On CDH fuzzy spaces

Abstract

We extend the concept of being countable dense homogeneous to include fuzzy topological spaces. Our extension is proved to be a good extension in the sense of Lowen. We study the relation between fuzzy CDH spaces and some ordinary topological spaces generated by these fuzzy spaces.

Keywords

Homogeneity countable dense homogeneity fuzzy topological spaces good extension

1 Introduction

Zadeh [34] introduced the notion of a fuzzy set in a set X as a function λ from X into the closed interval [0,1]. Chang [10] introduced the notion of fuzzy topology 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. The concept of induced fuzzy topological space was introduced by Weiss [32]. Since then many authors have investigated such spaces and much attention has been given to the extension of the separation notions to fuzzy topological spaces. Dealing with uncertain data is a very hot research topic [35, 36].

In 1920, Sierpinski [27] introduced the notion of a homogeneous space. Seven years earlier, Brouwer had shown that if A and B are two countable dense subsets of the n-dimensional Euclidean space $ℝ^{n}$ , then there is an autohomeomorphism on $ℝ^{n}$ that takes A to B. He needed this result in his development of dimension theory.

In 1972, Bennett [9] began the abstract study, who called such spaces countable dense homogeneous spaces. In 1954, Ford [17] introduced a new concept called strongly locally homogeneous and he denoted it by SLH. Fitzpatrick, Fletcher, McCoy, Ungar, Van Mill, and Lauer extended the study to new types of homogeneity.

Mathematicians generalized many concepts of ordinary topological spaces to include fuzzy topological spaces such as: separation axioms, connectedness, compactness, countability axioms. Several fuzzy homogeneity concepts were discussed in [1–6 , 16]. Since countable dense homogeneity is of importance in general topology and still a hot area of research, as appears in [7 , 30] and other papers, we found it necessary to extend it to include fuzzy topological spaces. One of the main goals of the present work is to show how the definition of countable dense homogeneous ordinary topological spaces can be modified in order to define a good extension of it in fuzzy topological spaces. We will see that the exact versions of some known results regarding countable dense homogeneous ordinary topological spaces are invalid in fuzzy topological spaces, and we will open the door for other researchers to study the validity or invalidity for many other related known results. The study will also deal with the relation between fuzzy CDH spaces and some ordinary topological spaces generated by these fuzzy spaces.

In this paper we shall follow [33] for the following notations: For λ, μ ∈ [0, 1] ^X, we write λ ⊆ μ iff λ (x) ≤ μ (x) for all x ∈ X. By λ = μ we mean that λ ⊆ μ and μ ⊆ λ, i.e., λ (x) = μ (x) for all x ∈X. If {λ_i : i∈ I } is a collection of fuzzy sets in X, then (⋃ λ_i) (x) = sup{ λ_i (x) : i ∈ I }, x ∈ X; and (⋂ λ_i) (x) = inf{ λ_i (x) : i ∈ I }, x ∈ X. Let r denote the fuzzy set given by r (x) = r for all x ∈ X, where 0≤ r ≤1 ; i.e., r denotes the “constant” fuzzy set of level r. The complement λ^c of a fuzzy set λ in X is given by λ^c (x) = 1 - λ (x), x ∈ X. If A ⊆ X, then χ_A denotes the characteristic function of A. If (X, τ) is an ordinary topological space, then the class of all lower semi-continuous functions from (X, τ) to ([0, 1] , τ_u), where τ_u is the usual Euclidean topology on [0, 1], is a fuzzy topology on X. This fuzzy topology is denotedby ω (τ).

In this paper we shall follow [33] for definitions of fuzzy point, the direct and the inverse images of a fuzzy set, and fuzzy continuous mapping. For instance, a fuzzy point x_a in a set X is given by x_a (y) = a for y = x (0 < a < 1) and x_a (y) = 0 for y ≠ x, x ∈ X is called the support of x_a and a the value (level) of x_a. However, we shall agree that a fuzzy crisp point x_a in X is a fuzzy set in X given by x_a (y) = 1 for y = x and x_a (y) = 0 for y ≠ x. We shall follow [28] for the definition of ’belonging to’. Namely: a fuzzy point x_a in X is said to belong to a fuzzy set λ in X (notation: x_a∈ λ) iff a < λ (x). Two fuzzy points x_a and y_b in X are said to be distinct iff their supports are distinct, i.e., x≠ y. A bijective map h : (X, ℑ ₁) → (Y, J₂); where (X, ℑ₁) and (Y, ℑ₂) are both fuzzy topological spaces; will be called fuzzy homeomorphism iff h and h^-1are both fuzzy continuous.

