Applied General Topology c© Universidad Politécnica de Valencia Volume 1, No. 3, 2002 pp. 13–23 A contribution to fuzzy subspaces Miguel Alamar and Vicente D. Estruch ∗ Abstract. We give a new concept of fuzzy topological subspace, which extends the usual one, and study in it the related concepts of interior, closure and conectedness. 2000 AMS Classification: 54A40 Keywords: Fuzzy connectedness, fuzzy topology, Q-neighborhood. 1. Introduction A simplest and at the same time a very important operation of General Topology is transition to a subspace because it let us consider hereditary and local properties, completation and compactification of subspaces, etc. When A is a subspace of a topological space X, the following assertion is satisfied: F is closed in A iff F = A∩T for a closed set T of X. This is not true, in general, in considering a similar problem for fuzzy topological spaces (fts). For this reason, in fuzzy research the consideration of subspaces in a fts (X,T ) is restricted by authors only to ordinary subsets of X. In this paper we extend the concept of a fuzzy topological subspace of X, to fuzzy sets A of X for which the above assertion is satisfied, and then we will be able to extend some concepts and results of the (fuzzy) topological spaces to these subspaces. The structure of the paper is as follows: After preliminaries, in section 3 we define and study the concept of (fuzzy) subspace, and in sections 4–5 we study in it the concepts of interior, closure and connectedness. 2. Preliminaries Throughout this paper, I will denote the unit real interval [0, 1]. For a non-empty set X, IX denotes the collection of all mappings from X into I. A member B of IX is called a fuzzy set of X. The set {x ∈ X : B(x) > 0} is called the support of B and is denoted by Supp B. If B takes only the values 0, ∗While working on this paper the authors have been partially supported by a grant from UPV: ”Incentivo a la Investigación/99” 14 M. Alamar and V. D. Estruch 1, B is called a crisp set in X. From now on, we shall not differentiate between a crisp set B in X and (the ordinary subset of X) Supp B. Nevertheless, the crisp set which always takes the value 1 (respectively, 0) on X is denoted by 1 (respectively, 0). The union and intersection of a family of fuzzy sets {Ai}i of X is ∨ i Ai and∧ i Ai, respectively. The complement of A ∈ I X, denoted by A′, is defined by the formula A′(x) = 1−A(x), x ∈ X. For A,B ∈ IX we write A ⊂ B or B ⊃ A if A(x) ≤ B(x), for each x ∈ X. The fuzzy set xλ of X given by xλ(y) = 0 if y 6= x, and xλ(x) = λ (λ ∈]0, 1]) is called a fuzzy point of X with support x [5]. The fuzzy point xλ is said to be contained in a fuzzy set A or belonging to A, denoted by xλ ∈ A, if λ ≤ A(x). A family T of fuzzy sets of X containing 0 and 1, is called a fuzzy topology on X [1] if it is closed under arbitrary unions and finite intersections. The pair (X,T ) is called a fuzzy topological space (fts). Each member of T is called an open (fuzzy) set. The complement of an open set is called a closed (fuzzy) set. T c denotes the family of all closed sets of (X,T ). 