# Separation Axioms in Intuitionistic Fuzzy Topological Spaces

Authors: Amit Kumar Singh; Rekha Srivastava
Journal: Advances in Fuzzy Systems
 (2012)

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## Abstract

In this paper we have studied separation axiomsTi, i=0,1,2 in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functors ℬ:IF-Top→BF-Top and 𝒟:BF-Top→IF-Top and observe that 𝒟 is left adjoint to ℬ.


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## Body

## 1.   Introduction

Fuzzy sets were introduced by Zadeh [1] in 1965 as follows: a fuzzy set A in a nonempty set X is a mapping from X to the unit interval [0,1], and A(x) is interpreted as the degree of membership of x in A. Atanassov [2] generalized this concept and introduced intuitionistic fuzzy sets which take into account both the degrees of membership and of nonmembership subject to the condition that their sum does not exceed 1. Çoker [3] subsequently initiated a study of intuitionistic fuzzy topological spaces.In this paper we have searched for appropriate definitions of the separation axiomsTi, i=0,1,2 in intuitionistic fuzzy topological spaces.Hausdorffness in an intuitionistic fuzzy topological space has been introduced earlier by Çoker [3], Bayhan and Çoker [4], and Lupianez [5]. In [4], the authors have given six possible definitions of Hausdorffness including that given in [3], and a comparative study has been done. In this paper we have introduced another definition which generalizes the corresponding definition in a fuzzy topological space given in [6]. Our definition is more general than those given in [3, 5], and it turns out to be equivalent to FT2(vi) in [4].T
1-ness in an intuitionistic fuzzy topological space has been defined earlier in [4] in six possible ways. Out of those, we have chosen FT1(ii) as it generalizes the most appropriate definition of T1-ness in a fuzzy topological space (cf. definition 5.1, [7]). We have also introduced a suitable definition of T0-ness in an intuitionistic topological space.The appropriateness of the definitions has been established by proving several basic desirable results; for example, they satisfy hereditary, productive, and projective properties. We have also shown that the functorℬ: IF-Top → BF-Top preserves these separation properties.


## 2. Preliminaries

ThroughoutX denotes a nonempty set, I denotes the unit interval [0,1], and I0 and I1 denote the intervals (0,1] and [0,1), respectively. A fuzzy set in X is a function from X to I. The collection of all fuzzy sets in X is denoted by IX. For any A∈IX, A′ denotes the fuzzy complement of A, and the constant fuzzy set in X, taking value α∈I, is denoted by α_. A crisp subset of X will be identified with its characteristic function. If Y⊆X, then A∈IY will be identified with the fuzzy set in X which takes the same value as A if x∈Y and zero if x∉Y.Definition 1 (Atanassov [2]).
LetX be a nonempty set. An intuitionistic fuzzy set (IFS, in short)  A is an ordered pair (μA,νA) of fuzzy sets in X. Here (μA,νA)(x) = (μA(x),νA(x)) and μA(x), νA(x), respectively, denote the degree of membership and the degree of nonmembership of x∈X to the set A and 0≤μA(x)+νA(x)≤1 for each x∈X.
We identify an ordinary fuzzy setA∈IX with the intuitionistic fuzzy set (A,A′).Definition 2 (Atanassov [2]).
LetX be a nonempty set and A,B be given by (μA,νA) and (μB,νB), respectively,(a)
A⊆BifμA(x)≤μB(x)andνA(x)≥νB(x)forallx∈X,(b)
A=BifA⊆BandB⊆A,(c)
A¯=(νA,μA),(d)
A∩B=(μA∩μB,νA∪νB),(e)
A∪B=(μA∪μB,νA∩νB).Definition 3 (Çoker [3]).
Let{Ai:i∈J} be an arbitrary family of IFSs in X. Then (a)
∩Ai=(∩μAi,∪νAi),(b)
∪Ai=(∪μAi,∩νAi),(c)
0~=(0_,1_), 1~=(1_,0_).Definition 4 (Çoker [3]).
LetX and Y be two nonempty sets and f:X→Y be a function. If A and B be IFSs in X and Y, respectively, then(a)
f(A)=(f(μA),(1-f(1-νA))),(b)
f-1(B)=(f-1(μB),f-1(νB)). It is easy to verify that f-1(∩Ai)=∩f-1(Ai) and f-1(∪Ai)=∪f-1(Ai).Definition 5 (Wong [8]).
A fuzzy pointxr in X is a fuzzy set in X taking value r∈(0,1) at x and zero elsewhere, and x and r are, respectively, called the support and value of x.
A fuzzy pointxr is said to belong to a fuzzy set A  (notation:xr∈A) if r<A(x) (cf. [6]).
Two fuzzy points are said to be distinct if their supports are distinct.Definition 6.