Definition 1.1. [15] Associated with a given fuzzy topological space (X, ℑ) and arbitrary ordinary subset M of X, we define the induced topology on M or the relative topology on M by: $ℑ_{M} = {λ ∣ M : λ \in ℑ}$

The corresponding pair (M, ℑ _M) is called a fuzzy open (closed) subspace iff the fuzzy set χ_M is fuzzy open (closed) in (X, ℑ).

Definition 1.2. [24] A property P_f of a fuzzy topological space is said to be a good extension of the property P in classical topology iff whenever the fuzzy topological space is topologically generated, say by (X, τ), then (X, ω (τ)) has property P_f iff (X, τ) has property P.

Definition 1.3. [14] A fuzzy topological space (X, ℑ) is said to be T₁ iff for any two distinct fuzzy points x_a, y_b in X, there exist open fuzzy sets μ₁ and μ₂ such that x_a ∈ μ₁, μ₁∩ y_b = 0 (i.e., μ₁ (y) = 0), μ₂ ∩ x_a = 0 (i.e., μ₂ (x) = 0) and y_b ∈ μ₂.

Definition 1.4. [15] A fuzzy topological space (X, ℑ) is called connected iff it has no clopen fuzzy sets other than the constant fuzzy sets.

If X is a set, then X will denote its cardinality. We write $ℚ$ to denote the set of all rational numbers. On a non-empty set X, τ_ind, τ_disc will denote the indiscrete and the discrete topologies on X, respectively. The closure of a fuzzy set λ in a fuzzy topological space (X, ℑ) will be denoted by $\bar{λ}$ or Cl (λ).

The following result obtained in [15] will be used in proving that our main new definition in this paper is a good extension.

Proposition 1.5. Let (X, τ₁) and (Y, τ₂) be two topological spaces. Then f : (X, τ₁) → (Y, τ₂) is continuous iff f : (X, ω (τ₁)) → (Y, ω (τ₂)) is fuzzy continuous.

2 Separability in fuzzy topological spaces

We start this section with the following two definitions:

Definition 2.1. [14] A collection D of fuzzy points in (X, ℑ) is said to be dense (I) if for every non-zero fuzzy open set λ there exists p∈ D such that p ∈ λ.

Definition 2.2. [14] A collection D of fuzzy points in (X, ℑ) is said to be dense (II) if Cl ⋃ _p∈D p = 1.

In 1989, Fora [14], proved that the above two definitions are independent of each other.

Depending on the above two different definitions of denseness, we define separability in fuzzy topological spaces in two ways as follows.

Definition 2.3. A fuzzy topological space (X, ℑ) is called separable (I) iff there exists a countable dense (I) collection of fuzzy points of X.

Definition 2.4. A fuzzy topological space (X, ℑ) is called separable (II) iff there exists a countable dense (II) collection of fuzzy points of X.

We will prove that Definitions 2.3 and 2.4 are equivalent. For this reason, we need the following notations and definitions.

Definition 2.5. Let P be a collection of fuzzy points in X. The support of P, denoted by S (P), is defined by $S (P) = {x : x_{a} \in P for some a} .$

Definition 2.6. Let A be a non-empty subset of X. Then $ℚ (A)$ will denote the set

$ℚ (A) = {x_{r} : x_{r} is a fuzzy point with x \in A$

and $r \in ℚ \cap (0, 1)} .$

The following result is important in proving our main result of this section.

Theorem 2.7. Let (X, ℑ) be a fuzzy topological space and let P be a collection of fuzzy points of X. Then we have the following:

If P is dense (I), then $ℚ (S (P))$ is dense (II).

If P is dense (II), then $ℚ (S (P))$ is dense (I).