3. Fuzzy subspaces Definition 3.1. Let (X,T ) be a fts and Y ∈ IX. The pair (Y,TY ) is called a fuzzy topological subspace of (X,T ) if the family TY = {G∩Y : G ∈T} satisfies the following conditions: (c1) For each H ∈TY there exist FH ∈T c such that Y −H = FH ∩Y . (c2) For each F ∈T c there exist GF ∈T such that Y −(F ∩Y ) = GF ∩Y . In this case, the members of TY will be called TY -open. If F ⊂ Y and Y −F ∈ TY , F will be called a TY -closed (fuzzy) set. For shortness we will say Y is a subspace of X. Notice that TY is closed under finite intersections and arbitrary unions and that 0 and Y (instead of X) are both TY -open and TY -closed. Also, if Y is an ordinary subspace of the topological space X, Y is a subspace of X in the sense of definition 3.1. Remark 3.2. The above conditions (c1) and (c2) establish that F is TY -closed iff F = T ∩Y for a T -closed set T of the fts (X,T ). Example 3.3. (a) (Ordinary subsets of a fts X are subspaces of X). Let (X,T ) be a fts and let Y an ordinary subset of X. If H ∈ TY then H = G∩Y for some G ∈T , thus Y −H = (1 −G) ∩A and (c1) is satisfied. If F ∈ T c then Y − (F ∩Y ) = (1 −F) ∩Y , where 1 −F ∈ T , and (c2) is satisfied. So, Y is a subspace of X. (b) Let X 6= ∅. We will denote by fc the constant function on X given by fc(x) = c, for each x ∈ X, with 0 ≤ c ≤ 1. Now, consider the Lowen indiscrete fuzzy topology T = {fc : c ∈ [0, 1]}, on X. (Notice that T = T c.) A contribution to fuzzy subspaces 15 Fix a real number k ∈]0, 1[, and choose a non-empty ordinary subset B of X. Now consider Y ∈ IX given by Y (x) = k if x ∈ B, and Y (x) = 0 if x /∈ B. We will see that Y is a subspace of (X,T ). In fact, H ∈ TY iff H(x) = m, for each x ∈ B and H(x) = 0 elsewhere, for some m ∈ [0,k]. Then Y − H is given by (Y − H)(x) = k − m if x ∈ B and (Y − H)(x) = 0 if x /∈ B. So, Y − H = fk−m ∩ Y and (c1) is satisfied since 1 −fk−m = f1−(k−m) ∈T . Now suppose F ∈ T c. Then, also F ∈ T , and thus F = fm, for some m ∈ [0, 1]. Hence, Y − (F ∩Y ) is given by (Y − (F ∩Y ))(x) = k− (m∧k) for x ∈ B and (Y − (F ∩Y ))(x) = 0 if x /∈ B, and clearly (c2) is satisfied. (c) (Construction of fuzzy topological subspaces). Fix a real number γ ∈]0, 1 2 [, and let L be a fuzzy topology on X such that G(x) ≤ γ, for each x ∈ X and G ∈L ∼ {1}. Take A ∈ IX such that A(x) ≤ 1 2 , for x ∈ X. Clearly, if L contains at least two proper open sets then A is not a subspace of (X,L). Now, denote L∗ = {G∗ : G∗ = 1 − (A− (A∩G)), G ∈L}. We will see that L∗ ∪{0} is a fuzzy topology on X: Consider a Family {G∗i : i ∈ I} of elements of L ∗, and suppose G∗i = 1 − (A− (A∩Gi)), where Gi ∈L, i ∈ I. We will see that ⋃ i G ∗ i ∈L ∗. We have ⋃ i G ∗ i = ⋃ i(1 − (A− (A∩Gi)) = (1 −A) + ⋃ i(A∩Gi) = (1 −A) + (A∩ ( ⋃ i Gi)) = (1 −A) + (A∩G) where G = ⋃ i Gi ∈L, and so ⋃ i G ∗ i ∈L ∗. Now, we will see L∗ is closed under finite intersection. Take G∗i = 1 − (A− (A∩Gi)), where Gi ∈L, i = 1, 2. We have G∗1 ∩G ∗ 2 = (1 − (A− (A∩G1))) ∩ (1 − (A− (A∩G2))) = (1 −A) + ((A∩G1) ∩ (A∩G2)) = (1 −A) + (A∩ (G1 ∩G2)) = 1 −A + (A∩G) where G = G1 ∩G2 ∈T , and thus G∗1 ∩G∗2 ∈L∗. Clearly 1 ∈L∗ and then L∗ ∪{0} is a fuzzy topology on X. Now, if G ∈L with G 6= 1, then G(x) < 1 2 for each x ∈ X, and if G∗ ∈ L∗ with G∗ 6= 0 we have G∗(x) ≥ 1 2 , and from these facts it is easy to verify that T = L∪L∗ is a fuzzy topology on X. Now we will see that A is subspace of (X,T ): Clearly, TA = LA ∪{A}. First we will see condition (c1) is satisfied. Let H ∈TA and suppose H = G∩A with G ∈T . We distinguish two possibilities: (i) G ∈L∗ ∪{1}. In this case H = G∩A = A and A−H = 0 = 0 ∩A. 16 M. Alamar and V. D. Estruch (ii) G = L ∼ {1}. In this case A−H = A− (G∩A) = (A− (G∩A)) ∩A since A− (G∩A) ⊂ A, with 1 − (A− (G∩A)) ∈L∗ ⊂T . Now, we will prove that condition (c2) is satisfied. Suppose F ∈ T c. We distinguish two possibilities: (i) 1 −F ∈L∗. In this case F = A− (A∩G), with G ∈L. So, A− [(A− (A∩G)) ∩A] = A− (A− (A∩G)) = A∩G ∈TA (ii) 1 − F ∈ L ∼ {1}. In this case F(x) ≥ 1 2 for each x ∈ X, and thus F ⊃ A. Now, A− (F ∩A) = 0 ∈TA. Definition 3.4. Let Y be a subspace of the fts (X,T ), and suppose B ⊂ Y . The pair (B,TB) is called a fuzzy topological subspace of (Y,TY ) if the family TB = {G∩B : G ∈TY} satisfy the following conditions (c1)’ For each H ∈TB it exists FH ∈T cY such that B −H = FH ∩B. (c2)’ For each F ∈T cY it exists GF ∈TY such that B − (F ∩B) = GF ∩B. The terminology TB (instead of TYB ) is justified in proposition 3.6. Otherwise, the elements of TB are called TB-open. If F ⊂ B and B−F ∈TB, F is called TB-closed. For shortness we will say B is a subspace of Y . Remark 3.5. The above conditions (c1)’ and (c2)’ establish that F is TB- closed iff F = B ∩T for a TY -closed set T of Y . Proposition 3.6. Suppose B is a subspace of Y and Y is a subspace of the fts (X,T ). Then, B is a subspace of (X,T ). Proof. It is an immediate consequence of remarks 3.2 and 3.5. � Proposition 3.7. Let Y be a subspace of the fts (X,T ). If Y is T -open (respectively, T -closed) then G ∈ TY iff G ∈ T (respectively, F ∈ T cY iff F ∈ T c). Proposition 3.8. Let B and Y the two subspaces of the fts (X,T ). If B ⊂ Y , then B is a subspace of Y . Proof. We have H ∈ TB iff there exists G∗ ∈ T such that H = G∗ ∩B. Now, G∩B = G∩Y ∩B, and since G = G∗∩Y ∈ TY , then TB = {G∩B : G ∈TY}. We will see that the family TB satisfies (c1)’ and (c2)’. Let H ∈TB; then there exists FH ∈T c such that B −H = FH ∩B = (FH ∩Y ) ∩B, where FH ∩Y ∈T cY , and so (c1)’ is satisfied. Now, let F ∈T cY ; then there exists GF ∈T such that B − (F ∩B) = GF ∩B = (GF ∩Y ) ∩B, where GF ∩Y ∈TY , and so (c2)’ is satisfied. � A contribution to fuzzy subspaces 17 Lemma 3.9. Let M, N and P be fuzzy sets of X. Then (M ∩N) − (P ∩ (M ∩N)) = (M − (P ∩M)) ∩ (N − (P ∩N)) Proof. Let x ∈ X and suppose M(x) ≥ N(x). We distinguish three possibili- ties: (1) N(x) ≤ P(x) ≤ M(x). In this case, the left hand of the above inequality becomes N(x) − N(x) = 0, and the right hand becomes (M(x) −P(x)) ∧ (N(x) −N(x)) = 0. (2) P(x) ≤ N(x) ≤ M(x). Now the left hand of the inequality becomes N(x) − P(x), and the right hand becomes (M(x) − P(x)) ∧ (N(x) − P(x)) = N(x) −P(x). (3) N(x) ≤ M(x) ≤ P(x). Now, the left hand of the inequality becomes N(x)−N(x) = 0, and the right hand becomes (M(x)−M(x))∧(N(x)− N(x)) = 0. Since the announced equality is symmetric respect M and N, the same argument is