Local β compactness as fuzzy predicates defined in Łukasiewicz logic

Abstract

In this paper, some characterizations of fuzzifying β-compactness are given, including characterizations in terms of nets and β-subbases. Several characterizations of locally β-compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.

Keywords

Łukasiewicz logic semantics fuzzifying topology fuzzifying compactness fuzzifying locally compactness

1 Introduction

In 1952, Rosser and Turquette [8] proposed the following unsolved problem: If there are many-valued theories beyond the level of predicate calculus, then what are the details of such theories? As an attempt to give a partial answer to this problem in the case of point set topology, Ying in 1991–1993 [13 –15] used a semantical method of continuous-valued logic to develop systematically fuzzifying topology. Briefly speaking, a fuzzifying topology on a set X assigns each crisp subset of X to a certain degree of being open, other than being definitely open or not. So far, there has been significant research on fuzzifying topologies [3 , 9–16]. For example, Ying [16] introduced the concepts of compactness and established a generalization of Tychonoff’s theorem in the framework of fuzzifying topology. In [12] the concept of local compactness in fuzzifying topology is introduced and some of its properties are established. The notion of β-open sets [1] was introduced by Abd El-Monsef, El-Deeb and Mahmoud in 1983 which was studied in Andrijevic [5] under the name semi-preopen sets. Dontchev and Przemiski [6] replaced the term semi-preopen by the term pre-semiopen. The concept of β-compact topological spaces was studied in [2, 4]. In [3] the concepts of fuzzifying β-open sets and fuzzifying β-continuity were introduced and studied. Also, Sayed in [10] introduced some concepts of fuzzifying β-separation axioms and clarified the relations of these axioms with each other as well as the relations with other fuzzifying separation axioms. Furthermore, Sayed and Abd-Allah [11] characterized the concepts of fuzzifying β-irresolute functions and used the finite intersection property to give a characterization of fuzzifying β-compact spaces. In this paper, the concepts of β-base and β-subbase of fuzzifying β-topology are introduced. Other characterizations of fuzzifying β-compactness are given, including characterizations in terms of nets and β-subbase. Several characterizations of locally β-compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained. Thus we fill a gap in the existing literature on fuzzifying topology. We use the terminologies and notations in [3 , 9–16] without any explanation. We note that the set of truth values is the unit interval and we do often not distinguish the connectives and their truth value functions and state strictly our results on formalization as Ying does. We will use the symbol ⊗ instead of the second “AND” operation $\underset{\cdot}{\land}$ as dot is hardly visible. This mean that [α]≤ [φ → ψ] ⇔ [α] ⊗ [φ] ≤ [ψ]. All of the contributions in General Topology in this paper which are not referenced may be original.

2 Preliminaries

We now give some definitions and results which are useful in the rest of the present paper.

Definition 2.1. [13] Let X be a set and τ ∈ ℑ (P (X)) is called a fuzzifying topology if it satisfies the following conditions

⊨X ∈ τ;

for any A, B ∈ P (X), ⊨ (A ∈ τ) ∧ (B ∈ τ) → ((A ∩ B ∈ τ);

for any {A_λ ∣ λ ∈ Λ}, ⊨ ∀ λ (λ ∈ Λ → A_λ ∈ τ) → ⋃ _λ∈ΛA_λ ∈ τ.

The pair (X, τ) is called a fuzzifying topological space.

The fuzzifying neighborhood system of a point x ∈ X is defined as $N_{x} (A) = \underset{x \in B \subseteq A}{⋁} τ (B) .$

The fuzzifying closure of a set A ⊆ X is defined as Cl (A) (x) =1 - N_x (X - A).

The fuzzifying interior of a set μ ∈ ℑ (X) is defined as Int (μ) = ⋃ {U ∈ P (X) ∣1_U ≤ μ} , where 1_U is a characteristic function.

