Base and subbase in intuitionistic I-fuzzy topological spaces

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1 Hacettepe Journal of Mathematics and Statistics Volume 43 (2) (204), Base and subbase in intuitionistic I-fuzzy topological spaces Chengyu Liang and Conghua Yan Abstract In this paper, the concepts of the base and subbase in intuitionistic I- fuzzy topological spaces are introduced, and use them to discuss fuzzy continuous mapping and fuzzy open mapping. We also study the base and subbase in the product of intuitionistic I-fuzzy topological spaces, and T 2 separation in product intuitionistic I-fuzzy topological spaces. Finally, the relation between the generated product intuitionistic I- fuzzy topological spaces and the product generated intuitionistic I- fuzzy topological spaces are studied. Keywords: Intuitionistic I-fuzzy topological space; Base; Subbase; T 2 separation; Generated Intuitionistic I-fuzzy topological spaces AMS Classification: 54A40. Introduction As a generalization of fuzzy sets, the concept of intuitionistic fuzzy sets was first introduced by Atanassov []. From then on, this theory has been studied and applied in a variety areas ([4, 4, 8], etc). Among of them, the research of the theory of intuitionistic fuzzy topology is similar to the the theory of fuzzy topology. In fact, Çoker [4] introduced the concept of intuitionistic fuzzy topological spaces, this concept is originated from the fuzzy topology in the sense of Chang [3](in this paper we call it intuitionistic I-topological spaces). Based on Çoker s work [4], many topological properties of intuitionistic I-topological spaces has been discussed ([5, 0,, 2, 3]). On the other hand, Šostak [7] proposed a new notion of fuzzy topological spaces, and this new fuzzy topological structure has been accepted widely. Influenced by Šostak s work [7], Çoker [7] gave the notion of intuitionistic fuzzy topological spaces in the sense of Šostak. By the standardized terminology introduced in [6], we will call it intuitionistic I-fuzzy Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 0008, PR China, liangchengyu87@63.com Corresponding Author, Institute of Math., School oh Math. Sciences, Nanjing Normal University, Nanjing, Jiangsu 20023, PR China,

2 232 topological spaces in this paper. In [5], the authors studied the compactness in intuitionistic I-fuzzy topological spaces. Recently, Yan and Wang [9] generalized Fang and Yue s work ([8, 2]) from I-fuzzy topological spaces to intuitionistic I-fuzzy topological spaces. In [9], they introduced the concept of intuitionistic I-fuzzy quasi-coincident neighborhood systems of intuitiostic fuzzy points, and construct the notion of generated intuitionistic I-fuzzy topology by using fuzzifying topologies. As an important result, Yan and Wang proved that the category of intuitionistic I-fuzzy topological spaces is isomorphic to the category of intuitionistic I-fuzzy quasi-coincident neighborhood spaces in [9]. It is well known that base and subbase are very important notions in classical topology. They also discussed in I-fuzzy topological spaces by Fang and Yue [9]. As a subsequent work of Yan and Wang [9], the main purpose of this paper is to introduce the concepts of the base and subbase in intuitionistic I-fuzzy topological spaces, and use them to discuss fuzzy continuous mapping and fuzzy open mapping. Then we also study the base and subbase in the product of intuitionistic I-fuzzy topological spaces, and T 2 separation in product intuitionistic I-fuzzy topological spaces. Finally, we obtain that the generated product intuitionistic I-fuzzy topological spaces is equal to the product generated intuitionistic I-fuzzy topological spaces. Throughout this paper, let I = [0, ], X a nonempty set, the family of all fuzzy sets and intuitionistic fuzzy sets on X be denoted by I X and ζ X, respectively. The notation pt(i X ) denotes the set of all fuzzy points on X. For all λ I, λ denotes the fuzzy set on X which takes the constant value λ. For all A ζ X, let A =< µ A, γ A >. (For the relating to knowledge of intuitionistic fuzzy sets and intuitionistic I-fuzzy topological spaces, we may refer to [] and [9].) 2. Some preliminaries 2.. Definition. ([20]) A fuzzifying topology on a set X is a function τ : 2 X I, such that () τ( ) = τ(x) = ; (2) A, B X, τ(a B) τ(a) τ(b); (3) A t X, t T, τ( t T A t) t T τ(a t). The pair (X, τ) is called a fuzzifying topological space Definition. ([, 2]) Let a, b be two real numbers in [0, ] satisfying the inequality a + b. Then the pair < a, b > is called an intuitionistic fuzzy pair. Let < a, b >, < a 2, b 2 > be two intuitionistic fuzzy pairs, then we define () < a, b >< a 2, b 2 > if and only if a a 2 and b b 2 ; (2) < a, b >=< a 2, b 2 > if and only if a = a 2 and b = b 2 ; (3) if < a j, b j > j J is a family of intuitionistic fuzzy pairs, then j J < a j, b j >=< j J a j, j J b j >, and j J < a j, b j >=< j J a j, j J b j >; (4) the complement of an intuitionistic fuzzy pair < a, b > is the intuitionistic fuzzy pair defined by < a, b > =< b, a >;

