Special Case of Intuitionistic Fuzzy Bitopological Spaces · Intuitionistic fuzzy . ã F. closed....
Transcript of Special Case of Intuitionistic Fuzzy Bitopological Spaces · Intuitionistic fuzzy . ã F. closed....
Special Case of Intuitionistic Fuzzy Bitopological Spaces
Alaa Saleh Abed1, Yiezi Kadhum Mahdi Al-talkany
2
1,2Department of mathematics, Faculty of Education for Girls, Iraq.
Abstract :
Our research concluding a study of a special case of intuitionistic fuzzy bitopological spaces
were used here to create space which is called (X , 𝜏 𝜏 ) intuitionistic fuzzy bitopological
space . After that a new intuitionistic fuzzy open set defined in this space which is called the
intuitionistic fuzzy 𝜆 open set in the intuitionistic fuzzy bitopological space (X , 𝜏 𝜏 ) and
denoted by IF𝜆 open set . Also the separation axiams in the intuitionistic fuzzy
bitopological space and the separation axiams in the special case are studied with some
theorems and properties
Keywords : Intuitionistic fuzzy bitopological space , Intuitionistic fuzzy 𝜆 open sets ,
Intuitionistic fuzzy 𝜆 closed sets and separation axioms in Intuitionistic fuzzy bitopological
space
1 – Introduction
After the introduced of fuzzy sets by zadeh [10] in 1965 and fuzzy topology by chang [4] in
1967 , there have been a number of generalizations of this fundamental concept . The notion
of intuitionistic fuzzy sets introduced by Atanassov [9] in 1983 .
Using the notion of intuitionistic fuzzy sets Coker [5] introduced the notion of Intuitionistic
fuzzy bitopological space . Coker and Demirc : [6,7] introduced the basic definitions and
properties of intuitionistic fuzzy topological space in Sastak's sense , which is generalized
form of fuzzy topological space developed by sastak [2,3] .
The notion of an Intuitionistic fuzzy bitopological spaces and the Intuitionistic fuzzy ideal
bitopological spaces studied by Mohammed [11] in 2015 . In this paper we introduce the
definition of IF 𝜆 open set ( IF 𝜆 closed set ) in Intuitionistic fuzzy bitopological spaces
(X , 𝜏 𝜏 ) , which is the special case of Intuitionistic fuzzy bitopological spaces ( X ,𝜏 𝜏 ) .
After that the separation axioms are studying with some theorems and properties about them
by using the definition of IF 𝜆 open set .
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2 – Preliminaries :-
Definition 2.1 [10] :-
Let X be a non – empty set and I = [0,1] be the closed interval of the real numbers . A fuzzy
subset 𝜇 of X is defined to be membership function , such that 𝜇 for every
. The set of all fuzzy subsets of X denoted by .
Definition 2.2 [9] :-
An intuitionistic fuzzy set (IFS , for short ) A is an object ham the form :
{ 𝜇 𝜈 } , where the function 𝜇 , 𝜈 denote the
degree of membership and the degree of non – membership of each element to the set A
respectively , and 𝜇 𝜈 , for each . The set of all intuitionistic fuzzy
sets in X denoted by IFS(X) .
Definition 2.3 [7] :-
are the intuitionistic sets corresponding to empty set and
the entire universe respectively .
Definition 2.4 [5] :-
Let X be a non – empty set . An intuitionistic fuzzy point ( IFP, for short ) denoted by is
an intuitionistic fuzzy set have the form {
, where is a fixed
point , and satisfy .
The set of all IFPs denoted by IFP(X) . If , we say that if and only if
𝜇 and 𝜈 , for each .
Definition 2.5 [7] :-
An intuitionistic fuzzy topology ( IFT , for short ) on a non – empty set X is a family 𝜏 of an
intuitionistic fuzzy sets in X such that .
𝜏
(ii) 𝜏
(iii) 𝜏 , for any arbitrary family { 𝜏} 𝜏 in this case the pair 𝜏 is called
an intuitionistic fuzzy topological space ( IFTS , for short ) .
