fuzzy spaces

ABSTRACT


INTRODUCTION
As defined in [1], the notion of a fuzzy set in a set X is a function from X into the closed interval [0,1]. Accordingly, Chang [2] introduced the notion of a fuzzy topological space on a non-empty set X as a collection of fuzzy sets on X, closed under arbitrary suprema and finite infima and containing the constant fuzzy sets 0 and 1. Mathematicians extended many topological concepts to include fuzzy topological spaces such as: separation axioms, connectedness, compactness and metrizability. Several fuzzy homogeneity concepts were discussed in [3][4][5][6][7][8][9][10][11]. A separable topological space (X, τ) is countable dense homogeneous (CDH) [12] if given any two countable dense subsets A and B of (X, τ) there is a homeomorphism f: (X, τ) → (X, τ) such that f(A) = B.
The study of CDH topological spaces and their related concepts is still a hot area of research, as appears in [13][14][15][16][17][18][19][20] and other papers. Recently, authors in [9] extended CDH topological property to include fuzzy topological spaces. They proved that their extension is a good extension in the sense of Lowen, and proved that a-cut topological space (X, ℑ a ) of a CDH fuzzy topological space (X, ℑ) is CDH in general only for a = 0. For the purpose of dealing with non-separable topological spaces, authors in [21] modified the definition of CDH topological spaces as follows: A subset A of a topological space (X, τ) is called a σ-discrete set if it is the union of countably many sets, each with the relative topology, being a discrete topological space. A topological space (X, τ) is densely homogeneous (DH) provided (X, τ) has a σ-discrete subset which is dense in (X, τ) and if A and B are two such σ-discrete subsets of (X, τ) there is a homeomorphism f: (X, τ) → (X, τ) such that f(A) = B. It is known that CDH and DH topological concepts are independent. The study of DH topological spaces is continued in [22][23][24][25][26][27][28] and other papers. As a main goal of the present work we will show how the definition of DH topological spaces can be modified in order to define a good extension of it in fuzzy topological spaces. We will give relationships between CDH and DH fuzzy.

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Throughout this paper, if is a set, then | | = will denote its cardinality. We write ℚ (resp. ℕ) to denote the set of all rational numbers (resp. natural numbers). The closure of a fuzzy set in a fuzzy topological space ( , ℑ) will be denoted by ( ). Associated with a given topological space ( , ) and arbitrary subset of , we denote the relative topology on by τ , the closure of by ( ) and the boundary of by ( ).topological spaces as well as we will deal with cut topological spaces.

PRELIMINARIES
In this paper we shall follow the notations and definitions of [29] and [30]. If (X, τ) is a topological space, then the class of all lower semi-continuous functions from ( , ) to ([0,1], ), where is the usual Euclidean topology on [0,1], is a fuzzy topology on . This fuzzy topology is denoted by ( ).
The following definitions and propositions will be used in the sequel: Definition 2.1. [9] Let be a non-empty set, be a non-empty subset of and be a collection of fuzzy points in . Then  ℚ( ) will denote the set ℚ( ) = { : is a fuzzy point with ∈ and r ∈ ℚ ∩ (0,1)}.  The support of , denoted by ( ), is defined by ( ) = { : ∈ for some }. Definition 2.2. [21] A subset of a topological space ( , ) is called a -discrete set if it is the union of countably many sets, each with the relative topology, being a discrete topological space. Definition 2.3. [21] A topological space ( , ) is called densely homogeneous (DH) iff  has a σ-discrete dense subset.  If and are two -discrete dense subsets of , then there is a homeomorphism ℎ: ( , ) → ( , ) such that ℎ( ) = . Definition 2.4. [31] Associated with a given fuzzy topological space (X, ℑ) and arbitrary subset of , we define the induced fuzzy topology on or the relative topology on by Definition 2.5. [9] A fuzzy topological space (X, ℑ) is said to be semi-discrete iff for any x ∈ X, there exists a fuzzy point or a fuzzy crisp point for some a with ∈ ℑ. Definition 2.6. [32] Let (X, ℑ) be a fuzzy topological space and let be a collection of fuzzy points of . Then is said to be  Dense(I) if for every non-zero fuzzy open set λ there exists ∈ such that ∈ .  Dense(II) if (⋃ ∈ ) = 1.
Proposition 2.11. [9] Let (X, ℑ) be a fuzzy topological space and let be a collection of fuzzy points of X. Then we have the following  If is dense (I), then ℚ(S( )) is dense(II).  If is dense (II), then ℚ(S( )) is dense(I).

DH FUZZY TOPOLOGICAL SPACES
In this section, we will define DH fuzzy topological spaces. We will prove that our new concept is a fuzzy topological property and a good extension of DH topological property in the sense of Lowen.