Definition 2.2. [3] The family of all fuzzifying β-open sets, denoted by τ_β ∈ ℑ (P (X)), is defined as

A ∈ τ_β : = ∀ x (x ∈ A → x ∈ Cl (Int (Cl (A)))), i. e., $τ_{β} (A) = \underset{x \in A}{⋀} Cl (Int (Cl (A))) (x))$ .

The family of all fuzzifying β-closed sets, denoted by Γ_β ∈ ℑ (P (X)), is defined as A ∈ Γ_β : = X - A ∈ τ_β.

The fuzzifying β-neighborhood system of a point x ∈ X is denoted by $N_{x}^{β^{X}} ($ or $N_{x}^{β}) \in ℑ (P (X))$ and defined as $N_{x}^{β} (A) = \underset{x \in B \subseteq A}{⋁} τ_{β} (B) .$

The fuzzifying β-closure of a set A ⊆ X, denoted by Cl_β ∈ ℑ (X), is defined as ${Cl}_{β} (A) (x) = 1 - N_{x}^{β} (X - A) .$

[9] If N (X) is the class of all nets in X, then the binary fuzzy predicates ⊳^β, ∝ ^β ∈ ℑ (N (X) × X) are defined as $S ⊳^{β} x : = \forall A (A \in N_{x}^{β} \to S \tilde{\subset} A),$ $S \propto^{β} x : = \forall A (A \in N_{x}^{β} \to S \tilde{≺} A),$ where “S ⊳ ^βx”, “S ∝ ^βx” stand for “S β-converges to x”, “x is an β-accumulation point of S”, respectively; and “ $\tilde{\subset}$ ”, “ $\tilde{≺}$ ” are the binary crisp predicates “almost in”, “often in”, respectively. The degree to which x is an β-adherence point of S is adh_βS (x) = [S ∝ ^βx].

[3, 11] If (X, τ) and (Y, σ) are two fuzzifying topological spaces and f ∈ Y^X, the unary fuzzy predicates C_β, I_β ∈ ℑ (Y^X) , called fuzzifying β-continuity, fuzzifying β-irresoluteness, are given as C_β (f) : = ∀ B (B ∈ σ → f^-1 (B) ∈ τ_β) , I_β (f) : = ∀ B (B ∈ σ_β → f^-1 (B) ∈ τ_β) , respectively.

Definition 2.3. Let Ω be the class of all fuzzifying topological spaces.

[10] A unary fuzzy predicate $T_{2}^{β} \in ℑ (Ω)$ , called fuzzifying β-Hausdorffness, is given $\begin{matrix} T_{2}^{β} (X, τ) = \forall x \forall y ((x \in X \land y \in X \land x \neq y) \\ \to \exists B \exists C (B \in N_{x}^{β} \land C \in N_{y}^{β} \land B \cap C \equiv φ)) . \end{matrix}$

[16] A unary fuzzy predicate Γ ∈ ℑ (Ω), called fuzzifying compactness, is given as $Γ (X, τ) : = (\forall R) (K_{\circ} (R, X) ⟶ (\exists ℘) ((℘ \leq R) \land K (℘, A) \otimes FF (℘)))$ and if A ⊆ X, then Γ (A) : = Γ (A, τ/A).

[16] A unary fuzzy predicate fI ∈ ℑ (ℑ (P (X))) , called fuzzy finite intersection property, is given as $fI (R) : = \forall ℘ ((℘ \leq R) \land FF (℘) \to \exists x \forall B (B \in ℘ \to x \in B))$ .

[11] A fuzzifying topological space (X, τ) is said to be fuzzifying β-topological space [11] if τ_β (A ∩ B) ≥ τ_β (A) ∧ τ_β (B).

[11] A binary fuzzy predicate K_β ∈ ℑ (ℑ (P (X)) × P (X)), called fuzzifying β-open covering, is given as $K_{β} (R, A) : = K (R, A) \otimes (R \subseteq τ_{β})$ .

