C-Closed Sets in LFuzzy Topological Spaces And

7/28/2019 C-Closed Sets in LFuzzy Topological Spaces And

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Turk J Math26 (2002) , 245 261.c TUBITAK

CClosed Sets in LFuzzy Topological Spaces and

Some of its Applications

Ali Ahmed Nouh

Abstract

We introduce and study the notion of Cclosed sets in Lfuzzy topologicalspaces. Then, Cconvergence theory for nets and ideals is established in terms ofCclosedness. Finally, we give a new concept of Ccontinuity on Lfuzzy topologicalspace by means of Lfuzzy Cclosedness and investigate some of its propertiesand its relationships with other Lfuzzy mappings introduced previously. Thenwe systematically study the characterizations of this notion with the aid of theCconvergence of Lfuzzy nets and Lfuzzy ideals.

Keywords and phrases. Lfuzzy topology, Qcompactness, Lfuzzy Cclosedset, Lfuzzy Ccontinuous mappings, Lfuzzy net, Lfuzzy ideal, Cconvergence.

1. Introduction

Continuity and its weaker forms constitute an important and intensely investigatedarea in the field of general topological spaces. For example, the notions of almostcontinuous, Ncontinuous, Hcontinuous, Ccontinuous, weakly continuous and semicontinuous have been introduced by different authors, and their interrelationships withother topological notions have been established. Most of these notions turn out to belocal properties; hence the pointwise approach is generally preferred in their studies anddefinitions. The concept of Ccontinuity in general topology was introduced by Gentry

and Hoyle [5] in 1970. The class of Cclosed sets (compact and closed) was defined by

Garg and Kumar [4] in 1989. Then several characterizations of Ccontinuous mappings

in terms of Cclosed sets are given. Recently, Dang, Behera and Nanda [3] extended

the concept to fuzzy topology, and introduced the notion of fuzzy Ccontinuous functionusing the fuzzy compactness given by Mukherjee and Sinha [8]. However, the fuzzy

compactness has some shortcomings, such as the Tychonoff product theorem does not

AMS Subject Classification 2000 : 54A20, 54A40, 54C08, 54H12.

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hold, and it is contradicts some kinds of separation axioms. Hence, the notion of fuzzy

Ccontinuous function in [3] is unsatisfactory. In this paper, we first define the concept

of Lfuzzy Cclosed sets by means of the concept of Lfuzzy Qcompactness in the

sense of Wang [11]. Then by making use of Lfuzzy Cclosed sets we introduce and

study the Cconvergence theory of Lfuzzy nets and Lfuzzy ideals. Finally, we give anew definition of fuzzy Ccontinuous which calls Lfuzzy Ccontinuity on the basis ofthe notion of Lfuzzy Cclosedness in Lfuzzy topology, and systematically discuss itscharacterizations and properties by making use of Cconverges theory of Lfuzzy netsand Lfuzzy ideals.

2. Preliminaries

Throughout this paper, L denotes a complete, completely distributive lattice; M(L)

denotes the set of all nonzero irreducible elements of L; and 0 and 1 denote the least

and greatest element in L, respectively. LX and LY denote the set of all Lfuzzy sets

on crisp sets X and Y, respectively. Write M(LX) = {x LX : x X, M(L)},

and call the elements in M(LX) molecules or Lfuzzy points on X. For LX , put

= { : }.

Let (LX , ) be an Lfuzzy topological space, briefly Lfts. For each LX , c(),

int() and will denote the closure, interior and the complement of , respectively. 0X

and 1X denote, respectively, the least and the greatest element of LX . If LX and

= int(c()), then it is called regular open. The complement of regular open is calledregular closed. The class of all Lfuzzy regular open (resp. regular closed) sets will be

denoted by RO(LX , ) (resp. RC(LX, )). Let (X, T) be a crisp topological space and

LX , if L, {x X : (x) } T, then we call a lower semicontinuous

function. The set of all these functions is denoted by L(T) and is an Lfuzzy topology

on X generated by T.

Definition 2.1 [10]: Let (LX , ) be an Lfts and x M(LX). is called a remoted

neighbourhood (Rnbd, for short) of x ifx . The set of all Rnbds of x is denoted

by Rx .

Definition 2.2 [1,10]: Let (LX

, ) be an Lfts and LX

.