RELATIONSHIPS BETWEEN DH AND CDH FUZZY TOPOLOGICAL SPACES
In this section, we will give some relationships between DH and CDH fuzzy topological spaces. The following useful lemma follows easily: Lemma 4.1. Let (X, ℑ) be a fuzzy topological space and P be a collection of fuzzy points of X with S(P) is countable and non-empty. Then P is σ-semi-discrete. Theorem 4.2. Let (X, ℑ) be a fuzzy topological space for which X is countable. Then (X, ℑ) is DH iff (X, ℑ) is semi-discrete. Proof. Since the result is obvious when |X| = 1, we will assume that |X|>1. Suppose that (X, ℑ) is DH and assume on the contrary that (X, ℑ) is not semi-discrete. Then there exists y ∈ X such that y a ∉ ℑ for all 0 < a ≤ 1. Set P = ℚ(X) and W = ℚ(X ∖ {y}). It is not difficult to see that P and W are dense (I). Also, by Lemma 4.1, P and W are σ-semi-discrete. So there is a fuzzy homeomorphism h: (X, ℑ) → (X, ℑ) such that h(S(P)) = S(W), therefore, h(X) = X ∖ {y} which is a contradiction since h is an onto map.
Conversely, suppose that (X, ℑ) is semi-discrete. Then by Proposition 2.15 (ii), (X, ℑ) is separable. Choose a countable dense (I) collection of fuzzy points P. Then S(P) is countable and by Lemma 4.1, P is σ-semi-discrete. Let P and W be any two σ-semi-discrete dense(I) collections of fuzzy points. Then by Proposition 2.15 (i), S(P) = S(W) = X and the identity fuzzy map completes the proof. Corollary 4.3. Let (X, ℑ) be a fuzzy topological space for which X is countable. Then (X, ℑ) is CDH iff (X, ℑ) is DH. Proof. Follows from Proposition 2.16 and Theorem 4.2. Theorem 4.4. If (X, ℑ) is separable and DH fuzzy topological space, then (X, ℑ) is CDH. Proof. Follows from the definitions and Lemma 4.1.
Recall that a fuzzy topological space (X, ℑ) is hereditarily separable if every subspace of (X, ℑ) is separable. Recall that a fuzzy topological space is second countable if it has a countable base. It is well known that second countable fuzzy topological spaces are hereditarily separable. Lemma 4.5. If (X, ℑ) is a hereditarily separable fuzzy topological space and P is a σ-semi-discrete collection of fuzzy points of (X, ℑ), with (A n , ℑ A n ) is semidiscrete for all n ∈ ℕ. Since (X, ℑ) is hereditarily separable, then for each n ∈ ℕ, (A n , ℑ A n )is separable and by Proposition 2.15 (ii) it follows that A n is countable. Thus, S(P) is countable. Theorem 4.6. If (X, ℑ) is hereditarily separable and CDH fuzzy topological space, then (X, ℑ) is DH. Proof. Since (X, ℑ) is hereditarily separable, then it is separable. So, there exists a countable dense (I) collection of fuzzy points P and by Lemma 4.1, P is σ-semi-discrete. Let P and W be two σ-semi-discrete dense(I) collections of fuzzy points. Then by Lemma 4.5, S(P) and S(W) are countable. By Proposition 2.11, ℚ(S(P)) and ℚ(S(W)) are countable dense(I). Since (X, ℑ) is CDH, there is a fuzzy homeomorphism h: (X, ℑ) → (X, ℑ) such that h(S(P)) = h(S(ℚ(S(P)))) = S(ℚ(S(W))) = S(W). Corollary 4.7. Let (X, ℑ) be a hereditarily separable fuzzy topological space. Then (X, ℑ) is CDH iff (X, ℑ) is DH. Proof. Follows from Theorems 4.4 and 4.6. Corollary 4.8. Let (X, ℑ) be a second countable fuzzy topological space. Then (X, ℑ) is CDH iff (X, ℑ) is DH.

CUT TOPOLOGICAL SPACES
In this section we will mainly show that a-cut topological space (X, ℑ a ) of a fuzzy topological (X, ℑ) is DH in general only if a = 0. Lemma 5.1. Let (X, ℑ) be a fuzzy topological space. Let B be non-empty subset of X and let P be a collection of fuzzy points of X. Then Proof. (i) Suppose that (B, ℑ B ) is semi-discrete and let x ∈ B. Then there exists a fuzzy point or a fuzzy crisp point x a for some a with x a ∈ ℑ B . Choose λ ∈ ℑ such thatx a = λ ∩ B . Then (λ ∩ B )(x) = min{λ(x), B (x)} > 0 and so, {x} = λ⁻¹(0,1] ∩ B ∈ (ℑ₀) B . Conversely, suppose that (ℑ₀) B is the discrete topology on B and let x ∈ B. Then there exists λ ∈ ℑ such that {x} = λ⁻¹(0,1] ∩ B. Now, λ ∩ B is the fuzzy or crisp point x λ(x) , on the other hand, λ ∩ B ∈ ℑ B. ii) Since B is σ-discrete in (X, ℑ₀), then ⋃ B n ∞ n=1 with (ℑ₀) B n is the discrete topology for all n ∈ ℕ. By (i), ℑ B n is semi-discrete for all n ∈ ℕ. Since S(ℚ(B)) = B, with ℑ A n is semidiscrete for all n ∈ ℕ. By (i), (ℑ₀) A n is the discrete topology on A n for all n ∈ ℕ. It follows that S(P) is σdiscrete in (X, ℑ₀). Theorem 5.2. If (X, ℑ) is a DH fuzzy topological space, then (X,ℑ₀) is DH.
Theorem 5.4. Let X be a countable set and let (X, ℑ) be a fuzzy topological space. Then the following are equivalent: Proof. Follows from Theorem 4.2, Lemma 5.1 (i) and Proposition 5.3. In fact if a > 0, then (X, ℑ) being DH does not imply, in general, that (X, ℑ a ) is DH. This will be explained in the following counterexample: Example 5.5. For fixed 0 < a < 1, let X = {x, y} and define ℑ = {0,1, x a/2 , y a/4 , x a/2 ∪ y a/4 }. It is clear that (X, ℑ) is semi-discrete and so by Theorem 4.2, it is DH. On the other hand, since ℑ a = {∅, X}, then (X, ℑ a ) is not DH.