[11] A unary fuzzy predicate Γ_β ∈ ℑ (Ω), called fuzzifying β-compactness, is given as $(X, τ) \in Γ_{β} : = (\forall R) (K_{β} (R, X) ⟶ (\exists ℘) ((℘ \leq R) \land K (℘, X) \otimes FF (℘)))$ and if A ⊆ X, then Γ_β (A) : = Γ_β (A, τ/A).

[12] A unary fuzzy predicate LC ∈ ℑ (Ω), called fuzzifying locally compactness, is given as LC (X, τ) : = (∀ x) (∃ B) ((x ∈ Int (B) ⊗ Γ (B, τ/B)).

3 Fuzzifying β-base and β-subbase

Definition 3.1. Let (X, τ) be a fuzzifying topological space and ß_β ⊂ τ_β. Then ß_β is called β-base of τ_β if ß_β fulfils the condition: $⊨ U \in N_{x}^{β} \to \exists V ((V \in ß_{β}) \land (x \in V \subseteq U)) .$

Remark 3.2. In above definition, we can obtain that τ_r = {U ∈ P (X) ∣ τ (U) ≥ r} is a classical topology for each r ∈ [0, 1], similarly, (ß _β) _r, (τ_β) _r . Then (ß _β) _r is a β-base of (τ_β) _r if for each $U \in (N_{x}^{β})_{r}$ , there exists V ∈ (ß _β) _r such that x ∈ V ⊆ U.

Example 3.3. Let X = {x, y, z} and we define a fuzzifying topology τ ∈ ℑ (P (X)) as follows: $\begin{matrix} τ (X) = τ (\emptyset) = 1, τ ({x}) = 0.8, \\ τ ({y}) = 0.6, τ ({z}) = 0.4, \\ τ ({x, y}) = 0.6, τ ({y, z}) = 0.6, τ ({z, x}) = 0.4 . \end{matrix}$ Since $N_{x} (A) = \underset{x \in B \subseteq A}{⋁} τ (B)$ , we have $\begin{matrix} N_{x} (X) = 1, \forall x \in X, \\ N_{x} ({x}) = N_{x} ({x, y}) = N_{x} ({x, z}) = 0.8, \\ N_{y} ({y}) = N_{y} ({y, z}) = 0.6, N_{y} ({x, y}) = 0.8, \\ N_{z} ({z}) = N_{z} ({x, z}) = 0.4, N_{z} ({y, z}) = 0.6 . \end{matrix}$ Since Cl (A) (x) =1 - N_x (X - A) , $\begin{matrix} Cl ({x}) (x) = 1 - N_{x} ({y, z}) = 1, \\ Cl ({x}) (y) = 1 - N_{y} ({y, z}) = 0.4, \\ Cl ({x}) (z) = 1 - N_{z} ({y, z}) = 0.4, \\ Cl ({y}) = (0.2, 1, 0.6), Cl ({z}) = (0.2, 0.2, 1), \\ Cl ({x, y}) = (1, 1, 0.6), Cl ({x, z}) = (1, 0.4, 1), \\ Cl ({y, z}) = (0.2, 1, 1), Cl (X) = (1, 1, 1), \\ Cl (\emptyset) = (0, 0, 0) . \end{matrix}$ Then Int (Cl (A)) = A and Cl (Int (Cl (A))) = Cl (A) for all A ∈ P (X). From Definition 2.2, we obtain a fuzzifying topology τ_β ∈ ℑ (P (X)) as follows: $\begin{matrix} τ_{β} (X) = τ_{β} (\emptyset) = 1, τ_{β} ({x}) = 0.4, \\ τ_{β} ({y}) = 0.2, τ_{β} ({z}) = 0.2 \\ τ_{β} ({x, y}) = 0.6, τ_{β} ({x, z}) = 0.4, τ_{β} ({y, z}) = 0.2 . \end{matrix}$

(1) We define $ß_{β} \in ℑ (P (X))$ as follows: $\begin{matrix} ß_{β} (X) = ß_{β} (\emptyset) = 1, ß_{β} ({x}) = 0.4, \\ ß_{β} ({y}) = 0.2, ß_{β} ({z}) = 0.2, \\ ß_{β} ({x, y}) = 0.6, ß_{β} ({x, z}) = ß_{β} ({y, z}) = 0 . \end{matrix}$ Since ß_β ⊂ τ_β and $N_{x}^{β} (U) \leq ⋁_{x \in V \subseteq U} ß_{β} (V)$ from Definition 3.1, then ß_β is a β-base of τ_β.