Proof. (i) Suppose that P is dense (I) and assume on the contrary that $ℚ (S (P))$ is not dense (II). Then Cl $⋃ {p : p \in ℚ (S (P))} \neq 1$ . So there exists x_∘ ∈ X such that $(Cl ⋃ {p : p \in ℚ (S (P))}) (x_{\circ}) < 1$ . Let μ = Cl $⋃ {p : p \in ℚ (S (P))}$ . Then μ^c is a non-zero fuzzy open set in X. Since P is dense (I), there exists y_b ∈P such that y_b ∈ μ^c, which implies μ (y) <1. On the other hand, y∈ $S (ℚ (S (P)))$ , so $(⋃ {p : p \in ℚ (S (P))}) (y) = 1$ . Therefore, μ (y) = 1, which is a contradiction.

(ii) Suppose that P is dense (II) and assume on the contrary that $ℚ (S (P))$ is not dense (I). Then there is a non-zero fuzzy open set λ such that p ∉ λ for each $p \in ℚ (S (P))$ . We are going to prove that ⋃{ p : p ∈ P } ⊆ λ^c. If $x \in S (ℚ (S (P)))$ , then λ (x) = 0, so λ^c (x) = 1 ≥ (⋃ { p : p ∈ P }) (x). If x∉ S (Q (S (P))), then $(⋃ {p : p \in ℚ (S (P))}) (x) = 0 \leq λ^{c} (x)$ . Therefore, ⋃{ p : p ∈ P } ⊆ λ^c, and hence, λ^c = 1. Thus λ = 0, which is a contradiction.

Following is the main result of this section:

Theorem 2.8. Let (X, ℑ) be a fuzzy topological space. Then (X, ℑ) is separable (I) iff (X, ℑ) isseparable (II).

Proof. If (X, ℑ) is separable (I), then there exists a countable dense (I) set P of fuzzy points of X. By Theorem 2.7 (i), $ℚ (S (P))$ is dense (II). But $ℚ (S (P))$ is clearly countable. Hence, (X, ℑ) is separable (II).

The other direction is similar to the above one.

The above result makes it clear that the two concepts of separability in fuzzy spaces are equivalent. Then from now on, we shall say separable to represent each of them.

3 CDH fuzzy topological spaces

Frechet and Brouwer have observed that n-dimensional Euclidean space $ℝ^{n}$ has the property that if A and B are two countable dense sets in $ℝ^{n}$ , then there is a homeomorphism $h : ℝ^{n}$ $\to ℝ^{n}$ taking A to B. They used this theorem in their development of dimension theory.

In 1962, Fort [18], proved that the Hilbert cube has the same property. In 1972, Bennett [9], gave the following definition:

Definition 3.1. [9] A space X is called countable dense homogeneous (CDH) iff

X is separable.

If A and B are two countable dense subsets of X, then there is a homeomorphism h : X → X such that h (A) = B.

In this section, we extend the above definition to include fuzzy spaces. For this reason, we need the following definition:

Definition 3.2. A fuzzy topological space (X, ℑ) is said to be countable dense homogeneous; denoted CDH; iff

X is separable.

If P₁and P₂ are two countable dense (I) collections of fuzzy points of X, then there is a fuzzy homeomorphism h : (X, ℑ) → (X, ℑ) such that h (S (P₁)) = S (P₂).

Now we shall give the following lemma which will be useful in showing that our Definition 3.2 is a good extension of the well-known Definition 3.1, according to Lowen [24].

Lemma 3.3. Let (X, τ) be a topological space, A ⊆X, and P be a collection of fuzzy points of X. Then we have the following:

If A is dense in (X, τ), then $ℚ (A)$ is dense (I) in (X, ω (τ)).

If P is dense (I) in (X, ω (τ)), then S (P) is dense in (X, τ).

Proof. (i) Let λ be any non-zero fuzzy open set of (X, ω (τ)). Then λ^-1 (0, 1] is a non-empty opensubset of X. Choose a ∈ λ^-1 (0, 1] ∩ A and choose $r \in ℚ$ such that 0 < r < λ (a). Consider the fuzzy point a_r. Then $a_{r} \in ℚ (A)$ and a_r ∈ λ.

(ii) Let U be any non-empty open subset of (X, τ). Then χ_Uis a non-zero fuzzy open set of (X, ω (τ)). Choose x_a ∈P such that x_a ∈χ_U, so x ∈ U ∩ S (P), which completes the proof.