valid for N(x) ≥ M(x), and then the equality is established. � Proposition 3.10. Let A and B be two subspaces of the fts (X,T ). Then A∩B is subspace of (X,T ). Proof. Consider the family TA∩B = {G∩ (A∩B) : G ∈ T}. We will see that TA∩B satisfies (c1) and (c2). Let H ∈TA∩B; then H = G∩ (A∩B) with G ∈T . Now, by lemma 3.9, (A∩B) −H = (A∩B) − (G∩ (A∩B)) = (A− (G∩A)) ∩ (B − (G∩B)). But A−(G∩A) is TA-closed and hence A−(G∩A) = FA∩A for some FA ∈T c. Also B − (G∩B) = FB ∩B for some FB ∈T c and therefore (A∩B) −H = (FA ∩A) ∩ (FB ∩B) = (FA ∩FB) ∩ (A∩B) and (c1) is satisfied since FA ∩FB is T -closed. Now, let F ∈T c. By lemma 3.9, (A∩B) − (F ∩ (A∩B)) = (A− (F ∩A)) ∩ (B − (F ∩B)) = GA ∩GB where GA = A−(F ∩A) ∈TA and GB = B−(F ∩B) ∈TB. So, there are G1, G2 ∈T such that GA = G1 ∩A and GB = G2 ∩B and therefore (A∩B) − (F ∩ (A∩B)) = (G1 ∩A) ∩ (G2 ∩B) = (G1 ∩G2) ∩ (A∩B) = G∩ (A∩B) where G = G1 ∩G2 ∈T , and (c2) is satisfied. � Since ordinary subsets in a fts (X,T ) are subspaces, we have the following corollary. 18 M. Alamar and V. D. Estruch Corollary 3.11. Let A be a subspace of the fts (X,T ) and Y an ordinary subset of X. Then A∩Y is a subspace of (X,T ). If A and B are two subspaces of X, in general A ∪ B is not a subspace of X, even if A∩B = 0, as shows the following example. Example 3.12. Let X = [0, 1] and choose three the real numbers a, b and c, such that 0 < c < a < b < 1. Consider X endowed with the Lowen indiscrete topology T of example 3.3 (b). Consider the fuzzy sets A and B of X given by A(x) = { a 0 ≤ x ≤ 1 2 0 1 2 < x ≤ 1 B(x) = { 0 0 ≤ x ≤ 1 2 b 1 2 < x ≤ 1 by (b) of the example 3.3, A and B are subspaces of X. Obviously A∩B = 0 but we will see A∪B is not a subspace of X. Consider the constant function fc on X defined by fc(x) = c, x ∈ X. We have that fc ∈T c and since (A∪B − (fc ∩ (A∪B)))(x) = (A∪B −fc)(x) = { a− c 0 ≤ x ≤ 1 2 b− c 1 2 < x ≤ 1 it is obvious that condition (c2) cannot be satisfied. 4. Interior and closure In this section, Y will be a subspace of the fts (X,T ). According with [1], the TY -interior, denoted by intY A, of a fuzzy set A contained in Y is the largest TY -open (fuzzy) set contained in A, and the TY - closure, denoted by clY A, is the smallest TY -closed (fuzzy) set containing A. Proposition 4.1. Let A ⊂ Y . Then (i) intX A = intY A∩ intX Y . (ii) clY A = clX A∩Y . Proof. It is similar to the classic case. � Through the notion of fuzzy point, it is possible to study the concepts of interior and cluster (adherence) point. According with [5], we give the following definition. Definition 4.2. A fuzzy set A in (X,TY ) is called a TY -neighborhood of the fuzzy point xλ if there exists B ∈TY such that xλ ∈ B ⊂ A. We also say that xλ is TY -interior of A. Then, xλ ∈ intY A iff xλ is TY -interior of A. In [3], is given the following definition. Definition 4.3. The fuzzy set point xλ is said to belong to B, written xλ ∼ ∈ B, iff B(x) > λ. According with [3], the following is a distinct definition of an interior point. A contribution to fuzzy subspaces 19 Definition 4.4. A fuzzy set A in (Y,TY ) is called a T ∼Y -neighborhood of the fuzzy point xλ if there exists B ∈TY such that