(2) We define c_β ∈ ℑ (P (X)) as follows: $\begin{matrix} c_{β} (X) = c_{β} (\emptyset) = 1, c_{β} ({x}) = 0.2, \\ c_{β} ({y}) = 0.2, c_{β} ({z}) = 0.2, \\ c_{β} ({x, y}) = 0.6, c_{β} ({x, z}) = c_{β} ({y, z}) = 0 . \end{matrix}$ We have c_β ⊂ τ_β and $N_{x}^{β} ({x, z}) = τ_{β} ({x, z}) = 0.4 ≰ ⋁_{x \in V \subseteq {x, z}} c_{β} (V) = 0.2$ . Hence c_β is not a β-base of τ_β. Moreover, (c_β) _0.3 = {X, ∅ , {x, y}} is not a β-base of (τ_β) _0.3 = {X, ∅ , {x} , {x, y} , {x, z}} in Remark 3.2.

Theorem 3.4. ß_β is an β-base of τ_β if and only if $τ_{β} = ß_{β}^{(\cup)},$ where $ß_{β}^{(\cup)} (U) = \underset{\underset{λ \in Λ}{\cup} V_{λ} = U}{\lor} \underset{λ \in Λ}{\land} ß_{β} (V_{λ}) .$

Proof. Suppose that ß_β is an β-base of τ_β . If $⋃_{λ \in Λ} V_{λ} = U$ , then from Theorem 3.1 (1) (b) in [3], $τ_{β} (U) = τ_{β} (⋃_{λ \in Λ} V_{λ}) \geq \underset{λ \in Λ}{⋀} τ_{β} (V_{λ}) \geq \underset{λ \in Λ}{⋀} ß_{β} (V_{λ}) .$ Consequently, $τ_{β} (U) \geq \underset{\underset{λ \in Λ}{\cup} V_{λ} = U}{\lor} \underset{λ \in Λ}{\land} ß_{β} (V_{λ}) .$

To prove that $τ_{β} (U) \leq \underset{\underset{λ \in Λ}{\cup} V_{λ} = U}{\lor} \underset{λ \in Λ}{\land} ß_{β} (V_{λ})$ , we first prove $τ_{β} (U) = \underset{x \in U}{⋀} \underset{x \in V \subseteq U}{⋁} τ_{β} (V) .$ (Indeed, assume δ_x = {V : x ∈ V ⊆ U} . Then for any $f \in \prod_{x \in U} δ_{x}$ , $⋃_{x \in U} f (x) = U,$ and furthermore $\begin{matrix} τ_{β} (U) & = & τ_{β} (⋃_{x \in U} f (x)) \geq \underset{x \in U}{⋀} τ_{β} (f (x)) \\ \geq & \underset{f \in \prod_{x \in U} δ_{x}}{\lor} \underset{x \in U}{\land} τ_{β} (f (x)) = \underset{x \in U}{\land} \underset{x \in V \subseteq U}{\lor} τ_{β} (V) . \end{matrix}$ Also $τ_{β} (U) \leq \underset{x \in U}{⋀} \underset{x \in V \subseteq U}{⋁} τ_{β} (V) .$ Therefore

$τ_{β} (U) = \underset{x \in U}{⋀} \underset{x \in V \subseteq U}{⋁} τ_{β} (V) .$