Theorem 3.4. Let (X, τ) be a topological space. Then (X, τ) is CDH iff (X, ω (τ)) is CDH.

Proof. If (X, τ) is CDH, then (X, τ) is separable, and so there exists a countable dense subset A of X. By Lemma 3.3 (i), $ℚ (A)$ is dense (I) in (X, ω (τ)), and so (X, ω (τ)) is separable. Let P₁ and P₂ be any two countable dense (I) collections of fuzzy points of (X, ω (τ)). Then by Lemma 3.3 (ii), S (P₁) and S (P₂) are two dense subsets of (X, τ), so there is a homeomorphism h : (X, τ) → (X, τ) such that h (S (P₁)) = S (P₂). Proposition 1.5 completes the proof of this direction.

The proof of the other direction is similar to the above one.

Corollary 3.5. CDH in fuzzy topological spaces is a good extension of CDH in ordinary topological spaces.

In ordinary topological spaces, it is well-known that the only countable CDH space is the discrete space. However, this is invalid in fuzzy spaces since the fuzzy space in the following example will be a countable non-discrete CDH fuzzy space.

Example 3.6. Let X ={ 1, 2 }, ℑ ={ 0, 1, x_a, y_b, λ }, where x_a and y_b are the two fuzzy points with x = 1, y = 2, $a = \frac{1}{4}$ , $b = \frac{1}{2}$ and $λ = {(1, \frac{1}{4}), (2, \frac{1}{2})}$ .

We are going to explore an alternative version of the above well-known fact about countable CDH fuzzy spaces. For this reason, we need the following definition and lemma:

Definition 3.7. 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 x_a for some a withx_a∈ ℑ.

Lemma 3.8. Let (X, ℑ) be a semi-discrete fuzzy topological space. Then we have the following:

If P is countable dense (I) in (X, ℑ), then X is countable.

(X, ℑ) is separable iff X is countable.

Proof. (i) Let x ∈ X. Then there is a fuzzy point or fuzzy crisp point x_a∈ ℑ. So there is y_b ∈P such that y_b ∈ x_a. Therefore, x = y and hence, x ∈ S (P).

(ii) If X is separable, then there is a countable dense (I) P of (X, ℑ), then by part (i) S (P) = X, but . Hence X is countable.

Conversely, if X is countable, then it is clear that $ℚ (X)$ is countable dense (I). Thus, (X, ℑ) isseparable.

The following result shows us that the only countable CDH fuzzy spaces on a countable set X are the semi discrete fuzzy spaces on X.

Theorem 3.9. Let X be a countable set and let (X, ℑ) be a fuzzy topological space. Then (X, ℑ) is CDH iff (X, ℑ) is a semi-discrete fuzzy topologicalspace.

Proof. If , then the result is obvious. If > 1, then suppose that (X, ℑ) is CDH and assume on the contrary that (X, ℑ) is not a semi-discrete fuzzy topological space. Then there exists x_∘ ∈ X such that x_a∉ ℑ for every fuzzy point or fuzzy crisp point x_a with x = x_∘. It is easy to see that $ℚ (X \ {x_{\circ}})$ is countable dense (I), so there is a fuzzy homeomorphism h : (X, ℑ) → (X, ℑ) such that $h (S (ℚ (X))) =$ $S (ℚ (X \ {x_{\circ}}))$ , therefore, h (X) = X\ { x_∘ } which is a contradiction since h is an onto map.

Conversely, if X is a semi-discrete fuzzy topological space, then by Lemma 3.8 (ii), (X, ℑ) is separable since X is countable. Let P₁ and P₂ be any two countable dense (I) in (X, ℑ). Then according to Lemma 3.8 (i), S (P₁) = S (P₂) = X. Therefore, the identity fuzzy homeomorphism completes the proof.

In ordinary topological spaces, we have the following definitions and results:

Definition 3.10. [27] An ordinary topological space X is said to be homogeneous if for any two points x₁, x₂ in X, there exists a homeomorphism h : X→ X such that h (x₁) = x₂.