(resp. RC(LX , )) is called an remoted (resp. regular closed remoted) neighbourhood

family of , briefly RF (resp. rcRF) of , if for each Lfuzzy point x , there is

such that Rx .

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Definition 2.3 [9,10]: Let (LX , ) be an Lfts. Then LX is called:

(i) Qcompact (resp. nearly Qcompact) if for any M(L) and every RF

(resp. rcRF) of there exists a finite subfamily of such that is an

RF of .

(ii) Strong Qcompact (resp. Strong nearly Qcompact) if it is Qcompact (resp.

nearly Qcompact) for all M(L).

If 1X is Qcompact (resp. nearly Qcompact, strong Qcompact, strong nearly Q

compact), then we say that (LX , ) is a Qcompact (resp. nearly Qcompact, strong

Qcompact, strong nearly Qcompact) space.

Definition 2.4 [7]: An (LX , ) is said to be :

(i) LFT2space (Lfuzzy Hausdorff space) iff (x, y M(LX), x = y)

( Rx)( Ry )( = 1X).

(ii) LFR2space (Lfuzzy regular space) iff (x M(LX))( Rx)

( Rx)( )( = 1X and = 0X).

(iii) Fully stratified if for all L.

(iv) Weakly induced if each nonempty element of is a lower semicontinuous mapping

from (X, []) to L.

(v) Induced if it is both fully stratified and weakly induced.

The family of all crisp open (resp. closed) sets in is denoted by [] (resp. []).

Obviously, (X, []) is a crisp topological space.

Theorem 2.5 [7]: A topological space (X, T) is a T2-space iff an Lfts (LX , L(T)) is a

LFT2-space.

Theorem 2.6 [6]: For fully stratified Lfts (LX , ) and LX , if for each M(L),

w [], then , where w = {x X : (x) and M(L)}.

Theorem 2.7 [9]: Each strong Qcompact Lfuzzy set in a fully stratified LF T2space

is Lfuzzy closed.

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Theorem 2.8 [9]: Every Lfuzzy closed set of a Qcompact (resp., strong Qcompact)

Lfts is Qcompact (resp., strong Qcompact).

Theorem 2.9 [9]: Let (X, T) be a topological space. Then Lfuzzy set LX is

Qcompact in (LX , L(T)) iffw is compact in (X, T), for all M(L).

Theorem 2.10 [9]: Let (LY , ) be an LF T2space and f : (LX , ) (LY , ) be an

Lfuzzy continuous mapping [12] and LX be a strong Qcompact in (LX, ), then

f() is a strong Qcompact Lfuzzy set in (LY , ).

Theorem 2.11 [9]: Let (LX , ) be an LF R2space. Then every strong nearly Q

compact set is strong Qcompact.

Theorem 2.12 [9]: Let (LX , ) be an induced Lfts. Then the concepts of Ncompactness

and strong Qcompactness are equivalent.

Definition 2.13 [12,15]: Let (LX , ) be an Lfts. An Lfuzzy net in (LX , ) is a mapping

S : D M(LX) denoted by S = {S(n), n D}, where D is a directed set. S is said to

be in LX if n D, S(n) .

Definition 2.14 [12,13]: The non empty family L LX is called an Lfuzzy ideal if, for

each 1, 2 LX the following satisfies:

(i) If 1 2 and 2 L, then 1 L.

(ii) If 1, 2 L, then 1 2 L.

(iii) 1X L.

Theorem 2.15 [12,13]: Let (LX , ) be an Lfts, LX and x M(LX). Then

x c() iff there exists an Lfuzzy net in (resp., an Lfuzzy ideal L not containing

) which converges to x (see Definitions 3.9 and 3.11).

Other unexplained notations and definitions in this paper can be found in [1,2,9,12,13].

3. Lfuzzy Cclosure and Cinterior operators.

In this section, we introduce and study the concepts of Cclosure operator and Cinterior operator by having the aid of the notion of Qcompactness and discuss theirproperties. Then we present the concepts of Climit and Ccluster points of Lfuzzy netsand Lfuzzy ideals.

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Definition 3.1: Let (LX , ) be an Lfts and LX . An Lfuzzy point x M(LX)

is called an Cadherent (resp. Nadherent) point of , written as x C.c() (resp.

x N.c()) iff for each CRx (resp. N

Rx), where CRx (resp.