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Definition 2.6 [7] :-
Let 𝜏 be an intuitionistic fuzzy topological space and { 𝜇 𝜈 }
be an intuitionistic fuzzy set in X . Then an intuitionistic fuzzy interior and intuitionistic fuzzy
closure of A are respectively defined by
Int(A) = = { I an IFos in X and }
Cl(A) = ̅ { is an IFcs in X and }
Definition 2.7 [12] :-
An IFS N in an IFTS 𝜏 is called an intuitionistic fuzzy neighborhood (IFN , in short ) of
an IFP if 𝜏 such that .
Proposition 2.8 [12] :-
Let 𝜏 be an IFTS . Then an IFS A in X is an IFOS iff A is an IFN of each IFP .
Definition 2.9 [8] :-
An IFS 𝜇 𝜈 in an IFTS 𝜏 is said to be an intuitionistic fuzzy open set (
IF OS in short ) if int(cl(int(A))) , while IFS A is said to be intuitionistic fuzzy
closed set ( IF CS in short) if cl(int(cl(A))) A .
Definition 2.10 [11] :-
Let 𝜏 and 𝜏 be two intuitionistic fuzzy topologies on a non – empty set X . The triple
𝜏 𝜏 is called an intuitionistic fuzzy bitopological space (IFBTS , for short) , every
member of 𝜏 is called 𝜏 intuitionistic fuzzy open set 𝜏 IFOS ) , { } and the
complement of 𝜏 IFOS is 𝜏 intuitionistic fuzzy closed set 𝜏 IFCS) , { } .
Definition 2.11 [11] :-
Let 𝜏 𝜏 be an IFBTS and . Then intuitionistic fuzzy interior and
intuitionistic fuzzy closure of A with respect to 𝜏 { } are defined by :
𝜏 int(A) = { 𝜏 }
𝜏 { 𝜏 } .
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Propoition 2.12 [11] :-
Let 𝜏 𝜏 be an IFBTS and A . Then we have :
𝜏 int(A) ; { } .
(ii) 𝜏 int(A) is a largest 𝜏 IFos contains in A .
(iii) A is a 𝜏 IFOS if and only if 𝜏 int(A) = A .
(iv) 𝜏 int(𝜏 int(A)) = 𝜏 int(A) .
(v) A 𝜏 cl(A) , { } .
(vi) 𝜏 cl(A) is a smallest 𝜏 IFCS contains A .
(vii) A is a 𝜏 IFCS if and only if 𝜏 cl(A) = A .
(viii) 𝜏 cl(𝜏 cl(A)) = 𝜏 cl(A) .
(ix) [𝜏 int(A) = 𝜏 cl { } .
(x) 𝜏 = 𝜏 int { } .
Proof :- clearly
3 – Main Results
This part of this paper including three section , in the first section we introduce a new relation
to define the 𝜏 intuitionistic fuzzy open set in the intuitionistic fuzzy bitopological space
which called intuitionitic fuzzy 𝜆 open set .
Section two includes some definitions of separation axioms with respect to intuitionistic fuzzy
bitopological space .
In section three we used the new definition of intuitionistic fuzzy open set to provid a new
definition of separation axioms .
3.1. Intuitionistic Fuzzy 𝝀 open Set
Remark (3.1.1) :-
IF is the intuitionistic fuzzy set of an intuitionistic fuzzy open set (IF OS for short )
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Example (3.1.2) :-
Let X = {a , b , c } , 𝜏 = { , A } where
A = { }
B = { } ,
= { } ,
= { } ,
And F = { } .
F is IF OS , since ( int(cl(int(A))) , using definition (2.9) .
And , A , B are IF OSs (by using definition (3.1) in [8] ) . Then
𝜏 { } .
Theorem 3.1.3 :-
If A is intuitionistic fuzzy 𝜏 open set (intuitionistic fuzzy set in the intuitionistic fuzzy
Topological space (X , 𝜏)) and B is intuitionistic fuzzy open set (The intuitionistic fuzzy
open set in the intuitionistic fuzzy topological space (X , 𝜏 ) . Then and is
intuitionistic fuzzy open set .
Proof :-
By theorem 3.5 in [8] . If A is T – intuitionistic fuzzy open set , then A is intuitionistic fuzzy
open set . Then for A B is intuitionistic fuzzy open set and by Lemma 3.4 in [8]
A B is intuitionistic fuzzy open set since A B is an intuitionistic fuzzy open set .