xλ ∼ ∈ B ⊂ A. Also, xλ is called a T ∼Y -interior point of A. Then, xλ ∼ ∈ intY A iff xλ is a T ∼Y -interior pointof A. Notice that if (intY A)(x) > 0 then x(intY A)(x) is TY -interior of A, but it is not T ∼Y -interior of A. Nevertheless, intY A = ⋃ {xλ : xλ is TY -interior of A} = ⋃ {xλ : xλ is T ∼Y -interior of A}. By (i) of proposition 4.1, we have the following corollary. Corollary 4.5. Let A be contained in Y . Then, a fuzzy point xλ ∈ intX Y (respectively, xλ ∼ ∈ intX Y ) is a TY -interior (respectively, a T ∼Y -interior) point of A iff it is a T -interior point of A. The following definitions and results are obvious generalizations of the ones given in [5]. Definition 4.6. A fuzzy point xλ is said to be Y -quasi-coincident with the fuzzy set A of X, denoted by xλ qY A, if λ + A(x) > Y (x). Definition 4.7. Let A,B ⊂ Y . A is said to be Y -quasi-coincident with B, denoted by A qY B, if there exists x ∈ X such that A(x) + B(x) > Y (x). Definition 4.8. A fuzzy set A in(X,TY ) is called a QY -neighborhood of xλ if there exists B ∈TY , B ⊂ A, such that xλ qY B. Proposition 4.9. Let A,B ⊂ G. Then A ⊂ B iff A and Y − B are not Y -quasi-coincident; particularly xλ ∈ A iff xλ is not Y -quasi-coincident with Y −A. Theorem 4.10. A fuzzy point xλ ∈ clY A iff each QY -neighborhood of xλ is Y -quasi-coincident with A. Definition 4.11. A fuzzy point xλ is called a TY -adherence point of a fuzzy set A if every QY -neighborhood of xλ is Y -quasi-coincident with A. According with [3] we give the following definition. Definition 4.12. The fuzzy point xλ is called a TY -cluster point of A if for each G ∈TY such that xY (x)−λ ∼ ∈ G implies G 6⊂ Y −A. Proposition 4.13. Let A ⊂ Y . The fuzzy point xλ is TY -cluster point of A iff it is a TY -adherence point of A. Proof. Suppose xλ is TY -cluster point of A. Let G ∈ TY a Q-neighborhood of xλ. Then, λ + G(x) > Y (x) and thus xY (x)−λ ∼ ∈ G, and G 6⊂ Y − A since xλ is a TY -cluster point of A. Therefore, there exists x ∈ X such that G(x) > Y (x) −A(x), and so G is Y -quasi-coincident with A. Suppose xλ is a TY -adherence fuzzy point of A. Let G ∈ TY such that xY (x)−λ ∼ ∈ G. Then Y (x) − λ < G(x) and thus G is a neighborhood of xλ. So, G is Y -quasi-coincident with A. and then there exists x ∈ X such that G(x) + A(x) > Y (x); therefore G 6⊂ Y −A and xλ is TY -cluster point of A. � 20 M. Alamar and V. D. Estruch By (ii) of proposition 4.1 we have the following corollary. Corollary 4.14. Let A be contained in Y . Then, a fuzzy point xλ ∈ Y is a TY -adherence (cluster) point of A iff it is a T -adherence (cluster) point of A. 5. Connectedness We will use the concepts of connectedness due to Pu and Liu [5], [6], but with terminology of [7]. Definition 5.1. A fuzzy set D in the fts (X,T ) is called C-disconnected (re- spectively O-disconnected) if there are A,B ∈ T c (respectively, A,B ∈ T ) such that A ∩ D 6= 0, B ∩ D 6= 0, A ∩ B ∩ D = 0 and A ∪ B ⊃ D. A fuzzy set is called C-connected (respectively O-connected) if it is not C-disconnected (respectively O-disconnected). In contrast to General Topology the use of closed and open fuzzy sets in definitions of connectedness of fuzzy