Now, since $N_{x}^{β} (U) \leq$ $\underset{x \in V \subseteq U}{⋁} ß_{β} (V),$ $\begin{matrix} τ_{β} (U) & = & \underset{x \in U}{⋀} \underset{x \in V \subseteq U}{⋁} τ_{β} (V) = \underset{x \in U}{⋀} N_{x}^{β} (U) \\ \leq & \underset{x \in U}{\land} \underset{x \in V \subseteq U}{\lor} α_{β} (V) = \underset{f \in \prod_{x \in U} δ_{x}}{\lor} {\underset{x \in U}{\land}}_{β} α_{β} (f (x)) . \end{matrix}$

Then $τ_{β} (U) \leq \underset{λ \in Λ ⋃ V_{λ} = U}{⋁} \underset{λ \in Λ}{⋀} ß_{β} (V_{λ}) .$ Therefore $τ_{β} (U) = \underset{λ \in Λ ⋃ V_{λ} = U}{⋁} \underset{λ \in Λ}{⋀} ß_{β} (V_{λ})$ .

Conversely, we assume $τ_{β} (U) = \underset{λ \in Λ ⋃ V_{λ} = U}{⋁} \underset{λ \in Λ}{⋀} ß_{β} (V_{λ})$ and we will show that ß_β is an β-base of τ_β, i.e., for any U ⊆ X, $N_{x}^{β} (U) \leq \underset{x \in V \subseteq U}{⋁} ß_{β} (V) .$ Indeed, if x ∈ V ⊆ U, $⋃_{λ \in Λ} V_{λ} = V,$ then there exists λ_∘ ∈ Λ such that x ∈ V_{λ
_∘} and $\underset{λ \in Λ}{⋀} ß_{β} (V_{λ}) \leq ß_{β} (V_{λ_{\circ}}) \leq \underset{x \in V \subseteq U}{⋁} ß_{β} (V) .$ Therefore $\begin{matrix} N_{x}^{β} (U) & = & \underset{x \in V \subseteq U}{⋁} τ_{β} (V) \\ = & \underset{x \in V \subseteq U}{⋁} \underset{λ \in Λ ⋃ V_{λ} = U}{⋁} \underset{λ \in Λ}{⋀} ß_{β} (V_{λ}) \\ \leq & \underset{x \in V \subseteq U}{⋁} ß_{β} (V) . \end{matrix}$

Theorem 3.5. Let $ß_{β} \in ℑ (P (X))$ . Then ß_β is an β-base for some fuzzifying β-topology τ_β if and only if it has the following properties:

$ß_{β}^{(\cup)} (X) = 1;$

⊨ (U ∈ ß _β) ∧ (V ∈ ß _β) ∧ (x ∈ U ∩ V) → ∃ W ((W ∈ ß _β) ∧ (x ∈ W ⊆ U ∩ V).

