Special Case of Intuitionistic Fuzzy Bitopological Spaces Alaa Saleh Abed 1 , Yiezi Kadhum Mahdi Al-talkany 2 1,2 Department 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 , ) intuitionistic fuzzy bitopological space . After that a new intuitionistic fuzzy open set defined in this space which is called the intuitionistic fuzzy open set in the intuitionistic fuzzy bitopological space (X , ) and denoted by IFopen 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 open sets , Intuitionistic fuzzy 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 open set ( IF closed set ) in Intuitionistic fuzzy bitopological spaces (X , ) , which is the special case of Intuitionistic fuzzy bitopological spaces ( X ,) . After that the separation axioms are studying with some theorems and properties about them by using the definition of IF open set . International Journal of Pure and Applied Mathematics Volume 119 No. 10 2018, 313-330 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu 313
18
Embed
Special Case of Intuitionistic Fuzzy Bitopological Spaces · Intuitionistic fuzzy . ã F. closed. sets and separation axioms in Intuitionistic fuzzy bitopological. space . 1 ± Introduction
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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 .
International Journal of Pure and Applied MathematicsVolume 119 No. 10 2018, 313-330ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu
313
2
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 ) .
International Journal of Pure and Applied Mathematics Special Issue
314
3
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 } .
International Journal of Pure and Applied Mathematics Special Issue
315
4
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 )
International Journal of Pure and Applied Mathematics Special Issue
316
5
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 ) .
International Journal of Pure and Applied Mathematics Special Issue
317
6
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))
International Journal of Pure and Applied Mathematics Special Issue
318
7
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
International Journal of Pure and Applied Mathematics Special Issue
319
8
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 :-
International Journal of Pure and Applied Mathematics Special Issue
320
9
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
International Journal of Pure and Applied Mathematics Special Issue
321
10
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 :-
International Journal of Pure and Applied Mathematics Special Issue
322
11
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 :
International Journal of Pure and Applied Mathematics Special Issue
323
12
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
International Journal of Pure and Applied Mathematics Special Issue
324
13
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 .
International Journal of Pure and Applied Mathematics Special Issue
325
14
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 .
International Journal of Pure and Applied Mathematics Special Issue
326
15
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 :-
International Journal of Pure and Applied Mathematics Special Issue
327
16
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