Definition 3.11. Let (X, τ) be a topological space. We define $\tilde{τ}$ on X as follows: For x₁, x₂ ∈ X, x₁ $\tilde{τ}$ x₂ iff there exists a homeomorphism h : X→ X such that h (x₁) = x₂.

The above relation is an equivalence relation. So this relation induces a partition on X into equivalence classes, and it leads us to the following definition:

Definition 3.12. A subset of a topological space (X, τ) which has the form $C_{x}^{τ} = {y \in X : x \tilde{τ} y}$ is called an h-component of (X, τ) at x.

Theorem 3.13. [12] Let X be a CDH space. Then every h-component $C_{x}^{τ}$ of X is clopen.

Theorem 3.14. [13] Every CDH space is T₁.

Theorem 3.15. [11] Every connected CDH space is homogeneous.

We are going to study the diversity of the above three results in fuzzy spaces. For this reason, we need the following definitions:

Definition 3.16. [16] A fuzzy topological space (X, ℑ) is said to be homogeneous if for any two points x₁, x₂ in X, there exists a fuzzy homeomorphism h : (X, ℑ) → (X, ℑ) such that h (x₁) = x₂.

Definition 3.17. [4] Let (X, ℑ) be a fuzzy topological space. We define the relation $\tilde{ℑ}$ on X as follows: for x₁, x₂∈ X, x₁ $\tilde{ℑ}$ x₂ iff there exists a fuzzy homeomorphism h : (X, ℑ) → (X, ℑ) such that h (x₁) = x₂.

It is easy to see that the above relation is an equivalence relation. So this relation induces a partition on X into equivalence classes, and it leads us to the following definition.

Definition 3.18. [4] A subset of a fuzzy topological space (X, ℑ) which has the form $C_{x}^{ℑ} = {y \in X : x \tilde{ℑ} y}$ is called an h-component of (X, ℑ) at x.

The exact versions of Theorems 3.13, 3.14 and 3.15 are invalid in fuzzy spaces, as one can easily see from Example 3.6.

Recall that a property P of fuzzy topological spaces is called a fuzzy topological property if whenever (X, ℑ ₁) possesses P and h : (X, ℑ ₁) → (Y, ℑ ₂) is a fuzzy homeomorphism, then (Y, ℑ ₂)possesses P.

The following two lemmas will be used in proving that the property CDH fuzzy spaces is a fuzzy topological property.

Lemma 3.19. Let f : X → Y be a bijective map and A be a collection of fuzzy points of X. Then S (f (A)) = f (S (A)).

Proof. Straightforward.

Lemma 3.20. If f : (X, ℑ ₁) → (Y, ℑ ₂) is a fuzzy homeomorphism map and D is a dense (I) collection of fuzzy points in (X, ℑ ₁), then f (D) is dense (I) in (Y, ℑ ₂).

Proof. Let λ be a non-zero fuzzy open set in Y. Then f^-1 (λ) ∈ ℑ ₁. Since f is surjective, then f^-1 (λ) ≠ 0 and so there exists d ∈ D such that d ∈ f^-1 (λ). Therefore, f (d) ∈ λ and hence, f (D) is dense (I) in (Y, ℑ ₂).

Theorem 3.21. In fuzzy spaces, “Being CDH” is a fuzzy topological property.

Proof. Assume (X, ℑ ₁) is a CDH fuzzy space and let f : (X, ℑ ₁) → (Y, ℑ ₂) be a fuzzy homeomorphism where (Y, ℑ ₂) is a fuzzy space. Choose a countable dense (I) collection of fuzzy points D of (X, ℑ ₁). By Lemma 3.20, f (D) is dense (I) in (Y, ℑ ₂) and hence, (Y, ℑ ₂) is separable. Let A and B be any two countable dense (I) collections of fuzzy points of (Y, ℑ ₂). Then by Lemma 3.20, f^-1 (A) and f^-1 (B) are two countable dense (I) collections of fuzzy points of (X, ℑ ₁). Since (X, ℑ ₁) is CDH, there is a fuzzy homeomorphism h : (X, ℑ ₁) → (X, ℑ ₁) such that h (S (f^-1 (A))) = S (f^-1 (B)). Define g : (Y, ℑ ₂) → (Y, ℑ ₂) by g = f ∘ h ∘ f^-1. Then g is a fuzzy homeomorphism. Using Lemma 3.19, it is not difficult to see that g (S (A)) = S (B).