NRx) is the family of all strong Qcompact (resp. strong nearly Qcompact) Rnbds

of x. C.c() (resp. N.c()) is said to be Cclosure (resp. Nclosure) of . If

C.c() (resp. N.c() ), then is called Lfuzzy Cclosed (resp. Nclosed).

The complement of an Lfuzzy Cclosed (resp. Nclosed) set is called Lfuzzy Copen

(resp. Nopen) set.

In [1], Chen and Wang have introduced the concept of Lfuzzy Nclosed sets by using

Ncompactness due to Zhao [14]. It is easy to see that every Lfuzzy Nclosed set is N

closed. So the properties and characterizations of Nclosed set and its related notionsare similar to those of Nclosed set.

Theorem 3.2: Let (LX , ) be an Lfts and , LX . Then the following statements

hold:

(i) c() N.c() C.c().

(ii) If , then C.c() C.c().

(iii) C.c(C.c()) = C.c().

(iv) C.c() = { LX : is a Cclosed set containing }.

Proof: It is similar to that of Theorem 3.1 in [2].

Theorem 3.3: Let (LX , ) be an Lfts. The following statements hold:

(i) 1X and 0X are both Cclosed.

(ii) Every strong Qcompact closed set is Cclosed.

(iii) The union of finite Cclosed sets is Cclosed.

(iv) The intersection of arbitrary Cclosed sets is Cclosed.

(v) LX is Cclosed iff there exists CRx such that for each x M(LX)

with x .

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Proof: It is similar to that of Theorem 3.2 in [2].

Theorem 3.4: Let (LX , ) be an Lfts and LX . Then the families

C = { LX : = C.c()} and N = { L

X : = N.c()}

of all Lfuzzy Copen and Nopen sets in X are Lfuzzy topologies on X associated with

. We call (LX , C) and (LX , N) Lfuzzy Cspace and Lfuzzy N

space, respectively,

induced by (LX , ).

Proof: It is an immediate consequnce of Definition 3.1 and Theorems 3.2 and 3.3.

Theorem 3.5: Let (LX , ) be an Lfts. Then

(i) C N .

(ii) If (LX , ) is strong Qcompact (resp. strong nearly Qcompact), then = C (resp.

= N).

(iii) If (LX , ) is LF R2space, then C = N .

(iv) If (LX , ) is induced Lfts, then N = N[1].

Proof: Follows from Theorems 2.11, 2.12 and 3.4.

Definition 3.6: Let (LX, ) be an Lfts, LX and C.int() = { LX : is an

Lfuzzy Copen set contained in }. We say that C.int() is the Cinterior of .

The following theorem shows the relationships between Cclosure operator and Cinterior operator.

Theorem 3.7: Let (LX , ) be an Lfts and LX . Then the following are true.

(i) is Copen iff = C.int().

(ii) C.int() int() .

(iii) C.int() = (C.c()).

(iii) If LX and , then C.int() C.int().

(iv) C.int(C.int()) = C.int().

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Dually, we have the following results.

Theorem 3.8: Let (LX , ) be an Lfts. The following statements hold:

(i) 1X and 0X are both Copen.

(ii) The intersection of finite Copen sets is Copen.

(iii) The union of arbitrary Copen sets is Copen.

Definition 3.9: Let S be an Lfuzzy net in an Lfts (LX , ) and x M(LX). Then

x is said to be a:

(i) limit point of S [12] or S converges to x, in symbol S x, if

( Rx)(n D)(m D, m n)(S(m) ).

(ii) Climit point of S or S Cconverges to x, in symbol SCx, if

( CRx)(n D)(m D, m n)(S(m) ).

The union of all limit (resp. C-limit) points of S is denoted by lim(S) (resp.

C.lim(S)).

Proposition 3.10: Suppose that S is an Lfuzzy net in (LX , ), LX and x

M(LX). Then the following results are true:

(i) x C.lim(S) iff SC

x.

(ii) lim(S) C.lim(S).

(iii) x C.c() iff there is an Lfuzzy net in which C-converges to x.

(iv) C.lim(S) is an Lfuzzy Cclosed set in LX .

Proof: (i) Let SCx, so by definition x C.lim(S). Conversely, let x C.lim(S)

and CRx. Since x , so we have C.lim(S) > (x). Thus C.lim(S) .