LetX be a nonempty set and x∈X a fixed element in X. If α∈(0,1) and β∈(0,1) are two fixed real numbers such that α+β≤1, then the IFS x(α,β)=(xα,1-x(1-β)) is called an intuitionistic fuzzy point (IFP, in short) in X, and x is called its support. Two IFPs are said to be distinct if their supports are distinct.
Letx(α,β) be an IFP in X and A=(μA,νA) be an IFS in X. Then x(α,β) is said to belong to A  (notation:x(α,β)∈A, in short) if α<μA(x), β>νA(x) (cf. [9]).We identify a fuzzy pointxr in X by the intuitionistic fuzzy point x(r,(1-r)) in X.Proposition 7.
An intuitionistic fuzzy setA in X is the union of all intuitionistic fuzzy points belonging to A.The proof is on similar lines as in [10, Theorem  2.4] and hence is omitted.Replacing fuzzy sets by intuitionistic fuzzy sets in Chang’s definition of a fuzzy topological space, we get the following.Definition 8 (Çoker [3]).
An intuitionistic fuzzy topology (IFT, in short) on a nonempty setX is a family τ of IFSs in X satisfying the following axioms:(1)
0~,1~∈τ,(2)
G1∩G2∈τ,forallGi∈τ,i=1,2,(3)
∪Gi∈τ for any arbitrary family {Gi∈τ:i∈J}.The pair (X,τ)  is called an intuitionistic fuzzy topological space (IFTS, in short), members of τ are called intuitionistic fuzzy open sets (IFOS, in short) in X, and their complements are called intuitionistic fuzzy closed sets (IFCS, in short).Definition 9.
Let(X,τ) be an IFTS. A subfamily ℬ⊆τ is called a base for τ if every U∈τ can be written as a union of members of ℬ.Proposition 10.
Let(X,τ) be an IFTS, and then a subfamily ℬ⊆τ is a base for τ if and only if for all U∈τ and intuitionistic fuzzy point x(α,β)∈U, ∃B∈ℬ such that x(α,β)∈B⊆U.The proof is easy omitted.Definition 11.
Let(X,τ) be an IFTS. Then a subfamily 𝒮⊆τ is called a subbase for τ if the family of finite intersections of members of 𝒮 forms a base for τ.
Given any collection𝒮 of IFSs in X, containing 0~ and 1~, the set τ consisting of arbitrary unions of finite intersections of members of 𝒮 forms an IFT on X. This is the smallest IFT on X containing 𝒮 and is called the IFT generated by 𝒮.Definition 12 (S. J. Lee and E. P. Lee [10]).
An IFSN in an IFTS (X,τ) is called an intuitionistic fuzzy neighborhood (IFN, in short) of an IFP x(α,β) if ∃U∈τ such that x(α,β)∈U⊆N.Proposition 13.
Let(X,τ) be an IFTS. Then an IFS A in X is an IFOS if and only if A is an IFN of each of IFP x(α,β)∈A.The proof is on similar lines as in([10], Theorem  2.6) and hence is omitted.Definition 14 (S. J. Lee and E. P. Lee [10]).
A mapf:(X,τ)→(Y,δ) between IFTSs is called intuitionistic fuzzy continuous if f-1(U)∈τ,forallU∈δ.Definition 15 (Abu Safia et al. [12]).
LetX be a nonempty set and τ1, τ2 be two fuzzy topologies on X. Then (X,τ1,τ2) is called a bifuzzy topological space (BFTS, in short).
A mapf:(X,τ1,τ2)→(Y,δ1,δ2) between two BFTSs is said to be FP continuous if f-1(Ui)∈τi,forallUi∈δi,i=1,2.Definition 16 (Bayhan and Çoker [4]).
LetA=(μA,νA) and B=(μB,νB) be IFSs in X and Y, respectively, and then A×B is the IFS in X×Y defined as follows
(1)A×B=(μA×μB,νA*νB),
where (μA×μB)(x,y)=min(μA(x),μB(y)), forall(x,y)∈X×Y and (νA*νB)(x,y)=max(νA(x),νB(y)), forall(x,y)∈X×Y.This definition can be extended to an arbitrary family of IFSs as follows.If{Ai=(μAi,νAi), i∈J} is a family of IFSs in Xi, then their product is defined as the IFS in ΠXi given by
(2)ΠAi=(ΠμAi,Π*νAi),
where ΠμAi(x)= inf μAi(xi), forallx=Πxi∈X and Π*νAi(x)=supνAi(xi),forallx=Πxi∈X.Definition 17 (Bayhan and Çoker [4]).