4 Topologies induced by fuzzy topologies

In this section, we shall study an important class of topological spaces that are induced by fuzzy topological spaces. We will study the relation between a CDH fuzzy topological space and its induced topological spaces. This class is introduced in the following definition:

Definition 4.1. [31] Let (X, ℑ) be a fuzzy topological space and a ∈ [0, 1). The topology {λ^-1 (a, 1] : λ∈ ℑ } on X is called will be denoted by ℑ_a. The ordinary space (X, ℑ _a) will be called a-cut topological space of (X, ℑ).

The following is an example of a non-fuzzy space (X, ℑ), for which (X, ℑ _a) is a topological space for all a ∈ [0, 1).

Example 4.2. Let X = {1, 2} and $ℑ = \{0, 1, x_{b}, y_{c}\}$ where x_b and y_b are two fuzzy points with x = 1, y = 2and $b = c = \frac{1}{2}$ . Then

$ℑ_{a} = \{\begin{array}{c} τ_{i n d}, & if a \geq \frac{1}{2}, \\ τ_{d i s c}, & 0 < a < \frac{1}{2}, \end{array}$

so ℑ_a is a topology on X for all a ∈ [0, 1). On the other hand, since x_b ∪ $y_{c} = \frac{1}{2}$ ∉ℑ, then ℑ is not a fuzzy topology on X.

Theorem 4.3. Let (X, ℑ) be a fuzzy topological space and let f : (X, ℑ) → (X, ℑ) be a fuzzy continuous (homeomorphism) map. Then f : (X, ℑ _a) → (X, ℑ _a) is continuous (homeomorphism) for all a ∈ [0, 1).

Proof. Straightforward.

The converse of the above result needs not be true in general as the following example shows.

Example 4.4. Let X = {1, 2} and ℑ = { 0, 1, λ₁, λ₂, λ₃, λ₄}, where λ₁ = { (1, 0) , (2, 1) } , λ₂ = { (1, 1) , (2, 0) } , λ₃ $= {(1, \frac{1}{2}), (2, 0)},$ $λ_{4} = {(1, \frac{1}{2}), (2, 1)}$ . Then it is easy to see that (X, ℑ) is a fuzzy space. Let a ∈ [0, 1), then $λ_{1}^{- 1} (a, 1] = {2}$ and $λ_{2}^{- 1} (a, 1] = {1}$ . Conversely, ℑ_a = τ_disc. Define h : (X, ℑ) → (X, ℑ) by h ={ (1, 2) , (2, 1) }. Then h^-1 (λ₃) $= {(1, 1), (2, \frac{1}{2})}$ ∉ℑ and hence, h is not fuzzy continuous (homeomorphism). On the other hand, h : (X, ℑ _a) → (X, ℑ _a) is obviously continuous (homeomorphism) since ℑ_a = τ_disc.

According to Theorem 4.3, one can easily prove the following result.

Theorem 4.5. Let (X, ℑ) be a fuzzy topological space. Then for a ∈ [0, 1), $C_{x}^{ℑ} \subseteq C_{x}^{ℑ a}$ for all x ∈ X.

In Theorem 4.5, the equality need not be true in general as can be deduced from Example 4.4. In fact $C_{1}^{ℑ_{\frac{1}{2}}} = X$ , but $C_{1}^{ℑ} = {1}$ . However, we have the following alternative result.

Theorem 4.6. Let (X, ℑ) be a fuzzy topological space and let a ∈ [0, 1). Then $C_{x}^{ℑ_{a}} = ⋃ {C_{y}^{ℑ} : y \in C_{x}^{ℑ_{a}}}$ .

Proof. For y $\in C_{x}^{ℑ_{a}}, y$ $\in C_{x}^{ℑ}$ . According to Theorem 4.5, $C_{x}^{ℑ} \subseteq$ $C_{x}^{ℑ a}$ . Therefore, $C_{y}^{ℑ} \subseteq$ $C_{x}^{ℑ a}$ for all y $\in C_{x}^{ℑ a}$ and hence, $C_{x}^{ℑ a} \subseteq ⋃ {C_{y}^{ℑ} : y \in C_{x}^{ℑ_{a}}} .$

Conversely,

if t $\in C_{x}^{ℑ a}$ , then t ∈ $C_{t}^{ℑ} \subseteq ⋃ {C_{y}^{ℑ} : y \in C_{x}^{ℑ_{a}}}$ .