Proof. If ß_β is an β-base for some fuzzifying β-topology τ_β, then $τ_{β} (X) = ß_{β}^{(\cup)} (X) .$ Clearly, $ß_{β}^{(\cup)} (X) = 1 .$ In addition, if x ∈ U ∩ V, then $\begin{matrix} ß_{β} (U) \land ß_{β} (V) \leq τ_{β} (U) \land τ_{β} (V) \leq τ_{β} (U \cap V) \\ \leq N_{x}^{β} (U \cap V) \leq \underset{x \in W \subseteq U \cap V}{⋁} ß_{β} (W) . \end{matrix}$ Conversely, if ß_β satisfies (1) and (2), then we have τ_β is a fuzzifying β-topology. In fact, τ_β (X) =1 . For any {U_λ : λ ∈ Λ } ⊆ P (X) , we set $δ_{λ} = {{V_{Φ_{λ}} : Φ_{λ} \in Λ_{λ}} : ⋃_{Φ_{λ} \in Λ_{λ}} V_{Φ_{λ}} = U_{λ}} .$ Then for any $f \in \prod_{λ \in Λ} δ_{λ},$ $⋃_{λ \in Λ} ⋃_{V_{Φ_{λ}} \in f (λ)} V_{Φ_{λ}} = ⋃_{λ \in Λ} U_{λ} .$ Therefore $\begin{matrix} τ_{β} (⋃_{λ \in Λ} U_{λ}) = \underset{Φ \in Λ ⋃ V_{Φ} = λ \in Λ ⋃ U_{λ}}{⋁} \underset{Φ \in Λ}{⋀} ß_{β} (V_{Φ}) \\ \geq \underset{f \in λ \in Λ \prod δ_{λ}}{⋁} \underset{λ \in Λ}{⋀} \underset{V_{Φ_{λ}} \in f (λ)}{⋀} ß_{β} (V_{Φ_{λ}}) \\ \geq \underset{λ \in Λ}{⋀} \underset{{V_{Φ_{λ}} : Φ_{λ} \in Λ_{λ}} \in δ_{λ}}{⋁} \underset{Φ_{λ} \in Λ_{λ}}{⋀} ß_{β} (V_{Φ_{λ}}) \\ = \underset{λ \in Λ}{⋀} τ_{β} (U_{λ}) . \end{matrix}$

Finally, we need to prove that τ_β (U ∩ V) ≥ τ_β (U) ∧ τ_β (V) . If τ_β (U) > t, τ_β (V) > t, then there exists {V_{λ
₁} : λ₁ ∈ Λ₁ } , {V_{λ
₂} : λ₂∈ Λ₂ } such that $⋃_{λ_{1} \in Λ_{1}} V_{λ_{1}} = U, ⋃_{λ_{2} \in Λ_{2}} V_{λ_{2}} = V$ and for any λ₁ ∈ Λ₁, ß _β (V_{λ
₁}) > t, for any λ₂ ∈ Λ₂, ß _β (V_{λ
₂}) > t . Now, for any x ∈ U ∩ V, there exists λ_1x ∈ Λ₁, λ_2x ∈ Λ₂ such that x ∈ V_{λ
_1x} ∩ V_{λ
_2x} . From the assumption, we know that $t < ß_{β} (V_{λ_{1 x}}) \land ß_{β} (V_{λ_{2 x}}) \leq \underset{x \in W \subseteq V_{λ_{1 x}} \cap V_{λ_{2 x}}}{⋁} ß_{β} (W)$ and furthermore, there exists W_x such that x ∈ W_x ⊆ V_{λ
_1x} ∩ V_{λ
_2x} ⊆ U ∩ V, ß _β (W_x) > t . Since $⋃_{x \in U \cap V} W_{x} = U \cap V,$ we have $t \leq \underset{x \in U \cap V}{⋀} ß_{β} (W_{x}) \leq \underset{λ \in Λ ⋃ V_{λ} = U \cap V}{⋁} \underset{λ \in Λ}{⋀} ß_{β} (V_{λ}) = τ_{β} (U \cap V) .$ Now, let τ_β (U) ∧ τ_β (V) = k . For any natural number n, we have $τ_{β} (U) > k - \frac{1}{n},$ $τ_{β} (V) > k - \frac{1}{n}$ and so $τ_{β} (U \cap V) \geq k - \frac{1}{n} .$ Therefore τ_β (U ∩ V) ≥ k = τ_β (U) ∧ τ_β (V).

Definition 3.6. φ_β ∈ ℑ (P (X)) is called an β-subbase of τ_β if $φ_{β}^{\cap}$ is an β-base of τ_β, where $φ_{β}^{\cap} (⋂_{λ \in Λ} V_{λ}) = \underset{λ \in Λ ⋂ V_{λ} = U}{⋁} \underset{λ \in Λ}{⋀} φ_{β} (V_{λ})$ , {V_λ : λ ∈ Λ } ⊂ _fP (X) , with “⊂_f” standing for “a finite subset of”.