Let(xi,τi),i=1,2 be two IFTSs, and then the product IFT τ1×τ2 on X1×X2 is defined as the IFT generated by {pi-1(Ui):Ui∈τi,i=1,2} where pi:X1×X2→Xi, i=1,2  are the projection maps, and the IFTS (X1×X2,τ1×τ2) is called the product IFTS.This definition can be extended to an arbitrary family of IFTSs as follows.Let{(Xi,τi):i∈J} be a family of IFTSs. Then the product intuitionistic fuzzy topology τ on X=ΠXi is the one having {pj-1(Uj):Uj∈τj,j∈J} as a subbase where pj:X→Xj is the jth projection map. (X,τ) is called the product IFTS of the family {(Xi,τi):i∈J}.Definition 18.
A fuzzy topological space(X,τ) is called(a)
T0 if forallx,y∈X, x≠y, ∃U∈τ such that either U(x)=1, U(y)=0, or U(y)=1, U(x)=0,(b)
T1 if forallx,y∈X, x≠y, ∃U,V∈τ such that U(x)=1, U(y)=0, V(y)=1, and V(x)=1,(c)
T2   (Hausdorff) if for all pair of distinct fuzzy points xr,ys in X, ∃U,V∈τ such that xr∈U, ys∈V, and U∩V=0_,(d)
q-T2   (q-Hausdorff) if for any pair of distinct fuzzy points xr, and ys  ∃U,V∈τ such that xr∈U, ys∈τ and U⊆V′. Here definitions (d), (c), (b), and (a) are from [5–7, 13], respectively.Definition 19.
Let(X,τ1,τ2) be a BFTS. Then it is called(a)
T0 if forallx,y∈X,x≠y, ∃U∈τ1∪τ2 such that U(x)=1, U(y)=0 or U(x)=0, U(y)=1,(b)
T1 if forallx,y∈X,x≠y, ∃U∈τ1 and V∈τ2 such that U(x)=1, U(y)=0 and V(x)=0, V(y)=1,(c)
T2 if for all pair of distinct fuzzy points xr,ys in X, ∃U∈τ1, V∈τ2 such that xr∈U, ys∈V and U∩V=0_.Here definitions (a) and (b) are from [14], and (c) is from [15].For the categorical concepts used here, we refer the reader to [16].


## 3.   Separation Axioms in Intuitionistic Fuzzy Topological Spaces

Definition 20.
An IFTS(X,τ) is called(a)
T0 if forallx,y∈X, x≠y, ∃U=(μU,νU), V=(μV,νV)∈τ such that (μU,νU)(x)=(1,0), (μU,νU)(y)=(0,1) or (μV,νV)(x)=(0,1), (μV,νV)(y)=(1,0),(b)
(Bayhan and Çoker [4]). T1 if forallx,y∈X, x≠y, ∃U=(μU,νU), V=(μV,νV)∈τ such that (μU,νU)(x)=(1,0), (μU,νU)(y)=(0,1), (μV,νV)(x)=(0,1) and (μV,νV)(y)=(1,0),(c)
T2   (Hausdorff) if for all pair of distinct intuitionistic fuzzy points x(α,β),y(γ,δ) in X, ∃U,V∈τ such that x(α,β)∈U, y(γ,δ)∈V and U∩V=0~,(d)
q-T2 if for every pair of distinct intuitionistic fuzzy points x(α,β),y(γ,δ) in X, ∃U and V∈τ such that x(α,β)∈U, y(γ,δ)∈V and U⊆V′.Example 21.
LetX={a,b} and let τ={0~,A,B,1~}, where A=〈x,(a/1,b/0),(a/0,b/1)〉 and B=〈x,(a/1,b/0),(a/0,b/1)〉, then (X,τ) is an IFTS, and it is T0, T1, T2 (Hausdorff) and q-T2.We haveT2⇒T1⇒T0 and T2⇒q-T2, but none of the implication are reversible.Now we associate a BFTS with an IFTS and vice versa on parallel lines as in Bayhan and Çoker [11].Let(X,τ) be an IFTS and τ1={μA∣∃νA∈IX such that (μA,νA)∈τ}, τ2={(1_-νA)∣∃μA∈IX such that (μA,νA)∈τ}. It is easy to see that (X,τ1) and (X,τ2) are fuzzy topological spaces in Chang’s sense.(
X
,
τ
1
,
τ
2
) is called the bifuzzy topological space associated with the IFTS (X,τ).Proposition 22.
Let(X,τ1,τ2) be a BFTS and ττ1,τ2={(U,V′)∣U∈τ1,V∈τ2 and U⊆V}. Then (X,ττ1,τ2) is an IFTS and (ττ1,τ2)1=τ1, (ττ1,τ2)2=τ2.Proof.
