Separation Axioms in Intuitionistic Fuzzy Topological Spaces

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 axioms Ti, 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]. T1-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 functor B : IF-Top → BF-Top preserves these separation properties.


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. C ¸oker [3] subsequently initiated a study of intuitionistic fuzzy topological spaces.
In this paper we have searched for appropriate definitions of the separation axioms T i , i = 0, 1, 2 in intuitionistic fuzzy topological spaces.
Hausdorffness in an intuitionistic fuzzy topological space has been introduced earlier by C ¸oker [3], Bayhan and C ¸oker [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 FT 2 (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 FT 1 (ii) as it generalizes the most appropriate definition of T 1 -ness in a fuzzy topological space (cf.definition 5.1, [7]).We have also introduced a suitable definition of T 0 -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 functor B : IF-Top → BF-Top preserves these separation properties.

Preliminaries
Throughout X denotes a nonempty set, I denotes the unit interval [0, 1], and I 0 and I 1 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 I X .For any A ∈ I X , 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 ∈ I Y will be identified with the fuzzy set in X which takes the same value as Definition 1 (Atanassov [2]).Let X be a nonempty set.An intuitionistic fuzzy set (IFS, in short) A is an ordered pair (μ A , ν A ) of fuzzy sets in X.
and μ A (x), ν A (x), respectively, denote the degree of membership and the degree of nonmembership of x ∈ X to the set A and 0 ≤ μ A (x) + ν A (x) ≤ 1 for each x ∈ X.
We identify an ordinary fuzzy set A ∈ I X with the intuitionistic fuzzy set (A, A ). Definition 2 (Atanassov [2]).Let X be a nonempty set and A, B be given by (μ A , ν A ) and (μ B , ν B ), respectively, Definition 3 (C ¸oker [3]).Let {A i : i ∈ J} be an arbitrary family of IFSs in X.Then (a Definition 4 (C ¸oker [3]).Let X 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 It is easy to verify that f −1 Definition 5 (Wong [8]).A fuzzy point x r 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 point x r is said to belong to a fuzzy set A (notation : x r ∈ A) if r < A(x) (cf.[6]).
Two fuzzy points are said to be distinct if their supports are distinct.Definition 6.Let X 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) = (x α , 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.
Let x (α,β) be an IFP in X and A = (μ A , ν A ) be an IFS in X.Then x (α,β) is said to belong to A (notation : We identify a fuzzy point x r in X by the intuitionistic fuzzy point x (r,(1−r)) in X.

Proposition 7. An intuitionistic fuzzy set A 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 (C ¸oker [3]).An intuitionistic fuzzy topology (IFT, in short) on a nonempty set X is a family τ of IFSs in X satisfying the following axioms: (3) ∪G i ∈ τ for any arbitrary family {G i ∈ 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 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 collection S 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.

Proposition 13. Let (X, τ) be an IFTS. Then an IFS A in X is an IFOS if and only if
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 map f : Definition 15 (Abu Safia et al. [12]).Let X be a nonempty set and τ 1 , τ 2 be two fuzzy topologies on X.Then (X, τ 1 , τ 2 ) is called a bifuzzy topological space (BFTS, in short).
A map f : (X, Definition 16 (Bayhan and C ¸oker [4]).Let A = (μ A , ν A ) and B = (μ B , ν B ) be IFSs in X and Y , respectively, and then A × B is the IFS in X × Y defined as follows where This definition can be extended to an arbitrary family of IFSs as follows.
If {A i = (μ Ai , ν Ai ), i ∈ J} is a family of IFSs in X i , then their product is defined as the IFS in ΠX i given by where Πμ Ai (x) = inf μ Ai (x i ), for all x = Πx i ∈ X and Definition 17 (Bayhan and C ¸oker [4]).Let (x i , τ i ), i = 1, 2 be two IFTSs, and then the product IFT This definition can be extended to an arbitrary family of IFTSs as follows.
Let {(X i , τ i ) : i ∈ J} be a family of IFTSs.Then the product intuitionistic fuzzy topology τ on X = ΠX i is the one having {p −1 j (U j ) : U j ∈ τ j , j ∈ J} as a subbase where p j : X → X j is the jth projection map.(X, τ) is called the product IFTS of the family {(X i , τ i ) : i ∈ J}.
(c) T 2 (Hausdorff) if for all pair of distinct fuzzy points x r , y s in X, ∃U, V ∈ τ such that x r ∈ U, y s ∈ V , and U ∩ V = 0, (d) q-T 2 (q-Hausdorff) if for any pair of distinct fuzzy points x r , and y s ∃U, V ∈ τ such that x r ∈ U, y s ∈ τ and U ⊆ V .
For the categorical concepts used here, we refer the reader to [16].
We have T 2 ⇒ T 1 ⇒ T 0 and T 2 ⇒ q-T 2 , 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 C ¸oker [11].
Let (X, τ) be an IFTS and It is easy to see that (X, τ 1 ) and (X, τ 2 ) are fuzzy topological spaces in Chang's sense.
Proposition 24.Let (X, τ 1 , τ 2 ) and (Y , δ 1 , δ 2 ) be two BFTSs and (X, τ τ1,τ2 ) and (Y , δ δ1,δ2 ) be the associated IFTSs respectively.Then f : (X, The converse follows from the previous Proposition 23 in view of the fact that (τ τ1,τ2 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 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): Therefore in view of Proposition 13, ({x} , {x}) is an IFOS.
The following theorem can be proved in a similar way.
Proof.Let (X, τ) be Hausdorff.To show that (X, τ 1 , τ 2 ) is Hausdorff, choose any two distinct fuzzy points x α , y 1−δ in X.Now choose β, γ ∈ (0, 1), and then x (α,β) and y (γ,δ) are distinct intuitionistic fuzzy points in and also we have From ( 8) we have x α ∈ μ U and y The proofs are easy and hence are omitted.
Proposition 34.The product IFTS (X, τ) of {(X j , τ j ) : j ∈ J} is initial with respect to the family of projections {p j : X → X j , 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 p j • g : (Y , η) → (X j , τ j ) is IF continuous for all j ∈ J.
Proof.Since projection maps are IF continuous and composition of IF-continuous maps are IF-continuous, so p j • g is IF continuous for all j ∈ J.
Proof.The product space (X, τ) is generated by {p −1 j (U j ) : U j ∈ τ j , j ∈ J} where p , j s are projection maps.Let U j = (μ Uj , ν Uj ), and then p −1 j (U j ) = (p −1 j (μ Uj ), p −1 j (ν Uj )).Now members of τ are of the form: by IF-Top.Now we define B : IF-Top → BF-Top as follows: B(X, τ) = (X, τ 1 , τ 2 ), B( f ) = f , for all morphism f and D : BF-Top → IF-Top as follows: D(X, τ 1 , τ 2 ) = (X, τ τ1,τ2 ), D( f ) = f , for all morphism f .It can be checked easily that B and D are covariant functors, and in view of Proposition 22 we have the following remark.Remark 25.B • D = Id BF-Top , the identity functor.Theorem 26.The functor D : BF-Top → IF-Top is left adjoint to the functor B : IF-Top → BF-Top.