Special Case of Intuitionistic Fuzzy Bitopological Spaces · Intuitionistic fuzzy . ã F. closed. sets and separation axioms in Intuitionistic fuzzy bitopological. space . 1 ± Introduction

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

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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 .


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) .


are the intuitionistic sets corresponding to empty set and

the entire universe respectively .


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 .


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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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 }


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 .


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 .


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) , { } .


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 = {< a , 0 , 1> , }

C = {< a , 0.1 , 0.2 > , }

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 .


If the intuitionistic fuzzy bitopological space (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space , then (X , 𝜏 𝜏 )

is IF𝜆 𝜏 space .

Proof :-


Theorem 3.3.9 :-

If (X , 𝜏 𝜏 ) is IF𝜆 𝜏 space , then it is hereditary property .

Proof :-


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


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.


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 = 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 .


(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 = , 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 .


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 = , 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 .

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Special Case of Intuitionistic Fuzzy Bitopological Spaces · Intuitionistic fuzzy . ã F. closed. sets and separation axioms in Intuitionistic fuzzy bitopological. space . 1 ± Introduction