Clearly members ofττ1,τ2 are intuitionistic fuzzy sets, and 0~ and 1~ belong to it. Now let (Ui,Vi′)∈ττ1,τ2, i=1,2 then (U1,V1′)∩(U2,V2′) = (U1∩U2,V1′∪V2′) = (U1∩U2,(V1∩V2)′)∈ττ1,τ2. Further let {(Ui,Vi′):i∈J, where J is arbitrary}⊆ττ1,τ2. Then ⋃‍{(Ui,Vi′)}=(⋃‍Ui,⋂‍Vi′)=(⋃‍Ui,(⋃‍Vi)′)∈ττ1,τ2. Thus (X,ττ1,τ2) is an IFTS.
Now letU∈τ1 then (U,ϕ)∈ττ1,τ2. Therefore τ1⊆(ττ1,τ2)1. Conversely let U∈(ττ1,τ2)1 then ∃V∈IX such that (U,V)∈ττ1,τ2⇒U∈τ1, so (ττ1,τ2)1⊆τ1. Thus (ττ1,τ2)1=τ1. Similarly we can show that (ττ1,τ2)2=τ2.The IFTS(X,ττ1,τ2) is called the IFTS associated with the BFTS (X,τ1,τ2).Proposition 23.
Let(X,τ) and (Y,δ) be two IFTSs and f:(X,τ)→(Y,δ) be IF-continuous. Then f:(X,τ1,τ2)→(Y,δ1,δ2) is FP-continuous (Here (X,τ1,τ2) and (Y,δ1,δ2) are BFTSs associated with (X,τ) and (Y,δ), resp.).Proof.
Letf:(X,τ)→(Y,δ) be IF-continuous. To show that f:(X,τ1,τ2)→(Y,δ1,δ2) is FP-continuous, take U∈δ1 then ∃V∈IY such that (U,V)∈δ. Hence since f is IF-continuous, f-1(U,V)∈τ, that is, (f-1(U),f-1(V))∈τ⇒f-1(U)∈τ1. Further take V1∈δ2 then ∃U1∈IY such that (U1,V1′)∈δ⇒f-1(U1,V1′)∈τ⇒(f-1(U1),f-1(V1′))∈τ⇒(f-1(U1),(f-1(V1))′)∈τ⇒f-1(V1)∈τ2. Thus f is FP-continuous.Proposition 24.
Let(X,τ1,τ2) and (Y,δ1,δ2) be two BFTSs and (X,ττ1,τ2) and (Y,δδ1,δ2) be the associated IFTSs respectively. Then f:(X,τ1,τ2)→(Y,δ1,δ2) is FP-continuous if and only if  f:(X,ττ1,τ2)→(Y,δδ1,δ2) is IF-continuous.Proof.
Letf:(X,τ1,τ2)→(Y,δ1,δ2) be FP-continuous. To show that f:(X,ττ1,τ2)→(Y,δδ1,δ2) is IF continuous, take any U=(μU,νU′)∈δδ1,δ2, where μU∈δ1, νU∈δ2 and μU⊆νU. Now using FP-continuity of f, f-1(μU)∈τ1 and f-1(νU)∈τ2. So (f-1(μU),(f-1(νU))′)∈ττ1,τ2⇒(f-1(μU),f-1(νU′))∈ττ1,τ2⇒f-1(μU,νU′)∈ττ1,τ2, that is,  f-1(U)∈ττ1,τ2, showing that f is IF-continuous.
The converse follows from the previous Proposition23 in view of the fact that (ττ1,τ2)1=τ1, (ττ1,τ2)2=τ2 and (δδ1,δ2)1=δ1 and (δδ1,δ2)2=δ2.The category of all BFTS together with FP-continuous functions will be denoted by BF-Top and the category of all IFTS together with IF-continuous function will be denoted by IF-Top.Now we defineℬ: IF-Top → BF-Top as follows:ℬ
(
X
,
τ
)
=
(
X
,
τ
1
,
τ
2
), ℬ(f)=f,forall morphism f  and 𝒟: BF-Top → IF-Top as follows:𝒟
(
X
,
τ
1
,
τ
2
)
=
(
X
,
τ
τ
1
,
τ
2
), 𝒟(f)=f,forall morphism f.It can be checked easily thatℬ and 𝒟 are covariant functors, and in view of Proposition 22 we have the following remark.Remark 25.
ℬ
∘
𝒟
=
I
d
BF-Top, the identity functor.Theorem 26.
The functor𝒟: BF-Top → IF-Top is left adjoint to the functor ℬ: IF-Top → BF-Top.The proof is on parallel lines as in([11], Theorem  3.10) and hence is omitted.Proposition 27.
The following statements are equivalent in an IFTS(X,τ):(1)
(X,τ) is T1,(2)
({x},{x}′) is intuitionistic fuzzy closed in (X,τ), for all x∈X.Proof.
