox0. On the other hand, the intersection f−1(y) ∩ E ∩ f−1(y)

is dense in

E ∩ f−1(y)and hence Ox0∩f−1(Oy) ⊃ f−1(y)∩Ox0 is dense in Ox0. Therefore,

f−1(Oy0) ∩ f−1(Oy) contains the non-empty open set Ox0. But this contradictsLemma 2.3.

2. Assume that the space X has a countable π-base and let (Un)∞n=1 be asequence of non-empty open subsets of X. Applying Lemma 2.4(3) to the mapf |U1 and the dense subset D ∩ U1, nd a point y0 ∈ f(D ∩ U1) such that foreach neighborhood Oy0 the preimage U1 ∩ f−1(Oy0) has non-empty interior. Byinduction, for every n ∈ N we shall nd a point yn ∈ f(D) \ yi : i < n such thatfor every neighborhood Oyn the set Un ∩ f−1(Oyn) has non-empty interior.

Assuming that for some n the points y0, . . . , yn−1 have being chosen, we shallnd a point yn. It follows that the intersection D∩Un is a countable dense subsetof Un such that f |D ∩ Un has no continuity point. Applying Lemma 2.5(1), wecan nd a dense subset Q ⊂ D ∩ Un \ f−1(y0, . . . , yn−1) in Un such that therestriction f |Q has no continuity point. Applying Lemma 2.4(3) to the map f |Un

and the dense subset Q ∩ Un of Un, nd a point yn ∈ f(Q) ⊂ f(D) \ yi : i < nsuch that for each neighborhood Oyn the preimage Un ∩ f−1(Oyn) has non-emptyinterior. This completes the inductive construction.

The space yn : n ∈ N, being innite and regular, contains an innite discretesubspace yn : n ∈ I. By induction, we can select pairwise disjoint open neigh-borhoods Wn ⊂ Y , n ∈ I, of the points yn. For every n ∈ I, the choice of thepoint yn guarantees that the set Un∩ f−1(Wn) contains a non-empty open set Vn.Then the set I ⊂ N and sequences (Vn)n∈I , (Wn)n∈I satisfy our requirements.

3. Assume that D ⊂ PS(X). Using the density of the countable set D andthe inclusion D ⊂ PS(X), we can show that the space X has a countable π-base.Applying Lemma 2.4(2), we get that D \AC(f) is dense in X.

By induction on the tree ω<ω we shall construct sequences (xs)s∈ω<ω of pointsof the set D \AC(f), and sequences (Vs)s∈ω<ω and (Us)s∈ω<ω , (Ws)s∈ω<ω of setsso that the following conditions hold for every nite number sequence s ∈ ω<ω:

109

(a) Vs is an open neighborhood of the point xs in X;(b) Ws ⊂ Us are open neighborhoods of f(xs) in Y ;(c) f(Vs) ⊂ Us;(d) Vsˆn ⊂ Vs and Usˆn ⊂ Us for all n ∈ ω;(e) the sequence (Vsˆn)n∈N converges to xs;(f) Ws ∩ Usˆn = ∅ = Usˆn ∩ Usˆm for all n 6= m in ω.

We start the induction letting V∅ = X, U∅ = Y and x∅ be any point of D \AC(f).

Assume that for a nite sequence s ∈ ω<ω the point xs ∈ D \AC(f) and opensets Vs ⊂ X and Us ⊂ Y with xs ∈ Vs and f(Vs) ⊂ Us have been constructed.Since f |Vs fails to be almost continuous at xs, there is a neighborhood Ws ⊂ Us off(xs) such that the closure of the preimage f−1(Ws) is not a neighborhood of xsin X. This fact and the Preiss-Simon property of X at xs allows us to construct asequence (V ′k)k∈ω of open subsets of Vs \clX

(f−1(Ws)

)that converges to xs in the

sense that each neighborhood of x contains all but nitely many sets V ′k. ApplyingLemma 2.5(2) to the map f |Vs : Vs → Us, we can nd an innite subset N ⊂ ωand a sequence (U ′k)k∈N of pairwise disjoint open sets of Us such that each setf−1(U ′k) ∩ V ′k, k ∈ N , has non-empty interior in X. Let N = kn : n ∈ ω be theincreasing enumeration of the set N .

For every n ∈ ω let Usˆn = U ′kn\ Ws, Vsˆn be a non-empty open subset in

f−1(Ukn) ∩ V ′kn

and xsˆn ∈ Vsˆn ∩ D \ AC(f) be any point (such a point existsbecause of the density of D \ AC(f) in X). One can check that the points xsˆn,n ∈ ω and sets Ws, Vsˆn, Usˆn, n ∈ ω satisfy the requirements of the inductiveconstruction.

After completing the inductive construction, consider the set Q = xs : s ∈ω<ω and note that it is rst countable, contains no non-empty nite open subsets,f |Q is bijective and f(Q) is a discrete subspace of Y .

3. Proof of Theorem 1.2

Given a function f : X → Y from a Preiss-Simon space X to a regular space Ywe need to prove the equivalence of the following conditions:

(1) f is weakly discontinuous;(2) f is resolvable;(3) f is open-resolvable.(4) f is base-resolvable.

The implication (1)⇒ (2) follows from Proposition 1.1 and (2)⇒ (3)⇒ (4) aretrivial. To prove that (4)⇒ (1), assume that the function f is base-resolvable butnot weakly discontinuous. By Lemma 2.1, f is not scatteredly continuous. Thespace X, being Preiss-Simon, has countable tightness. Then Lemma 2.2 yields anon-empty countable set D ⊂ X such that the restriction f |D has no continuitypoints. Applying Lemma 2.5(3) to the restriction f |D, we can nd a countablerst-countable subset Q ⊂ D without nite open sets such that f |Q is bijectiveand f(Q) is a discrete subspace of Y . It is clear that f |Q has no continuity point.The space X being second countable and without nite open sets, can be writtenas the union Q = Q1 ∪Q2 of two disjoint dense subsets of Q.

110

Let B be a countably additive base for Y such that for every B ∈ B thepreimage f−1(B) is resolvable in X. Since the set f(Q) is countable and discrete,for every x ∈ Q we can select a neighborhood Of(x) ∈ B of f(x) so small that thefamily Of(x) : x ∈ Q is disjoint. The countable additivity of the base B impliesthat for every i ∈ 1, 2 the set Wi =

⋃x∈Qi

Of(x) belongs to B. Consequently,

the preimage f−1(Wi) is a resolvable subset of X and hence Q ∩ f−1(Wi) ⊃ Qi

is a dense resolvable subset of Q. So, the space Q contains two disjoint denseresolvable subsets Q ∩ f−1(W1) and Q ∩ f−1(W2), which contradicts Lemma 2.3.This contradiction completes the proof of the implication (4)⇒ (1).

References

[1] A.V. Arkhangelskii, B.M. Bokalo, Tangency of topologies and tangential properties

of topological spaces, Tr.Mosk.Mat. Ob-va 54 (1992), 160185, (in Russian); Englishtransl.: Trans. Mosk. Math. Soc. 54 (1993),139-163.

[2] T. Banakh, B. Bokalo, On scatteredly continuous maps between topological spaces,Topology Appl. 157:1 (2010), 108122.

[3] T. Banakh, S. Kutsak, V. Maslyuchenko, O. Maslyuchenko, Direct and inverse prob-

lems of the Baire classications of integrals dependent on a parameter, Ukr. Mat.Zhurn. 56:11 (2004), 14431457 (in Ukrainian).

[4] B. Bokalo, O. Malanyuk, On almost continuous mappings, Matem. Studii. 9:1 (1995),9093 (in Ukrainian).

[5] Á. Császár, M.Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar.10 (1975), 463472.

[6] Á. Császár, M.Laczkovich, Some remarks on discrete Baire classes, Acta Math. Acad.Sci. Hungar. 33 (1979), 5170.

[7] J. Jayne, C.A. Rogers, First level Borel functions and isomorphisms, J. Math. PuresAppl.(9) 61:2 (1982), 177205.

[8] O. Karlova, V. Mykhaylyuk, On composition of Baire functions, Topology Appl. 216(2017) 824.

[9] B. Kirchheim, Baire one star functions, Real Analysis Exchange, 18:2 (1992/93),385399.

[10] K. Kuratowski, Topology, I, PWN, Warszawa, 1966.[11] R. O'Malley, Baire∗ 1, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187

192.[12] D. Preiss, P. Simon, A weakly pseudocompact subspace of Banach space is weakly

compact, Comment. Math. Univ. Carol. 15 (1974), 603609.[13] S. Solecki, Decomposing Borel sets and functions and the structure of Baire clas 1

functions, J. Amer. Math. Soc. 11:3 (1998), 521550.[14] V.A. Vinokurov, Strong regularizability of discontinuous functions, Dokl. Akad. Nauk

SSSR 281 (1985), no. 2, 265269 (in Russian).

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 111 125

A study of the quasi covering dimension for nitespaces through the matrix theory

D. N. Georgiou∗†, A. C. Megaritis‡ and F. Sereti

Abstract

We use matrices to study the dimension function dimq, calling quasicovering dimension, for nite topological spaces, which is always greaterthan or equal to the classical covering dimension dim. In particular,we present algorithms in order to compute the dimq(X) of an arbitrarynite topological space X.

Keywords: Covering dimension, quasi covering dimension, quasi cover, denseset.

2000 AMS Classication: 54F45, 65F30

1. Preliminaries and notations

In this section we recall the notion of the topological covering dimension. Werefer to [3, 6] for more details.

A cover of a topological space X is a non-empty set of subsets of X, whoseunion is X. A cover c of X is said to be open (closed) if all elements of c are open(closed). A family r of subsets of X is said to be a renement of a family c ofsubsets of X if each element of r is contained in an element of c.

In what follows, we consider two symbols, −1" and ∞", for which we supposethat:

(1) −1 < k <∞ for every k ∈ 0, 1, . . ..(2) ∞+k = k+∞ =∞, −1+k = k+(−1) = k for every k ∈ 0, 1, . . .∪−1,∞.∗University of Patras, Department of Mathematics, 265 00 Patras, Greece,

Email: georgiou@math.upatras.gr†Corresponding Author.‡Technological Educational Institute of Western Greece, Department of Accounting and Fi-

nance, 302 00 Messolonghi, Greece,Email: thanasismeg13@gmail.com

University of Patras, Department of Mathematics, 265 00 Patras, Greece,Email: seretifot@gmail.com

Doi : 10.15672/HJMS.2016.403

112

We dene the order of a family r of subsets of a space X as follows:

(a) ord(r) = −1 if and only if r consists the empty set only.

(b) ord(r) = k, where k ∈ 0, 1, . . ., if and only if the intersection of any k + 2distinct elements of r is empty and there exist k + 1 distinct elements of r,whose intersection is not empty.

(c) ord(r) = ∞, if and only if for every k ∈ 1, 2, . . . there exist k distinctelements of r, whose intersection is not empty.

We denote by dim the function, calling covering dimension, with domain theclass of all topological spaces and range the set 0, 1, . . .∪−1,∞, satisfying thefollowing conditions:

(1) dim(X) 6 k if and only if for every nite open cover c of the space X thereexists a nite open cover r of X, renement of c, such that ord(r) 6 k.

(2) dim(X) = k, if dim(X) 6 k and dim(X) k − 1.

(3) dim(X) =∞, if dim(X) 6 k does not hold for every k = −1, 0, 1, 2, . . .In study [5], we insert a topological dimension, calling quasi covering dimension

and we prove that it is always greater than or equal to the classical coveringdimension.

1.1. Denition. [5] A quasi cover of X is a non-empty set of subsets of X, whoseunion is dense in X. A quasi cover c of X is said to be open if all elements of c areopen in the space X. Moreover, two quasi covers c1 and c2 are said to be similar(in short c1 ∼ c2) if their unions are the same dense subset of X.

For every topological space X the relation ∼ is an equivalence relation on theset of all quasi covers of X. The collection of all equivalence classes under ∼ willbe denoted by QC(X,∼).1.2. Denition. [5] We denote by dimq the function, calling quasi coveringdimension, with domain the class of all topological spaces and range the set0, 1, . . . ∪ −1,∞, satisfying the following conditions:

(1) dimq(X) 6 k if for every nite open quasi cover c of X there exists a niteopen quasi cover r ofX such that r ∼ c, r is a renement of c, and ord(r) 6 k.

(2) dimq(X) = k if dimq(X) 6 k and dimq(X) 66 k − 1.

(3) dimq(X) =∞ if dimq(X) 6 k does not hold for every k = −1, 0, 1, 2, . . .In this paper we shall consider only nite topological spaces. Let

X = x1, x2, . . . , xnbe a nite topological space and let Ui be the smallest open subset of X whichcontains the point xi, for i = 1, 2, . . . , n. We give some notations which will beused in the rest of our study (see [1, 2]).

The n× n matrix TX = (tij), where

tij =

1, if xi ∈ Uj

0, otherwise

is called the incidence matrix of the space X. We denote by c1, c2, . . . , cn the ncolumns of the matrix TX and by 1 the n × 1 matrix which has all the elements

113

equal to one, that is

1 =

11...1

.

Let i1, i2, . . . , im be distinct elements of the set 1, . . . , n. By ai1i2···im andbi1i2···im we denote respectively the n× 1 matrices

ai1i2···im =

a1i1i2···ima2i1i2···im

...ani1i2···im

and bi1i2···im =

b1i1i2···imb2i1i2···im

...bni1i2···im

,

where

aii1i2···im =

1, if i ∈ i1, i2, . . . , im0, otherwise

and

bii1i2···im =

0, if tii1 = tii2 = . . . = tiim = 0

1, otherwise.

Let

ci =

c1ic2i...cni

and cj =

c1jc2j...cnj

be two n × 1 matrices. Then, by max(ci) we denote the maximum of the setc1i, c2i, . . . , cni and by ci + cj the n× 1 matrix

ci + cj =

c1i + c1jc2i + c2j

...cni + cnj

.

Also, we write ci 6 cj if only if csi 6 csj , for each s = 1, . . . , n.The rest of the paper is organized as follows. In section 2 we give an algorithm

to compute the dimension dimq of a space X through a characterization of openand dense subsets of X. In section 3 we present a new algorithm to compute thedimension dimq using the notion of quasi covers. Finally, in section 4 we presentremarks concerning to this dimension.

2. An algorithm to compute the dimension dimq(X) through a

characterization of open and dense subsets of X

In this section we are going to characterize the open and dense subsets of axed nite topological space X = x1, x2, . . . , xn using matrices.

114

2.1. Proposition. Let i1, . . . , im be distinct elements of the set 1, . . . , n. Then,xi1 , . . . , xim = Uj1 ∪ . . . ∪Ujl , for some j1, . . . , jl ∈ i1, . . . , im if and only ifai1i2···im = bj1j2···jl .

Proof. Let xi1 , . . . , xim = Uj1 ∪ . . . ∪ Ujl , for some j1, . . . , jl ∈ i1, . . . , im.We prove that ai1i2···im = bj1j2···jl . For every i ∈ 1, . . . , n in the i-row of thesematrices we have the following cases:

(1) aii1i2···im = 1⇔ i ∈ i1, . . . , im ⇔ xi ∈ xi1 , . . . , xim⇔ there exists r ∈ 1, . . . , l such that xi ∈ Ujr

⇔ tijr = 1⇔ bij1j2···jl = 1.

(2) aii1i2···im = 0⇔ i /∈ i1, . . . , im ⇔ xi /∈ xi1 , . . . , xim⇔ xi /∈ Ujr , for each r ∈ 1, . . . , l⇔ tijr = 0, for each r ∈ 1, . . . , l ⇔ bij1j2···jl = 0.

We conclude that ai1i2···im = bj1j2···jl .Conversely, assume that ai1i2···im = bj1j2···jl , for some j1, . . . , jl ∈ i1, . . . , im.

We prove that xi1 , . . . , xim = Uj1 ∪ . . . ∪ Ujl . Let i ∈ i1, . . . , im. Then,aii1i2···im = 1. By assumption, bij1j2···jl = 1. Therefore, there exists r ∈ 1, . . . , lsuch that tijr = 1 or equivalently xi ∈ Ujr . Hence, xi1 , . . . , xim ⊆ Uj1∪. . .∪Ujl .Let xi ∈ Uj1 ∪ . . . ∪ Ujl . Then, there exists r ∈ 1, . . . , l such that xi ∈ Ujr

or equivalently tijr = 1. Thus, bij1j2···jl = 1. By assumption, aii1i2···im = 1 and,therefore, xi ∈ xi1 , . . . , xim. Hence, Uj1 ∪ . . . ∪ Ujl ⊆ xi1 , . . . , xim. Thus,xi1 , . . . , xim = Uj1 ∪ . . . ∪Ujl .

2.2. Corollary. Let i1, . . . , im be distinct elements of the set 1, . . . , n. Then,xi1 , . . . , xim = Uir , for some r ∈ 1, . . . ,m if and only if ai1i2···im = cir .

Proof. Follows from Proposition 2.1 and by the fact that bir = cir , for everyr ∈ 1, . . . ,m.

2.3. Proposition. Let j1, . . . , jl be distinct elements of the set 1, . . . , n. Theset Uj1 ∪ . . . ∪Ujl is dense in X if and only if max(bj1j2···jl + cj) = 2, for eachj ∈ 1, . . . , n \ j1, . . . , jl.

Proof. Suppose thatUj1∪. . .∪Ujl is dense inX and let j ∈ 1, . . . , n\j1, . . . , jl.We set k = max(bj1j2···jl + cj) and prove that k = 2. Clearly, k > 0 and bythe denitions of the matrices TX and bj1j2···jl we have that either k = 1 ork = 2. Since Uj1 ∪ . . . ∪Ujl is dense in X, there exists q ∈ 1, . . . , l such thatUjq ∩Uj 6= ∅. Therefore, ti0jq = ti0j = 1, for some i0 ∈ 1, . . . , n, which means

that bi0j1j2···jl + ti0j = 1 + 1 = 2. Thus, k = 2.

Conversely, let max(bj1j2···jl + cj) = 2, for each j ∈ 1, . . . , n \ j1, . . . , jl.We shall prove that the set Uj1 ∪ . . . ∪Ujl is dense in X. Assume that the setUj1 ∪ . . .∪Ujl is not dense in X. Then, there exists an open set U in X such that

U ∩ (Uj1 ∪ . . . ∪Ujl) = ∅.(2.1)

Therefore, there exists µ ∈ 1, . . . , n such that Uµ ⊆ U and xµ /∈ Uj1 ∪ . . .∪Ujl .Hence, µ /∈ j1, . . . , jl. Since max(bj1j2···jl + cµ) = 2, there exists i0 ∈ 1, . . . , n

115

such that bi0j1j2···jl = ti0µ = 1. Thus, xi0 ∈ Ujq ∩Uµ, for some q ∈ 1, . . . , l, whichcontradicts the relation (2.1).