Definition 3.1.4 :-
Let 𝜏 be an intuitionistic fuzzy topological space and (X , 𝜏 ) is the intuitionistic fuzzy
topology on X . Then (X , 𝜏 𝜏 ) is an intuitionistic fuzzy bitopological space .
Intuitionistic fuzzy subset A of X is said to be IF 𝜆 open set if and only if there exist U , is
IF OS , such that and 𝜏 cl(U) . Where 𝜏 cl(U) is the closure with respect to
the intuitionistic fuzzy topological space (X , 𝜏 ) .
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Remark 3.1.5 :-
An intuitionistic fuzzy subset A of X is said to be IF 𝜆 closed set if and only if its
complement of IF 𝜆 open set .
Theorem 3.1.6 :-
The family of all intuitionistic fuzzy 𝜆 open sets is an intuitionistic fuzzy topological space .
Proof :-
Since is 𝜏 IFOS by theorem 3.5 in [8] is 𝜏 –IF OS
such that and and 𝜏 int( ) and 𝜏 – int( ) .
Now to prove the intersection and the arbitrary union is IF𝜆 open sets .
Let A and B are two IF𝜆 open sets , then there exist U , W are 𝜏 IF open sets such that
A U , B W and A 𝜏 int(U) and B 𝜏 int(W)
A B 𝜏 –int(U) 𝜏 – int(W) = 𝜏 – int(U W) then A B I IF𝜆 open sets .
Let is IF𝜆 open sets , , I is arbitrary , then there exist are 𝜏 IF open sets for
each I such and 𝜏 – int( ) , then 𝜏 – int( ) 𝜏 int( ) and then
is IF𝜆 open sets . From the above discussion we get (X , IF𝜆 open set) is
intuitionistic fuzzy topological space .
Theorem 3.1.7 :-
Let (X , 𝜏 𝜏 ) be an intuitionistic fuzzy bitopological space then every IF – open set is
IF𝜆 open set
Proof :-
Let A is 𝜏 IF – open set (by definition 3.1 in [8] ) ,
then A is IF open set ,
Then A = 𝜏 int(A) (proposition (2.12))
And A 𝜏 cl(A) (proposition (2.12))
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A is IF open set and A 𝜏 – cl(A)
A is IF𝜆 open set (by definition (3.1.4))
Definition 3.1.8 :-
Let (X , 𝜏 𝜏 ) be an intuitionistic fuzzy bitopological space and let Y be an intuitionistic
fuzzy subset of X , then the intuitionistic fuzzy relrelatively topology of Y with respect to 𝜏 and
𝜏 defined by ;-
{ 𝜏 }
{ 𝜏 }
Remark 3.1.11 :-
We can define IF𝜆 open set with respect to the intuitionistic fuzzy subspacey in this way
IF𝜆
open = { U , U is IF𝜆 open set}
Theorem 3.1.10 :-
Let (X , 𝜏 𝜏 ) be intuitionistic fuzzy bitopological space , and Y be an intuitionistic fuzzy
subset of X if A Y is IF𝜆 open set (IF𝜆 closed set ) in X . Then A is IF𝜆 open set (IF𝜆
closed set ) inY .
Proof :-
We use the same proof in intuitionistic fuzzy topological space with replacing the
intuitionistic fuzzy open set by intuitionistic fuzzy 𝜆 open set .
Theorem 3.1.11 :-
Let (X , 𝜏 𝜏 ) be an intuitionistiv fuzzy bitopological space and Y is intuitionistic fuzzy
subset of X then :
i – An intuitionistic fuzzy subset A of Y is IF𝜆 closed set in Y if and only if there exist K is
IF𝜆 closed set in X such that : A = K Y
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ii – For every A Y , 𝜏 (A) = 𝜏 (A) ,
Proof :-
We use the same proof in intuitionistic fuzzy topological space with replacing the
intuitionistic fuzzy open set by the IF𝜆 open set .