sets results in two distinct concepts. Nev- ertheless, we will see that these concepts agree in subspaces. According with definition 5.1, we give the following definition. Definition 5.2. A subspace (Y,TY ) of the fts X will be called C-disconnected (respectively O-disconnected) if there are two non-empty TY -closed (respec- tively TY -open) sets A and B such that A∩B = 0 and Y = A∪B. The following proposition shows that connected properties are absolute prop- erties in subspaces. Proposition 5.3. If Y is a subspace of the fts X, then the fuzzy set Y is C-connected (respectively O-connected) iff the subspace (Y,TY ) is C-connected (respectively O-connected). Proof. It is straightforward. � It is clear that a fts (X,T ) is O-connected iff it is C-connected. Now this fact is extendable to subspaces in the next proposition. Proposition 5.4. Let Y be a subspace of the fts (X,T ). Then Y is C- connected iff it is O-connected. Proof. Suppose Y is not O-connected. Then there are two sets G,H ∈T such that (1) G∩Y 6= 0, H ∩Y 6= 0, G∩H ∩Y 6= 0 and (2) Y = (G∩Y ) ∪ (H ∩Y ). Now, by (1), for x ∈ X, (G ∩ Y )(x) 6= 0 iff (H ∩ Y )(x) = 0 and also (H∩Y )(x) 6= 0 iff (G∩Y )(x) = 0, and thus Y −(G∩Y ) 6= 0, Y −(H∩Y ) 6= 0, and (Y − (G∩Y )) ∪ (Y − (H ∩Y )) = Y . Also, by (2), for x ∈ X if G(x) < Y (x) then H(x) ≥ Y (x), and if H(x) < Y (x) then G(x) ≥ Y (x), and thus (Y − (G ∩ Y )) ∩ (Y − (H ∩ Y )) = 0, and then (Y,TY ) is not C-connected. The converse is showed with a similar argument. � A contribution to fuzzy subspaces 21 Definition 5.5 (P1). Two fuzzy sets A1 and A2 in a fts (X,T ) are said to be Q-separated if there exist two T -closed sets Hi (i = 1, 2) such that Hi ⊃ Ai (i = 1, 2), and H1 ∩ A2 = H2 ∩ A1 = 0. It is obvious that A1 and A2 are Q-separated iff cl A1 ∩A2 = cl A2 ∩A1 = 0. Note 5.6. A fuzzy set D is C-disconnected [5] iff there exist two non-empty sets A and B, both two contained in Supp D, such that A and B are Q-separated, and D = A∪B. According with this, we give the following definition. Definition 5.7. A fuzzy subspace (Y,TY ) of X is here called disconnected if there exists two non-empty sets, both two contained in Y , such that A and B are Q-separated and Y = A∪B. Y is called connected if it is not disconnected. Theorem 5.8. Let Y be a subspace of (X,T ). They are equivalent: (i) Y is connected. (ii) Y is C-connected. (iii) Y is O-connected. Proof. By proposition 5.4 we only have to prove that (i) and (ii) are equivalent. Suppose Y is disconnected. Then there exist two sets A,B ⊂ Y such that clX A∩B = 0, clX B ∩A = 0 and Y = A∪B. Now, by (ii) of proposition 4.1, we have clY A = Y ∩ clXA = (A∪B) ∩ clXA = (A∩ clXA) ∪ (B ∩ clXA) = A, and hence A is TY -closed. Similarly, B is TY -closed and then Y is C-dis- connected. Suppose now that Y is C-disconnected. Then, there are two non-empty TY - closed sets A and B, such that A∩B = 0 and Y = A∪B. Now, otherwise by (ii) of proposition 4.1, we have A∩ clXB = (A∩Y ) ∩ clXB = A∩ clY B = A∩B = 0. Similarly, B ∩ clX A = 0, and then A and B are Q-separated. � Lemma 5.9. If A and B are Q-separated in the fts X and Y is a connected subspace of X, with Y ⊂ A∪B, then