Since for every open subset U = xi1 , . . . , xim of X there exist elementsj1, . . . , jl ∈ i1, . . . , im such that U = Uj1 ∪ . . . ∪ Ujl , from Propositions 2.1and 2.3 we have the following corollary.

2.4. Corollary. Let i1, . . . , im be distinct elements of the set 1, . . . , n. Then,the set xi1 , . . . , xim is open and dense in X if and only if the following conditionshold:

(1) There exist j1, . . . , jl ∈ i1, . . . , im such that ai1i2···im = bj1j2···jl .

(2) max(bj1j2···jl + cj) = 2, for each j ∈ 1, . . . , n \ j1, . . . , jl.2.5. Example. Let X = x1, x2, x3, x4, x5. We consider on X the topologywhich has as a basis the family x1, x1, x2, x1, x3, x1, x4, x1, x3, x4, x5.The incidence matrix TX of X is the 5× 5 matrix

TX =

1 1 1 1 10 1 0 0 00 0 1 0 10 0 0 1 10 0 0 0 1

,

where U1 = x1, U2 = x1, x2, U3 = x1, x3, U4 = x1, x4 and U5 =x1, x3, x4, x5.

For the subset x1 of X we have

a1 =

10000

= b1 = c1.

Hence, this set is open in X and by Corollary 2.2 we have that x1 = U1.Moreover,

b1 + c2 =

21000

, b1 + c3 =

20100

, b1 + c4 =

20010

, b1 + c5 =

20111

.

Therefore, max(b1 + cj) = 2, for j = 2, 3, 4, 5. By the Corollary 2.4 we have thatthe set x1 is open and dense in X.

For the subset x2, x3 of X we have

a23 =

01100

.

116

Since a23 6= b2 =

11000

, a23 6= b3 =

10100

, a23 6= b23 =

11100

, by Proposition

2.1 the set x2, x3 is not open in X.For the subset x1, x3, x4 of X we have

a134 =

10110

= b34.

Hence, this set is open in X and by Proposition 2.1 we have that x1, x3, x4 =U3 ∪U4. Moreover,

b34 + c1 =

20110

, b34 + c2 =

21110

, b34 + c5 =

20221

.

Therefore, max(b34 + cj) = 2, for j = 1, 2, 5. By the Corollary 2.4 we have thatthe set x1, x3, x4 is open and dense in X.

2.6. Proposition. [5] For the space X we have

dimq(X) = maxdim(D) : D is an open and dense subset of X.From Corollary 2.4 we get the following proposition.

2.7. Proposition. The quasi covering dimension dimq(X) is equal to the maxi-mum of all dim(xi1 , . . . , xim) with the properties:

(1) There exist j1, . . . , jl ∈ i1, . . . , im such that ai1i2···im = bj1j2···jl .

(2) max(bj1j2···jl + cj) = 2, for each j ∈ 1, . . . , n \ j1, . . . , jl.In the study [2] it was presented an algorithm of polynomial order for comput-

ing the covering dimension of the space X = x1, . . . , xn. More precisely, thealgorithm consists of the following n− 1 steps:

2.8. Algorithm.

Step 1: Read the n columns c1, . . . , cn of the incidence matrix TX of X. If somecolumn is equal to 1, then print dim(X) = 0. Otherwise, go to Step 2.

Step 2: Find the sums cj11 + cj21 + . . .+ cj(n−1)1, for each

j11, j21, . . . , j(n−1)1 ⊆ 1, . . . , n.If there exists j011, j021, . . . , j0(n−1)1 ⊆ 1, . . . , n such that

cj011 + cj021 + . . .+ cj0(n−1)1

> 1,

then go to Step 3. Otherwise, print

dim(X) = max(c1 + c2 + . . .+ cn)− 1.

117

Step 3: Find the sums cj12 + cj22 + . . .+ cj(n−2)2, for each

j12, j22, . . . , j(n−2)2 ⊆ j011, j021, . . . , j0(n−1)1.If there exists j021, j022, . . . , j0(n−2)2 ⊆ j011, j021, . . . , j0(n−1)1 such that

cj012 + cj022 + . . .+ cj0(n−2)2

> 1,

then go to Step 4. Otherwise, print

dim(X) = max(cj011 + cj021 + . . .+ cj0(n−1)1

)− 1.

. . .

. . .

. . .

Step n− 2: Find the sums cj1(n−3)+ cj2(n−3)

+ cj3(n−3), for each

j1(n−3), j2(n−3), j3(n−3) ⊆ j01(n−4), j02(n−4), j03(n−4), j04(n−4).If there exists j01(n−3), j02(n−3), j03(n−3) ⊆ j01(n−4), j02(n−4), j03(n−4), j04(n−4) suchthat

cj01(n−3)

+ cj02(n−3)

+ cj03(n−3)

> 1,

then go to Step n− 1. Otherwise, print

dim(X) = max(cj01(n−4)

+ cj02(n−4)

+ cj03(n−3)

+ cj04(n−4)

)− 1.

Step n− 1: Find the sums cj1(n−2)+ cj2(n−2)

, for each

j1(n−2), j2(n−2) ⊆ j01(n−3), j02(n−3), j03(n−3).If there exists j01(n−2), j02(n−2) ⊆ j01(n−3), j02(n−3), j03(n−3) such that

cj01(n−2)

+ cj02(n−2)

> 1,

then print

dim(X) = max(cj01(n−2)

+ cj02(n−2)

)− 1.

2.9. Remark. It was proved that an upper bound on the number of iterations ofthe Algorithm 2.8 is 1

2n2 + 3

2n− 3.

Now, we are going to give an algorithm for computing the quasi covering di-mension of the space X = x1, . . . , xn.2.10. Algorithm.

Step 0: Read the n columns c1, . . . , cn of the incidence matrix TX of X.

Step 1: Find k1 = dim(X) (Algorithm 2.8).

Step 2: Find the set P1 of all subsets i11, . . . , i(n−1)1 of 1, . . . , n with theproperties:

(1) There exist j11, . . . , jl1 ∈ i11, . . . , i(n−1)1 such thatai11i21···i(n−1)1

= bj11j21···jl1 .

(2) max(bj11j21···jl1 + cj) = 2, for each j ∈ 1, . . . , n \ j11, . . . , jl1.

118

If P1 = ∅, then put k2 = 0 and go to the step 3. Otherwise, use Algorithm 2.8 tond

k2 = max(dim(xi11 , . . . , xi(n−1)1) : i11, . . . , i(n−1)1 ∈ P1)

and go to the Step 3.

Step 3: Find the set P2 of all subsets i12, . . . , i(n−2)2 of 1, . . . , n with theproperties:

(1) There exist j12, . . . , jl2 ∈ i12, . . . , i(n−2)2 such thatai12i22···i(n−2)2

= bj12j22···jl2 .

(2) max(bj12j22···jl2 + cj) = 2, for each j ∈ 1, . . . , n \ j12, . . . , jl2.If P2 = ∅, then put k3 = 0 and go to the step 4. Otherwise, use Algorithm 2.8 tond

k3 = max(dim(xi12 , . . . , xi(n−2)2) : i12, . . . , i(n−2)2 ∈ P2)

and go to the Step 4.

. . .

. . .

. . .

Step n: Find the set Pn−1 of all subsets i1(n−1) of 1, . . . , n with the propertyai1(n−1)

= bi1(n−1)= ci1(n−1)

. If Pn−1 = ∅, then put kn = 0 and go to the stepn+ 1. Otherwise, use Algorithm 2.8 to nd

kn = max(dim(xi1(n−1)) : i1(n−1) ∈ Pn−1)

and go to the Step n+ 1.

Step n+ 1: Print dimq(X) = maxk1, k2, . . . , kn.2.11. Example. Let X be the space of Example 2.5. We use Algorithm 2.10 tocompute dimq(X).

Step 0. The 5 columns of the incidence matrix TX are

c1 =

10000

, c2 =

11000

, c3 =

10100

, c4 =

10010

, c5 =

10111

.

Step 1. Using Algorithm 2.8 we nd k1 = dim(X) = 1.

Step 2. We have P1 = 1, 2, 3, 4, 1, 3, 4, 5. Using Algorithm 2.8 we nddim(x1, x2, x3, x4) = 2 and dim(x1, x3, x4, x5) = 0. Therefore, k2 = 2.

Step 3. We have P2 = 1, 2, 3, 1, 2, 4, 1, 3, 4. Using Algorithm 2.8 we nddim(x1, x2, x3) = dim(x1, x2, x4) = dim(x1, x3, x4) = 1. Therefore, k3 = 1.

Step 4. We have P3 = 1, 2, 1, 3, 1, 4. Using Algorithm 2.8 we nddim(x1, x2) = dim(x1, x3) = dim(x1, x4) = 0. Therefore, k4 = 2.

Step 5. We have P4 = 1. Using Algorithm 2.8 we nd dim(x1) = 0.Therefore, k5 = 0.

Step 6. Print dimq(X) = maxk1, k2, k3, k4, k5 = 2.

119

3. An algorithm to compute the dimension dimq(X) using the no-

tion of quasi cover

In what follows, we consider a xed nite topological spaceX = x1, x2, . . . , xn.For every c ∈ QC(X,∼) we denote by c(X) the set of all subsets xi1 , . . . , ximof X such that the family Ui1 , . . . ,Uim ∈ c. Also by c we dene a relation onthe set c(X) as follows:

xi1 , . . . , xim1 c xi′1 , . . . , xi′m2

if and only if

Ui1 , . . . ,Uim1 ⊆ Ui′1 , . . . ,Ui′m2

.This relation is a preorder on the set c(X).

3.1. Denition. Let c ∈ QC(X,∼). Every minimum element of (c(X),c) iscalled a c-minimal family.

3.2. Remark. (1) For the nite topological space X and for every c ∈ QC(X,∼)there exist c-minimal families on the set c(X) (see Proposition 3.4).(2) If xi1 , . . . , xim1

and xi′1 , . . . , xi′m2 are two c-minimal families, for some

c ∈ QC(X,∼) then Ui1 , . . . ,Uim1 = Ui′1 , . . . ,Ui′m2

.(3) It is known that a nite space X is T0 if and only if Ui = Uj implies xi = xjfor every i, j. We note that, if the nite space X is T0, then the relation c is anorder. Also, in this case there exists exactly one minimal family on the set c(X).

3.3. Proposition. Let c ∈ QC(X,∼). If the family xi1 , . . . , xim ∈ c(X) isnot a c-minimal family, then there exist i′1, . . . , i

′m−1 ∈ i1, . . . , im such that

xi′1 , . . . , xi′m−1 ∈ c(X).

Proof. Suppose that the family xi1 , . . . , xim ∈ c(X) is not c-minimal. Then,there exists xr1 , . . . , xrµ ∈ c(X) such that xi1 , . . . , xim c xr1 , . . . , xrµ orequivalently Ui1 , . . . ,Uim * Ur1 , . . . ,Urµ. Let α ∈ 1, . . . ,m such thatUiα /∈ Ur1 , . . . ,Urµ. Since Ur1 , . . . ,Urµ ∈ c, there exists β ∈ 1, . . . , µ suchthat xiα ∈ Urβ . By the fact that Uiα is the smallest open set of X containingthe point xiα we have that Uiα ⊆ Urβ . Also, since Uiα /∈ Ur1 , . . . ,Urµ, wehave Uiα 6= Urβ . Therefore, Uiα ⊂ Urβ . Since Ui1 , . . . ,Uim ∈ c, there existsγ ∈ 1, . . . ,m such that xrβ ∈ Uiγ . By the fact that Urβ is the smallest open setof X containing the point xrβ we have that Urβ ⊆ Uiγ . Hence, Uiα ⊂ Uiγ and,therefore, the family Ui1 , . . . ,Uim \ Uiα ∈ c has m− 1 elements. 3.4. Proposition. Let c ∈ QC(X,∼),ν = minm ∈ 1, 2, . . . : there exist j1, . . . , jm such that xj1 , . . . , xjm ∈ c(X),and xj1 , . . . , xjν ∈ c(X). Then, xj1 , . . . , xjν is a c-minimal family.

Proof. Suppose that the family xj1 , . . . , xjν is not c-minimal. By Proposition3.3, there exists an element of c(X) with ν − 1 elements, which is a contradictionby the choice of ν. 3.5. Proposition. Let c ∈ QC(X,∼) and xi1 , . . . , xim be a c-minimal family.If ord(Ui1 , . . . ,Uim) = k > 0, then for every xr1 , . . . , xrµ ∈ c(X) we haveord(Ur1 , . . . ,Urµ) > k.

120

Proof. Let xr1 , . . . , xrµ ∈ c(X). Then, xi1 , . . . , xim c xr1 , . . . , xrµ and,therefore, Ui1 , . . . ,Uim ⊆ Ur1 , . . . ,Urµ. Since ord(Ui1 , . . . ,Uim) = k, wehave ord(Ur1 , . . . ,Urµ > k. 2

3.6. Proposition. Let k ∈ 0, 1, . . .. Then, dimq(X) 6 k if and only if for everyc ∈ QC(X,∼) there exists Ui1 , . . . ,Uim ∈ c such that ord(Ui1 , . . . ,Uim) 6 k.Proof. Let dimq(X) 6 k and c ∈ QC(X,∼). We set

ν = minm ∈ 1, 2, . . . : there exist i1, . . . , im such that Ui1 , . . . ,Uim ∈ cand c = Ui1 , . . . ,Uiν ∈ c. Since dimq(X) 6 k, there exists an open quasi coverr = V1, . . . , Vµ of X such that r ∼ c, r is a renement of c, and ord(r) 6 k. Forthe proof of the proposition it suces to prove that c ⊆ r. We suppose that thereexists α ∈ 1, . . . , ν such that Uiα /∈ r. Since r ∼ c, there exists β ∈ 1, . . . , µsuch that xiα ∈ Vβ . By the fact that Uiα is the smallest open set of X containingthe point xiα we have that Uiα ⊆ Vβ . Also, since Uiα /∈ r, we have Uiα 6= Vβ .Therefore, Uiα ⊂ Vβ . Since r is a renement of c, there exists γ ∈ 1, . . . , ν suchthat Vβ ⊆ Ujγ . Hence,

Uiα ⊂ Uiγ .

We observe that the family c\Uiα ∈ c has ν−1 elements, which is a contradictionby the choice of ν. Thus, c ⊆ r.

Conversely, suppose that for every c ∈ QC(X,∼) there exists Ui1 , . . . ,Uim ∈c such that ord(Ui1 , . . . ,Uim) 6 k. We prove that dimq(X) 6 k. Let c be anarbitrary nite open quasi cover of the space X. Then, there exists c ∈ QC(X,∼)such that c ∈ c. Let r = Ui1 , . . . ,Uim ∈ c such that ord(Ui1 , . . . ,Uim) 6 k.Then, r ∼ c. It suces to prove that the open quasi cover Ui1 , . . . ,Uim of X isa renement of c. Indeed, since r ∼ c, for each q ∈ 1, . . . ,m there exists Vq ∈ csuch that xiq ∈ Vq. Hence, Uiq ⊆ Vq, for every q ∈ 1, . . . ,m.

3.7. Proposition. Let k ∈ 0, 1, . . .. Then, dimq(X) 6 k if and only if forevery c ∈ QC(X,∼) there exists a c-minimal family xj1 , . . . , xjν such thatord(Uj1 , . . . ,Ujν) 6 k.Proof. Let dimq(X) 6 k and c ∈ QC(X,∼). By Proposition 3.6 there existsxi1 , . . . , xim ∈ c(X) with ord(Ui1 , . . . ,Uim) 6 k. Let xj1 , . . . , xjν ∈ c(X)be a c-minimal family (see Proposition 3.4). If ord(Uj1 , . . . ,Ujν) > k, then byProposition 3.5, ord(Ui1 , . . . ,Uim) > k, which is a contradiction. Therefore,ord(Uj1 , . . . ,Ujν) 6 k.

Conversely, suppose that for every c ∈ QC(X,∼) there is a c-minimal familyxj1 , . . . , xjν such that ord(Uj1 , . . . ,Ujν) 6 k. Then, Uj1 , . . . ,Ujν ∈ c andby Proposition 3.6 we have dimq(X) 6 k.

3.8. Proposition. [1] Let ci1 , . . . , cim be m columns of the incidence matrix TXand k = max(ci1 + . . .+ cim). Then, ord(Ui1 , . . . ,Uim) = k − 1.

3.9. Proposition. For every c ∈ QC(X,∼) let xic1 , . . . , xicm ∈ c(X) be a c-minimal family. Then,

dimq(X) = maxmax(cic1 + . . .+ cicm)− 1 : c ∈ QC(X,∼).

121

Proof. Let kc = max(cic1 + . . .+ cicm), for every c ∈ QC(X,∼) andk = maxkc − 1 : c ∈ QC(X,∼).

By Proposition 3.8 we have

ord(Uic1, . . . ,Uicm

) = kc − 1, c ∈ QC(X,∼).(3.1)

Therefore, by Proposition 3.7, dimq(X) 6 k. We prove that dimq(X) = k. Sup-pose that dimq(X) < k. Let c0 ∈ QC(X,∼) such that k = kc0

−1. By Proposition3.6 there exists Ur1 , . . . ,Urµ ∈ c0 such that ord(Ur1 , . . . ,Urµ) < k. By rela-tion (3.1) we have ord(Ui

c01, . . . ,Ui

c0m) = kc0 −1 = k. Therefore, by Proposition

3.5, ord(Ur1 , . . . ,Urµ) > k which is a contradiction. Thus, dimq(X) = k. The proof of the following proposition is a straightforward verication from the

denitions.

3.10. Proposition. The quasi covers Ui1 , . . . , Uik1 and Uj1 , . . . , Ujk2 of Xare similar if and only if bi1i2···ik1 = bj1j2···jk2 .

Using the notion of the quasi cover, Proposition 2.3 can be written as follows.

3.11. Proposition. Let i1, . . . , im be distinct elements of the set 1, . . . , n. Theset Ui1 , . . . ,Uim is a quasi cover of X if and only if max(bi1i2···im + cj) = 2, foreach j ∈ 1, . . . , n \ i1, . . . , im.3.12. Proposition. Let i1, . . . , im be distinct elements of the set 1, . . . , n suchthat max(bi1i2···im + cj) = 2, for each j ∈ 1, . . . , n \ i1, . . . , im. If for everyset i′1, . . . , i′m−1 ⊆ i1, . . . , im we have bi′1i′2···i′m−1

6= bi1i2···im , then the family

xi1 , . . . , xim is a c-minimal family, where Ui1 , . . . ,Uim ∈ c.