(
1
)
⇒
(
2
) We show that ({x}′,{x}) is intuitionistic fuzzy open in X. Choose any IFP y(α,β) in ({x}′,{x}) then, y≠x. Hence ∃ IFOSs U,V∈τ such that U(x)=(μU(x),νU(x))=(1,0), U(y)=(μU(y),νU(x))=(0,1) and V(x)=(μV(x),νV(x))=(0,1), V(y)=(μV(y),νV(y))=(1,0). Now y(α,β)∈V⊆({x}′,{x}). Therefore in view of Proposition 13, ({x}′,{x}) is an IFOS.
(
2
)
⇒
(
1
) Choose x,y∈X such that x≠y. Then ({x},{x}′) and ({y},{y}′) are IFCSs and hence U=({x}′,{x}), V=({y}′,{y}) are IFOSs such that U(x)=(0,1), U(y)=(1,0), V(y)=(0,1) and V(x)=(1,0) showing that (X,τ) is T1.Proposition 28.
If an IFTS(X,τ) is T1, then its associated BFTS is T1.Proof.
Let(X,τ) be T1. Then forallx≠y, ∃U=(μU,νU) and V=(μV,νV)∈τ such that (μU,νU)(x)=(1,0), (μU,νU)(y)=(0,1), (μV,νV)(x)=(0,1), and (μV,νV)(y)=(1,0). Now (μU,νU)(x)=(1,0)⇒μU(x)=1, νU(x)=0.
Similarly,μU(y)=0,νU(y)=1, μV(x)=0,νV(x)=1, and μV(y)=1,νV(y)=0. Now μU∈τ1, (1_-νV)∈τ2 and further μU(x)=1, μU(y)=0, (1_-νV)(x)=0, and (1_-νV)(y)=1 showing that (X,τ1,τ2) is a T1 space.Similarly it can be shown that if an IFTS(X,τ) is T0, then its associated BFTS is T0.Theorem 29.
Let{(Xi,τi):i∈J} be a family of T1 IFTSs and (X,τ) be their product IFTS. Then (X,τ) is T1 if and only if  (Xi,τi) is T1, for all i∈J.Proof.
Let(Xi,τi) be T1  foralli∈J. To show that (X,τ) is T1, choose x,y∈X, x≠y. Let x=Πxi, y=Πyi then ∃j∈J such that xj≠yj. Now since (Xj,τj) is T1, ∃Uj=(μUj,νUj) and Vj=(μVj,νVj)∈τj such that μUj(xj)=1, νUj(xj)=0, μUj(yj)=0, νUj(yj)=1, and μVj(yj)=1, νVj(yj)=0, μVj(xj)=0, νVj(xj)=1. Now consider the basic IFOSs ΠUk and ΠVk where Uk=Vk=(1_,0_) for k∈J,k≠j and Uk=UJ and Vk=VJ when k=j. Then ΠUk(x)=(infμUk(xk), supνUk(xk))=(1,0), ΠUk(y)=(infμUk(yk), supνUk(yk))=(0,1). Similarly we can show that ΠVk(y)=(1,0) and ΠVk(x)=(0,1). Therefore (X,τ) is T1.
Conversly let(X,τ) be T1. To show that (Xj,τj) is T1, choose xj,yj∈Xj such that xj≠yj. Now consider x=Πxi, y=Πyi where xi=yi, i≠j and the jth coordinate of x,y are xj and yj, respectively. Then x≠y, therefore since (X,τ) is T1, ∃U=(μU,νU) and V=(μV,νV)∈τ such that U(x)=(1,0), U(y)=(0,1), V(x)=(0,1), and V(y)=(1,0). Now consider the intuitionistic fuzzy points x(α,1-α)∈U and y(γ,1-γ)∈V. Then ∃ basic IFOSs ΠUiα and ΠViγ in X such that x(α,1-α)∈ΠUiα⊆U and y(γ,1-γ)∈ΠViγ⊆V:
(3)x(α,1-α)∈ΠUiα⊆U⇒α<infiμUiα(xi),1-α>supiνUiα(xi)⇒α<μUiα(xi),(1-α)>νUiα(xi),∀i∈J.
Similarly we can show that
(4)y(γ,1-γ)∈ΠViγ⊆V⇒γ<μViγ(yi),(1-γ)>νViγ(yi),∀i∈J.
Now Vj=∪{UJγ:γ∈(0,1)} is such that μVj(yj)=1, νVJ(yj)=0. Further, since xi=yi for i≠j, from (3) and (4), it follows that(5)α<μUiα(yi),(1-α)>νUiα(yi),∀i∈J,i≠j,α∈(0,1),(6)γ<μViγ(xi),(1-γ)>νViγ(xi),∀i∈J,i≠j,γ∈(0,1).