3 – 2 Separation Axioms in the Intuitionistic Fuzzy Bitopological Space (X , 𝝉 𝝉 )
Definition 3.2.1 :-
The intuitionistic fuzzy bitopolgical space (X , 𝜏 , 𝜏 ) is said to be intuitionistic fuzzy
𝜏 - (IF𝜏 space for short ) if
𝜇 𝜈 , V = (𝜇 𝜈 ) 𝜏 𝜏 such that
(𝜇 𝜈 )(x) = (1,0) , (𝜇 𝜈 )(y) = (0,1) or
(𝜇 𝜈 )(x) = (0,1) , (𝜇 𝜈 (y) = (1,0)
Definition 3.2.2 :-
The intuitionistic fuzzy bitopological space (X , 𝜏 𝜏 ) is said to be intuitionistic fuzzy
𝜏 space (IF𝜏 space for short) if , 𝜇 𝜈 ;
V = (𝜇 𝜈 ) 𝜏 𝜏 such that (𝜇 𝜈 )(x) = (1,0) , (𝜇 𝜈 and
(𝜇 𝜈 (x) = (0,1) , (𝜇 𝜈 (y) = (1,0)
Definition 3.2.3 :-
The intuitionitic fuzzy bitopological space (X , 𝜏 𝜏 ) is said to be intuitionistic fuzzy
𝜏 space (IF𝜏 space for short) . If pair of distinct intuitionistic fuzzy points
in X , 𝜇 𝜈 and V = (𝜇 𝜈 such that and
U V =
Definition 3.2.4 :-
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An IFBTS (X , 𝜏 𝜏 ) is called intuitionistic fuzzy regular space (IFRS for short) if for each
and each IFCS C in 𝜏 such that there exist IFOSs M and N in
𝜏 𝜏 such that and C N .
An IFBTS (X , 𝜏 𝜏 ) is called intuitionistic fuzzy 𝜏 space (IF𝜏 space for short) if it is
IF𝜏 space and IFR – space .
Definition 3.2.5 :-
An IFBTS (X , 𝜏 𝜏 ) is called intuitionistic fuzzy normal space if for each pair of IFCSs
and in 𝜏 such that , there exits IFOSs and in 𝜏 𝜏 such that
and .
An IFBTS (X , 𝜏 𝜏 ) is called intuitionistic fuzzy space (IF𝜏 space for short) if it is
IF𝜏 space and IF – normal space .
3 – 3 Separation Axioms in The Intuitionistic Fuzzy Bitopological Space (X 𝝉 𝝉 )
In this part we will define all the above separation axioms with respect to IF𝜆 open set as
follows
Definition 3.3.1 :-
The intuitionistic fuzzy bitopological space (X , 𝜏 𝜏 ) is said to be IF𝜆 𝜏 if and only if for
each there exist IF𝜆OSs U and V where U = (𝜇 𝜈 ) and V = (𝜇 𝜈 ) ,
then (𝜇 𝜈 𝜇 𝜈 or (𝜇 𝜈 𝜇 𝜈
Example 3.3.2 :-
Let (X , 𝜏 𝜏 ) is IFBTS , then (X , 𝝉 𝝉 ) is IF𝜆 𝜏 space
Let X = { a , b , c , d } , 𝜏 { A , B} , 𝜏 { A , B , C} ,where
A = {< a , 1 , 0 > , < b , 0 , 1 >}
B = {< a , 0 , 1> , < b , 1 , 0 >}
C = {< a , 0.1 , 0.2 > , < b , 0.2 , 0.1>}
Then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space
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Proposition 3.3.3 :-
If the intuitionistic fuzzy topological space (X , 𝜏 ) is IF𝜏 space , then (X , 𝜏 𝜏 ) is
IF𝜆 𝜏 space
Proof :-
Let , Since (X , 𝜏 ) is IF 𝜏 space then there exist IFOSs (by the definition
3.1 in [1] )
U = (𝜇 𝜈 ) , V = (𝜇 𝜈 𝜏 such that (𝜇 𝜈 𝜇 𝜈 or
(𝜇 𝜈 𝜇 𝜈 by theorem (3.1.7) U and V are IF𝜆 open sets
then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Proposition 3.3.4:-
If (X , 𝜏 𝜏 ) I IF𝜆 𝜏 space , then it is hereditary property .
Proof :-
The proof exist by definition .