3 233 In the following, for convenience, we will use the symbols and 0 denote the intuitionistic fuzzy pairs <, 0 > and < 0, >. The family of all intuitionistic fuzzy pairs is denoted by A. It is easy to find that the set of all intuitionistic fuzzy pairs with above order forms a complete lattice, and, 0 are its top element and bottom element, respectively Definition. ([4]) Let X, Y be two nonempty sets and f : X Y a function, if B = {< y, µ B (y), γ B (y) >: y Y } ζ Y, then the preimage of B under f, denoted by f (B), is the intuitionistic fuzzy set defined by f (B) = {< x, f (µ B )(x), f (γ B )(x) >: x X}. Here f (µ B )(x) = µ B (f(x)), f (γ B )(x) = γ B (f(x)). [6]). (This notation is from If A = {< x, µ A (x), γ A (x) >: x X} ζ X, then the image A under f, denoted by f (A) is the intuitionistic fuzzy set defined by f (A) = {< y, f (µ A )(y), ( f ( γ A ))(y) >: y Y }. Where { f supx f (µ A )(y) = (y) µ A (x), if f (y), 0, if f (y) =. f ( γ A )(y) = { infx f (y) γ A (x), if f (y),, if f (y) = Definition. ([7]) Let X be a nonempty set, δ : ζ X A satisfy the following: () δ(< 0, >) = δ(<, 0 >) = ; (2) A, B ζ X, δ(a B) δ(a) δ(b); (3) A t ζ X, t T, δ( t T A t) t T δ(a t). Then δ is called an intuitionistic I-fuzzy topology on X, and the pair (X, δ) is called an intuitionistic I-fuzzy topological space. For any A ζ X, we always suppose that δ(a) =< µ δ (A), γ δ (A) > later, the number µ δ (A) is called the openness degree of A, while γ δ (A) is called the nonopenness degree of A. A fuzzy continuous mapping between two intuitionistic I-fuzzy topological spaces (ζ X, δ ) and (ζ Y, δ 2 ) is a mapping f : X Y such that δ (f (A)) δ 2 (A). The category of intuitionistic I-fuzzy topological spaces and fuzzy continuous mappings is denoted by II-FTOP Definition. ([6,, 2]) Let X be a nonempty set. An intuitionistic fuzzy point, denoted by x (α,β), is an intuitionistic fuzzy set A = {< y, µ A (y), γ A (y) >: y X}, such that and µ A (y) = γ A (y) = { α, if y = x, 0, if y x. { β, if y = x,, if y x.