The following lemma is needed in the following result.

Lemma 4.7. Let(X, ℑ) be a fuzzy topological space. Let A be a subset of X and let P be a collection of fuzzy points of X. Then we have the following:

If A is dense in (X, ℑ ₀), then $ℚ (A)$ is dense in (X, ℑ).

If P is dense in (X, ℑ), then S (P) is dense in (X, ℑ ₀).

Proof. Mimic the proof of Lemma 3.3.

The following is our other main result:

Theorem 4.8. If (X, ℑ) is a CDH fuzzy topological space, then (X, ℑ ₀) is CDH.

Proof. Since (X, ℑ) is CDH, then there is a countable dense (I) collection of fuzzy points P of (X, ℑ). Then according to Lemma 4.7 (ii), S (P) is dense in (X, ℑ ₀), but S (P) is clearly countable. Thus, (X, ℑ ₀) is separable. Let A and B be any two countable dense subsets of (X, ℑ ₀), then by Lemma 4.7 (i), $ℚ (A)$ and $ℚ (B)$ are two dense (I) in (X, ℑ). Then there is a fuzzy homeomorphism h : (X, ℑ) → (X, ℑ) such that $h (S (ℚ (A)))$ $= S (ℚ (B))$ , so h (A) = B. According to Theorem 4.3, h : (X, ℑ ₀) → (X, ℑ ₀) is a homeomorphism, which completes the proof.

In fact if a > 0, then (X, ℑ) being CDH does not imply, in general, that (X, ℑ _a) is CDH. This will be explained in the following counterexample.

Example 4.9. For fixed 0< a <1, let X ={ 1, 2 } and define ℑ ={ 0, 1, x_b, y_c, λ }, where x_b and y_c are fuzzy points with $x = 1, y = 2, b = \frac{a}{2},$ $c = \frac{a}{4}$ , and λ = ${(1, \frac{a}{2}), (2, \frac{a}{4})}$ . Then ℑ_a = τ_ind which is not CDH. On the other hand, one can easily show that (X, ℑ) is a CDH fuzzy space.

5 Conclusion

In this paper, as an ordinary topological property countable dense homogeneity is extended to include fuzzy topological spaces. It is proved that this extension is a good extension in the sense of Lowen. A characterization of countable dense homogeneous fuzzy topological spaces on a countable set is obtained. The exact versions of four known results related to countable dense homogeneity in ordinary topological spaces are proved to be invalid in fuzzy topological spaces. It is proved that a-cut topological space (X, ℑ _a) of a CDH fuzzy space (X, ℑ) is CDH in general only for a = 0. In our future study concerning CDH fuzzy spaces, the following topics could be considered: 1) To define several modifications of CDH fuzzy spaces such as DH; 2) To define the degrees of CDH property of fuzzy sets in a similar way to that used in [23]; 3) To show how the definition of CDH spaces can be modified in order to define CDH generalized L-topological spaces [8].

Footnotes

Acknowledgments

The authors are very grateful to the referees for their valuable comments and suggestions.

References

Al Ghour

, Homogeneity in fuzzy spaces and their induced spaces, Questions Answers Gen Topology 21(2) (2003), 185–195.

Al Ghour

, SLH fuzzy spaces, Afr Diaspora J Math 2(2) (2004), 61–67.

Al Ghour

and Fora

, Minimality and homogeneity in fuzzy spaces, J Fuzzy Math 12(3) (2004), 725–737.

Al Ghour

, Local homogeneity in fuzzy topological spaces, Int J Math Math Sci (2006), 14. Art. ID 81497.

Al Ghour

, Some generalizations of minimal fuzzy open sets, Acta Math Univ Comenian (N.S.) 75(1) (2006), 107–117.

Al Ghour

and Al-Zoubi

, On some ordinary and fuzzy homogeneity types, Acta Math Univ Comenian (N.S.) 77(2) (2008), 199–208.