Proof. By Proposition 3.11 the set Ui1 , . . . ,Uim is a quasi cover of X. Let cbe the element of QC(X,∼) for which Ui1 , . . . ,Uim ∈ c. Suppose that thefamily xi1 , . . . , xim is not a c-minimal family. By Proposition 3.3, there existi′1, . . . , i

′m−1 ∈ i1, . . . , im such that xi′1 , . . . , xi′m−1

∈ c(X). By Proposition

3.10, bi′1i′2···i′m−1= bi1i2···im which is a contradiction.

The proof of the following proposition is straightforward verication of thePropositions 3.9 and 3.12.

3.13. Proposition. The quasi covering dimension dimq(X) is equal to the max-imum of all max(ci1 + . . .+ cim)− 1 with the properties:

(1) max(bi1i2···im + cj) = 2, for each j ∈ 1, . . . , n \ i1, . . . , im.(2) For every i′1, . . . , i′m−1 ⊆ i1, . . . , im we have bi′1i′2···i′m−1

6= bi1i2···im .

3.14. Algorithm.

Let X = x1, . . . , xn be a nite space. Our intended algorithm contains thefollowing n+ 1 steps:

Step 0. Read the n columns c1, . . . , cn of the matrix TX .

Step 1. Find the set S1 of all i11 ⊆ 1, . . . , n satisfying the property:max(bi11 + cj) = 2, for each j ∈ 1, . . . , n \ i11.If S1 = ∅, then put k1 = 0 and go to the Step 2. Otherwise, put

k1 = maxmax(ci11)− 1 : i11 ∈ S1

122

and go to the Step 2.

Step 2. Find the set S2 of all i12, i22 ⊆ 1, . . . , n satisfying the properties:

(1) max(bi12i22 + cj) = 2, for each j ∈ 1, . . . , n \ i12, i22.(2) For every i′12 ⊆ i12, i22 we have bi′12 6= bi12i22 .

If S2 = ∅, then put k2 = 0 and go to the Step 3. Otherwise, put

k2 = maxmax(ci11 + ci22)− 1 : i11, i12 ∈ S2

and go to the Step 3.

. . .

. . .

. . .

Step n − 2. Find the set Sn−2 of all i1(n−2), . . . , i(n−2)(n−2) ⊆ 1, . . . , nsatisfying the properties:

(1) max(bi1(n−2)i2(n−2)···i(n−2)(n−2)+ cj) = 2, for each

j ∈ 1, . . . , n \ i1(n−2), . . . , i(n−2)(n−2).(2) For every i′1(n−2), . . . , i′(n−3)(n−2) ⊆ i1(n−2), . . . , i(n−2)(n−2) we have

bi′1(n−2)

i′2(n−2)

···i′(n−3)(n−2)

6= bi1(n−2)i2(n−2)···i(n−2)(n−2).

If Sn−2 = ∅, then put kn−2 = 0 and go to the Step n− 1. Otherwise, put

kn−2 = maxmax(ci1(n−2)+. . .+ci(n−2)(n−2)

)−1 : i1(n−2), . . . , i(n−2)(n−2) ∈ Sn−2

and go to the Step n− 1.

Step n − 1. Find the set Sn−1 of all i1(n−1), . . . , i(n−1)(n−1) ⊆ 1, . . . , nsatisfying the properties:

(1) max(bi1(n−1)i2(n−1)···i(n−1)(n−1)+ cj) = 2, for each

j ∈ 1, . . . , n \ i1(n−1), . . . , i(n−1)(n−1).(2) For every i′1(n−1), . . . , i′(n−2)(n−1) ⊆ i1(n−1), . . . , i(n−1)(n−1) we have

bi′1(n−1)

i′2(n−1)

···i′(n−2)(n−1)

6= bi1(n−1)i2(n−1)···i(n−1)(n−1).

If Sn−1 = ∅, then put kn−1 = 0 and go to the Step n. Otherwise, put

kn−1 = maxmax(ci1(n−1)+. . .+ci(n−1)(n−1)

)−1 : i1(n−1), . . . , i(n−1)(n−1) ∈ Sn−1

and go to the Step n.

Step n. If for every i′1n, . . . , i′(n−1)n ⊆ 1, . . . , n we have bi′1ni′2n···i′(n−1)n6= 1,

then put

kn = max(c1 + . . .+ cn)− 1

and go to the Step n+ 1. Otherwise, put kn = 0 and go to the Step n+ 1.

Step n+ 1. Print dimq(X) = maxk1, k2, . . . , kn.

3.15. Example. Let X be the space of Example 2.5. We use Algorithm 3.14 tocompute dimq(X).

123

Step 0. The 5 columns of the incidence matrix TX are

c1 =

10000

, c2 =

11000

, c3 =

10100

, c4 =

10010

, c5 =

10111

.

Step 1. We have S1 = 1, 2, 3, 4, 5 andk1 = maxmax(ci)− 1 : i = 1, . . . , 5 = 0.

Step 2. We have S2 = 2, 3, 2, 4, 2, 5, 3, 4 andmax(c2 + c3)− 1 = max(c2 + c4)− 1 = max(c2 + c5)− 1 = max(c3 + c4)− 1 = 1.

Hence, k2 = 1.

Step 3. We have S3 = 2, 3, 4 and k3 = max(c2 + c3 + c4)− 1 = 2.

Step 4. We have S4 = ∅ and k4 = 0.

Step 5. We have b2345 = 1 and k5 = 0.

Step 6. Print dimq(X) = maxk1, k2, k3, k4, k5 = 2.

4. Remarks on the quasi covering dimension

In this section we present some remarks with respect to quasi covering dimensionand the algorithms of sections 2 and 3.

4.1. Remark. Let A = (αij) be a n×n matrix and B = (βij) be a m×m matrix.The Kronecker product of A and B (see, for instance, [4]) is the mn×mn matrix

A⊗B =

α11B . . . α1nB...

. . ....

αn1B . . . αmnB

.

Let X = x1, x2, . . . , xn and Y = y1, y2, . . . , ym be two nite spaces withincidence matrices TX and TY , respectively. It is known that the incidence matrixof the space X × Y is the kronecker product TX ⊗ TY of TX and TY (see, [7]).

Here, we give an example from which we may conclude that the inequality

dimq(X × Y ) 6 dimq(X) + dimq(Y )

does not hold for every nite topological spaces X and Y .

4.2. Example. Let X = x1, x2, x3 and Y = y1, y2, y3, y4 with the topologies

τX = ∅, x2, x1, x2, x2, x3, Xand

τY = ∅, y1, y1, y2, y1, y3, y1, y4, y1, y2, y3, y1, y2, y4, y1, y3, y4, Y .

124

The incidence matrices of X and Y are

TX =

1 0 01 1 10 0 1

and TY =

1 1 1 10 1 0 00 0 1 00 0 0 1

.

Therefore, the incidence matrix TX×Y of the product space X × Y is

TX×Y = TX ⊗ TY =

1 1 1 1 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 10 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 10 0 0 0 0 0 0 0 1 1 1 10 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

.

In study [1], we have compute that dim(X ×Y ) = 5. Thus, by Proposition 2.6 wehave that dimq(X × Y ) > 5. Also, for the topological spaces X and Y , followingone of the Algorithms 2.10 and 3.14, we have that dimq(X) = 1 and dimq(Y ) = 2.From the above we may conclude that dimq(X × Y ) dimq(X) + dimq(Y ).

4.3. Remark. Let X = x1, x2, . . . , xn be a nite space.

(a) Algorithm 2.10: From the Step 1 up to Step n we appoint all the open anddense subsets xi1 , xi2 , . . . , xim of X and we compute their covering dimen-sions (based on the Algorithm 2.8). So, we have to apply the Algorithm 2.10

(n

n

)+

(n

n− 1

)+ . . .+

(n

2

)+

(n

1

)= 2n − 1 times.

(b) Algorithm 3.14: We do not need to use Algorithm 2.8. From the Step 1 upto Step n we nd all the numbers max(ci1 + . . . + cim) − 1 of the subsetsi1, . . . , im of 1, . . . , n which satisfy the conditions of Proposition 3.13.Therefore, the number of iterations the algorithm performs in Steps 1, 2, . . . , nis 2n − 1.

References

[1] Georgiou, D. N. and Megaritis, A. C. Covering dimension and nite spaces, Applied Math-ematics and Computation, 218 (2011), 31223130.

[2] Georgiou, D. N. and Megaritis, A. C. An algorithm of polynomial order for computing the

covering dimension of a nite space, Applied Mathematics and Computation, 231 (2014),276283.

[3] Engelking, R. Theory of dimensions, nite and innite, Sigma Series in Pure Mathematics,10. Heldermann Verlag, Lemgo, 1995. viii+401 pp.

[4] Eves, H. Elementary matrix theory, Reprint of the 1966 edition, Dover Books on AdvancedMathematics, Dover Publications, Inc., New York, 1980. xvi+325 pp.

125

[5] Georgiou, D. N., Megaritis, A. C. and Sereti, F. A topological dimension greater than or

equal to the classical covering dimension, accepted for publication in Houston Journal ofMathematics.

[6] Pears, A. R. Dimension Theory of General Spaces, Cambridge University Press, Cambridge,England-New York-Melbourne, 1975. xii+428 pp.

[7] Shiraki, M. On nite topological spaces, Rep. Fac. Sci. Kagoshima Univ. 1 1968 18.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 127 147

U(k)- and L(k)-homotopic properties ofdigitizations of nD Hausdor spaces

Sang-Eon Han∗

Abstract

For X ⊂ Rn let (X,EnX) be the usual topological space induced by

the nD Euclidean topological space (Rn, En). Based on the upperlimit (U -, for short) topology (resp. the lower limit (L-, for brevity)topology), after proceeding with a digitization of (X,En

X), we obtaina U - (resp. an L-) digitized space denoted by DU (X) (resp. DL(X))in Zn [16]. Further considering DU (X) (resp. DL(X)) with a digitalk-connectivity, we obtain a digital image from the viewpoint of digitaltopology in a graph-theoretical approach, i.e. Rosenfeld model [25],denoted by DU(k)(X) (resp. DL(k)(X)) in the present paper. Sincea Euclidean topological homotopy has some limitations of studyinga digitization of (X,En

X), the present paper establishes the so calledU(k)-homotopy (resp. L(k)-homotopy) which can be used to study ho-motopic properties of both (X,En

X) and DU(k)(X) (resp. both (X,EnX)

and DL(k)(X)). The goal of the paper is to study some relationshipsamong an ordinary homotopy equivalence, a U(k)-homotopy equiva-lence, an L(k)-homotopy equivalence and a k-homotopy equivalence.Finally, we classify (X,En

X) in terms of a U(k)-homotopy equivalenceand an L(k)-homotopy equivalence. This approach can be used to studyapplied topology, approximation theory and digital geometry.

Keywords: U(k)-digitization, L(k)-digitization, U - and L-localized neighbor-hood, U(k)- and L(k)-homotopy.

2000 AMS Classication: 54A10, 54C05, 55R15, 54C08, 54F65, 68U05, 68U10

∗Department of Mathematics Education, Institute of Pure and Applied Mathematics Chon-buk National University, Jeonju-City Jeonbuk, 54896, Republic of KoreaEmail : sehan@jbnu.ac.krThe author was supported by Basic Science Research Program through the National Re-search Foundation of Korea(NRF) funded by the Ministry of Education, Science andTechnology(2016R1D1A3A03918403).

Doi : 10.15672/HJMS.2016.404

128

1. Introduction

In relation to the digitizations of nD Euclidean spaces [3, 5, 14], the presentpaper uses two kinds of local rules associated with the upper limit (U -, for short)and the lower limit (L-, for brevity) topology [23]. These local rules are used to U -and L-digitize Euclidean nD subspace so that we obtain digital images from theviewpoint of digital topology in the graph-theoretical approach proposed in [25].

Let Z (resp. N) represent the set of integers (resp. natural numbers), and Zn

the set of points in the Euclidean nD space with integer coordinates. In digitaltopology there are several approaches [1, 18, 25, 28] and so forth. Since the paperuses both digital graph theory on Zn and topology on the nD Euclidean space,we need to recall the graph-theoretical approach to digital topology. Rosenfeld[25] introduced a digital image X ⊂ Zn with k-adjacency, denoted by (X, k),and a (k0, k1)-continuous map f : (X, k0) → (Y, k1) of which f maps every k0-connected subset of (X, k0) into a k1-connected subset of (Y, k1). We denote byDTC the category of digital images (X, k) as Ob(DTC) and (k0, k1)-continuousmaps between every pair of digital images (X, k0) and (Y, k1) in Ob(DTC) asMor(DTC) [7, 9].

Let (Rn, En) be the nD real space with Euclidean topology [23]. For X ⊂ Rn

we consider the subspace (X,EnX) induced by (Rn, En). In this paper we denoted

by ETC the category of Euclidean topological spaces [27] consisting of the followingtwo sets:• the set of spaces (X,En

X) as objects, denoted by Ob(ETC);• for every ordered pair of objects (X,En

X) and (Y,EnY ), the set of (Euclidean

topologically) continuous maps as morphisms denoted by Mor(ETC).To digitize (X,En

X) into a space in Zn in a certain digital topological approach,we have often used graph theory and locally nite topological structures and soforth [1, 4, 5, 11, 16, 20, 21, 22, 24, 28]. Hereafter, based on the U -topology andthe L-topology, after proceeding with a digitization of (X,En

X) [16], we obtain aU - (resp. an L-) digitized space denoted by DU (X) (resp. DL(X)) in Zn [16].Further considering DU (X) (resp. DL(X)) with a k-adjacency, we obtain a digitalimage denoted by DU(k)(X) := (DU (X), k) (resp. DL(k)(X) := (DL(X), k)) inthe present paper.

Since we have some diculty in digitizing an ordinary map f ∈ Mor(ETC)(see Lemma 6.1 in the present paper), the present paper develops both a U(k)-map and an L(k)-map and (see Denitions 11 and 12). The present paper provesthat each of these maps is stronger than an ordinary map in ETC (see Lemma6.1) but suitable for digitizing nD Euclidean spaces based on the graph-theoreticalapproach (see Theorem 6.5). Besides, we establish a category, denoted by UDC(resp. LDC), consisting of the sets of subspaces (X,En

X) and U(k)-maps (resp.L(k)-maps) (see Section 5).

Let f : (X,EnX)→ (Y,En

Y ) be a map inMor(ETC). LetDU(k)(f) : DU(k)(X)→DU(k)(Y ) be a k-continuous map induced by the map f (see Denition 11) andlet DL(k)(f) : DL(k)(X) → DL(k)(Y ) be a k-continuous map induced by the mapf (see Denition 12).

129

To study some homotopic properties of among (X,EnX) in Ob(ETC), DU(k)(X)

and DL(k)(X) in Ob(DTC), the present paper develops a U(k)-homotopy in UDC(see Denition 15) and an L(k)-homotopy in LDC (see Denition 16). In relationto these homotopies, we may pose the following queries:Assume two Euclidean topological spaces (X,En

X) and (Y,EnY ). Let DU(k)(X) and

DU(k)(Y ) in Ob(DTC) be their U(k)-digitized spaces (or U(k)-spaces for short)and let DL(k)(X) and DL(k)(Y ) in Ob(DTC) be their L(k)-digitized spaces (orL(k)-spaces for brevity).

Assume that f, g : (X,EnX) → (Y,En

Y ) are homotopic in ETC. Then we havethe following queries (Q1)-(Q2) (see also the properties (4.1) and (4.2) and De-nitions 13, 15, and 16):

(Q1) Are DU(k)(f) and DU(k)(g) k-homotopic ?

(Q2) Are DL(k)(f) and DL(k)(g) k-homotopic ?

Let us investigate homotopic properties of maps inMor(UDC) andMor(LDC).(Q3) In case f, g : (X,En

X) → (Y,EnY ) are U(k)-homotopic in UDC, are

DU(k)(f) and DU(k)(g) k-homotopic ?

(Q4) In case f, g : (X,EnX)→ (Y,En

Y ) are L(k)-homotopic in LDC, areDL(k)(f)and DL(k)(g) k-homotopic ?

More generally, we have the following:(Q5) What are relationships among an ordinary homotopy equivalence in ETC, aU(k)-homotopy equivalence in UDC and an L(k)-homotopy equivalence in LDC ?

The present paper shall address these issues in Sections 4-7. Roughly saying,both the rst and the second question can be answered negatively and both thethird and the fourth question can be answered armatively.

The rest of the paper proceeds as follows: Section 2 provides some basic notionson digital topology and various notions in UDC and LDC. Section 3 investigatessome properties of a U - and an L-local rule of (X,En

X) to establish a local neigh-borhood. Section 4 proposes a U(k)- and an L(k)-digitization of (X,En

X). Section5 develops two maps such as a U(k)-map and an L(k)-map and proves that thesemaps are not compatible with a map in Mor(ETC) but suitable for studying adigitization of a map f ∈ Mor(ETC). Section 6 develops a U(k)-homotopy andan L(k)-homotopy and investigates their properties. Section 7 investigates somerelationships among a homotopy equivalence in ETC, a U(k)-homotopy equiva-lence in UDC and an L(k)-homotopy equivalence in LDC. Section 8 concludesthe paper with a remark.

2. Preliminaries

This section recalls basic notions of the graph-theoretical approach to digitaltopology. A digital picture is usually represented as a quadruple (Zn, k, k,X),where n ∈ N, a black points set X ⊂ Zn is the set of points we regard as belonging

130

to the image depicted, k represents as an adjacency relation for X and k representsan adjacency relation for white points set Zn \X [25]. We say that the pair (X, k)is a digital image in a quadruple (Zn, k, k,X) [25]. Thus, motivated by 4- and8-adjacencies of a 2D digital image and, 6-, 18-, and 26-adjacencies of a 3D digitalimage [20, 25], the k-adjacency relations of Zn can be established to study a multi-dimensional digital image. Indeed, these are induced by the following operator [6](see also [7]): for a natural number m with 1 ≤ m ≤ n, two distinct points

p = (p1, p2, ..., pn), q = (q1, q2, ..., qn) ∈ Zn,

are k(m,n)-(for brevity, k- or km-) adjacent if

at most m of their coordinates dier by ± 1, and all others coincide. (2.1)

The number k of the k(m,n)-adjacency is the number of points q which arek-adjacent to a given point p according to the number m in (2.1) [7] (see also[9], for more details, see [10]). Concretely, the k-adjacency relations of Zn can berepresented, as follows:

k := k(m,n) =

n−1∑

i=n−m2n−iCn

i , where Cni =

n!