4 234 Where x X is a fixed point, the constants α I 0, β I and α + β. The set of all intuitionistic fuzzy points x (α,β) is denoted by pt(ζ X ) Definition. ([2]) Let x (α,β) pt(ζ X ) and A, B ζ X. We say x (α,β) quasi-coincides with A, or x (α,β) is quasi-coincident with A, denoted x (α,β)ˆqa, if µ A (x) + α > and γ A (x) + β <. Say A quasi-coincides with B at x, or say A is quasi-coincident with B at x, AˆqB at x, in short, if µ A (x) + µ B (x) > and γ A (x) + γ B (x) <. Say A quasi-coincides with B, or A is quasi-coincident with B, if A is quasi-coincident with B at some point x X. Relation does not quasi-coincides with or is not quasi-coincident with is denoted by ˆq. It is easily to know for x (α,β) pt(ζ X ), x (α,β)ˆq <, 0 > and x (α,β) ˆq < 0, > Definition. ([9]) Let (X, δ) be an intuitionistic I-fuzzy topological space. For all x (α,β) pt(ζ X ), U ζ X, the mapping Q δ x (α,β) : ζ X A is defined as follows { δ(v ), x (α,β) q U; Q δ x (α,β) (U) = x (α,β) q V U 0, x (α,β) q U. The set of Q δ = {Q δ x (α,β) : x (α,β) pt(ζ X )} is called intuitionistic I-fuzzy quasicoincident neighborhood system of δ on X Theorem. ([9]) Let (X, δ) be an intuitionistic I-fuzzy topological space, Q δ = {Q δ x (α,β) : x (α,β) pt(ζ X )} of maps Q δ x (α,β) : ζ X A defined in Definition 2.7 satisfies: U, V ζ X, () Q δ x (α,β) (, 0 ) =, Q δ x (α,β) ( 0, ) = 0 ; (2) Q δ x (α,β) (U) > 0 x (α,β) q U; (3) Q δ x (α,β) (U V ) = Q δ x (α,β) (U) Q δ x (α,β) (V ); (4) Q δ x (α,β) (U) = Q δ y (λ,ρ) (V ); (5) δ(u) = x (α,β) q U x (α,β) q V U y (λ,ρ) q V Q δ x (α,β) (U) Lemma. ([2]) Suppose that (X, τ) is a fuzzifying topological space, for each A I X, let ω(τ)(a) = r I τ(σ r(a)), where σ r (A) = {x : A(x) > r}. Then ω(τ) is an I-fuzzy topology on X, and ω(τ) is called induced I-fuzzy topology determined by fuzzifying topology τ Definition. ([9]) Let (X, τ) be a fuzzifying topological space, ω(τ) is an induced I-fuzzy topology determined by fuzzifying topology τ. For each A ζ X, let Iω(τ)(A) =< µ τ (A), γ τ (A) >, where µ τ (A) = ω(τ)(µ A ) ω(τ)( γ A ), γ τ (A) = µ τ (A).We say that (ζ X, Iω(τ)) is a generated intuitionistic I-fuzzy topological space by fuzzifying topological space (X, τ). 2.. Lemma. ([9]) Let (X, τ) be a fuzzifying topological space, then () A X, µ τ (< A, A c >) = τ(a). (2) A =< α, β > ζ X, Iω(τ)(A) =.

5 Lemma. ([9]) Suppose that (ζ X, δ) is an intuitionistic I-fuzzy topological space, for each A X, let [δ](a) = µ δ (< A, A c >). Then [δ] is a fuzzifying topology on X Lemma. ([9]) Let (X, τ) be a fuzzifying topological space and (X, Iω(τ)) a generated intuitionistic I-fuzzy topological space. Then [Iω(τ)] = τ. 3. Base and subbase in Intuitionistic I-fuzzy topological spaces 3.. Definition. Let (X, τ) be an intuitionistic I-fuzzy topological space and B : ζ X A. B is called a base of τ if B satisfies the following condition τ(u) = B(B λ ), U ζ X. B λ =U 3.2. Definition. Let (X, τ) be an intuitionistic I-fuzzy topological space and ϕ : ζ X A, ϕ is called a subbase of τ if ϕ ( ) : ζ X A is a base, where ϕ ( ) (A) = ϕ(b λ ), for all A ζ X with ( ) standing for finite intersection. {B λ :}=A 3.3. Theorem. Suppose that B : ζ X A. Then B is a base of some intuitionistic I-fuzzy topology, if B satisfies the following condition () B(0 ) = B( ) =, (2) U, V ζ X, B(U V ) B(U) B(V ). Proof. For A ζ X, let τ(a) = B(B λ ). To show that B is a base B λ =A of τ, we only need to prove τ is an intuitionistic I-fuzzy topology on X. For all U, V ζ X, τ(u) τ(v ) = ( ) ( ) B(A α ) B(B β ) α K β K 2 = α K A α=u A α=u, B β =V α K β K 2 α K,β K 2 (A αb β )=UV C λ =UV = τ(u V ). B(C λ ) β K 2 B β =V For all {A λ : λ K} ζ X, Let B λ = {{B δλ : δ λ K λ } : τ( A λ ) = B δ = A λ δ K (( ) ( )) B(A α ) B(B β ) α K β K 2 ( ) B(A α B β ) α K,β K 2 δ K B(B δ ). δ λ K λ B δλ = A λ }, then