Therefore
(7)U(y)=(0,1)⇒ΠUiα(y)=(0,1)⇒infiμUiα(yi)=0,supiνUiα(yi)=1,∀α∈(0,1)⇒μUjα(yj)=0,νUjα(yj)=1,∀α∈(0,1)(inviewof(5))⇒supαμUjα(yj)=0,infανUjα(yj)=1.
ThusμUJ(yj)=0 and νUj(yj)=1.  That is, Uj(yj)=(0,1). Similarly it can be shown that VJ(xJ)=(0,1). Hence (Xj,τj) is T1.The following theorem can be proved in a similar way.Theorem 30.
Let{(Xi,τi):i∈J} be a family of T0 IFTSs and (X,τ) be their product IFTS. Then (X,τ) is T0 if and only if  (Xi,τi) is T0, for all i∈J.Proposition 31.
An IFTS(X,τ) is Hausdorff, and then its associated BFTS (X,τ1,τ2) is Hausdorff.Proof.
Let(X,τ) be Hausdorff. To show that (X,τ1,τ2) is Hausdorff, choose any two distinct fuzzy points xα, y1-δ in X. Now choose β,γ∈(0,1), and then x(α,β) and y(γ,δ) are distinct intuitionistic fuzzy points in X. Since (X,τ) is Hausdorff, ∃U,V∈τ such that x(α,β)∈U, y(γ,δ)∈V and U∩V=0~. Let U=(μU,νU), V=(μV,νV), then
(8)x(α,β)∈U⇒α<μU(x),β>νU(x)y(γ,δ)∈V⇒γ<μV(y),δ>νV(y)⟺1-δ<1-νV(y),
and also we have μU+νU⊆1_ and μV+νV⊆1_.
Now(9)U∩V=0~⟺μU∩μV=0_,νU∪νV=1_.
From (8) we have xα∈μU and y1-δ∈1_-νV. Now we show that μU∩(1_-νV)=0_ as follows.
We have,νU(x)<1⇒ν_V(x)=1  (in view of (9)), ⇒1-νV(x)=0⇒(μU∩(1_-νV))(x)=0.
Further,νV(y)<1⇒νU(y)=1  (in view of (9)), ⇒μU(y)=0 since μU(y)+νU(y)≤1⇒(μU∩(1_-νV))(y)=0.
Now takez∈X such that z≠x,y. If μU(z)=0, then obviously (μU∩(1_-νV))(z)=0, and if μU(z)≠0, then νU(z)<1  (since μU+νU⊆1_)⇒νV(z)=1⇒1-νV(z)=0⇒(μU∩(1_-νV))(z)=0.
Thus we haveμU∈τ1, (1_-νV)∈τ2 such that xα∈μU, y1-δ∈(1_-νV) and μU∩(1_-νV)=0_, and hence (X,τ1,τ2) is Hausdorff.Definition 32.
Let(X,τ) be an IFTS and Y⊆X. Then (Y,τ∣Y) is called a subspace of (X,τ) where τ∣Y={U∣Y=(μU∣Y,νU∣Y):U∈τ}.Proposition 33.
If an IFTS(X,τ) is Ti, i=0,1,2, then its subspace,  (Y,τ∣Y) is also Ti, i=0,1,2.The proofs are easy and hence are omitted.Proposition 34.
The product IFTS(X,τ) of {(Xj,τj):j∈J} is initial with respect to the family of projections {pj:X→Xj,j∈J}, that is, for any IFTS (Y,η), a map g:(Y,η)→(X,τ) is IF continuous if and only if the map pj∘g:(Y,η)→(Xj,τj) is IF continuous for all  j∈J.Proof.
Since projection maps are IF continuous and composition of IF-continuous maps are IF-continuous, sopj∘g is IF continuous forallj∈J.
Conversely, ifpj∘g is IF-continuous forallj∈J, then (pj∘g)-1(Uj)=g-1(pj-1(Uj)) is IFO in (Y,η)  forallUj∈τj,j∈J showing that inverse image of every subbasic IFOS in X is IFO in Y which implies that g is IF continuous.Proposition 35.
Let{(Xj,τj):j∈J} be a family of IFTSs, (Xj,(τj)1,(τj)2) be the BFTS associated with (Xj,τj), and (X,τ1,τ2) be the BFTS associated with the product IFTS (X,τ). Then τ1=Π(τj)1 and τ2=Π(τj)2.Proof.
The product space(X,τ) is generated by {pj-1(Uj):Uj∈τj,j∈J} where pj,s are projection maps. Let Uj=(μUj,νUj), and then pj-1(Uj)=(pj-1(μUj),pj-1(νUj)).