(n− i)!i! . (2.2)

For instance, (n,m, k) ∈ (4, 1, 8), (4, 2, 32), (4, 3, 64), (4, 4, 80); (5, 1, 10), (5, 2, 50),(5, 3, 130), (5, 4, 210), (5, 5, 242) [6, 7, 9].Owing to the digital k-connectivity paradox of a digital image (X, k) [20], we re-mind the reader that k 6= k except for the case (Z, 2, 2, X). For a, b ⊂ Z witha < b, [a, b]Z = a ≤ n ≤ b |n ∈ Z is considered in (Z, 2, 2, [a, b]Z) [20]. However,the present paper is not concerned with the k-adjacency of Zn \X. To follow thegraph-theoretical approach to the study of nD digital images [26, 7], we use thek-adjacency relations of Zn (see the property (2.2)), a digital k-neighborhood andso forth [25].

Nk(p) := q | p is k-adjacent to q.Furthermore, we often use the notation [20]

N∗k (p) := Nk(p) ∪ p.We say that two subsets (A, k) and (B, k) of (X, k) are k-adjacent to each other ifA∩B = ∅ and there are points a ∈ A and b ∈ B such that a and b are k-adjacentto each other [20]. We say that a set X ⊂ Zn is k-connected if it is not a union oftwo disjoint non-empty sets that are not k-adjacent to each other [20].

For a k-adjacency relation of Zn, a simple k-path with l + 1 elements in Zn

is assumed to be an injective sequence (xi)i∈[0,l]Z ⊂ Zn such that xi and xj arek-adjacent if and only if | i− j | = 1 [20]. If x0 = x and xl = y, then the length ofthe simple k-path, denoted by lk(x, y), is the number l. A simple closed k-curve

with l elements in Zn, denoted by SCn,lk [20, 6] (see Fig.2(a),(b)), is the simple

k-path (xi)i∈[0,l−1]Z , where xi and xj are k-adjacent if and only if |i−j| = 1(mod l)[20].

For a digital image (X, k), we dene the digital k-neighborhood of x0 ∈ X withradius ε to be the following set [6]: Nk(x0, ε) := x ∈ X | lk(x0, x) ≤ ε ∪ x0,

131

where lk(x0, x) is the length of a shortest simple k-path from x0 to x and ε ∈ N.Concretely, for X ⊂ Zn we obtain [11]

Nk(x, 1) = Nk(x) ∩X. (2.3)

The paper [25] established the notion of digital continuity. Motivated by thiscontinuity, we can represent the digital continuity of maps between digital images,as follows:

2.1. Proposition. [6, 7, 10] Let (X, k0) and (Y, k1) be digital images in Zn0 and

Zn1 , respectively. A function f : X → Y is (k0, k1)-continuous if and only if for

every x ∈ X f(Nk0(x, 1)) ⊂ Nk1(f(x), 1).

In Proposition 2.1 in case n0 = n1 and k0 = k1, we call it k0-continuous.Besides, the digital continuity of Proposition 2.1 has the transitive property.

Since the digital image (X, k) is considered to be a set X ⊂ Zn with one of theadjacency relations of (2.2), we use the terminology a “(k0, k1)-isomorphism" asused in [8] rather than a “(k0, k1)-homeomorphism" as proposed in [2].

2.2. Denition. [2] (see also [8]) For two digital images (X, k0) in Zn0 and (Y, k1)in Zn1 , a map h : X → Y is called a (k0, k1)-isomorphism if h is a (k0, k1)-continuous bijection and further, h−1 : Y → X is (k1, k0)-continuous.

In Denition 1, in case n0 = n1 and k0 = k1, we call it a k0-isomorphism.

3. Some properties of a U and an L-local rule

When digitizing a space (X,EnX) into a digital image, it is required that the

connectedness of the given spaces is preserved (see Lemma 6.4 in the presentpaper). To do this work, this section uses two types of local rules which areused to formulate special kinds of neighborhoods of the given point p ∈ Zn. Andthe structures of the neighborhoods depend on the digital topological structuresrelated to the local rules. The U -topology on R, denoted by (R, EU ), is inducedby the set (a, b] | a, b ∈ R and a < b as a base [23]. Then we obtain the producttopology on Rn, denoted by (Rn, En

U ), induced by (R, EU ). Based on (Rn, EnU ),

we use a U -local rule [16] which is used to digitize (Rn, EnU ) into (Zn, Dn), where

(Zn, Dn) is the discrete topology on Zn.

3.1. Denition. [16] Under (Rn, EnU ), for a point p := (pi)i∈[1,n]Z ∈ Zn we dene

NU (p) := (xi)i∈[1,n]Z |xi ∈ (pi − 12 , pi + 1

2 ] and we call NU (p) the U -localizedneighborhood of p associated with (Rn, En

U ).

For instance, we see NU (p) in Fig.1(b) for a 2D case.

In relation to the digitization of (Rn, EnU ), let us consider the following relation.

3.2. Denition. [16] For two points x, y ∈ Rn, x is related to y if x, y ∈ NU (p)for some point p ∈ Zn, denoted by ‘x ∼U y'. Then we say that (Rn,∼U ) is arelation set associated with (Rn, En

U ).

3.3. Lemma. [16] The relation ‘ ∼U ' of Denition 3 is an equivalence relation.

132

3.4. Remark. [16] Since Rn =⋃

p∈Zn NU (p) and further, for two points p, q in

Zn with p 6= q, NU (p) ∩NU (q) = ∅, we conclude that the set NU (p) | p ∈ Zn isa partition of Rn.

By Lemma 3.1, we conclude that Zn is the space obtained by identifying thepoints of Rn which belong to the same equivalence class of p. Namely, we mayconclude NU (p) = [p], where [p] is the equivalence class of the point p.

Concretely, based on (Rn, EnU ) associated with the U -topology, we can digitize

Rn according to the U -topology in such a way

(Rn, EnU )→ (Zn, Dn) given by NU (p)→ p. (3.1)

It is obvious that the process (3.1) is continuous.

Meanwhile, we may proceed the process of (3.1) in such a way:

(Rn, En)→ (Zn, Dn) given by NU (p)→ p.

Then this process cannot be continuous in topological sense. This approach willbe used in Section 4.

Let us now recall the L-local rule in [16]. The L-topology on R, denotedby (R, EL), is induced by the set of closed open intervals in R, [a, b) | a, b ∈R and a < b, as a base [23]. Then we obtain the product topology on Rn,denoted by (Rn, En

L), induced by (R, EL).Let us consider the L-local rule associated with the L-topology.

3.5. Denition. [16] Under (Rn, EnL), for a point p := (pi)i∈[1,n]Z ∈ Zn we

dene NL(p) := (xi)i∈[1,n]Z |xi ∈ [pi− 12 , pi + 1

2 ). We call NL(p) the L-localizedneighborhood of p associated with (Rn, En

L).

For instance, we see NL(p) in Fig.1(a) for a 2D space.

In relation to the digitization of (Rn, EnL), let us consider the following relation:

3.6. Denition. [16] For two points x, y ∈ Rn, we say that x is related to y iffor some point p ∈ Zn, x, y ∈ NL(p), denoted by ‘x ∼L y'. Then we say that(Rn,∼L) is a relation set associated with (Rn, En

L).

3.7. Lemma. [16] The relation ‘ ∼L' of Denition 5 is an equivalence relation.

3.8. Remark. [16] The set NL(p) | p ∈ Zn is a partition of Rn.

By Lemma 3.3, we observe that the set Zn can be considered on the spaceobtained by identifying the points of Rn which belong to the same equivalenceclass of p. By Lemma 3.3 and Remark 3.4, we may assume NL(p) = [p]. Finally,based on (Rn, En

L), we can digitize Rn according to the L-topology in such a way

(Rn, EnL)→ (Zn, Dn) given by NL(p)→ p. (3.2)

It is obvious that the process (3.2) is continuous.

Meanwhile, we may proceed the process of (3.2) in such a way:

(Rn, En)→ (Zn, Dn) given by NL(p)→ p.

133

p p

(a) N (p)L (b) N (p)

U

P1 - 1

2 P2( , )

P1 -P2( , )1

2

P1+ 1

2 P2( , )

P1 +P2( , )1

2

Figure 1. [16] Conguration of the local rules if the given point pin the 2D Euclidean space in terms of the L-topology (a) and the U -topology (b), where p := (p1, p2)

Then this process cannot be continuous in topological sense. This approach willbe used in Section 4.

4. Establishments of a U(k)- and an L(k)-digitization

This section recalls two types of digitizations associated with the U - and theL-topology. By using the local rule proposed in Denitions 3 and 4, we establishthe following:

4.1. Denition. [16] Let X be a subspace in (Rn, EnU ) (resp. (Rn, En

L)). TheU - (resp. L-) digitization of X, denoted by DU (X) (resp. DL(X)), is dened asfollows:

DU (X) = p ∈ Zn |NU (p) ∩X 6= ∅ ;DL(X) = p ∈ Zn |NL(p) ∩X 6= ∅

with a k-adjacency of Zn of (2.1) depending on the situation.

4.2. Remark. [16] For a set X ⊂ Rn, we say that for two points x, y ∈ X, x is∼U (resp. ∼L) related to y according to U - (resp. L-) topology, as follows:

(1) x ∼U y, if x, y ∈ NU (p) for some point p ∈ Zn such that X ∩ NU (p) 6= ∅.The relation “ ∼U" is an equivalence relation (relative to X).

(2) x ∼L y, if x, y ∈ NL(p) for some point p ∈ Zn such that X ∩ NL(p) 6= ∅.The relation “ ∼L" is an equivalence relation (relative to X).

Motivated by Remark 3.2, we obviously obtain the following:

4.3. Corollary. [16] For a non-empty nD Euclidean space (X,EnX), there is a

partition of Rn associated with the space (X,EnX):

NU (p),Rn \ ∪p∈DU (X)NU (p) | p ∈ DU (X).4.4. Denition. [16] For a space (X,En

X) and two points p, q ∈ X, we say thatthe point p is related to q if there is a point x ∈ DU (X) such that p, q ∈ NU (x). Inthis case we use the notation (p, q) ∈ LX and further, the relation set is denotedby (X,LX).

134

It is clear that the relation LX in the set (X,LX) of Denition 7 is an equivalencerelation [16].

After digitizing X in the U - and the L-topological approach (see Lemmas 3.1and 3.3), we dene the following:

4.5. Denition. (1) We say that DU(k)(X) is the set DU (X) with a k-adjacency.(2) We say that DL(k)(X) is the set DL(X) with a k-adjacency.

Using the local rule in Denition 2, we dene the following:

4.6. Denition. LetDU(k) : (Rn, En)→ (Zn, k) be the map dened byDU(k)(x) =p, where x ∈ NU (p) and the k-adjacency depends on the situation. Then we saythat DU(k) is a U(k)-digitization operator.

Indeed, the U(k)-digitization operator DU(k) is represented as follows: under(Rn, En

U ), for a point x = (x1, ..., xn) ∈ Rn let DU : (Rn, EnU ) → Zn be a map

dened by DU ((xi)i∈[1,n]Z) = (pi)i∈[1,n]Z := p ∈ Zn satisfying that for all i ∈[1, n]Z, xi = pi + di, where − 1

2 < di ≤ 12 (see Lemma 3.1) [16]. Furthermore, on

Zn consider one of the k-adjacency relations of Zn of (2.1). Finally, we obtain thefollowing process.

(Rn, En)→ (Rn, EnU )→ (Zn, Dn)→ (Zn, k) (∗1)

Using the local rule of Denition 4, we dene the following:

4.7. Denition. LetDL(k) : (Rn, En)→ (Zn, k) be the map dened byDL(k)(x) =p, where x ∈ NL(p), p ∈ Zn and the k-adjacency depends on the situation. Thenwe say that DL(k) is an L(k)-digitization operator.

Indeed, an L(k)-digitization operator DL(k) is represented as follows: under(Rn, En

L), for a point x = (x1, ..., xn) ∈ Rn let DL(k) : (Rn, EnL) → (Zn, k) be

the map dened by DL((xi)i∈[1,n]Z) = (pi)i∈[1,n]Z := p ∈ Zn satisfying that for all

i ∈ [1, n]Z, xi = pi + di, where − 12 ≤ di < 1

2 (see Lemma 3.3) [16]. Besides, on Zn

consider one of the k-adjacency relations of Zn of (2.1).Finally, we obtain the following process.

(Rn, En)→ (Rn, EnL)→ (Zn, Dn)→ (Zn, k) (∗2)

For a non-empty set X ⊂ Rn, let us now investigate some properties of a U(k)-and an L(k)-digitization.

The digitizations DU(k)(X) and DL(k)(X) of a Euclidean subspace X are pro-ceeded according to the following algorithms.

Algorithms for the U(k)- and the L(k)-digitizing process from ETC to

DTCFor (X,En

X) ∈ ETC we write the following algorithms for digitizing a space(X,En

X) from ETC to DTC in such two ways [16]:(Case 1): In case of the U -digitization of (X,En

X):(Step 1) Read the points p ∈ DU (X).(Step 2) For each point p ∈ DU (X) take NU (p) ∩X.(Step 3) Put NU (p) ∩X := p.

135

(Step 4) Consider the set DU (X) with a certain k-adjacency so that we obtain(DU (X), k) ∈ DTC.Finally, according to (∗1) and (3.1), we obtain the map DU(k) : ETC → DTCgiven by

DU(k)((X,EnX)) = (DU (X), k) ∈ Obj(DTC) (4.1)

which is called a U(k)-digitization operator for (X,EnX).

(Case 2): In case of the L(k)-digitization of (X,EnX):

(Step 1) Read the points p ∈ DL(X).(Step 2) For each point p ∈ DL(X) take NL(p) ∩X.(Step 3) Put NL(p) ∩X := p.(Step 4) Consider the set DL(X) with a certain k-adjacency so that we obtain(DL(X), k) ∈ DTC.Finally, according to (∗2) and (3.2), we obtain the map DL(k) : ETC → DTCgiven by

DL(k)((X,EnX)) = (DL(X), k) ∈ Obj(DTC) (4.2)

which is called an L(k)-digitization operator for (X,EnX).

4.8. Proposition. For a (X,EnX) in Ob(ETC), DU(k)(X) is dierent from DL(k)(X).

Proof: As shown in Fig.2, given a space (X,EnX) in Ob(ETC), take points

p := (pi)i∈[1,n]Z ∈ Zn such that there is a point x := (xi)i∈[1,n]Z ∈ X satisfying

xi = pi + di, where di = 12 or di = −1

2 . Then, depending on the choice of a U -or an L-local rule of x, the point x is recognized to be a dierent point. HenceDU(k)(X) is dierent from DL(k)(X). 2

4.9. Example. Consider the space (X,E2X) in Fig.2. After obtaining DU(k)(X)

and DL(k)(X), we can see some dierence between them, k ∈ 4, 8

5. Developments of a U(k)- and an L(k)-map

Combining a U -localized neighborhood of Denition 2 with a k-continuousmap, we dene the following map which can be used to study both (X,En

X) andDU(k)(X).

5.1. Denition. Consider the map F : (X,EnX)→ (Y,En

Y ) in ETC.(1) We dene DU(k)(F ) := f : DU(k)(X)→ DU(k)(Y ) given by

for p ∈ DU(k)(X), f maps p to qi,

where qi ∈ Zn | NU (qi) ∩ F (NU (p) ∩X) 6= ∅ ⊂ DU(k)(Y ).

Then DU (F ) is called a U -digitization of F .

(2) DU(k)(F ) := f is a k-continuous map satisfying that for any pointp ∈ DU(k)(X), F (NU (p) ∩X) ⊂ NU (f(p)) ∩ Y.

Then we say that the map F is a lattice-based U(k)-continuous map (a U(k)-map, for short).

136

D (X) L

X

D (X) U

Figure 2. Comparison between DU(k)(X) and DL(k)(X), k ∈ 4, 8.

The paper denotes by UDC the category consisting of the following two sets:(∗ 1) the set of spaces (X,En

X) := X as objects of UDC denoted by Ob(UDC);(∗ 2) the set of U(k)-maps of every ordered pair of elements in Ob(UDC) asmorphisms of UDC denoted by Mor(UDC).

5.2. Example. In Fig.3(a), put X = X1 ∪ X2 ∪ X3, where X1 := (−12 ,−14 ]2,

X2 := (−14 ,14 ]2 and X3 := ( 1

4 ,12 )2. Besides, put Y = Y1 ∪ Y2 ∪ Y3, where Y1 = X1,

Y2 := (−18 ,18 ]2, X3 = Y3 and p = (0, 0). Then consider the map F : (X,E2

X) →(Y,E2

Y ) given by F (p) = p, F (Xi) ⊂ Yi, i ∈ 1, 2, 3. Then F is a U(k)-map,k ∈ 4, 8.

By using the method similar to the establishment of a U(k)-map, we can estab-lish an L(k)-map: Combining an L-localized neighborhood of Denition 4 with ak-continuous map, let us now dene the following map which can be used to studyboth (X,En

X) and DL(k)(X).

5.3. Denition. Consider the map F : (X,EnX)→ (Y,En

Y ) in ETC.(1) We dene DL(k)(F ) := f : DL(k)(X)→ DL(k)(Y ) given by

for p ∈ DL(k)(X), f maps p to qi,

where qi ∈ Zn | NL(qi) ∩ F (NL(p) ∩X) 6= ∅ ⊂ DL(k)(Y ).

Then DL(F ) is called an L-digitization of F .

137

p F

1 X

3 X

2 X

3 Y

1 Y

2 Y X Y p (a)

(b) p G 3 Z

2 Z

3 W

1 W

2 W Z W p

1 Z

Figure 3. (a) Conguration of a U(k)-map in UDC; (b) congurationof an L(k)-map in LDC.

(2) DL(k)(F ) := f is a k-continuous map satisfying that for any pointp ∈ DL(k)(X), F (NL(p) ∩X) ⊂ NL(f(p)) ∩ Y.

Then we say that the map F is a lattice-based L(k)-continuous map (an L(k)-map, for short).

The paper denotes by LDC the category consisting of the following two sets:(∗ 1) the set of spaces (X,En

X) := X as objects of LDC denoted by Ob(LDC);(∗ 2) the set of L(k)-maps of every ordered pair of elements in Ob(LDC) as mor-phisms of LDC denoted by Mor(LDC).

5.4. Example. In Fig.3(b), put Z = Z1 ∪ Z2 ∪ Z3, where Z1 := (−12 ,−14 )2,

Z2 := [−14 ,14 )2 and Z3 := [14 ,

12 )2. Besides, put W = W1 ∪ W2 ∪ W3, where

W1 = Z1, W2 := [−18 ,18 )2, Z3 = W3 and p = (0, 0). Then consider the map

G : (Z,E2Z) → (W,E2

W ) given by G(p) = p,G(Zi) ⊂ Wi, i ∈ 1, 2, 3. Then G isan L(k)-map, k ∈ 4, 8.