6 236 For all f B λ, we have Therefore, Similarly, we have Hence µ τ( A λ ) = B δλ f(λ) B δλ = A λ. µ B δ = B(Bδ ) A λ δ K δ K µ f B(Bδλ ) B λ B δλ f(λ) = µ B(Bδλ ) {B δλ :δ λ K λ } B λ δ λ K λ = µ τ(aλ ). γ τ( τ( A λ ) A λ ) γ τ(aλ ). τ(a λ ). This means that τ is an intuitionistic I-fuzzy topology on X and B is a base of τ Theorem. Let (X, τ), (Y, δ) be two intuitionistic I-fuzzy topology spaces and δ generated by its subbase ϕ. The mapping f : (X, τ) (Y, δ) satisfies ϕ(u) τ(f (U)), for all U ζ Y. Then f is fuzzy continuous, i.e., δ(u) τ(f (U)), U ζ Y. Proof. U ζ Y, δ(u) = This completes the proof. A λ =U {B µ:µ K λ }=A λ A λ =U {B µ:µ K λ }=A λ A λ =U A λ =U = τ(f (U)). τ(f (A λ )) τ(f ( A λ )) µ K λ ϕ(b µ ) µ K λ τ(f (B µ ))

7 Theorem. Suppose that (X, τ), (Y, δ) are two intuitionistic I-fuzzy topology spaces and τ is generated by its base B. If the mapping f : (X, τ) (Y, δ) satisfies B(U) δ(f (U)), for all U ζ X. Then f is fuzzy open, i.e., W ζ X, τ(w ) δ(f (W )). Proof. W ζ X, Therefore, f is open. τ(w ) = B(A λ ) A λ =W A λ =W A λ =W = δ(f (W )). δ(f (A λ )) δ(f ( A λ )) 3.5. Theorem. Let (X, τ), (Y, δ) be two intuitionistic I-fuzzy topology spaces and f : (X, τ) (Y, δ) intuitionistic I-fuzzy continuous, Z X. Then f Z : (Z, τ Z ) (Y, δ) is continuous, where (f Z )(x) = f(x), (τ Z )(A) = {τ(u) : U Z = A}, for all x Z, A ζ Z. Proof. W ζ Z, (f Z ) (W ) = f (W ) Z, we have (τ Z )((f Z ) (W )) = {τ(u) : U Z = (f Z ) (W )} τ(f (W )) δ(w ). Then f Z is intuitionistic I-fuzzy continuous Theorem. Let (X, τ) be an intuitionistic I-fuzzy topology space and τ generated by its base B, B Y (U) = {B(W ) : W Y = U}, for Y X, U ζ Y. Then B Y is a base of τ Y. Proof. For U ζ X, (τ Y )(U) = τ(v ) = B(A λ ). It remains to show the following equality B(A λ ) = V Y =U A λ =V V Y =U V Y =U A λ =V B λ =U In one hand, for all V ζ X with V Y = U, and A λ Y = U. Put B λ = A λ Y, clearly B λ = U. Then B λ =U W Y =B λ B(W ) W Y =B λ B(W ). A λ B(A λ ). = V, we have