Therefore there exists y M(LX) such that SCy , but y and so CRy. Hence

(n D)(m D, m n)(S(m) ). Thus SC

x.(ii) Let x lim(S) and CRx. Since CRx Rx, then Rx . And since

x lim(S), then ( Rx)(n D)(m D, m n)(S(m) ) and so S(m) .

Hence x C.lim(S). So lim(S) C.lim(S).

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(iii) Let x C.c(). Then ( CRx)( ) and so there exists (, ) L\{0}

such that x(,) and x(,) . Since the pair (CRx, ) is a directed set, we can

define an Lfuzzy net S : CRx M(LX) given by S() = x(,), CRx. Then Sis

an Lfuzzy net in . Now let CRx such that , so we have the situation in which

there exists S() = x(,) > . Then x(,) . So SCx. Conversely, let S be an

Lfuzzy net in with SCx. Then ( CRx)(n D)(m D, m n)(S(m) ).

Since S is an Lfuzzy net in , then S(m) > . Hence ( )( CRx). So

x C.c().

(iv) Let x C.c(C.lim(S)) and CRx. Then C.lim(S) . So there

exists y M(LX) such that y C.lim(S) and y . Then ( CRy)(n

D)(m D, m n)(S(m) ) and so S(m) . Hence x C.lim(S). Thus

C.c(C.lim(S)) C.lim(S) and so C.lim(S) is a Lfuzzy Cclosed set.

Definition 3.11: Let L be an Lfuzzy ideal in an Lfts (LX , ) and x M(LX). Then

x is said to be:

(i) a limit point of L [13] or L converges to x, in symbol L x, if Rx L.

(ii) Climit point of L or L Cconverges to x, in symbol LCx, if CRx L.

The union of all limit points (resp., Climit points) of L is denoted by lim(L) (resp.

C.lim(L)).

Proposition 3.12: Suppose that L is an Lfuzzy ideal in (LX , ), LX and x

M(LX). Then the following results are true:

(i) x C.lim(L) iff LCx.

(ii) lim(L) C.lim(L).

(iii) x C.c() iff there is an Lfuzzy ideal L which C-converges to x and L.

(iv) C.lim(L) is an Lfuzzy Cclosed set in LX

.

Proof: The proof of the statements (i), (ii) and (iv) are similar to the correspondence

statements of Proposition 3.10.

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(iii) Let x C.c(). Let L(CRx) = { LX : CRx }. It easy

to show that L(CRx) is an Lfuzzy ideal. Now we show that / L(CRx). Since

x C.c(), then for each CRx, . So by definition of L(CRx) we have

/ L(CRx). Finally, we show that LCx. Let CRx and since , then

L(CRx). So CRx L(CRx). Thus LCx. Conversely, let L be an Lfuzzy

ideal, / L and LCx. Then for each CRx , L. Since L, / L, then

and so x C.c().

4. Lfuzzy Ccontinuous mappings.

Definition 4.1: An Lfuzzy mapping f : (L

X

, ) (L

Y

, ) is said to be :

(i) An Lfuzzy Ccontinuous iff1() for each strong Qcompact Lfuzzy closed

set in LY .

(ii) An Lfuzzy Ccontinuous at Lfuzzy point x M(LX) if f1() Rx for each

CRf(x).

Theorem 4.2: A mapping f : (X, T1) (Y, T2) is C-continuous iff an Lfuzzy mapping

f : (LX , L(T1)) (LY , L(T2)) is Lfuzzy C-continuous.

Proof: Let f : (LX , T1) (LY , T2) be C-continuous and let L

Y be strong

Qcompact Lfuzzy closed. Then by Theorem 3.2 in [6] and Theorem 2.9, we have

w Y is compact and closed in (Y, T2), M(L). Since f1(w ) = (f

1())w ,

then f1(w ) T

1 for each M(L) and so f1() L(T1) = (L(T1))

.

Thus f : (LX , L(T1)) (LY , L(T2)) is Lfuzzy C-continuous. Conversely; let f :

(LX , L(T1)) (LY , L(T2)) be Lfuzzy C-continuous and let A Y be compact

and closed. Then, by Theorem 2.9, 1A LY is Qcompact and Lfuzzy closed in

(LY , L(T2)). Since 1f1(A) = f1(1A) L(T1) so f

1(A) T1. Hence f : (X, T1)

(Y, T2) is C-continuous.