Owing to Proposition 4.3, we obtain the following:

5.5. Proposition. For a given map f : (X,EmX ) → (Y,En

X) in Mor(ETC), aU(k)-map is dierent from an L(k)-map.

5.6. Denition. (1) Let f : (X,EmX ) → (Y,En

Y ) be a U(k)-map. Then considera map DU(k)(f) : DU(k)(X) → DU(k)(Y ) induced by the given map f . Then wesay that DU(k)(f) is a U(k)-digitization induced by the map f of Denition 11.

(2) Let f : (X,EmX )→ (Y,En

Y ) be an L(k)-map. Then consider a mapDL(k)(f) :DL(k)(X) → DL(k)(Y ) induced by the given map f . Then we say that DL(k)(f)is an L(k)-digitization induced by the map f of Denition 12.

6. U(k)- and L(k)-homotopic properties in UDC and LDC

This section addresses the questions (Q1) and (Q2) posed in Section 1. Firstof all, let us investigate a relation among f ∈ Mor(ETC), a U(k)-map and anL(k)-map, as follows:

138

6.1. Lemma. A map f ∈Mor(ETC) need not induce a U(k)-map and an L(k)-map.

Proof: By using a counterexample, we prove the assertion (see Fig.4(a)). PutX := (t, 1) | 0 t ≤ 1 ∪ (1, t) | 0 t ≤ 1 and Y := (t, 2) | 0 t ≤ 2 ∪(2, t) | 0 t ≤ 2 (see Fig.4(a)).

Let us consider the map f : (X,E2X)→ (Y,E2

Y ) given by

f((t, 1)) = (4t, 2) if 0 t 1

2,

f((t, 1)) = (2, 2) if1

2≤ t ≤ 1,

f((1, t)) = (2, 4t) if 0 t 1

2,

f((1, t)) = (2, 2) if1

2 t ≤ 1.

Then the map f is a continuous map in Mor(ETC). But it is clear that themap f is neither a U(k)-map nor an L(k)-map, k ∈ 4, 8.

To be specic, based on the given map f , we cannot have its U(k)- and L(k)-maps which are denoted by DU(k)(f) and DL(k)(f) induced by the map f , respec-tively. contrary to the properties of Denitions 11 and 12, respectively. Namely,DU(k)(f) : DU(k)(X)→ DU(k)(Y ) is not a k-continuous map, k ∈ 4, 8;DL(k)(f) : DL(k)(X)→ DL(k)(Y ) is not a k-continuous map, k ∈ 4, 8.

For instance, we observe that DU(k)(f) (resp. DL(k)(f)) is not a U(k)-map (resp.an L(k)-map) at the point (0, 0), k ∈ 4, 8. 26.2. Remark. (1) Unlike the given map f in Lemma 6.1, as shown in Fig.4(c),it is clear that the given map g : Z → W given by g(t) = 2t is a (Euclideantopologically) continuous map, where Z := (0, 14 ) and W := (0, 12 ). But we seethat its digitization DU(2)(g) (resp. DL(2)(g)) is a U(2)-map (resp. an L(2)-map).

(2) By Lemma 6.1 and Proposition 5.3, it turns out that none of a map f ∈Mor(ETC), a U(k)-map in Mor(UDC) and an L(k)-map in Mor(LDC) impliesthe other.

In view of Lemma 6.1, we need to propose a certain homotopy suitable forstudying a U(k)- and an L(k)-digitization. To do this work, rst of all, we needto recall the notion of a k-homotopy [2]. For a space X ∈ Ob(DTC) let B be asubset of X. Then (X,B) is called a digital image pair [7]. Furthermore, if B isa singleton set x0, then (X,x0) is called a pointed digital image in Ob(DTC).To study homotopic properties of DU(k)(X), in this section we use the notionsof a k-homotopy relative to a subset B ⊂ X [10] and a k-homotopy equivalence[6, 15]. Based on the pointed digital homotopy in [2], the following notion of ak-homotopy relative to a subset A ⊂ X is often used in studying a k-homotopicthinning and a strong k-deformation retract of a digital image (X, k) in Zn [9].

6.3. Denition. [9] (see also [10]) Let ((X,A), k0) and (Y, k1) be a digital imagepair and a digital image, respectively. Let f, g : X → Y be (k0, k1)-continuousfunctions. Suppose there exist m ∈ N and a function F : X × [0,m]Z → Y such

139

X Y f

(0, 1)

(2, 2)

(1, 1)

(1, 0) (2, 0)

D (X)

(0, 2) (2, 2)

(1, 1)

(1, 0) (2, 0)

U(4) D (Y)

U(4)

(1,2)

(2, 1)

D (f) U(4)

(a)

(b)

(c) 0

Z W g

0

(0, 2)

(0, 1)

Figure 4. Comparison among a map f ∈ Mor(ETC), a U(k)-mapand an L(k)-map

that• for all x ∈ X,F (x, 0) = f(x) and F (x,m) = g(x);• for all x ∈ X, the induced function Fx : [0,m]Z → Y given byFx(t) = F (x, t) for all t ∈ [0,m]Z is (2, k1)-continuous;• for all t ∈ [0,m]Z, the induced function Ft : X → Y given by Ft(x) = F (x, t) forall x ∈ X is (k0, k1)-continuous.Then we say that F is a (k0, k1)-homotopy between f and g [2].• Furthermore, for all t ∈ [0,m]Z, Ft(x) = f(x) = g(x) for all x ∈ A.Then we call F a (k0, k1)-homotopy relative to A between f and g, and we saythat f and g are (k0, k1)-homotopic relative to A in Y , f '(k0,k1)relA g in symbols.

In Denition 14, if A = x0 ⊂ X, then we say that F is a pointed (k0, k1)-homotopy at x0 [2]. When f and g are pointed (k0, k1)-homotopic in Y , we usethe notation that f '(k0,k1) g. In addition, if k0 = k1 and n0 = n1, then we saythat f and g are pointed k0-homotopic in Y and we use the notation that f 'k0

gand f ∈ [g] which denotes the k0-homotopy class of g.

Based on this digital k-homotopy, to study some relations between DU(k)(X)and (X,En

X) from the viewpoint of homotopy theory, after combining an ordinary

140

homotopy in ETC and a k-homotopy in DTC, we develop the following U(k)-homotopy.

6.4. Denition. Consider (X,EnX) := X and (Y,En

Y ) := Y and (B,EnB) := B

which is a subspace of (X,EnX). Let f, g : X → Y be U(k)-maps. Suppose there

exist m ∈ N and a function F : X × [0,m]Z → Y such that• for all x ∈ X,F (x, 0) = f(x) and F (x,m) = g(x);• for all x ∈ X, the induced function Fx : [0,m]Z → Y given byFx(t) = F (x, t) for all t ∈ [0,m]Z is a U(k)-map;• for all t ∈ [0,m]Z, the induced function Ft : X → Y given by Ft(x) = F (x, t) forall x ∈ X is a U(k)-map.Then we say that F is a U(k)-homotopy between f and g.• Furthermore, for all t ∈ [0,m]Z, assume that Ft(x) = f(x) = g(x) for all x ∈ B.Then we call F a U(k)-homotopy relative to B between f and g, and we say thatf and g are U(k)-homotopic relative to B in Y , f 'U(k)rel.B g in symbol.

To study some relations between DL(k)(X) and (X,EnX) from the viewpoint of

homotopy theory, combining an ordinary homotopy in ETC and a k-homotopy inDTC, we develop the following L(k)-homotopy.

6.5. Denition. Consider (X,EnX) := X and (Y,En

Y ) := Y and (B,EnB) := B

which is a subspace of (X,EnX). Let f, g : X → Y be L(k)-maps. Suppose there

exist m ∈ N and a function F : X × [0,m]Z → Y such that• for all x ∈ X,F (x, 0) = f(x) and F (x,m) = g(x);• for all x ∈ X, the induced function Fx : [0,m]Z → Y given byFx(t) = F (x, t) for all t ∈ [0,m]Z is an L(k)-map;• for all t ∈ [0,m]Z, the induced function Ft : X → Y given by Ft(x) = F (x, t) forall x ∈ X is an L(k)-map.Then we say that F is an L(k)-homotopy between f and g.• Furthermore, for all t ∈ [0,m]Z, assume that Ft(x) = f(x) = g(x) for all x ∈ B.Then we call F an L(k)-homotopy relative to B between f and g, and we say thatf and g are L(k)-homotopic relative to B in Y , f 'L(k)rel.B g in symbol.

Owing to Lemma 6.1 and Remark 6.2, we obtain the following related to thequeries (Q1)-(Q2):

6.6. Proposition. An ordinary homotopy in ETC does not induce a U(k)-homotopy

in UDC and an L(k)-homotopy in LDC

Let us now investigate relations among a U(k)-homotopy, an L(k)-homotopyand a k-homotopy. To do this work, we recall some notions related to a U(k)- andan L(k)-digitization. The paper [16] studied the following:

6.7. Lemma. [16] If (X,EnX) is connected, then both DU(k)(X) and DL(k)(X) are

(3n − 1)-connected.

Let us prove that a U(k)- and an L(k)-homotopy induces a k-homotopy inDTC, as follows:

6.8. Theorem. Consider two U(k)-maps f, g : (X,EnX) → (Y,En

Y ) and their

U(k)-digitizations DU(k)(f), DU(k)(g) : DU(k)(X) → DU(k)(Y ), where (X,EnX)

and (Y,EnY ) are connected. If there is a U(k)-homotopy between f and g, then we

141

obtain a k-homotopy between DU(k)(f) and DU(k)(g) induced by the given U(k)-homotopy.

Proof: Assume a U(k)-homotopy H in UDC between two U(k)-maps f, g :(X,En

X)→ (Y,EnY ), i.e.

H : X × [0,m]Z → Y such that H(x, 0) = f(x) and H(x,m) = g(x)

satisfying the property of Denition 15. By Remark 3.2, we obtain

DU(k)(H) : DU(k)(X)× [0,m]Z → DU(k)(Y ) such that

• for all x′ ∈ DU(k)(X)DU(k)(H)(x′, 0) = DU(k)(f)(x′) and

DU(k)(H)(x′,m) = DU(k)(g)(x′);

• for all x′ ∈ DU(k)(X), the induced function Hx′ : [0,m]Z → DU(k)(Y ) givenby Hx′(t) = H(x′, t) for all t ∈ [0,m]Z is a (2, k)-continuous map;

• for all t ∈ [0,m]Z, the induced function Ht : DU(k)(X)→ DU(k)(Y ) given byHt(x

′) = H(x′, t) for all x′ ∈ DU(k)(X) is a k-continuous map,which implies that H is a k-homotopy between the above k-continuous mapsDU(k)(f) and DU(k)(g). 2

By Denition 16 and Remark 3.4, by using the method similar to Theorem 6.5,we obtain the following:

6.9. Corollary. Consider two L(k)-maps f, g : (X,EnX) → (Y,En

Y ) and their

L(k)-digitizations DL(k)(f), DL(k)(g) : DL(k)(X)→ DL(k)(Y ). If there is an L(k)-homotopy between f and g, then we obtain a k-homotopy between DL(k)(f) and

DL(k)(g) induced by the given L(k)-homotopy.

6.10. Remark. In view of Propositions 5.3 and 6.3, it turns out that none of anordinary homotopy in ETC, a U(k)- and an L(k)-homotopy implies the other.

In view of Theorem 6.5 and Corollary 6.6, we can answer the questions (Q3)-(Q4) armatively. Finally, it turns out that both a U(k)- and an L(k)-homotopycan play an important role in studying both (X,En

X) and its U(k)-digitized spaceDU(k)(X) and its L(k)-digitized space DL(k)(X).

7. A comparison among an ordinary homotopy equivalence, a

U(k)- and an L(k)-homotopy equivalence and a k-homotopy

equivalence

In this section, after proposing the notions of a U(k)- and an L(k)-homotopyequivalence, we compare among an ordinary homotopy equivalence, a U(k)- andan L(k)-homotopy equivalence, and a k-homotopy equivalence.

7.1. Denition. [6](see also [15]) In DTC, for two spaces X and Y , if there arek-continuous maps h : X → Y and l : Y → X such that l h is k-homotopic to 1Xand h l is k-homotopic to 1Y , then the map h : X → Y is called a k-homotopyequivalence. Then we use the notation X 'k·h·e Y .

142

7.2. Theorem. [6] The composition also preserves a k-homotopy equivalence in

DTC. Namely, if X 'k·h·e Y and Y 'k·h·e Z, then X 'k·h·e Z.

Motivated by several types of digital versions of homotopy equivalences in [6,9, 10], let us propose the notion of a U(k)-homotopy equivalence in UDC.

7.3. Denition. For two spaces (X,EnX) and (Y,En

Y ), if there are U(k)-mapsh : X → Y and l : Y → X such that l h is U(k)-homotopic to 1X and h lis U(k)-homotopic to 1Y , then the map h : X → Y is called an U(k)-homotopyequivalence. Then we use the notation X 'U(k)·h·e Y .

7.4. Example. Consider the two spaces (X,E2X) and (Y,E2

Y ) in Fig.5(a) and (b),where p := (−12 ,

32 ) and q := (−12 ,

12 ) in Fig.5. In addition, the spaces X and Y are

assumed to contain the point p and do not have the point q, respectively. Whilethey are quite dierent from each other up to an ordinary homotopy equivalence,they are U(k)-equivalent, k ∈ 4, 8. Indeed, in this case we see DU(k)(X) =DU(k)(Y ).

X

(a) D (X)

U

D (X) L (0,0)

(2,1)

(1, 2)

(1, 2)

(2,1)

(0,0)

(-1,1)

Y

(b)

(0,0)

(2,1)

p p

D (Y) U

D (Y) L

q q

Figure 5. Comparison among a homotopy equivalence in ETC, aU(k)-, an L(k)-and a k-homotopy equivalence.

Comparing a U(k)-homotopy equivalence and an ordinary homotopy equiva-lence in [27], we can observe that a U(k)-homotopy equivalence has some meritsin approximation theory.

7.5. Theorem. The composition also preserves a U(k)-homotopy equivalence in

UDC. Namely, if X 'U(k)·h·e Y and Y 'U(k)·h·e Z, then X 'U(k)·h·e Z.

Proof: It is clear.By using the method similar to that of Denition 18, we now establish the

notion of an L(k)-homotopy equivalence in LDC.

7.6. Denition. For two spaces (X,EnX) and (Y,En

Y ), if there are L(k)-mapsh : X → Y and l : Y → X such that l h is L(k)-homotopic to 1X and h lis L(k)-homotopic to 1Y , then the map h : X → Y is called an L(k)-homotopyequivalence. Then we use the notation X 'L(k)·h·e Y .

7.7. Example. Consider the two spaces (X,E2X) and (Y,E2

Y ) in Fig.5 (a) and(b). While they are quite dierent from each other up to an ordinary homotopyequivalence, they are L(k)-homotopy equivalent, k ∈ 4, 8. Indeed, in this casewe see DL(k)(X) = DL(k)(Y ), k ∈ 4, 8.

143

Let us now compare among an ordinary homotopy equivalence in ETC, a U(k)-and an L(k)-homotopy equivalence.

7.8. Theorem. None of a homotopy equivalence in ETC and a U(k)-homotopy

equivalence in UDC implies the other.

Proof: Consider two Euclidean topological spaces (X,EnX) and (Y,En

Y ) in Fig.6and their U(k)-spaces DU(k)(X) and DU(k)(Y ). In addition, we assume bothX and Y contain the point p and they do not have the point q. Besides, inX, we assume p := (−12 ,

78 ) and q := (−78 ,

12 ); in Y , we assume p := (−12 ,

12 )

and q := (−12 ,−12 ) First of all, by Lemma 6.1 and Remark 6.2, it is clear that

none of a homotopy equivalence between (X,EnX) and (Y,En

Y ) in ETC and ak-homotopy equivalence in DTC implies the other. For instance, consider thespaces (X,E2

X) in Fig.6(a) and (Y,E2Y ) in Fig.6(b). While they are homotopy

equivalent to each other, they are not U(k)-homotopy equivalent, k ∈ 4, 8. Tobe specic, comparing DU (X) in Fig.6(a) and DU (Y ) in Fig.6(b), we obviouslysee that (DU (X), k) in Fig.6(a) is not k-homotopy equivalent to (DU (Y ), k) inFig.6(b), k ∈ 4, 8. Hence the given space (X,En

X) cannot be U(k)-homotopyequivalent to (Y,En

Y ), k ∈ 4, 8.Conversely, consider the spaces (Y,E2

Y ) in Fig.6(b) and (Z,E2Z) in Fig.6(c).

While the U(k)-spaces DU(k)(Y ) in Fig.6(a) and DU(k)(Z) in Fig.6(c) are 8-

homotopy equivalent to each other, it is clear that the space (Y,E2Y ) is not homo-

topy equivalent to (Z,E2Z) in ETC, which means that a U(k)-homotopy equiva-

lence of (Y,E2Y ) and (Z,E2

Z) in UDC does not imply their homotopy equivalencein ETC. 2

Let us now compare between a U(k)-homotopy equivalence in UDC and a k-homotopy equivalence in DTC.

7.9. Theorem. A U(k)-homotopy equivalence between (X,EnX) and (Y,En

Y ) in

UDC implies a k-homotopy equivalence between DU(k)(X) and DU(k)(Y ) in DTC.

Proof: Consider two topological spaces (X,EnX) and (Y,En

Y ) in Ob(UDC) andtheir U(k)-spaces DU(k)(X) and DU(k)(Y ). By Theorem 6.5, we conclude thatan U(k)-homotopy equivalence between (X,En

X) and (Y,EnY ) in UDC implies a

k-homotopy equivalence between DU(k)(X) and DU(k)(Y ) in DTC. 2By the method similar to Theorem 7.5, we obtain the following:

7.10. Corollary. None of a homotopy equivalence in ETC and an L(k)-homotopy

equivalence in LDC implies the other.

By the method similar to Theorem 7.6, we obtain the following:

7.11. Corollary. An L(k)-homotopy equivalence between (X,EnX) and (Y,En

Y ) inLDC implies a k-homotopy equivalence between DL(k)(X) and DL(k)(Y ) in DTC.

In view of Proposition 5.3, we obtain the following:

7.12. Proposition. None of U(k)- and L(k)-homotopy equivalence implies the

other.

Proof: By Proposition 4.3, the proof is trivial. 2

144

(0, 1)

(1,0)

(0, -1)

(-1,0) X

D (X) U

(a)

(0, 1)

(1,0)

(0, -1)

(-1,0)

(0, 1)

(1,0)

(0, -1)

(-1,0) D (X) L

(0, 1)

(2,0)

(0, -1)

(-1,0) Y

D (Y) U

(b)

(0, 1)

(2, 0)

(0, -1)

(-1,0)

(0, 1)

(0, -1)

(-1,0) D (Y) L

p

q

(0, 1)

(2,0)

(1, -1)

(-1,0) z

D (Z) U

(c) (2,0) (-1,0)

D (Z) L (1, -1)

(0, 1)

p

q

(2, 0)

Figure 6. Comparison among the homotopy equivalence in ETC, theU(k)- and L(k)-homotopy equivalence, and the k-homotopy equiva-lence.