8 238 Thus, V Y =U A λ =V B(A λ ) On the other hand, a (0, ], a < family of {B λ : λ K} ζ Y, such that () B λ = U; B λ =U B λ =U W Y =B λ B(W ). W Y =B λ µ B(W ), there exists a (2) λ K, there exists W λ ζ X with W λ Y = B λ such that a < µ B(Wλ ). Let V = W λ, it is clear V Y = U and µ B(Wλ ) a. Then V Y =U A λ =V By the arbitrariness of a, we have µ B(Aλ ) V Y =U A λ =V Similarly, we may obtain that γ B(Aλ ) So we have V Y =U A λ =V V Y =U A λ =V B(A λ ) µ B(Aλ ) a. B λ =U B λ =U B λ =U W Y =B λ µ B(W ). W Y =B λ γ B(W ). W Y =B λ B(W ). Therefore, B(A λ ) = V Y =U A λ =V B λ =U W Y =B λ B(W ). This means that B Y is a base of τ Y Theorem. Let {(X α, τ α )} be a family of intuitionistic I-fuzzy topology X α, spaces and P β : X α X β the projection. For all W ζ ϕ(w ) = τ α (U). Then ϕ is a subbase of some intuitionistic I-fuzzy topology τ, Pα (U)=W here τ is called the product intuitionistic I-fuzzy topologies of {τ α : α J} and denoted by τ = τ α.

9 239 Proof. We need to prove ϕ ( ) is a subbase of τ. ϕ ( ) ( ) = ϕ(b λ ) = {B λ :}= τ α (U) {B λ :}= Pα (U)=B λ =. Similarly, ϕ ( ) (0 ) = X α,. For all U, V ζ we have ϕ ( ) (U) ϕ ( ) (V ) = ( ) ( ) ϕ(b α ) ϕ(c β ) {B α:α E }=U α E {C β :β E 2}=V β E 2 ( ) = ( ϕ(b α )) ( ϕ(c β )) {B α:α E }=U {C β :β E 2}=V α E β E 2 ϕ(b λ ) {B λ :}=UV = ϕ ( ) (U V ). Hence, ϕ ( ) is a base of τ, i.e., ϕ is a subbase of τ. And by Theorem 3.3 we have τ(a) = ϕ ( ) (B λ ) = = B λ =A ϕ(c ρ ) B λ =A {C ρ:ρ E}=B λ ρ E B λ =A {C ρ:ρ E}=B λ ρ E P α (V )=Cρ τ α (V ). By the above discussions, we easily obtain the following corollary Corollary. Let ( X α, τ α ) be the product space of a family of intuitionistic I-fuzzy topology spaces {(X α, τ α )}. Then P β : ( X α, τ α ) (X β, τ β ) is continuous, for all β J. Proof. U ζ X β, τ(p β (U)) = B λ =Pβ τ β (U) (U) {C ρ:ρ E}=B λ ρ E P α (V )=Cρ τ α (V ) Therefore, P β is continuous.

10 Applications in product Intuitionistic I-fuzzy topological space 4.. Definition. Let (X, τ) be an intuitionistic I-fuzzy topology space. The degree to which two distinguished intuitionistic fuzzy points x (α,β), y (λ,ρ) pt(ζ X )(x y) are T 2 is defined as follows T 2 (x (α,β), y (λ,ρ) ) = (Q x(α,β) (U) Q y(λ,ρ) (V )). UV =0 The degree to which (X, τ) is T 2 is defined by T 2 (X, τ) = { T2 (x (α,β), y (λ,ρ) ) : x (α,β), y (λ,ρ) pt(ζ X ), x y } Theorem. Let (X, Iω(τ)) be a generated intuitionistic I-fuzzy topological space by fuzzifying topological space (X, τ) and T 2 (X, Iω(τ)) µ T2(X,Iω(τ)), γ T2(X,Iω(τ)). Then µ T2(X,Iω(τ)) = T 2 (X, τ). Proof. For all x, y X, x y, and each a < { ) : UV =0 ( µqx(α,β) (U)µ Qy(λ,ρ) (V ) x (α,β), y (λ,ρ) pt(ζ X ), x y }, there exists U, V ζ X with U V = 0 such that a < µ Qx(,0) (U), a < µ Qy(,0) (V ). Then there exists U, V ζ X, such that x (,0) q U U, a < ω(τ)(µ U ), y (,0) q V V, a < ω(τ)(µ V ). Denote A = σ 0 (µ U ), B = σ 0 (µ V ), it is clear that x A, y B. From the fact U V = 0, it implies µ U µ V = 0. Then we have σ 0 (µ U ) σ 0 (µ V ) =, i.e.,a B =. Thus a < ω(τ)(µ U ) = r I a < τ(σ r (µ U )) τ(σ 0 (µ U )) = τ(a). x U A τ(u) = N x (A). Similarly, we have a < N y (B). Hence a < (N x (A) N y (B)). Then Therefore, a { A B= A B= (N x (A) N y (B)) : x, y X, x y }. { ( ) µqx(α,β) (U) µ Qy(λ,ρ) (V ) : x(α,β), y (λ,ρ) pt(ζ X ), x y } UV =0 { (N x (A) N y (B)) : x, y X, x y }. A B= On the other hand, for all x (α,β), y (λ,ρ) pt(ζ X ), x y, and a < { A B= (N x (A) N y (B)) : x, y X, x y }, there exists A, B 2 X, A B =, such that a < N x (A), a < N y (B). Then there exists A, B 2 X, such that x A A, a < τ(a ),