Theorem 4.3: Let f : (LX , ) (LY , ) be an Lfuzzy mapping. Then the following

are equivalent:

(i) f is Lfuzzy Ccontinuous;

(ii) f is Lfuzzy Ccontinuous at x for each Lfuzzy point x M(LX);

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(iii) For each with is strong Qcompact, then f1() .

These statements are implied by

(iv) If LY is strong Qcompact, then f1() .

Moreover, if (LY , ) is fully stratified LF T2space, all the statements are equivalent.

Proof : (i) = (ii) Suppose that f is Lfuzzy Ccontinuous, x M(LX) and

CRf(x), then f1() . Since f(x) is equivalent to x f1(), so

f1() Rx , and hence f is Lfuzzy Ccontinuous at x.

(ii) = (i) Let f be an Lfuzzy Ccontinuous at x for each x M(LX).

If f is not Lfuzzy Ccontinuous, then there is C-closed Lfuzzy set LY withc(f1()) f1(). Then there exists x M(L

X) such that x c(f1())

and x f1(). Since x f1() implies that f(x) , so CRf(x). But

f1() / Rx, a contradiction. Therefore, f must be Lfuzzy Ccontinuous.

(i) (iii) Follows straightforward from Definition 4.1.

(iv) = (iii) Let with is strong Qcompact. By (iv), we have f1() .

Thus, f1() = (f1()) .

Now suppose that (LY , ) is fully stratified LF T2space.

(iii) = (iv) Let LY be strong Qcompact set. Since (LY , ) is fully

stratified LF T2space, then

and so

. By (iii), f

1

(

) . Thusf1() = (f1()) .

By view of Theorems 4.2 and 4.3 the following example shows that LFT2 is necessary

when showing (i) implies (iii) in the above Theorem.

Example 4.4: Let X = {1, 2, 3}, Y = R, = L(S), where S = {X, , {3}, {2, 3}} and

= L(T), where T be a topology on Y generated by {(, r)(r, ) : r Y}. Then

the mapping f : (X, S) (Y, T) defined by f(x) = x for each x X is Ccontinuous (See,

Example 1 in [5]). Hence by Theorem 4.2, the mapping f : (LX, L(S)) (LY , (L(T))

is Lfuzzy Ccontinuous but does not satisfy statement (iii) in Theorem 4.3.

Theorem 4.5: Let f : (LX , ) (LY , ) be an surjective Lfuzzy mapping. Then the

following conditions are equivalent:


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(ii) For each LX , f(c()) C.c(f()).

(iii) For each LY , c(f1()) f1(C.c()).

(iv) For each LY , f1(C.int()) int(f1()).

(v) f1() is Lfuzzy open in LX , for each Lfuzzy C-open set in LY.

(vi) f1() is Lfuzzy closed in LX, for each Lfuzzy C-closed set in LY .

Proof: (i) = (ii) Let LX and x M(LX) with x c(). Then f(x)

f(c()). Let CRf(x). So by (i), f

1() Rx . Since x c() and f

1() Rx,then f1(). Thus f() and CRf(x) and so f(x) C.c(f()). Hence

f(c()) C.c(f()).

(ii) = (iii) Let LY . Then f1() LX . By (ii), we have f(c(f1())

C.c(ff1()) C.c() and so f(c(f1()) C.c() Thus f1f(c(f1()))

f1(C.c()). Since c(f1()) f1f(c(f1())), then c(f1()) f1(C.c()).

(iii) = (iv) Let LY . By (iii), c(f1()) f1(C.c()). Since c(f1()) =

c(f1()) = (int(f1())) and f1(C.c()) = (f1(C.int())). So, (int(f1()))

(f1(C.int())) and by the complement, int(f1()) f1(C.int()).

(iv) = (v) Let be C-open in LY

. Then f1

() = f1

(C.int()) and by (iv),f1(C.int()) int(f1()), so f1() int(f1()). Thus f1() .

(v) = (vi) Let be C-closed in LY . By (v), f1() . Then (f1()) =

f1() . So f1() .