Definition 3.3.5 :-
The intuitionistic fuzzy bitopological space (X , 𝜏 𝜏 ) is said to be intuitionistic fuzzy
𝜆 𝜏 space (IF𝜆 𝜏 space , for short ) if and only if for each there
exist IF𝜆OSs U,V, where U = (𝜇 𝜈 V = (𝜇 𝜈 , such that (𝜇 𝜈
(𝜇 𝜈 and (𝜇 𝜈 𝜇 𝜈 .
Example 3.3.6:-
(X , D , ) is an IF𝜆 𝜏 space , where . D is the intuitionitic fuzzy discrete topology .
Theorem 3.3.7:-
If the inintuitionitic fuzzy topological space (X , 𝜏 ) is the IF𝜏 pace , then (X , 𝜏 𝜏 ) is
IF𝜆 𝜏 space .
Proof :-
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Let x , y , since (X , 𝜏 ) is IF𝜏 space , (by using definition (3.1) in [1] ) There
exist IFOSs U = (𝜇 𝜈 V = (𝜇 𝜈 𝜏 such that (𝜇 𝜈 , (𝜇 𝜈
and (𝜇 𝜈 𝜇 𝜈
By theorem (3.1.7) U,V are IF𝜆 open set , then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Proposition 3.3.8 :-
If the intuitionistic fuzzy bitopological space (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space , then (X , 𝜏 𝜏 )
is IF𝜆 𝜏 space .
Proof :-
The proof exist by definition .
Theorem 3.3.9 :-
If (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space , then it is hereditary property .
Proof :-
The proof exist by definition .
Theorem 3.3.10 :-
The intuitionistic fuzzy bitoological space (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space if and only if every
intuitionistic fuzzy singleton subset A of X is IF𝜆 closed set .
Proof :-
Let a , b X , such that and let A = { 𝜇 𝜈 : a X}
B = { 𝜇 𝜈 : b X } are IF singleton set and IF𝜆 closed sets then X – A
and X – B are IF𝜆 open sets such that
(𝜇 𝜈 𝜇 𝜈 and
(𝜇 𝜈 𝜇 𝜈 . then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 pace
Conversely :
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Let and and , since (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space then there exist
IF𝜆 open set U such that
𝜇 𝜈 𝜇 𝜈 , then , then X – A is IF𝜆 open set ,
then A is IF𝜆 closed set .
Definition 3.3.11 :-
The intuitiopnistic fuzzy bitopological space (X , 𝜏 𝜏 ) is said to be IF𝜆 𝜏 space if and
only if each pair of distinct intuitionistic fuzzy points , in X , IF𝜆OSs U and V ,
where U = (𝜇 𝜈 𝜇 𝜈 , such that and .
Example 3.3.12:-
Clearly that in example (3.3.2) X is IF𝜆 𝜏 space and IF𝜆 𝜏 space and IF𝜆
𝜏 space .
Example 3.3.13 :-
(X , ) is IF𝜆 𝜏 space , where D is the intuitionistic fuzzy discrete topology
Proposition 3.3.14 :-
If (X , 𝜏 ) is IF𝜏 space , then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Proof :-
Since (X , 𝜏 ) is IF𝜏 space (by using definition 3.1 in [1] )
pair of distinct intuitionistic fuzzy points in X , IFOSs U , V 𝜏 such that
and .
By theorem (3.1.7) U , V are IF𝜆 open sets
Then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Theorem 3.3.15 :-
Let IFBTS (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space is hereditary property
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Proof :-
The proof exist by definition.
Definition 3.3.16 :-
An IFBTS (X , 𝜏 𝜏 ) will be called intuitionistic fuzzy 𝜆 regular space (IF𝜆 R – space ,
for short) if for each IFP X and each 𝜏 IFCS C such that , there
exist IF𝜆 open sets M, N such that and C . An IFBTS (X , 𝜏 𝜏 ) is called
intuitionistic fuzzy 𝜆 𝜏 space (IF𝜆 𝜏 space , for short) if and only if it is IF𝜆 𝜏
space and IF𝜆 R – space .
Theorem 3.3.17 :-
If (X , 𝜏 ) is IF𝜏 space , then (X , 𝜏 𝜏 ) is IF𝜆 R – space .