11 24 y B B, a < τ(b ). Let U = A, A c, V = B, B c, where A c is the complement of A, then x (α,β) q U, y (λ,ρ) q V. In fact, A (x) = > α, A c (x) = 0 < β. Thus x (α,β) q U. Similarly, we have y (λ,ρ) q V. By A B =, we have A B =. Then for all z X, we obtain Hence Since r I, σ r ( A ) = A, we have ω(τ)( A ) = ( A B )(z) = A (z) B (z) = 0, ( A c B c )(z) = A c (z) B c (z) =. A B = 0, A c B c =. r I τ(σ r ( A )) = τ(a ). By A c = A, and a < τ(a ), we have a < ω(τ)( A ) ω(τ)( A c ) = ω(τ)(µ U ) ω(τ)( γ U ). So, a < x (α,β) q W U (ω(τ)(µ W ) ω(τ)( γ W )) = µ Qx(α,β) (U). Similarly, we have a < µ Qy(λ,ρ) (V ). This deduces that a < ( ) µqx (U) µ Qy(λ,ρ) (V ) (α,β). UV =0 Furthermore, we may obtain a { ( ) µqx (U) µ Qy(λ,ρ) (V ) (α,β) : x(α,β), y (λ,ρ) pt(ζ X ), x y }. Hence UV =0 This means that { y } = { { ( ) µqx(α,β) (U) µ Qy(λ,ρ) (V ) : x(α,β), y (λ,ρ) pt(ζ X ), x y } UV =0 { (N x (A) N y (B)) : x, y X, x y }. A B= A B= UV =0 ( µqx(α,β) (U)µ Qy(λ,ρ) (V ) ) : x(α,β), y (λ,ρ) pt(ζ X ), x (N x (A) N y (B)) : x, y X, x y }. Therefore we have µ T2(X,Iω(τ)) = T 2 (X, τ) Lemma. Let ( X j, τ j ) be the product space of a family of intuitionistic j J j J I-fuzzy topology spaces {(X j, τ j )} j J. Then τ j (A j ) ( τ j )(Pj (A j)), for all j J, A j ζ Xj. j J