(vi) = (i) Let be strongly Qcompact and closed set in LY. Then by Theorem

3.3 (ii), we have is Cclosed set in LY. Hence by (vi), f1() . Hence f is Lfuzzy

C-continuous.

Theorem 4.6: Every Lfuzzy continuous mapping in the sense of Wang [12] is Lfuzzy

Ccontinuous.

Proof : Straightforward.

By view of Theorems 4.2 and 4.6 the following example shows that not every LfuzzyCcontinuous mapping is Lfuzzy continuous.

Example 4.7: Let R be the set of reals with the usual topology TU and define f :

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(R, TU) (R, TU) by

f(x) =

1x

: x = 012

: x = 0.

Then f is Ccontinuous but not continuous (See, Example 2 in [5]). Hence by Theo-

rem 4.2, f : (LR, L(TU)) (LR, L(TU)) is Lfuzzy Ccontinuous but not Lfuzzy

continuous.

In the following two Theorems we discuss the conditions which the Lfuzzy Ccontinuity is equivalent to the Lfuzzy continuity.

Theorem 4.8: A mapping f : (LX , ) (LY, C) is Lfuzzy continuous iff it is Lfuzzy

Ccontinuous.

Proof: Since C , then necessity is evident. Now, suppose that f is Lfuzzy C

continuous and C. Then by Theorem 4.5 (iii) we have f1() = f1(C.c())

c(f1()) and so f1() .

Theorem 4.9: Let f : (LX , ) (LY , ) be an Lfuzzy mapping and (LY , ) be strong

Qcompact space. Then f is Lfuzzy continuous ifff is Lfuzzy Ccontinuous.

Proof: By Theorem 4.6 we need only to investigate the sufficiency. Let . Since

(LY , ) is strong Qcompact then, by Theorem 2.8, is strong Qcompact and so is

Lfuzzy Cclosed set. By Lfuzzy Ccontinuity of f, we have f1() . Hence f is

Lfuzzy continuous.

In [1] Chen and Wang have introduced and studied the concept of Lfuzzy N

continuous mapping by using nearly Ncompactness due to Chen and Wang [1]. Here we

redefine this concept by using strong nearly Qcompactness due to Nouh [9]. However,

its detailed treatment is beyond the scope of this paper and will be dealt elsewhere.

Definition 4.10: An Lfuzzy mapping f : (LX , ) (LY , ) is said to be :

(i) An Lfuzzy Ncontinuous iff1() for each strong nearly Qcompact Lfuzzy

closed set in LY.

(ii) An Lfuzzy Ncontinuous at Lfuzzy point x M(LX) iff1() Rx for each

NRf(x).

Theorem 4.11: Every Lfuzzy Ncontinuous mapping is Lfuzzy Ncontinuous in the

sense of Chen and Wang [1].

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Proof: Follows from the fact that every Ncompact Lfuzzy set is strong Qcompact.

The converse of Theorem 4.11 is not true in general as can be seen from Example 1

in [15]. However, we have the following result.

Theorem 4.12: Let f : (LX , ) (LY, ) be an Lfuzzy mapping and (LY, ) be

induced Lfts. Then f is Lfuzzy Ncontinuous iff f is Lfuzzy Ncontinuous.

Proof: Follows from Theorems 2.12 and 4.11.

Theorem 4.13: Every Lfuzzy Ncontinuous mapping is Lfuzzy Ccontinuous.

Proof: Follows from the fact that every strong Qcompact set is strong nearly Qcompact.

Theorem 4.14: Let f : (LX , ) (LY, ) be an Lfuzzy mapping and (LY, ) beLFR2space. Then f is Lfuzzy N

continuous iff f is Lfuzzy Ccontinuous.


Remark 4.15: For an Lfuzzy mapping f : (LX , ) (LY , ), we obtain the following

implications:

Lfuzzy continuity Lfuzzy Ncontinuity Lfuzzy Ccontinuity.

The following counterexample shows that none of these implications are reversible.

Counterexample 4.16: Let (LX , ) and (LX , ) be two Lftss, where (LX , ) is fully

stratified LF T2 and (LX , ) is not LF R2. Let f : (L

X , ) (LX , ) be the identity

mapping. Then :

(i) If is strictly finer than and (LX , ) is LF R2, then f is Lfuzzy Ncontinuous

but not Lfuzzy continuous.