Arhangel’skii

A.V.

and van Mill

, On the cardinality of countable dense homogeneous spaces, Proc Amer Math Soc 141(11) (2013), 4031–4038.

Bai

S.Z.

, Generalized L-topological spaces, Journal of Intelligent and Fuzzy Systems 28(1) (2015), 301–309.

Bennett

, Countable dense homogeneous spaces, Fund Math 74(3) (1972), 189–194.

10.

Chang

C.L.

, Fuzzy topological spaces, J Math Anal Appl 24 (1968), 182–190.

11.

Fitzpatrick

and Lauer

N.F.

, Densely homogeneous spaces (I), Houston J Math 13(1) (1987), 19–25.

12.

Fitzpatrick

and Zhou

H.X.

, Countable dense homogeneity and the Baire property, Topology Appl 43(1) (1992), 1–14.

13.

Fitzpatrick

, White

J.M.S.

and Zhou

H.X.

, Homogeneity and σ-discrete sets, Topology Appl 44(1-3) (1992), 143–147.

14.

Fora

, Separation axioms for fuzzy spaces, Fuzzy Sets and Systems 33(1) (1989), 59–75.

15.

Fora

, Separation axioms, subspaces and product spaces in fuzzy topology, Arab Gulf J Sci Res 8(3) (1990), 1–16.

16.

Fora

and Al Ghour

, Homogeneity in fuzzy spaces, Questions Answers Gen Topology 19(2) (2001), 159–164.

17.

Ford

L.R.

, Homeomorphism groups and coset spaces, Trans Amer Math Soc 77 (1954), 490–497.

18.

Fort

M.K.

, Homogeneity of infinite products of manifolds with boundary, Pacific J Math 12 (1962), 879–884.

19.

Hernandez-Gutiérrez

, Countable dense homogeneity and the double arrow space, Topology Appl 160(10) (2013), 1123–1128.

20.

Hernandez-Gutiérrez

and Hrušak

, Non-meager P-filters are countable dense homogeneous, Colloq Math 130(2) (2013), 281–289.

21.

Hernandez-Gutiérrez

, Hrušak

and van Mill

, Countable dense homogeneity and λ-sets, Fund Math 226(2) (2014), 157–172.

22.

Hrusak

and van Mill

, Nearly countable dense homogeneous spaces, Canad J Math 66(4) (2014), 743–758.

23.

Liang

, Degrees of countable compactness and the Lindelof property of L-fuzzy sets, Journal of Intelligent and Fuzzy Systems 28(3) (2015), 1099–1107.

24.

Lowen

, A comparison of different compactness notions in fuzzy topological spaces, J Math Anal Appl 64(2) (1978), 446–454.

25.

Medvedev

S.V.

, Metrizable DH-spaces of the first category, Topology and its Applications 179(1) (2015), 171–178.

26.

Repov

, Zdomskyy

and Zhang

, Countable dense homogeneous filters and the Menger covering property, Fund Math 224(3) (2014), 233–240.

27.

Sierpenski

, Sur uni proprit’te’ topologique des ensembles de’nombrable dense en soi, Fund Math 1 (1920), 11–28.

28.

Srivastava

, Lal

S.N.

and Srivastava

A.K.

, Fuzzy Hausdorff topological spaces, J Math Anal Appl 81(2) (1981), 497–506.

29.

van Mill

, On countable dense and n-homogeneity, Canad Math Bull 56(4) (2013), 860–869.

30.

van Mill

, Countable dense homogeneous rimcompact spaces and local connectivity, Filomat 29(1) (2015), 179–182.

31.

Wang

G.J.

, Theory of L-fuzzy topological space, Shanxi Normal University Press, Xian, 1988 (in Chinese).

32.

Weiss

M.D.

, Fixed points, separation, and induced topologies for fuzzy sets, J Math Anal Appl 50 (1975), 142–150.

33.

Wong

C.K.

, Fuzzy points and local properties of fuzzy topology, J Math Anal Appl 46 (1974), 316–328.

34.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

35.

Zhan

, Liu

and Davvaz

, A new rough set theory: Rough soft hemirings, Journal of Intelligent and Fuzzy Systems 28 (2015), 1687–1697.

36.

Zhan

, The Uncertainties of Ideal Theory on Hemirings, Science Press, 2015.