Proof :-
Let a,b such that and A = < a , 𝜇 𝜈 ,
B = < b , 𝜇 , 𝜈 > are IF singleton sets . Sine (X , 𝜏 ) is IF𝜏 space , then by
theorem (3.3.7) (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space . And by theorem (3.3.10) A and B are
IF𝜆 closed sets and since (X , 𝜏 ) is IF – regular space there exist IF 𝜏 – open sets M , N
such that and and by theorem (3.1.7) M , N IF𝜆 open set , then (X , 𝜏 𝜏 ) is
IF𝜆 regular space .
Proposition 3.3.18 :-
(X , 𝜏 𝜏 ) IF𝜆 regular space is hereditary property .
Proof :-
Let (X , 𝜏 𝜏 ) be an IF𝜆 regular space and be an intuitionistic fuzzy subset of X , let
A be IF 𝜏 closed set and and , then , since (X , 𝜏 𝜏 ) is IF𝜆
regular space there exist IF𝜏 closed set F such that , then
since (X , 𝜏 𝜏 ) is IF𝜆 regular space there exist IF𝜆 open sets M , N such that ,
. Then (Y , 𝜏 𝜏
) is IF𝜆 regular space .
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Theorem 3.3.19 :-
If (X , 𝜏 ) is IF𝜏 space , then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Proof :-
If (X , 𝜏 ) is IF𝜏 space by theorem (3.3.7) (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space and also by
theorem (3.3.17) (X , 𝜏 𝜏 ) is IF𝜆 regular space , then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Theorem 3.3.20 :-
If (X 𝜏 ) is IF𝜏 space , then every IF𝜆 𝜏 space (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Proof :-
Let such that and A = < a , 𝜇 𝜈 > ,
B = < b , 𝜇 𝜈 > , since (X 𝜏 ) is IF𝜏 space , then A , B are IF 𝜏 closed sets
and . Since (X , 𝜏 𝜏 ) is IF𝜆 regular space there exist U , V are IF𝜆 open
sets such that and and therefor (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Theorem 3.3.21 :-
If (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space , then it is hereditary property .
Proof :-
Let (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space and Y is IF subset of X , since (X , 𝜏 𝜏 ) is IF𝜆 𝜏
space and IF𝜆 regular space are hereditary property . Then (Y , 𝜏 ) is hereditary
property .
Definition 3.3.22 :-
An IFBTS (X , 𝜏 𝜏 ) will be called intuitionistic fuzzy 𝜆 normal space (IF𝜆 normal space
, for short) if for each 𝜏 IFCSs and , such that there exist IF𝜆 open
sets and such that ( i = 1,2) and .
And IFBTS (X , 𝜏 𝜏 ) is called intuitionistic fuzzy 𝜆 𝜏 space (IF𝜆 𝜏 space , for
short ) if and only if it is IF𝜆 𝜏 space and IF𝜆 normal space .
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Theorem 3.3.23 :-
If (X , 𝜏) is IF normal space , then (X , 𝜏 𝜏 ) is IF𝜆 normal space .
Proof :-
Let , such that A = < a , 𝜇 𝜈 > , B = < b , 𝜇 , 𝜈 > , then A
, B are IF𝜏 closed sets since every IF𝜏 open set is IF𝜏 open set A , B are IF Ss and
, since (X , 𝜏 ) is IF – normal space there exist U , V are IF𝜆 open sets satisfies
and and , since every IF𝜏 – open set is IF𝜆 open set , then (X ,
𝜏 𝜏 ) is IF –𝜆 normal space .
Theorem 3.3.24 :-
In the IFBTS (X , 𝜏 𝜏 ) IF𝜆 normal space is hereditary property .
Proof :-
Let (X , 𝜏 𝜏 ) is IF𝜆 normal space and intuitionitic fuzzy subset Y of X , let A , B are
IF𝜏 closed sets such that , then there exist IF𝜏 closed sets F and K such
that A = , B = and . Since (X , 𝜏 𝜏 ) is IF𝜆 normal
space there exist U , V are IF𝜆 open sets such that and .
From that we have where and are IF𝜆 open
sets and . then therefore (Y , 𝜏 𝜏
) is
IF𝜆 normal space .