Now members ofτ are of the form:
(10)⋃l∈L(arbitrary)‍⋂k∈K(finite)‍plk-1(Ulk)=⋃l‍⋂k‍plk-1(μUlk,νUlk)=⋃l‍⋂k‍(plk-1(μUlk),plk-1(νUlk))=(⋃l‍⋂k‍plk-1(μUlk),⋂l‍⋃kplk-1(νUlk)‍).
So, members of τ1 are of the form ⋃l‍⋂k‍plk-1(μUlk), hence τ1=Π(τj)1, and members of τ2 are of the form 1_-⋂l‍⋃k‍plk-1(νUlk)=⋃l‍⋂k‍plk-1(1_-νUlk), hence τ2=Π(τj)2.Theorem 36.
An IFTS(X,τ) is Hausdorff if and only if (ΔX,ΔX′) is IFCS in (X×X,τ×τ).Proof.
We show that(ΔX′,ΔX) is IFOS in X. Choose any (x,y)(α,β)∈(ΔX′,ΔX) then x≠y and α<ΔX′(x,y), β>ΔX(x,y). Now x(α,β) and y(α,β) are distinct intuitionistic fuzzy points in X. Since (X,τ) is Hausdorff, ∃ IFOSs U=(μU,νU) and V=(μV,νV) in τ such that x(α,β)∈U, y(α,β)∈V and U∩V=0~, that is, (μU∩μV,νU∪νV)=(0_,1_).
Now considerU×V=(μU×μV,νU*νV), and then (x,y)(α,β)∈U×V⊆(ΔX′,ΔX) as shown below:
(
x
,
y
)
(
α
,
β
)
∈
(
μ
U
×
μ
V
,
ν
U
*
ν
V
) as α<(μU×μV)(x,y)= inf (μU(x),μV(y))  (since α<μU(x),α<μV(y) both) and β>(νU*νV)(x,y)= sup (νU(x),νV(y))   (since β>νU(x),β>νV(y) both).
Further,U×V⊆(ΔX′,ΔX) since U×V=(μU×μV,νU*νV) and μU×μV⊆ΔX′ as (μU×μV)(x,x)=inf(μU(x),μV(x))=0,forallx∈X and νU*νV⊇ΔX as (νU×νV)(x,x)=sup(νU(x),νV(x))=1,forallx∈X.
Conversely, let(ΔX,ΔX′) be IFCS in (X,τ). Then (ΔX′,ΔX) is IFOS in X.  To show that (X,τ) is Hausdorff choose any two distinct intuitionistic fuzzy points x(α,β), y(γ,δ) in X then x≠y. Let max (α,γ)  =  α and min (β,δ)  =  β. Consider the intuitionistic fuzzy point (x,y)(α,β) in X×X. Then (x,y)(α,β)∈(ΔX′,ΔX) and hence ∃ a basic IFOS U×V in (X×X,τ×τ) such that (x,y)(α,β)∈U×V⊆(ΔX′,ΔX).
Now,(x,y)(α,β)∈U×V⇒α<(μU×μV)(x,y)  = inf (μU(x),μV(y))≤μU(x), and β>(νU*νV)(x,y)  = sup (νU(x),νV(y))≥νU(x) implies that x(α,β)∈U, similarly y(α,β)∈V which implies that y(γ,δ)∈V. Now we show that U∩V= 0~. Since U×V⊆(ΔX′,ΔX), (μU×μV)(x,x)≤ΔX′(x,x)  =  0⇒ inf (μU(x),μV(x))= 0, forallx∈X and (νU×νV)(x,x)≥ΔX(x,x)  =1⇒ sup (νU(x),νV(x))  =1, forallx∈X. Thus U∩V = (μU∩μV,νU∪νV)=(0_,1_) =0~.Definition 37.
A BFTS(X,τ1,τ2) is called q-T2 if for any two distinct fuzzy points xr and ys there exists U∈τ1, V∈τ2 such that xr∈U, ys∈V and U⊆V′.Proposition 38.
An IFTS(X,τ) is q-T2, and then its associated BFTS (X,τ1,τ2) is q-T2.Proof.
Letxα and y1-δ be any two distinct fuzzy points in X. Since (X,τ) is q-T2 and x(α,β), y(γ,δ) are distinct intuitionistic fuzzy points in X, there exist G1=(μG1,νG1) and G2=(μG2,νG2) such that x(α,β)∈G1, y(γ,δ)∈G2 and μG1⊆μG2′, νG1⊇νG2′. ⇒α<μG1(x) and δ>νG2(y)⇒xα∈μG1 and y1-δ∈(1_-νG2). Thus for fuzzy points xα and y1-δ in X, ∃μG1∈τ1 and (1_-νG2)∈τ2. Further, since μG1+νG1⊆1_, we have μG1⊆(1_-νG1)⊆νG2  (since νG1⊇νG2′). Hence (X,τ1,τ2) is q-T2.Theorem 39.