(ii) If = and (LX , ) is not LF R2, then f is Lfuzzy Ccontinuous but not Lfuzzy

Ncontinuous.

However, if (LY , ) is strong Qcompact (resp. LF R2) space, then Theorem 4.9 (resp.

Theorem 4.14) implies that the concepts of Lfuzzy continuity (resp. Ncontinuity) and

Lfuzzy Ccontinuity are equivalent.

Definition 4.17 [1]: Let f : (LX , ) (LY , ) be an Lfuzzy mapping and A X.

Define an Lfuzzy mapping f|A : LA LY as follows:

(f|A)() = f() 1A = f(), for each LA

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And call f|A the restriction of f on A. Where denote the extension of in LX ,

that is for each x X,

(x) =

(x) : x A

0 : x / A

Theorem 4.18: If f : (LX , ) (LY, ) is an Lfuzzy Ccontinuous and A X, then

f|A : (LA, A) (LY , ) is an Lfuzzy Ccontinuous mapping.

Proof: Let LY be Cclosed. Since f is Lfuzzy Ccontinuous, so f1()

and (f|A)1() = f1() 1A A. Hence f|A : (LA, A) (LY , ) is Lfuzzy C

continuous.

The composition of two Lfuzzy Ccontinuous mappings need not be Lfuzzy C

continuous (See, Example 3.14 in [3]). However, we have the following result.

Theorem 4.19: If f : (LX , 1) (LY, 2) is Lfuzzy continuous mapping and g :

(LY , 2) (LZ, 3) is Lfuzzy Ccontinuous mapping, then g f : (L

X , 1) (LZ, 3) is

Lfuzzy Ccontinuous.

Proof: Obvious.

Theorem 4.20: If (LX , ), (LY, ) are Lftss and 1X = 1A 1B, where 1A and 1B are

closed of LX and f : (LX , ) (LY , ) is Lfuzzy mapping such that f|A and f|B are

Lfuzzy Ccontinuous, then f is Lfuzzy Ccontinuous.

Proof: Let 1A, 1B . Let LY be Cclosed. Then (f|A)1() (f|B)1() =

(f1() 1A) (f1() 1B) = f1() (1A 1B) = f1() 1X = f1(). Hence

f1() = (f|A)1() (f|B)1() . So f : (LX , ) (LY, ) is Lfuzzy C

continuous.

5. More Characterizations of Lfuzzy Ccontinuous mappings

Theorem 5.1: Let f : (LX , ) (LY, ) be Lfuzzy Ccontinuous and (LY , ) be a

fully stratified LF T2space. If f(1X) is contained in some strong Qcompact set of LY ,

then f is Lfuzzy continuous.

Proof: Let LY be a strong Qcompact containing f(1X) and let . Since

is strong Qcompact in (LY , ) which is fully stratified LF T2-space, so .

Thus . Hence by Theorem 2.8, LY is strong Qcompact. Thus

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LY is Cclosed. Since f is Lfuzzy Ccontinuous, then f1( ) . But,

f1( ) = f1() f1() = f1() 1X = f1(). So f1() . Hence f is

Lfuzzy continuous.

Theorem 5.2: Let (LX , ) be an Lfts and (LY , ) be fully stratified LF T2space. Iff :

(LX , ) (LY , ) is a bijective and Lfuzzy continuous, then f1 : (LY , ) (LX , )

is Lfuzzy Ccontinuous.

Proof: Let LX be strong Qcompact. Since f is Lfuzzy continuous, then by

Theorem 2.10, f() is strong Qcompact. Since (LY , ) is fully stratified LF T2-space,

then f() . Hence by Theorem 4.3, f1 is Lfuzzy Ccontinuous.

Corollary 5.3: Let (LX , ) be a strong Qcompact space and (LY , ) be fully stratifiedLF T2-space. Iff : (L

X , ) (LY , ) is a bijective and Lfuzzy continuous, then f is an

Lfuzzy homeomorphism.


Theorem 5.4: Let f : (LX , ) (LY , ) be an surjective Lfuzzy mapping. Then the

following conditions are equivalent:


(ii) For each x M(LX) and each Lfuzzy net S in LX , f(S)C

f(x) if S x.

(iii) f(lim(S)) C.lim(f(S)), for each Lfuzzy net S in LX .