Theorem 3.3.25 :-
If (X , 𝜏) is IF𝜏 space , then (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Proof :-
Since (X , 𝜏 ) is IF𝜏 space and IF – normal space . By theorem (3.3.7) , (X , 𝜏 𝜏 ) is
IF𝜆 𝜏 space and by theorem (3.3.23) , (X , 𝜏 𝜏 ) is IF𝜆 normal space , then by
definition (3.3.22) (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Theorem 3.3.26 :-
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If (X 𝜏 ) is IF𝜏 space , then every IF𝜆 𝜏 space (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space .
Proof :-
Let (X , 𝜏 𝜏 ) be IF𝜆 𝜏 space by assume , let F is IF 𝜏 closed sets and
Since (X , 𝜏 ) is IF𝜆 𝜏 space then A = < a , 𝜇 𝜈 > is IF𝜏 closed set such
that , since (X , 𝜏 𝜏 ) is IF𝜆 normal space , then there exist U , V are IF𝜆
open sets such that and and therefore (X , 𝜏 𝜏 ) is IF𝜆 𝜏
space .
Theorem 3.3.27 :-
IF𝜆 𝜏 space of theIFBTS (X , 𝜏 𝜏 ) is hereditary property .
Proof :-
Let (X , 𝜏 𝜏 ) be an IF𝜆 𝜏 space and Y be IF subset of X . By definition (3.3.22) , (X ,
𝜏 𝜏 ) is IF𝜆 𝜏 space and IF𝜆 normal space are hereditary property , then Y is
IF𝜆 𝜏 space and IF𝜆 normal space , they (Y , 𝜏 𝜏
) is IF𝜆 𝜏 space .
Reference
1 – Amitkumer singh and Rekhasrivastava , " Separation Axioms in Intuitionistic Fuzzy
Topological Space " , Advances in Fuzzy Systems , volume 2012 (2012) , Article ID
604396 , 7 pages .
2 – A. Sostak , " On a Fuzzy Topological structure " , Rend. Circ. Mat Palermo (2) supp 1. ,
no. 11 , 89 – 103 , 1985 .
3 – A. Sostak , " on Compactness and connectedness degress of fuzzy set in fuzzy topological
Space " , General Topology and It's Relations to Modern Analysis and Algebra , VICP
rague , 1986. Re . EXP. Math. , Vol . 16, Heldermann, Berlin , PP. 519 – 532 , 1988
4 – C.L chang , " Fuzzy Topological space " , J. Math. Anal. App 1. 24, 182 – 190 , (1968)
5 – D. Coker and M. Demirci , " on intuitionistic fuzzy points " . NIFSI . 2, 79 – 84 . (1995) .
International Journal of Pure and Applied Mathematics Special Issue
328
17
6 – D. Coker and M. Demirci , " An intuitionistic to intuitionistic fuzzy topological space in
Sostak's sense " , BUSEEAL 67 , 67 – 76 , 1996 .
7 – D. Coker , " An introduction to intuitionistic fuzzy topological space " , fuzzy sets and
systems . 88, 81 – 89 . (1997) .
8 – Joungkonjeon, Youngbaejun, " andjinhan park ; intuitionistic fuzzy alpha – continuity and
intuitionistic fuzzy pre continuity " , International Journal of Mathematics and
Mathematical Sciences 2005 : 19(2005) . 3091 – 3101 .
9 – K. Atanassov , " Intuitionistic fuzzy sets " , Fuzzy sets Systems. 20 , 87 – 96 . (1986) .
10 – L. A. Zadeh , " Fuzzy Sets Information and Control " . 8 , 338 – 353 . (1965) .
11 – Mohammed Jassim Tuaimah , Mohammed Jassim Mohammed " on intuitionistic fuzzy
ideals bitopological space " , journal of advances in mathematics Vol. 11 , No. 6 , ISSN
2347 – 1921 (2015) .
12 – S. J. Lee , E. P – Lee , " The Category of intuitionistic fuzzy topological spaces " , Bull .
Korean Math . Soc. 37 , 63 – 76 (2000) .
International Journal of Pure and Applied Mathematics Special Issue
329
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