12 242 Proof. Let τ j = δ, x (α,β) q f (U) f (x (α,β) ) q U. Then for all j J, A j j J ζ Xj, we have δ(p j (A j )) = This completes the proof. = Q δ x (α,β) (Pj (A j )) x (α,β) q Pj (Aj) Q τj P x (α,β) q Pj (Aj) j (x (α,β)) (A j) Q τj P Pj (x j (x (α,β)) (A j) (α,β)) q A j (A j ) x j (α,β) q Aj Q τj x j (α,β) = τ j (A j ) Theorem. Let ( X j, τ j ) be the product space of a family of intuitionistic I-fuzzy topology spaces {(X j, τ j )} j J. Then T 2 (X j, τ j ) T 2 ( X j, τ j j J j J ). X j j J Proof. For all g (α,β), h (λ,ρ) pt(ζ ) and g h. Then there exists j0 J such that g(j 0 ) h(j 0 ), where g(j 0 ), h(j 0 ) X j0. j J For all U j0, V j0 ζ Xj 0 with U j0 V j0 = 0 Xj 0, we have P j 0 (U j0 ) P j 0 (V j0 ) = P j 0 (U j0 V j0 ) = 0 X j j J. Then Q g(j0) (α,β) (U j0 ) Q g(α,β) (Pj 0 (U j0 )). In fact, if g(j 0 ) (α,β) q U j0, then g (α,β) q Pj 0 (U j0 ). For all V U j0, we have Pj 0 (V ) Pj 0 (U j0 ). On account of Lemma 4.3, we have τ j0 (V ) )( τ j )(Pj 0 (V )) g(j 0) (α,β) q V U j0 g (α,β) q Pj (V )P 0 j (U j0 j J 0 τ j )(G), g (α,β) q GP j 0 (U j0 )( j J i.e., Q g(j0) (α,β) (U j0 ) Q g(α,β) (Pj 0 (U j0 )). Thus, (Q g(j0) (α,β) (U) Q h(j0) (λ,ρ) (V )) UV =0 X j 0 X j Pj (U)P j J 0 j (V )=0 0 X j j J GH=0 j J j J (Q g(α,β) (P j 0 (U))Q h(λ,ρ) (P j 0 (V ))) (Q g(α,β) (G) Q h(λ,ρ) (H)).

13 243 So we have Thus T 2 (g(j 0 ) (α,β), h(j 0 ) (λ,ρ) ) T 2 (g (α,β), h (λ,ρ) ). T 2 (X j0, τ j0 ) T 2 ( X j, τ j ). j J j J Therefore, j J T 2 (X j, τ j ) T 2 ( j J X j, j J 4.5. Lemma. Let (X, Iω(τ)) be a generated intuitionistic I-fuzzy topological space by fuzzifying topological space (X, τ). Then () Iω(τ)(A) =, for all A = α, β ζ X ; (2) B X, τ(b) = µ Iω(τ) ( B, B c ). τ j ). Proof. By Lemma 2., 2.2 and 2.3, it is easy to prove it Lemma. Let (X, δ) be a stratified intuitionistic I-fuzzy topological space (i.e., for all < α, β > A, δ(< α, β >) = ). Then for all A ζ X µ δ ( σr(µ A ), (σr(µ A )) c ) µ δ(a). r I Proof. For all A ζ X, and for any a < µ δ ( σr(µ A ), (σr(µ A )) c ), y (α,β) pt(ζ X ) with y (α,β) q A, clearly µ A (y) > α. Then there exists δ > 0 such that µ A (y) > α + δ. Thus y σ α+δ (µ A ). So we have Then Therefore, r I y (α,β) q σ α+δ (µ A ), (σ α+δ (µ A )) c. a < µ δ ( σ α+δ (µ A ), (σ α+δ (µ A )) c ) = µ(q z(α,β) ( σ α+δ (µ A ), (σ α+δ (µ A )) c )). z (α,β) q σ α+δ (µ A ), (σ α+δ (µ A )) c a < µ(q y(α,β) ( σ α+δ (µ A ), (σ α+δ (µ A )) c )). Since (X, δ) is a stratified intuitionistic I-fuzzy topological space, we have Q y(α,β) ( α + δ, α δ ) =. Moreover, it is well known that the following relations hold α + δ σ α+δ (µ A ) µ A, So we have α δ (σ α+δ (µ A )) c µ A γ A. a < µ(q y(α,β) ( α + δ σ α+δ (µ A ), α δ (σ α+δ (µ A )) c )) µ(q y (α,β) (A)).