Proof: (i) = (ii) Let x M(LX) and S = {xnn : n D} be an Lfuzzy net in

LX which converges to x. Let CRf(x), by (i) f1() Rx . Since S x, then

(n D)(m D, m n)(S(m) f1()). Then f(S(m)) ff1() = . Thus

f(S(m)) . Hence f(S)C

f(x).

(ii) = (iii) Let x lim(S), then f(x) f(lim(S)), by (ii) f(x)

C.lim(f(S)). Thus f(lim(S)) C.lim(f(S)).

(iii) = (i) Let LY be Lfuzzy Cclosed and x

M(LX) with x

c(f1()),

by Theorem 2.15, there exists an Lfuzzy net S in f1() which converges to x. Thus

x lim(S) and so f(x) f(lim(S)). By (iii), f(x) f(lim(S)) C.limf(S) and

so, f(S)C

f(x). Since S is Lfuzzy net in f1(), then for each n D, S(n) f1()

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and so f(S(n)) ff1() . Hence f(S(n)) for each n D. Thus f(S) is

Lfuzzy net in . So we have f(S)C

f(x) and f(S) is Lfuzzy net in so by Proposition

3.10 (iii), f(x) C.c(). But since is Cclosed, so = C.c(). Thus f(x) .

Hence x f1(). So c(f1()) f1(). Hence f1() . Then f is Lfuzzy

Ccontinuous.


conditions are equivalent:


(ii) For each x M(LX) and each Lfuzzy ideal L in LX which converges to x in

LX , f(L) C-converges to f(x), where f(L) = { LY : L such that for

any x M(LX), f(x) if x } is an Lfuzzy ideal in LY .

(iii) f(lim(L)) C.lim(f(L)), for each Lfuzzy ideal L in LX .

Proof: The proof is similar to that of Theorem 5.4.

Similarly, we have the following result.


conditions are equivalent:


(ii) For each x M(LX) and each Lfuzzy ideal L in LX which converges to x in

LX , then (f(L)) C-converges to f(x).

(iii) f(lim(L)) C.lim((f(L))), for each Lfuzzy ideal L in LX .

References

[1] Chen, S.L., Wang, X.G.: Lfuzzy Ncontinuous mappings. J. Fuzzy Math., 4 (3). 621629

(1996).

[2] Chen, S.L., Chen, S.T.: A new extension of fuzzy convergence. Fuzzy Sets and Systems,109, 199204 (2000).

[3] Dang, S., Behera, A., Nanda, S.: Fuzzy Ccontinuous functions. J. Fuzzy Math., 2 (1),187196 (1994).

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[4] Garg, G. L., Kumar, B.: Cclosed functions. Math. Vesnik, 41, 233238 (1989).

[5] Gentry, K. R., Hoyle, H. B.: III, C-continuous functions. Yokohama Math. J., 18 (2), 7176

(1970).

[6] Liu, Y. M., Luo, M. K.: Fuzzy StoneCechtype compactifications. Fuzzy Sets and Systems,33, 355372 (1989).

[7] Liu, Y. M., Luo, M. K.: Separation in latticevalued induced spaces. Fuzzy Sets andSystems, 36, 5566 (1990).

[8] Mukherjee, M. N., Sinha, S. P.: Almost compact fuzzy sets in fuzzy topological spaces.Fuzzy Sets and Systems, 38, 386396 (1990).

[9] Nouh, A. A.: On Qcompactness in Lfuzzy topological spaces. Submitted

[10] Wang, G. J.: A new fuzzy compactness defined by fuzzy nets. J. Math. Anal. Appl., 94,

5967 (1983).

[11] Wang, G. J.: Generalized topological molecular lattices. Scientia Sinica, (Ser. A) 27 (8),

785793 (1984).

[12] Wang, G. J.: Theory of Lfuzzy topological spaces. Shaanxi Normal University Press. Xian1988.

[13] Yang, Z. Q.: Ideal in topological molecular lattices. Acta Mathematica Sinica, 29 (2), 276279 (1986).

[14] Zhao, D. S.: The N-compactness in Lfuzzy topological space. J. Math. Anal. Appl., 128,

6479 (1987).

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Ali Ahmed NOUHMathematics Department,Faculty of Science,Mansoura University-EGYPT

Received 22.06.2001

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