Let{(Xi,τi):i∈J} be a family of IFTs and (X,τ) be their product IFTS. Then (X,τ) is Hausdorff if and only if each coordinate space (Xi,τi) is Hausdorff.Proof.
Letx(α,β) and y(γ,δ) be two distinct intuitionistic fuzzy points in X. Let x=Πxi, y=Πyi then x≠y and hence ∃k∈J such that xk≠yk. Consider the distinct intuitionistic fuzzy points (xk)(α,β) and (yk)(γ,δ) in (Xk,τk). Since (Xk,τk) is Hausdorff, ∃ disjoint Uk,Vk in τk such that (xk)(α,β)∈Uk, (yk)(γ,δ)∈Vk and Uk∩Vk=0~. Let Uk=(μUk,νUk) and Vk=(μVk,νVk), and then α<μUk(xk), β>νUk(xk) and γ<μVk(yk), δ>νVk(yk). Equivalently, we have, α<pk-1(μUk)(x), β>pk-1(νUk)(x), that is, x(α,β)∈pk-1(Uk) and γ<pk-1(μVk)(y), δ>pk-1(νVk)(y), that is, y(γ,δ)∈pk-1(Vk) and pk-1(Uk)⋂‍pk-1(Vk)=0~, since Uk⋂‍Vk=0~.Conversely, let(X,τ) be Hausdorff. To show that (Xj,τj) is Hausdorff, and consider two distinct intuitionistic fuzzy points (xj)(α,β) and (yj)(γ,δ) in Xj, then xj≠yj. Now consider x=Πxi and y=Πyi in X where xi=yi for i≠j and the jth coordinate of x,y are xj and yj, respectively. Since (X,τ) is Hausdorff, ∃U,V∈τ such that x(α,β)∈U, y(γ,δ)∈V and U∩V=0~. Now consider the basic IFOSs ΠUi and ΠVi in X such that x(α,β)∈ΠUi⊆U and y(γ,δ)∈ΠVi⊆V. Here the IFS Ui=(1_,0_) except for finitely many i’s say i1,i2,...,ik and Vi=(1_,0_) except for finitely many i’s say l1,l2,…,lk1.Now,x(α,β)∈ΠUi⊆U⇒α< inf {μUi1(xi1),μUi2(xi2),…,μUik(xik)},
(11)β>sup{νUi1(xi1),νUi2(xi2),…,νUik(xik)}⇒α<μUim(xim),β>νUim(xim),∀m=1,2,…,k.
Similarly,
(12)y(γ,δ)∈ΠVi⊆V⇒γ<μVln(yln),δ>νVln(yln),∀n=1,2,…,k1.
We claim that j∈{i1,i2,…,ik} and also j∈{l1,l2,…,lk1}. If it is not so, then xim=yim and xln=yln, forallm=1,2,…,k and foralln=1,2,…,k1 and hence from (11) and (12) we have
(13)α<μUim(xim),β>νUim(xim),∀m=1,2,…,k,γ<μVln(xln),δ>νVln(xln),∀n=1,2,…,k1
implying that ΠμUi(x)>0, ΠμVi(x)>0, therefore (ΠμUi∩ΠμVi)≠0_⇒μU∩μV≠0_ which is a contradiction to the fact that U∩V=0~.So,α<μUj(xj), β>νUj(xj), γ<μVj(yj) and δ>νVj(yj) in view of (11) and (12) showing that (xj)(α,β)∈Uj, (yj)(γ,δ)∈Vj.Now it remains to show thatUj∩Vj=0~, that is, μUj∩μVj=0_ and νUj∪νVj=1_. Let ∃zj1 and zj2 in Xj such that (μUj∩μVj)(zj1)>0 and (νUj∪νVj)(zj2)<1. Consider z1=Πzi in X where zi=xi(=yi) for i≠j and zj=zj1.Now(14)(μUj∩μVj)(zj1)>0⇒μUj(zj1)>0,μVj(zj1)>0.
In view of (11), (12), and (14) we have (ΠμUi∩ΠμVi)(z1)>0 implying that U∩V≠0~ which is a contradiction. Hence μUj∩μVj=0_. Now let z2=Πzi where zi=xi(=yi) for i≠j and zj=zj2. Since (νUj∪νVj)(zj2)<1,We have(15)νUj(zj2)<1,νVj(zj2)<1.
Therefore from (11), (12), and (15) we have (ΠνUj∪ΠνVj)(z2)<1 implying that U∩V≠0~, again a contradiction. Hence (Xj,τj) is Hausdorff.



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*Source: 604396-2012-12-26.xml*