14 244 Then a µ δ (A). Therefore, µ δ ( σr(µ A ), (σr(µ A )) c ) µ δ(a). r I 4.7. Theorem. Let ( X α, τ α ) be the product space of a family of fuzzifying topological space {(X α, τ α )}. Then ( Iω(τ α ))(A) = Iω( τ α )(A). Proof. Let ( Iω(τ α ))(A) = µ Iω(τ α))(a), γ Iω(τ α))(a). For all a < µ Iω(τ α))(a), there exists {Uj a} j K such that Uj a = A, for each U j a, there exists {Aa λ,j } j K such that A a λ,j = U j a, where E is an finite index set. In addition, for every λ E, there exists α α(λ) J and W α ζ Xα with Pα (W α ) = A a λ,j such that a < µ(iω(τ α )(W α )). Then we have a < ω(τ α )(µ Wα ), a < ω(τ α )( γ Wα ). Thus for all r I, we have a < τ α (σ r (µ Wα )) ( τ α )(Pα (σ r (µ Wα ))) = ( τ α )(σ r (Pα (µ Wα ))) = ( τ α )(σ r (µ A a λ,j )). Hence Furthermore a ( = ( τ α )( σ r (µ A a λ,j )) τ α )(σ r ( µ A a λ,j )) = ( τ α )(σ r (µ U a j )). a ( = ( τ α )( σ r (µ U a j )) j K τ α )(σ r ( µ U a j )) j K = ( τ α )(σ r (µ A )).

15 245 So a ( τ α )(σ r (µ A )) r I = ω( τ α )(µ A ). Similarly, we have a ω( τ α )( γ A ). Hence a µ(iω( µ(iω( τ α )(A)). τ α )(A)). By the arbitrariness of a, we have µ(( Iω(τ α ))(A)) On the other hand, for a < µ(iω( τ α )(A)), we have a < ω( τ α )(µ A ) = ( τ α )(σ r (µ A )) r I and a < ω( τ α )( γ A ). Then for all r I, we have a < ( Thus there exists {U a j,r } j K X satisfies τ α )(σ r (µ A )). j K there exists {A a λ,j,r }, where E is an finite index set, such that U a j,r = σ r(µ A ), and for all j K, A a λ,j,r = U a j,r. For all λ E, there exists α(λ) J, W α ζ Xα, such that P α (W α ) = A a λ,j,r. By Lemma 4.5 we have a < τ α (W α ) = µ Iω(τα)( Wα, W c α ) µ( = µ( = µ( µ( = µ( = µ( Iω(τ α ))(P α ( Wα, W c α )) Iω(τ α ))( P α (W α), P α (W c α ) ) Iω(τ α ))( A a λ,j,r, (A a λ,j,r ) c ) Iω(τ α ))( A a λ,j,r, (A a λ,j,r ) c ) Iω(τ α ))( A a, λ,j,r (A a )c ) λ,j,r Iω(τ α ))( U a j,r, (U a j,r ) c ).

16 246 Then a µ( Iω(τ α ))( U a, j,r ( U a )c ) j,r j K j K = µ( Iω(τ α ))( σr(µ A ), (σr(µ A )) c ). By Lemma 4.6 we have a µ( Iω(τ α ))( σr(µ A ), (σr(µ A )) c ) r I µ(( Iω(τ α ))(A)). Then Hence Then Therefore, µ(( Iω(τ α ))(A)) µ(iω( τ α )(A)). µ(( Iω(τ α ))(A)) = µ(iω( τ α )(A)). γ(( Iω(τ α ))(A)) = γ(iω( τ α )(A)). ( Iω(τ α ))(A) = Iω( τ α )(A). 5. Further remarks As we have shown, the notions of the base and subbase in intuitionistic I-fuzzy topological spaces are introduced in this paper, and some important applications of them are obtained. Specially, we also use the concept of subbase to study the product of intuitionistic I-fuzzy topological spaces. In addition, we have proved that the functor Iω preserves the product. There are two categories in our paper, the one is the category FYTS of fuzzifying topological spaces, and the other is the category IFTS of intuitionistic I-fuzzy topological spaces. It is easy to find that Iω is the functor from FYTS to IFTS. We discussed the property of the functor Iω in Theorem 4.7. A direction worthy of further study is to discuss the the properties of the functor Iω in detail. Moreover, we hope to point out that another continuation of this paper is to deal with other topological properties of intuitionistic I-fuzzy topological spaces. References [] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(986), [2] K.T. Atanassov, Intuitionistic Fuzzy Sets, Springer, Heidelberg, 999. [3] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24(968), [4] D. Çoker, An introduction to intuitionistic fuzzy topological space, Fuzzy Sets and Systems, 88(997), 8 89.

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