7.13. Example. Consider the spaces (Y,E2Y ) in Fig.6(b) and (Z,E2

Z) in Fig.6(c).Then, while (Y,E2

Y ) is U(8)-homotopy equivalent to (Z,E2Z), they are not L(8)-

homotopy equivalent.

7.14. Remark. In view of Theorem 6.5, we obtain the following:(1) the notion of a U(k)-homotopy equivalence in UDC can be used to study both(X,En

X) and its U(k)- space DU(k)(X) from the viewpoint of homotopy theory.(2) In view of Corollary 6.6, the notion of an L(k)-homotopy equivalence in

LDC can be used to study both (X,EnX) and its L(k)-space DL(k)(X) from the

viewpoint of homotopy theory.

8. Summary and further works

Comparing with the usual topology on Rn, we found that the U - and the L-topology has some merits of digitizations of (X,En

X). Thus we have studied vari-ous properties of an L(k)-homotopy and an L(k)-homotopy equivalence. Besides,comparing a Euclidean topological continuous map with an L(k)-map, we observed

145

NO NO

H.E. in ETC --> k-H.E

H.E. in ETC --> U(k)-

H.E

k-H.E. --> H.E. in

ETC

U(k)-H.E --> H.E in

ETC

U(k)-H.E --> k-H.E

NO NO YES

NO NO

H.E. in ETC --> k-H.E

H.E. in ETC --> L(k)-

H.E

k-H.E. --> H.E. in

ETC

L(k)-H.E --> H.E in

ETC

L(k)-H.E --> k-H.E

NO NO YES

Figure 7. Comparison among a homotopy equivalence in ETC, aU(k)- an L(k)-and a k-homotopy equivalence.

that an L(k)-map has strong merits of digitizing (X,EnX). Furthermore, compar-

ing a Euclidean homotopy with both a U(k)-homotopy and an L(k)-homotopy, weconcluded that a U(k)-homotopy and an L(k)-homotopy are suitable homotopiesfor studying both ETC, UDC and LDC. Besides, the paper investigated somerelations between subspaces (X,En

X) and their U(k)-spaces DU(k)(X) in terms ofan U(k)-homotopy equivalence and a k-homotopy equivalence (see Fig.7).

Recently, the paper[13] improved the LMA-map in [14] as follows: Let us nowdevelop the notion of a generalized LMA-map as follows:

8.1. Denition. [13] Consider the map F : (X,E2X)→ (Y,E2

Y ) such thatDMA(F ) :=f : DMA(X) → DMA(Y ) is an MA-map, where DMA(F ) := f is induced by Fsatisfying that for any point p ∈ DMA(X)

F (NM (p) ∩X) ⊂ NM (f(p)) ∩ Y, andf maps p to qi, where

qi ∈ Z2 | NM (qi) ∩ F (NM (p) ∩X) 6= ∅ ⊂ DMA(Y ).

Then we say that the map F is a generalized LMA-map.

It turns out that [13] this version is both a kind of a generalization of an LMA-map in [14] and an improved and corrected version of an LMA-map in [14]. Thusthe LMA-map of the paper [14] can be replaced by the current generalized LMA-map. Hereafter, we will call the map F in Denition 20 an LMA-map instead ofa generalized LMA-map[13].Besides, the paper[13] also improved the LA-map in [12] as follows:

8.2. Denition. [13] Consider the map F : (X,EnX)→ (Y,En

Y ) such thatDKA(F ) :=f : DKA(X)→ DKA(Y ) is an A-map, where DKA(F ) := f is induced by F satis-fying that for any point p ∈ DKA(X)

146

F (NK(p) ∩X) ⊂ NK(f(p)) ∩ Y and

f maps p to qi, where

qi ∈ Zn | NK(qi) ∩ F (NK(p) ∩X) 6= ∅ ⊂ DKA(Y ).

Then we say that the map F is a generalized LA-map.

It turns out that [13] this version is both a kind of a generalization of an LA-map in [12] and an improved and corrected version of an LA-map in [12].Hereafter, we will call the map F in Denition 21 an LA-map instead of a gen-

eralized LA-map[13]. Thus the LA-map of the paper [12] can be replaced by thecurrent generalized LA-map.As a further work, we can compare among digitizations based on several kinds ofdigital topological structures in terms of the above LMA-map, LA-map, U(k)-map, and L(k)-map and further, nd their own features and utilities.

References

[1] P. Alexandor, Diskrete Räume, Mat. Sb. 2 (1937) 501-518.[2] L. Boxer, A classical construction for the digital fundamental group, Jour. of Mathematical

Imaging and Vision 10 (1999) 51-62.[3] V. E. Brimkov and, R. P. Barneva, Plane digitization and related combinatorial problems,

Discrete Applied Mathematics 147 (2005) 169-186.[4] U. Eckhardt, L. J. Latecki, Topologies for the digital spaces Z2 and Z3, Computer Vision

and Image Understanding 90(3) (2003) 295-312.[5] A. Gross and L. J. Latecki, A Realistic Digitization Model of Straight Lines, Computer

Vision and Image Understanding 67(2) (1997) 131-142.[6] S.-E. Han, On the classication of the digital images up to a digital homotopy equivalence,

The Jour. of Computer and Communications Research 10 (2000) 194-207.[7] S.-E. Han, Non-product property of the digital fundamental group, Information Sciences

171(1-3) (2005) 73-91.[8] S.-E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical

Journal 27(1) (2005) 115-129.[9] S.-E. Han, The k-homotopic thinning and a torus-like digital image in Zn, Journal of Math-

ematical Imaging and Vision 31 (1)(2008) 1-16.[10] S.-E. Han, KD-(k0, k1)-homotopy equivalence and its applications, J. Korean Math. Soc. 47

(2010) 1031-1054.[11] S.-E. Han, Homotopy equivalence which is suitable for studying Khalimsky nD spaces,

Topology Appl. 159 (2012) 1705-1714.[12] S.-E. Han, A digitization method of the Euclidean nD space associated with the Khalimsky

adjacency structure, Computational & Applied Mathematics (2015), DOI 10.1007/s40314-015-0223-6 (in press).

[13] S.-E. Han and Sik Lee, Some properties of lattice-based K- and M -maps, Honam Mathe-matical Journal 38(3) (2016) 625-642.

[14] S.-E. Han and Wei Yao, An MA-Digitization of Hausdor spaces by using a connectednessgraph of the Marcus-Wyse topology, Discrete Applies Mathematics, 201 (2016) 358-371.

[15] S.-E. Han and B.G. Park, Digital graph (k0, k1)-homotopy equivalence and its applications,http://atlas-conferences.com/c/a/k/b/35.htm(2003).

[16] J.-M. Kang, S.-E. Han, K.-C. Min, Digitizations associated with several types of digitaltopological approaches, Computational & Applied Mathematics (2015), DOI10.1007/s40314-015-0245-0.

[17] E. Khalimsky, R. Kopperman, P. R. Meyer, Computer graphics and connected topologieson nite ordered sets, Topology and Its Applications 36(1) (1991) 1-17.

[18] E. Khalimsky, Motion, deformation, and homotopy in nite spaces, Proceedings IEEE In-ternational Conferences on Systems, Man, and Cybernetics (1987) 227-234.

147

[19] C. O. Kiselman, Digital geometry and mathematical morphology, Lecture Notes, UppsalaUniversity, Department of Mathematics, available at www.math.uu.se/ kiselman (2002).

[20] R. Klette and A. Rosenfeld, Digital straightness, Discrete Applied Mathematics 139 (2004)197-230.

[21] G. Largeteau-Skapin, E. Andres, Discrete-Euclidean operations, Discrete Applied Mathe-matics 157 (2009) 510-523.

[22] E. Melin, Continuous digitization in Khalimsky spaces, Journal of Approximation Theory150 (2008) 96-116.

[23] James R. Munkres, Topology, Prentice Hall, Inc. (2000).[24] C. Ronse, M. Tajinea, Discretization in Hausdor space, Journal of Mathematical Imaging

and Vision 12 (2000) 219-242.[25] A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), 76-87.[26] A. Rosenfeld, Digital straight line segments, IEEE Trans. Comput, 23(12) (1974) 1264-1269.[27] E. H. Spanier, Algebraic Topology, McGraw-Hill Inc., New York, 1966.[28] F. Wyse and D. Marcus et al., Solution to problem 5712, Amer. Math. Monthly 77(1970)

1119.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 149 162

Ideal convergence in 2-fuzzy 2-normed spaces

Mohammad H.M. Rashid ∗ and Ljubi²a D.R. Ko£inac†

Abstract

In this paper we introduce the notion of I-convergence and I-Cauchyness of sequences in 2-fuzzy 2-normed spaces and establishedsome basic results related to these notions. Further, we dene I-limitand I-cluster points of a sequence in a 2-fuzzy 2-normed linear spaceand investigate the relations between these concepts.

Keywords: 2-fuzzy 2-norm, ideal convergence, ideal Cauchy sequence

2000 AMS Classication: Primary 40A35; Secondary 46A70, 54A20, 54A40

1. Introduction

Convergence (of sequences) is one of the basic and most important concepts inmathematics. It was generalized in several directions.

The notion of statistical convergence of sequences of real numbers was intro-duced independently by H. Fast [20] and H. Steinhaus [42], although the rst ideaof statistical convergence, under the name almost convergence, have appeared in1935 in the rst edition of the famous Zygmund's monograph [46]. It is based onthe notion of asymptotic density of a subset of the set N of natural numbers. ForA ⊂ N and n ∈ N, let A(n) := k ∈ A : k ≤ n and let |A(n)| denote cardinalityof A(n). The asymptotic (or natural) density of A is dened by

δ(A) = limn→∞|A(n)|n .

∗Department of Mathematics & Statistics, Faculty of Science, P.O.Box 7, Mu'tah University,Al-Karak, JordanEmail : malik_okasha@yahoo.com†Faculty of Sciences and Mathematics, University of Ni², P.O. Box 224, Vi²egradska 33, 18000

Ni², SerbiaEmail : lkocinac@gmail.com Corresponding author.

Doi : 10.15672/HJMS.2016.406

150

Statistical convergence has many applications in dierent elds of mathematics(see, for example, [17, 22] and references therein). Let us mention that statisticalconvergence in function spaces was studied in [7, 11].

In 1970, Bernstein [8] introduced convergence of sequences with respect to alter F on N. Using the concept of an ideal, Kostyrko et al. [30] (see also [31])introduced the notion of ideal convergence which is a common generalization ofordinary convergence and statistical convergence. The ideal convergence providesa general framework to study the properties of various types of convergence. LetS be a non empty set. Then a family of sets I ⊂ 2S (2S is the power set of S)is said to be an ideal on S if for each A,B ∈ I we have A ∪ B ∈ I, and for eachA ∈ I and each B ⊂ A, we have B ∈ I. A non empty family of sets F ⊂ 2S issaid to be lter on S if ∅ /∈ F, for each A,B ∈ F we have A ∩B ∈ F and for eachA ∈ F and each B ⊃ A, we have B ∈ F. An ideal I on S is called non-trivialif I 6= ∅ and S /∈ I. It is clear that I ⊂ 2S is an non-trivial ideal on S if andonly if F = F(I) =: S \ A : A ∈ I is a lter on S. A non-trivial ideal I ⊂ 2S

is called an admissible ideal if I ⊃ x : x ∈ S. In this paper we considerthe case S = N. An admissible ideal I ⊂ 2N is said to have the property (AP)[13, 30] if for any sequence A1, A2, · · · of mutually disjoint sets of I, there is asequence B1, B2, · · · of subsets of N such that each symmetric dierence Ai∆Bi

(i = 1, 2, · · · ) is nite and∞⋃

i=1

Bi ∈ I.

We will need the following two lemmas concerning ideals with property (AP).

1.1. Lemma. ([30]) Let Ai∞i=1 be a countable collection of subsets of N suchthat Ai ∈ F(I) for each i, where I is an admissible ideal with the property (AP).Then there exists a set A ⊂ N such that A ∈ F(I) and the set A \ Ai is nite forall i.

1.2. Lemma. ([30]) Let I ⊂ 2N be an admissible ideal with the property (AP)and (X, ρ) be a metric space. Then I- lim

k→∞xk = x0 if and only if there exists a set

P ∈ F(I), P = p1 < p2 < · · · < pk < · · · such that limk→∞

xpk = x0.

On the other hand, the fuzzy theory has emerged as one of the most active areaof research in many branches of mathematics and engineering. This new theorywas introduced by Zadeh [44] in 1965 and since then a large number of researchpapers have appeared by using the concept of fuzzy sets/numbers and fuzzicationof many classical theories has also been made.

The idea of fuzzy norm was initiated by Katsaras [29]. Felbin [21] dened a fuzzynorm on a linear space whose associated fuzzy metric is of Kaleva and Seikkala type[25]. Cheng and Mordeson [12] introduced an idea of a fuzzy norm on a linear spacewhose associated metric is Kramosil and Michalek type [32]. Bag and Samanta in[4] gave a denition of a fuzzy norm in such a manner that the corresponding fuzzymetric is of Kramosil and Michalek type [32]. They also studied some properties ofthe fuzzy norm in [5] and [6]. Bag and Samanta discussed the notions of convergentsequence and Cauchy sequence in fuzzy normed linear space in [4]. They also madein [6] a comparative study of the fuzzy norms dened by Katsaras [32], Felbin [21],and Bag and Samanta [4]. The concept of 2-normed spaces was initially introduced

151

by Gähler [19] in the 1960s. Since then, this concept has been studied by manyauthors (see, for instance, [4, 9, 10, 14, 18, 37]).

Karaku³ et al. [26] dened statistical convergence in intuitionistic fuzzy normedspaces and Mursaleen et al. [34] investigated statistical convergence of double se-quences in intuitionistic fuzzy normed spaces. Quite recently, in [27, 28], Karakayaet al dened and studied statistical and ideal convergence of sequences of functionsin intuitionistic fuzzy normed spaces. The concept of statistical limit inferior andlimit superior of sequences of fuzzy numbers found in [2, 3]. Let us mention thatstatistical and ideal convergence have been studied in fuzzy context (relations withfuzzy numbers and fuzzy normed linear spaces) in many papers (see, for example,[1, 23, 24, 33, 36, 3841]).

In this paper we investigate ideal convergence in the fuzzy settings, more pre-cisely in 2-fuzzy 2-normed spaces. The paper is organized as follows: In the sec-ond section, we present some preliminary denitions and results related to fuzzynormed spaces and 2-fuzzy 2-normed spaces. In the third section, we introduce thenotion of I-convergent sequence and I∗-convergence in a 2-fuzzy 2-normed spaceand some basic results are obtained. In fourth section, we introduce the notion ofI-Cauchy and I∗-Cauchy sequences in a 2-fuzzy 2-normed space. In Section 5 theconcepts of I-limit points and I-cluster points of a sequence in a 2-fuzzy 2-normedspace are dened and relations between these concepts are investigated.

2. Denitions and preliminaries

By R we denote the set of real numbers. All linear spaces are assumed to beover R.

For the sake of completeness, we reproduce some denitions due to Gähler [19],Bag and Samanta [4], Somasundaram and Beaula [43], and Zhang [45].

2.1. Denition. ([19]) Let X be a real linear space of dimension s, where 2 ≤s <∞. A 2-norm on X is a function ‖., .‖ : X ×X → R which satises

(i) ‖x, y‖ = 0 if and only if x and y are linearly dependent;(ii) ‖x, y‖ = ‖y, x‖ for all x, y ∈ X;(iii) ‖cx, y‖ = |c| ‖x, y‖ for all x, y ∈ X and c ∈ R ;(iv) ‖x+ y, z‖ ≤ ‖x, z‖+ ‖y, z‖ for all x, y, z ∈ X.

The pair (X, ‖., .‖) is then called a 2-normed space.

An example of a 2-normed space is the set X = R2 equipped with the 2-norm

‖x, y‖ = |x1y2 − x2y1|, x = (x1, x2), y = (y1, y2),

i.e. ‖x, y‖ is the area of the parallelogram spanned by the vectors x and y.

2.2. Denition. ([4]) Let X be a linear space over R. A fuzzy subset N of X×Ris called a fuzzy norm on X if for all x, y ∈ X and c ∈ R.(FN1) For all t ∈ R with t ≤ 0, N(x, t) = 0;(FN2) for all t ∈ R with t > 0, N(x, t) = 1, if and only if x = 0;(FN3) for all t ∈ R with t > 0, N(cx, t) = N(x, t/|c|), if c 6= 0,(FN4) for all s, t ∈ R, x, y ∈ X, N(x+ y, s+ t) ≥ minN(x, s), N(y, t),(FN5) N(x, .) is a non decreasing function of R and lim

t→∞N(x, t) = 1.

152

The pair (X,N) will be referred to as a fuzzy normed linear space.

The following denition is actually a decomposition theorem from [4].

2.3. Denition. ([4]) Let (X,N) be a fuzzy normed linear space. Assume furtherthat

(FN6) N(x, t) > 0 for all t > 0 implies x = 0.

Dene ‖x‖α = inft : N(x, t) ≥ α, α ∈ (0, 1). Then ‖.‖α : α ∈ (0, 1) is anascending family of norms on X or α-norms on X corresponding to the fuzzynorm on X.

2.4. Denition. ([45]) Let X be a non-empty set and F(X) be the set of all fuzzysets on X. For U, V ∈ F(X) and k ∈ R dene

U + V = (x+ y, λ ∧ µ) : (x, λ) ∈ U, (y, µ) ∈ V ,and kU = (kx, λ) : (x, λ) ∈ U.

2.5. Denition. ([43]) A fuzzy linear space X = X × (0, 1] over R where theaddition and scalar multiplication operation on X are dened by (x, λ) + (y, µ) =

(x + y, λ ∧ µ), k(x, λ) = (kx, λ) is a fuzzy normed space if for every (x, λ) ∈ Xthere is associated a non-negative real number, ‖(x, λ)‖ , called the fuzzy norm of(x, λ), in such a way that

(1) ‖(x, λ)‖ = 0 ⇐⇒ x = 0 the zero element of X, λ ∈ (0, 1],

(2) ‖k(x, λ)‖ = |k| ‖(x, λ)‖ for all (x, λ) ∈ X and all k ∈ R,(3) ‖(x+ y, λ+ µ)‖ ≤ ‖(x, λ+ µ)‖+ ‖(y, λ+ µ)‖ for all (x, λ), (y, µ) ∈ X,(4) ‖(x,∨tλt)‖ = ∧t ‖(x, λt)‖ for λt ∈ (0, 1].