Y ⊂ A or Y ⊂ B. Proof. If A and B are Q-separated in the fts X, then A∩Y , and B∩Y are also Q-separated in X, and Y = (A∩Y ) ∪ (B ∩Y ), then A∩Y = 0 or B ∩Y = 0, i.e., Y ⊂ B or Y ⊂ A. � As in the classic case, theorem 5.8 and lemma 5.9 provide with some neat ways of proving a given space X is connected. Theorem 5.10. Let X be a fts. (a) If X = ⋃ α Xα, where each Xα is a connected subspace of X and⋂ α Xα 6= 0, then X is connected. (b) If each pair p,q of fuzzy points of X lies in some connected subspace Ep,q of X, then X is connected. 22 M. Alamar and V. D. Estruch (c) If X = ⋃∞ n=1 Xn where each Xn is a connected subspace of X and Xn−1 ∩Xn 6= 0, for each n ≥ 2, then X is connected. Proof. The proofs are slight modifications of the classic cases. � Next theorem is a generalization of theorem 10.1 of [5]. Theorem 5.11. Let Y be a subspace of the fts (X,T ) and let D be a C- connected fuzzy set in (X,T ). If D ⊂ Y then clY D is also C-connected. Proof. Suppose clY D is C-disconnected. Then, there are two T -closed sets A and B such that A∩clY D 6= 0, B∩clY D 6= 0, A∩B∩Y = 0 and A∪B ⊃ clY D. By the connectedness of D, we may assume that A∩D = 0, that is D ⊂ B. It follows that clY D ⊂ B and thus A∩ clY D = 0, which is a contradiction. � Definition 5.12. Two fuzzy sets A1 and A2 contained in a subspace (Y,TY ) of the fts X is said Q-separated in Y if there exist TY -closed sets Hi (i = 1, 2) such that Hi ⊃ Ai (i = 1, 2) and H1 ∩A2 = H2 ∩A1 = 0. Theorem 5.13. Let A be a family of C-connected fuzzy sets in fts X. Suppose⋃ A is a fuzzy subspace of X. If no two members of A are Q-separated in ⋃ A, then ⋃ A is connected. Proof. The same proof as theorem 10.2 of [5], but replacing Supp ⋃ A by ⋃ A, proves that ⋃ A is C-connected, and by theorem 5.8 ⋃ A is connected. � 5.14 Final considerations. One can extend in a natural way the Ti-fuzzy separation axioms of [5] in fts to subspaces, in such manner that were hereditary properties. Notice that there are many definitions of T2-fts in the literature (see [2]), but are particularly interesting the fuzzy separation axioms given in [4], through the concept of R-neighborhood. References [1] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182–190 [2] D. R. Cuttler, I.L. Reilly, A comparison of some Haussdorf notions in fuzzy topological spaces, Computers Math. Applic., Vol. 19, N. 11, (1990) 97–104 [3] Z. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl. 86 (1982), 74–95 [4] W. Guojun, A new fuzzy compactness defined by fuzzy sets, J. Math. Anal. Appl. 94 (1983), 1–23 [5] P.M. Pu, Y.M. Liu, Fuzzy Topology. I: Neighborhood stucture of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), 571–599 [6] P.M. Pu, Y.M. Liu, Fuzzy topology. II: Product and quotient spaces, J. Math. Anal. Appl. 77 (1980), 20–37 [7] A.P. Shostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Math. Surveys, 44: 6 (1989), 125–186 Received March 2001 A contribution to fuzzy subspaces 23 M. Alamar and V.D. Estruch Dep. de Matemática Aplicada Escuela Politécnica Superior de Gandia Universidad Politécnica de Valencia 46022 Valencia Spain E-mail address : malamar@mat.upv.es, vdestruc@mat.upv.es