2.6. Denition. ([43]) Let X be a non-empty and F(X) be the set of all fuzzysets in X. If f ∈ F(X), then f = (x, µ) : x ∈ X and µ ∈ (0, 1]. Clearly f is abounded function for |f(x)| ≤ 1. F(X) is a linear space over the eld R, where theaddition and scalar multiplication are dened by

f + g = (x, µ) + (y, ν) = (x+ y, µ ∧ ν) : (x, µ) ∈ f and (y, ν) ∈ gand

kf = (kx, µ) such that (x, µ) ∈ fwhere k ∈ R.

The linear space F(X) is said to be a normed space if for every f ∈ F(X) thereis associated a non-negative real number ‖f‖ called the norm of f in such a waythat

(1) ‖f‖ = 0 if and only if f = 0.(2) ‖kf‖ = |k| ‖f‖, k ∈ R.(3) ‖f + g‖ ≤ ‖f‖+ ‖g‖ for every f, g ∈ F(X).

Then (F(X), ‖.‖) is a normed linear space.

2.7. Denition. ([43]) A 2-fuzzy set on X is a fuzzy set on F(X).

2.8. Denition. ([43]) Let F(X) be a linear space over R. A fuzzy subset N ofF(X)× F(X)× R is called a 2-fuzzy 2-norm on X (or fuzzy 2-norm on F(X)) if

(F2N1) for all t ∈ R with t ≤ 0 N(f1, f2, t) = 0;

153

(F2N2) for all t ∈ R with t > 0, N(f1, f2, t) = 1, if and only if f1 and f2 arelinearly dependent;

(F2N3) N(f1, f2, t) is invariant under any permutation of f1, f2;(F2N4) for all t ∈ R, with t > 0, N(f1, cf2, t) = N(f1, f2, t/|c|) if c 6= 0, c ∈ R;(F2N5) for all s, t ∈ R, N(f1, f2 + f3, s+ t) ≥ minN(f1, f2, s), N(f1, f3, t);(F2N6) N(f1, f2, .) : (0,∞)→ [0, 1] is continuous,(F2N7) lim

t→∞N(f1, f2, t) = 1.

Then (F(X), N) is a fuzzy 2-normed linear space or (X,N) is a 2-fuzzy 2-normedlinear space.

2.9. Lemma. ([43, Theorem 3.2]) Let (F(X), N) be a fuzzy 2-normed linear space.Assume that

(F2N8) N(f1, f2, t) > 0 for all t > 0 implies f1 and f2 are linearly dependent,

dene‖f1, f2‖α = inft : N(f1, f2, t) ≥ α, α ∈ (0, 1).

Then ‖., .‖α : α ∈ [0, 1] is an ascending family of 2-norms on F(X). These2-norms are called α-2-norms on F(X) corresponding to the fuzzy 2-norms.

3. I-convergence in 2-fuzzy 2-normed spaces

In this section we introduce the notion of I-convergence and I∗-convergenceof sequences in a 2-fuzzy 2-normed space X, i. e. in a fuzzy 2-normed space(F(X), N), and present some basic results.

3.1. Denition. Let (F(X), N) be fuzzy 2-normed linear space. A sequence fkin F(X) is said to be I-convergent to f in F(X) with respect to the α-2-norms onF(X) if for each ε > 0, α ∈ [0, 1] and each g ∈ F(X), the set A(ε) = k ∈ N :

‖fk − f, g‖α ≥ ε belongs to I. In this case we write fkI−→ f . The element f is

called the I-limit of fk in F(X).

The usual interpretation of the above denition is the following:

fkI−→ f ⇐⇒ I- lim

k→∞‖fk − f, g‖α = 0, for all g ∈ F(X) and α ∈ [0, 1].

3.2. Lemma. Let (F(X), N) be fuzzy 2-normed linear space and I be an admis-sible ideal of N. If a sequence fk in F(X) is I-convergent with respect to theα-2-norm on F(X), then I-limit is unique.

Proof. Suppose that fkI−→ f and fk

I−→ g and f 6= g. Since ‖., .‖α is an α-2-norm,we get for each h ∈ F(X),

(3.1)

‖f − g, h‖α = ‖fk − fk + f − g, h‖α ≤ ‖fk − f, h‖α+‖fk − g, h‖α , for all k ∈ N.Put

A(ε) = k ∈ N : ‖f − g, h‖α ≥ ε,B(

ε

2) = k ∈ N : ‖fk − f, h‖α ≥

ε

2,

C(ε

2) = k ∈ N : ‖fk − g, h‖α ≥

ε

2.

154

By assumption, we get that B( ε2 ) and C( ε2 ) belong to I, so we have B( ε2 )∪C( ε2 ) ∈ I.From (3.1), it follows that A(ε) ⊆ B( ε2 )∪C( ε2 ). This implies that A(ε) ∈ I, whichachieves the proof.

3.3. Theorem. Let (F(X), N) be a fuzzy 2-normed linear space and I be anadmissible ideal of N. Let fk and hk be two sequences in F(X) such that

fkI−→ f and hk

I−→ h, where f, h ∈ F(X). Then

(i) fk + hkI−→ f + h;

(ii) fkhkI−→ fh;

(iii) cfkI−→ cf for c ∈ R.

Proof. (i) Suppose that fkI−→ f and hk

I−→ h. Since ‖., .‖α is an α-2-norm, we getfor each ε > 0, g ∈ F(X) and α ∈ [0, 1],

(3.2) ‖(fk + hk)− (f + h), g‖α ≤ ‖fk − f, g‖α + ‖hk − h, g‖α , for all k ∈ N.

Put

A(ε) = k ∈ N : ‖(fk + hk)− (f + h), g‖α ≥ ε,B(

ε

2) = k ∈ N : ‖fk − f, g‖α ≥

ε

2,

C(ε

2) = k ∈ N : ‖hk − h, g‖α ≥

ε

2.

By assumption, B( ε2 ) and C( ε2 ) belong to I, and thus B( ε2 ) ∪ C( ε2 ) ∈ I. From(3.2), it follows that A(ε) ⊆ B( ε2 ) ∪ C( ε2 ). This implies that A(ε) ∈ I.

(ii) Since fkI−→ f , we have

A(1) = k ∈ N : ‖fk − f, g‖α < 1 ∈ F(I).

Now being α-2-norm, we get

‖fkhk − fh, g‖α ≤ ‖fk, g‖α ‖hk − h, g‖α + ‖h, g‖α ‖fk − f, g‖α .For k ∈ A(1), we have ‖fk, g‖α ≤ ‖f‖α + 1 and it follows that

(3.3) ‖fkhk − fh, g‖α ≤ (‖f‖α + 1) ‖hk − h, g‖α + ‖h, g‖α ‖fk − f, g‖α .Let ε > 0 and g ∈ F(X) be given. Choose η > 0 such that

(3.4) 0 < 2η <ε

‖f‖α + ‖h‖α + 1.

Since fkI−→ f and hk

I−→ h, the sets

B(η) = k ∈ N : ‖fk − f, g‖α < η and C(η) = k ∈ N : ‖hk − h, g‖α < η.belong to F(I). Thus we have B(η), C(η) ∈ F(I).

Obviously, A(1) ∩B(η) ∩C(η) ∈ F(I) and for each k ∈ A(1) ∩B(η) ∩C(η), wehave from (3.3) and (3.4),

‖fkhk − fh, g‖α < ε.

This implies that k ∈ N : ‖fkhk − fh, g‖α ≥ ε ∈ I, i.e., fkgkI−→ fh.

155

(iii) It is trivial for c = 0. Now let c 6= 0, ε > 0, α ∈ [0, 1] and g ∈ F(X). Since

‖., .‖α is an α-2-norm, we get ‖cf, g‖α = |c| ‖f, g‖. Since fk I−→ f , therefore the set

A = k ∈ N : ‖fk − f, g‖α ≥ εbelongs I. Let B(ε) = k ∈ N : ‖cfk − cf, g‖α ≥ ε. We need to show that B(ε)is contained in A(ε1), for some ε1 > 0. Let m ∈ B(ε); then ε ≤ ‖cfm − cf, g‖α =|c| ‖fm − f, g‖α. This implies that ‖fm − f, g‖α ≥ ε

|c| = ε1. Therefore, m ∈ A(ε1).

Then we have B(ε) ⊂ A(ε1). By the denition of I, we get B(ε) ∈ I.The theorem is proved.

3.4. Theorem. Let I be an admissible ideal with the property (AP). Let (F(X), N)be a fuzzy 2-normed space and fk be a sequence in F(X). Then fk is an I-convergent sequence in F(X) if and only if there is a sequence hk converging tof and such that k ∈ N : fk 6= hk ∈ I.

Proof. Suppose fkI−→ f . For each k ∈ N and each g ∈ F(X), let

An = k ∈ N : ‖fk − f, g‖α <1

n,

Then An ∈ F(I) for each n ∈ N.Since I is admissible ideal with the property (AP), by Lemma 1.1, there is

A ⊂ N such that A ∈ F(I) and the set A \ An is nite for each n. Observe thatfk →(A) f , i.e., for each ε > 0, there exists n0 = n0(ε) ∈ N such that k ≥ n0 andk ∈ A imply ‖fk − f, g‖α < ε.

Dene a sequence hk in F(X) as

hk =

fk, for k ∈ A;f, for k ∈ N \A.

The sequence hk is convergent to f with respect to the α-2-norm on F(X). Thuswe have k ∈ N : fk 6= hk ∈ I.

Conversely, suppose that k ∈ N : fk 6= hk ∈ I and hk → f . Let ε > 0 andg ∈ F(X) be given. Then for each n, we can write

(3.5) k ≤ n : ‖fk − f, g‖α ≥ ε ⊆ k ≤ n : fk 6= hk∪k ≤ n : ‖hk − f, g‖α > ε.Since the rst set on the right side of (3.5) belongs to I and the second set iscontained in a nite subset of N, it belongs to I. This implies that k ∈ N :‖fk − f, g‖α ≥ ε belongs to I. This achieves the proof.

Now we prove a decomposition theorem for I-convergent sequences.

3.5. Theorem. Let fk be a sequence in a fuzzy 2-normed space (F(X), N) andlet I be an admissible ideal with the property (AP). Then the following assertionsare equivalent:

(i) fkI−→ f ;

(ii) There exist hk and qk in F(X) such that fk = hk + qk, hk → h, andsupp(qk) = k ∈ N : qk 6= θ ∈ I, where θ is the zero element of the linearspace F(X).

156

Proof. (i)=⇒ (ii) Suppose fkI−→ f . Then, by Lemma 1.2, we there is a set H ∈

F(I), H = km : k1 < k2 < · · · ⊂ N such that fkm → f .Dene the sequence hk in F(X) as

(3.6) hk =

fk, for k ∈ H;f, for k ∈ N \H.

It is clear that hk → f . Further, we set qk = fk − hk, for each k ∈ N. Sincek ∈ N : fk 6= hk ∈ N \ H ∈ I, we have k ∈ N : qk 6= θ ∈ I. It follows thatsupp(qk) ∈ I and by (3.6), we get fk = hk + qk.

(ii)=⇒ (i) Suppose that there exist two sequences hk and qk in F(X) such

that fk = hk+qk; hk → f and supp(qk) ∈ I. We prove that fkI−→ f. Let H = km

be a subset of N such that H = k ∈ N : qk = θ. Since supp(qk) = m ∈ N : qm 6=θ ∈ I, we have H ∈ F(I), therefore fk = hk, if k ∈ H. Thus, we conclude thatthere exists a set H = km : k1 < k2 < · · · ⊂ N, H ∈ F(I), such that fkm → f .

By Lemma 1.2, it follows that fkI−→ f .

3.6. Denition. Let (F(X), N) be a fuzzy 2-normed space. We say that a se-quence fk in F(X) is I∗-convergent to f ∈ F(X) with respect to the α-2-normon F(X) if there exists a subset

K = km : k1 < k2 < · · · ⊂ N

such that K ∈ I and limm→∞ ‖fkm − f, g‖α = 0 for each g ∈ F(X).

In this case we write fkI∗−→ f .

3.7. Theorem. Let (F(X), N) be a fuzzy 2-normed space and I be an admissible

ideal. If fkI∗−→ f , then fk

I−→ f .

Proof. Suppose that fkI∗−→ f . Then, by denition, there exists

K = km ∈ N : k1 < k2 < · · · ∈ F(I)

such that

(3.7) limm→∞

‖fkm − f, g‖α = 0, for all g ∈ F(X).

Let ε > 0 and g ∈ F(X) be given. By (3.7, there exists an integer n0 ∈ N suchthat ‖fkm − f, g‖α < ε for every km ∈ K,km ≥ n0.

Let A = k1, k2, · · · , kn0. Since K ∈ F(I), there exists a set B ∈ I such that

K = N \B. It is clearA1(ε) = k ∈ N : ‖fk − f, g‖α ≥ ε ⊆ A ∪B.

As I is admissible ideal, A ∈ I. This implies that A ∪ B ∈ I and so A1(ε) ∈ I.

Thus we have fkI−→ f . This completes the proof.

3.8. Theorem. Let I be an admissible ideal with the property (AP) and (F(X), N)

be a fuzzy 2-normed space and fk be a sequence in F(X). Then fkI−→ f if and

only if fkI∗−→ f.

157

Proof. If fkI∗−→ f, then fk

I−→ f by Theorem 3.7.

Conversely, let fkI−→ f . Then by denition, for each ε > 0 and each g ∈ F(X),

there exists an integer n = n(ε) such that

B(ε) = k ∈ ‖fk‖α ≥ ε ∈ I.

For m ∈ N, we dene the set Pm as follows:

P1 = k ∈ N : ‖fk − f, g‖α ≥ 1,

Pm = k ∈ N :1

m≤ ‖fk − f, g‖α <

1

m− 1, for m ≥ 2 in N.

It is clear that P1, P2, · · · is a countable family of mutually disjoint sets be-longing to I. Then by the property (AP) of I, there is a countable family ofsets Q1, Q2, · · · in I such that Pj∆Qj is a nite set for each j ∈ N and Q =⋃∞j=1Qj ∈ I. Since Q ∈ I, we have B = N \ Q ∈ F(I). To prove the result it is

sucient to show that fk →(B) f . Let ξ > 0 be given. Choose an integer p such

that ξ > 1p+1 . Thus, we have

(3.8) k ∈ N : ‖fk − f, g‖α ≥ ξ ⊂ k ∈ N : ‖fk − f, g‖α ≥1

p+ 1 =

p+1⋃

m=1

Pm.

Since Pm∆Qm is a nite set for eachm = 1, · · · , p+1, therefore there exists k0 ∈ Nsuch that(

p+1⋃

m=1

Qm

)∩ k ∈ N : k ≥ k0 =

p+1⋃

m=1

Pm ∩ k ∈ N : k ≥ k0.

If k ≥ k0 and k ∈ Q, then k /∈p+1⋃

m=1

Qm and so k /∈p+1⋃

m=1

Pm. Thus for every k ≥ k0

and k ∈ B, from (3.8), we get ‖fk − f, g‖α < ξ. This shows fk →(B) f whichcompletes the proof.

The proof of the following theorem follows from the decomposition theorem(Theorem 3.5).

3.9. Theorem. Let fk be a sequence in a fuzzy 2-normed space (F(X), X) andI be an admissible ideal. If there exist two sequences hk and qk in F(X) such

that fk = hk + qk; hk → f and supp(qk) = k ∈ N : qk 6= θ ∈ I, then fkI∗−→ f .

4. I-Cauchy and I∗-Cauchy sequences in 2-fuzzy 2-normed spaces

In this section we study the concepts of I-Cauchy and I∗-Cauchy sequences infuzzy 2-normed spaces (F(X), N). Also, we will study the relations between theseconcepts. For statistical Cauchy sequences and I-Cauchy sequences see [15, 16, 35].

4.1. Denition. Let (F(X), N) be a fuzzy 2-normed space and I be an admissibleideal of N. A sequence fk in F(X) is said to be I-Cauchy with respect to theα-2-norm on F(X) if for each ε > 0 there exist a positive integer n = n(ε) andg, h ∈ F(X) which are linearly independent such that k ∈ N : ‖fk − fn, g‖α ≥ εand k ∈ N : ‖fk − fn, h‖α ≥ ε belong to I.

158

4.2. Denition. Let (F(X), N) be a fuzzy 2-normed space and I be an admissibleideal of N. A sequence fk in F(X) is said to be I∗-Cauchy with respect to theα-2-norm on F(X) if there exists a set

K = km : k1 < k2 < · · · ⊂ N

such that K ∈ F(I) and fkm is an ordinary Cauchy sequence in F(X).

The next theorem gives that each I∗-Cauchy sequence is I-Cauchy sequence.

4.3. Theorem. Let I be an admissible ideal and (F(X), N) be a fuzzy 2-normedspace. Then every I-convergent sequence is an I-Cauchy sequence.

Proof. Let fkI−→ f . Then for each ε > 0 and each φ ∈ F(X), we have

A(ε) = k ∈ N : ‖fk − f, φ‖α ≥ ε ∈ I.

Since I is an admissible ideal, there exists an k0 ∈ N such that k0 /∈ A(ε).Let B(ε) = k ∈ N : ‖fk − fk0 , φ‖α ≥ 2ε. Since ‖., .‖α is an α-2-norm, we get

‖fk − fk0 , φ‖α ≤ ‖fk − f, φ‖α + ‖fk0 − f, φ‖α .We observe that if k ∈ B(ε), then ‖fk − f, φ‖α + ‖fk0 − f, φ‖α ≥ 2ε.

On the other hand, since k0 /∈ A(ε), we have

‖fk0 − f, φ‖α < ε.

So we conclude that ‖fk − f, φ‖α ≥ ε, hence k ∈ A(ε).This implies that B(ε) ⊂ A(ε), for each ε > 0. This gives B(ε) ∈ I. Since

φ ∈ F(X) was arbitrary we can take g, h ∈ F(X) which are linearly independent,such that the sets k ∈ N : ‖fk − fk0 , g‖α ≥ 2ε and k ∈ N : ‖fk − fk0 , h‖α ≥ 2εbelong to I, i.e., fk is an I-Cauchy sequence.

4.4. Theorem. Let (F(X)), N) be a fuzzy 2-normed space and I be an admissibleideal of N. If fk is I∗-Cauchy sequence, then it is an I-Cauchy sequence.

Proof. Let fk be an I∗-Cauchy sequence. Then for ε > 0 and each φ ∈ F(X),there are

K = km : k1 < k2 < · · · ∈ F(I)

and a number n0 ∈ N such that∥∥fkm − fkp , φ

∥∥α< ε

for every m, p ≥ n0. Now, x p = kn0+1. Then for every ε > 0 and each φ ∈ F(X),we have

‖fkm − fp, φ‖α < ε for every m ≥ n0.Let H = N \K. It is obvious that H ∈ I and

A(ε) = k ∈ N : ‖fkm − fp, φ‖α ≥ ε ⊂ H ∪ k1 < k2 < · · · < kn0 ∈ I.

Therefore, for every ε > 0, we can nd p ∈ N and g, h ∈ F(X) such that the sets

k ∈ N : ‖fkm − fp, g‖α ≥ ε and k ∈ N : ‖fkm − fp, h‖α ≥ εbelong to I, i.e., fk is an I-Cauchy sequence.

159

Now we prove that I∗-convergence implies I-Cauchy condition in a fuzzy 2-normed space (F(X), N).

4.5. Theorem. Let (F(X), N) be a fuzzy 2-normed space and I be an admissibleideal of N. If fk is I∗-convergent, then it is an I-Cauchy sequence.

Proof. By assumption there exists a set K = km : k1 < k2 < · · · such thatK ∈ F(I) and limm ‖fkm − f, φ‖α = 0 for each φ ∈ F(X), i.e., there exists n0 ∈ Nsuch that ‖fkm − f, φ‖α < ε for every ε > 0, each φ ∈ F(X) and m > n0. Since‖., .‖α is an α-2-norm, we have

∥∥fkm − fkp , φ∥∥α≤ ‖fkm − f, φ‖α +

∥∥fkp − f, φ∥∥α< 2ε

for every ε > 0, each φ ∈ F(X) and m, p > n0, we have∥∥fkm − fkp , φ

∥∥α< 2ε for

every m, p > n0 and each φ ∈ F(X), i.e., fk is an I∗-Cauchy sequence in F(X).Therefore, it follows from Theorem 4.4 that fk is an I-Cauchy sequence.

4.6. Theorem. Let I be an admissible ideal. Let fk be a sequence in a fuzzy2-normed space (F(X), N) and denote A(ε) = k ∈ N : ‖fk − fn, g‖α ≥ ε, wheren ∈ N and g ∈ F(X). If fk is an I-Cauchy sequence, then for every ε > 0 andg ∈ F(X) there exists B ∈ I such that ‖fl − fk, g‖α < ε, for all k, l /∈ B.

Proof. Let ε > 0 and g ∈ F(X) be given. Set B = An(ε/2), where n ∈ N. Sincefk is an I-Cauchy sequence, we have B ∈ I and for all l, k /∈ B, we get

‖fk − fn, g‖α <ε

2and ‖fl − fn, g‖α <

ε

2.

Because ‖., .‖ is an α-2-norm, by the triangle inequality we have ‖fk − fl, g‖α < ε,for all l, k /∈ B.

5. I-limit points and I-cluster points

In this section we introduce the notion of I-limit point and I-cluster point ofreal sequences in 2-fuzzy 2-normed linear spaces.

5.1. Denition. Let fk be a sequence in a fuzzy 2-normed space (F(X), N).An element ψ ∈ F(X) is said to be an I-limit point of fk provided there is a setK = k1 < k2 < · · · < km < · · · ⊂ N such that K /∈ I and limm ‖fkm − ψ, g‖α =0, for every g ∈ F(X).

5.2. Denition. Let fk be a sequence in a fuzzy 2-normed space (F(X), N).An element φ ∈ F(X) is said to be an I-cluster point of fk if for every ε > 0 andeach g ∈ F(X), the set k ∈ N : ‖fk − φ, g‖α < ε /∈ I.

We denote LIF(X)(fk) and CI

F(X)(fk) the set of of all I-limit points and I-cluster

points of a sequence fk in (F(X), N), respectively.

5.3. Theorem. Let I be an admissible ideal. Then for any fk in a fuzzy 2-normed space (F(X), N), we have

LIF(X)(fk) ⊂ CI

F(X)(fk).

160

Proof. Suppose that ψ ∈ LIF(X)(fk). Then there exists a set K = k1 < k2 < · · · <

km < · · · ⊂ N such that K /∈ I and

(5.1) limm→∞

‖fk − ψ, g‖α = 0 for each g ∈ F(X).

Let ε > 0 and g ∈ F(X) be given. According to (5.1), there exists an integern0 = n0(ε) ∈ N such that for k ≥ n0, we have ‖fk − ψ, g‖α < ε. Thus we have

K \ k1, k2, · · · , kn0 ⊂ k ∈ N : ‖fk − ψ, g‖α < ε.This implies that k ∈ N : ‖fk − ψ, g‖α < ε /∈ I. Therefore, ψ ∈ CI

F(X)(fk).

5.4. Theorem. Let fk be a sequence in a fuzzy 2-normed space (F(X), N). If

fkI−→ f , then

LIF(X)(fk) = CI

F(X)(fk) = f.

Proof. Suppose that fkI−→ f . Then for each ε > 0 and g ∈ F(X), we have

k ∈ N : ‖fk − f, g‖α ≥ ε ∈ I, i.e. k ∈ N : ‖fk − f, g‖α < ε /∈ I,

which implies that f ∈ CIF(X)(fk).

We assume that there exists at least one h ∈ CIF(X)(fk) such that h 6= f . Then

there exists ε > 0 such that

k ∈ N : ‖fk − f, g‖α ≥ ε ⊃ k ∈ N; ‖fk − h, g‖α < ε.But k ∈ N : ‖fk − f, g‖α ≥ ε ∈ I implies that k ∈ N : ‖fk − h, g‖α < ε ∈ I,

which contradicts that h ∈ CIF(X)(fk). Thus we have CI

F(X)(fk) = f.On the other hand, from fk

I−→ f , by Theorem 3.4 and Denition 5.2, we havef ∈ CI

F(X)(fk). By Theorem 5.3, we have LIF(X)(fk) = CI

F(X)(fk) = f.

References

[1] S. Aytar, Statistical limit points of sequences of fuzzy numbers, Inform. Sci. 165 (2004),129138.

[2] S. Aytar, M. Mammadov, S. Pehlivan, Statistical limit inferior and limit superior for se-quences of fuzzy numbers, Fuzzy Sets Syst. 157:7 (2006), 976985.

[3] S. Aytar, S. Pehlivan, Statistical cluster and extreme limit points of sequences of fuzzynumbers, Inform. Sci. 177 (2007), 32903296.

[4] T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11:3(2003), 687705.

[5] T. Bag, S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst. 151:3 (2005),513547.

[6] T. Bag, S.K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy SetsSyst. 159 (2008), 670684.

[7] M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence forsequences of functions, J. Math. Anal. Appl. 328 (2007), 715729.

[8] A.I. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1970),185193.

[9] H. Çakalli, S. Ersan, Strongly lacunary ward continuity in 2-normed spaces, Sci. World J.2014 (2014), Art. ID 479679, 5 pp.

[10] H. Çakalli, S. Ersan, New types of continuity in 2-normed spaces, Filomat 30:3 (2016),525532.

[11] A. Caserta, G. Di Maio, Lj.D.R. Ko£inac, Statistical convergence in function spaces, Abstr.Appl. Anal. 2011 (2011), Article ID 420419, 11 pages.

161

[12] S.C. Cheng, J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull.Cal. Math. Soc 86 (1994), 429436.

[13] P. Das, S.K. Ghosal, Some further results on I-Cauchy sequences and condition (AP), Com-put. Math. Appl. 59 (2010), 25972600.

[14] P. Das, S. Pal, S.K. Ghosal, Further investigations of ideal summability in 2-normed spaces,Appl. Math. Letters 24 (2011), 3943.

[15] P. Das, E. Sava³, On I-statistically pre-Cauchy sequences, Taiwanese J. Math. 18:1 (2014),115126.

[16] K. Dems, On I-Cauchy sequences, Real Anal. Exch. 30:1 (2004-2005), 123128.[17] G. Di Maio, Lj.D.R. Ko£inac, Statistical convergence in topology, Topology Appl. 156:1

(2008), 2845.[18] S. Ersan, H. Çakalli, Ward continuity in 2-normed spaces, Filomat 29:7 (2015), 15071513.[19] S. Gähler, 2-metrishe Räume und ihr topologishe struktur, Math. Nachr. 26 (1963), 115148.[20] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241244.[21] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48 (1992), 239

248.[22] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301313.[23] B. Hazarika, On ideal convergent sequences in fuzzy normed linear spaces, Afrika Matem-

atika 25:4 (2014), 987999.[24] B. Hazarika, E. Sava³, Some I-convergent lamda-summable dierence sequence spaces of

fuzzy real numbers dened by a sequence of Orlicz, Math. Comp. Modelling 54 (2011),29862998.

[25] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst. 12 (1984), 215229.[26] S. Karaku³, K. Demirci, O. Duman, Statistical convergence on intuitionistic fuzzy normed

spaces, Chaos, Solitons and Fractals 35 (2008), 763769.[27] V. Karakaya, N.N. im³ek , M. Ertürk, F. Gürsoy, Statistical A-convergence of sequences

of functions in intuitionistic fuzzy normed spaces, Abstr. Appl. Anal. 2012 (2012).[28] V. Karakaya, N.N. im³ek , M. Ertürk, F. Gürsoy, On ideal convergence of sequences of

functions in intuitionistic fuzzy normed spaces, Appl. Math. Inf. Sci. 8:5 (2014), 23072313.[29] A.K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets Syst. 12 (1984), 143154.[30] P. Kostyrko, T. alát, W. Wilczy«ski, I-convergence, Real Anal. Exchange, 26:2 (2000-2001),

669686.[31] P. Kostyrko, M. Ma£aj, T. alát, M. Sleziak, I-convergence and extremal I-limit points,

Math. Slovaca 55 (2005), 443464.[32] A.K. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11

(1975), 326334.[33] V. Kumar, K Kumar, On the ideal convergence of sequences of fuzzy numbers, Inform. Sci.

178 (2008), 46704678.[34] M. Mursaleen, S.A. Mohiuddine, Statistical convergence of double sequences in intuitionistic

fuzzy normed spaces, Chaos Solitons and Fractals 41 (2009), 24142421.[35] A. Nabin, S. Pehlivan, M. Gürdal, On I-Cauchy sequences, Taiwanese J. Math. 11:2 (2007),

569576.[36] F. Nuray, E. Sava³, Statistical convergence of sequences of fuzzy numbers, Math. Slovaca

45 (1995), 269273.[37] W. Raymond, Y. Freese, J. Cho, Geometry of linear 2-normed spaces, N.Y. Nova Science

Publishers, Huntington, 2001.[38] E. Sava³, On statistically convergent sequence of fuzzy numbers, Inform. Sci. 137 (2001),

272282.[39] E. Sava³, On some new sequence spaces in 2-normed spaces using ideal convergence and an

Orlicz function, J. Inequal. Appl. 2010 (2010), Art. ID 482392, 8 pp.[40] E. Sava³, Some I-convergent sequence spaces of fuzzy numbers dened by innite matrix,

Math. Comp. Appl. 18:2 (2013), 8493.[41] C. ençimen, S. Pehlivan, Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets

Syst. 159 (2008), 361370.[42] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math.

2 (1951), 7334.

162

[43] R.M. Somasundaram, T. Beaula, Some aspects of 2-fuzzy 2-normed linear spaces, Bull.Malays. Math. Sci. Soc. (2) 32:2 (2009), 211221.

[44] L.A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965), 338353.[45] J. Zhang, The continuity and boundedness of fuzzy linear operators in fuzzy normed space,

J. Fuzzy Math. 13:3 (2005), 519536.[46] A. Zygmund, Trigonometric Series, 2nd edition, Cambridge University Press, Cambridge,

1979.

Hacettepe Journal of Mathematics and StatisticsVolume 46 (1) (2017), 163 164

A man of words

In memory of Professor L. Michael Brown

Ivan Reilly ∗† and Bill Barton‡

Abstract

This note highlights an often overlooked aspect of Michael Brown′sscholarship: his interest in language, especially as it relates to mathe-matics.

Keywords: language and mathematics, topology.

In 1978 the rst author met Michael Brown for the rst time, at an internationaltopology conference in Budapest, Hungary. They were aware of their common re-search interests in bitopological spaces and related topics because each of themhad read some of the other′s early published papers. Their professional relation-ship was bolstered by the appointment of the rst author by the University ofGlasgow to be an external examiner of Brown′s 1980 doctoral dissertation. Thisappointment was a secret for perhaps a decade - they enjoyed smiling about itlater in their friendship. Over the years they met about a dozen times, usually atinternational conferences.

In 2004, the rst author nally visited Ankara and Hacettepe University for therst time, to attend the Do§an Çoker memorial conference. The second authormet Michael Brown at that 2004 Çoker conference. We were kindly hosted byMichael and his colleagues after requesting access to a Turkish-speaking topologycommunity for our research into language eects on advanced abstract mathemat-ics.

Our visit to Hacettepe University in 2004 was crucial for our research on lan-guage and topology [1]. We were able to discuss ideas with, and seek information

∗Department of Mathematics, The University of Auckland, PB 92019, Auckland 1140, NewZealand,Email: i.reilly@auckland.ac.nz†Corresponding Author.‡Department of Mathematics, The University of Auckland, PB 92019, Auckland 1140, New

Zealand,Email: b.barton@auckland.ac.nz

Doi : 10.15672/HJMS.2016.401

164

from, several Turkish topologists as well as Michael Brown. For us this was aunique situation: a community who studied and worked in topology in a languageother than English, French, or Spanish. Michael gave us books, dictionaries, andthe whole group was able to speak to us about the development of a modern math-ematical register in Turkish . Our debt to these consultants is explicit in our paper[1].

Prior to our 2004 visit to Hacettepe University, Michael Brown had been incorrespondence with us concerning these matters. Let us quote some of Michael′sown words [2].

In general terms both [English and Turkish] seem capable of expressing mathe-matical concepts and arguments with equal precision. But having said that I cannothelp but feel that the structure of English is somewhat better suited to mathematicsthan that of Turkish. One point . . . is the position of the verb at the end of thesentence. Whereas in English one would write There exists a continuous functionf . . . which established from the beginning that it is the existence of somethingthat is involved, in Turkish one would say something like having the property ofcontinuity, a function f there is giving the property (continuity) rst, of what (thefunction) second, and its existence last. Longer examples can have you describingquite complex properties of things before it comes clear what it is that has theseproperties. Of course the end result is no less exact in an absolute sense, and onegets used to having things this way round, so perhaps it is just a question of whatone is used to. However, there are ways of forcing a word-order more similar toEnglish by using an equivalent of such that (the result not being considered good Turkish ). [This is] often resorted to by speakers used to lecturing inEnglish and (often) by research students, so perhaps the eort required to producea well structured sentence in such cases is something that even native speakers ofTurkish nd noticeable. Turkish is quite an expressive language, and the use ofsuxes means one can pack a lot of meaning into a single word, so it is often veryeconomic. In some areas it is well supplied with synonyms, but not in all, so itis sometimes dicult to name new concepts similar, but not identical to, knownones.

It is clear that Michael Brown was the right person to consult on the similaritiesand dierences between Turkish and English as languages for the expression oftopological ideas and concepts. He had, with his wife and colleagues, producedthe canonical dictionary of mathematical terms [3]. It was our good fortune to beable to consult with him in the course of our research project.

References

[1] Barton, B., Lichtenberk, F. and Reilly, I.L. The language of topology: A Turkish case study,, Applied General Topology 6 (6), 107-117, 2005.

[2] Brown L.M., private communication, 2003.[3] Hacsaliho§lu, H., Hacyev, A., Kalantarov, V., Sabuncuo§lu A., Brown, L.M., bikli, E.,

and Brown, S. Matematik Terimleri Sozlugu, Ankara: Hacettepe University, 2000.

Hacettepe Journal ofADVICE TO AUTHORS

Mathematics and Statistics

Hacettepe Journal of Mathematics and Statistics publishes short to medium lengthresearch papers and occasional survey articles written in English. All papers arerefereed to international standards.

Address for Correspondence

Editorial Oce,Hacettepe Journal of Mathematics and Statistics,Hacettepe University,Faculty of Science,Department of Mathematics,06532 Beytepe,Ankara,Turkey.

E-mail: hjms@hacettepe.edu.trTel: Editors: + 90 312 297 7898

+ 90 312 297 7850 / 122

Associate Editor: + 90 312 297 7880

Fax : + 90 312 299 2017

Web Page : http//www.hjms.hacettepe.edu.tr

Advise to Authors : The style of articles should be lucid but concise. Thetext should be preceded by a short descriptive title and an informative abstract ofnot more than 100 words. Keywords or phrases and the 2010 AMS Classicationshould also be included.The main body of the text should be divided into num-bered sections with appropriate headings. Items should be numbered in the form2.4. Lemma, 2.5. Denition. These items should be referred to in the text usingthe form Lemma 2.4., Denition 2.5. Figures and tables should be incorporatedin the text. A numbered caption should be placed above them. References shouldbe punctuated according to the following examples, be listed in alphabetical or-der according to in the (rst) author' s surname, be numbered consecutively andreferred to in the text by the same number enclosed in square brackets. Onlyrecognized abbreviations of the names of journals should be used.

[1 ] Banaschewski, B. Extensions of topological spaces, Canad. Math. Bull.7 (1), 122, 1964.

[2 ] Ehrig, H. and Herrlich, H. The construct PRO of projection spaces: its

internal structure, in: Categorical methods in Computer Science, LectureNotes in Computer Science 393 (Springer-Verlag, Berlin, 1989), 286293.

[3 ] Hurvich, C. M. and Tsai, C. L. Regression and time series model selection

in small samples, Biometrika 76 (2), 297307, 1989.[4 ] Papoulis, A. Probability random variables and stochastic process (McGraw-

Hill, 1965).

Hacettepe Journal ofINSTRUCTIONS FOR AUTHORS

Mathematics and Statistics

Preparation of Manuscripts : Manuscripts will be typeset using the LATEXtypesetting system. Authors should prepare the article using the HJMS stylebefore submission by e-mail. The style bundle contains the styles for the article andthe bibliography as well as example of formatted document as pdf le. The authorsshould send their paper directly to hjms@hacettepe.edu.tr with the suggestion ofarea editor and potential reviewers for their submission. The editorial oce maynot use these suggestions, but this may help to speed up the selection of appropriatereviewers.

Copyright : No manuscript should be submitted which has previously been pub-lished, or which has been simultaneously submitted for publication elsewhere. Thecopyright in a published article rests solely with the Faculty of Science of HacettepeUniversity, and the paper may not be reproduced in whole in part by any meanswhatsoever without prior written permission.

Notice : The content of a paper published in this Journal is the sole responsibilityof the author or authors, and its publication does not imply the concurrence ofthe Editors or the Publisher.