Remark 3.7. From Remark 3.2 and Definition 3.6, since (X, τ_r) is a classical topology for each r ∈ [0, 1] and (τ_β) _r is the collection of all β-open sets in X, then a β-subbase of (τ_β) _r is a collection (φ_β) _r of β-open sets such that every β-open set of (τ_β) _r is the union of sets that are finite intersections of {V_λ : λ ∈ Λ } ⊂ (φ_β) _r with a finite index Λ.

Theorem 3.8. φ_β ∈ ℑ (P (X)) is an β-subbase of some fuzzifying β-topology if and only if $φ_{β}^{(\cup)} (X) = 1 .$

Proof. We only demonstrate that $φ_{β}^{\cap}$ satisfies the second condition of Theorem 3.5, and others are obvious. In fact $\begin{matrix} φ_{β}^{\cap} (U) \land φ_{β}^{\cap} (V) = (\underset{λ_{1} \in Λ_{1} ⋂ V_{λ_{1}} = U}{⋁} \underset{λ_{1} \in Λ_{1}}{⋀} φ_{β} (V_{λ_{1}})) \\ \land (\underset{λ_{2} \in Λ_{2} ⋂ V_{λ_{2}} = V}{⋁} \underset{λ_{2} \in Λ_{2}}{⋀} φ_{β} (V_{λ_{2}})) \\ = \underset{λ_{1} \in Λ_{1} ⋂ V_{λ_{1}} = U}{⋁} \underset{λ_{2} \in Λ_{2} ⋂ V_{λ_{2}} = V}{⋁} (\underset{λ_{1} \in Λ_{1}}{⋀} φ_{β} (V_{λ_{1}})) \\ \land (\underset{λ_{2} \in Λ_{2}}{⋀} φ_{β} (V_{λ_{2}})) \\ \leq \underset{λ \in Λ ⋂ V_{λ} = U \cap V}{⋁} (\underset{λ \in Λ}{⋀} φ_{β} (V_{λ})) \\ = φ_{β}^{\cap} (U \cap V) . \end{matrix}$

Therefore if x ∈ U ∩ V, then $φ_{β}^{\cap} (U) \land φ_{β}^{\cap} (V) \leq φ_{β}^{\cap} (U \cap V) \leq \underset{x \in W \subseteq U \cap V}{⋁} φ_{β}^{\cap} (W) .$

4 Fuzzifying β-compact spaces

Theorem 4.1. Let (X, τ) be a fuzzifying topological space, φ_β be an β-subbase of τ_β, and

$β_{1} : = (\forall R) (K_{φ_{β}} (R, X) \to \exists ℘ ((℘ \leq R) \land K (℘, X) \otimes FF (℘))),$

where $K_{φ_{β}} (R, X) : = K (R, X) \otimes (R \subseteq φ_{β});$

β₂ : = (∀ S) ((S is a universal net in X)→ ∃ x ((x ∈ X) ∧ (S ⊳ ^βx)) ;

β₃ : = (∀ S) ((S ∈ N (X) → (∃ T) (∃ x) ((T < S) ∧ (x ∈ X) ∧ (T ⊳ ^βx)) ,

where “T < S” stands for “T is a subnet of S”;

β₄ : = (∀ S) ((S ∈ N (X) → ¬ (adh_βS ≡ φ)) ;

$β_{5} : = (\forall R) (R \in ℑ (P (X)) \land R \subseteq Γ_{β} \otimes fI (R) \to \exists x \forall A (A \in R \to x \in A)) .$

Then ⊨ (X, τ) ∈ Γ_β ↔ β_i , i = 1, 2, . . . , 5.

Proof. It is similar to the proof of Theorem 2.2 [16].

As a sense in Remarks 3.2 and 3.7, the above theorem is a generalization of the following corollary.

Corollary 4.2. The following are equivalent for a topological space (X, τ).

X is an β-compact space.

Every cover of X by members of an β-subbase of τ_β has a finite subcover.

Every universal net in X β-converges to a point in X.

Each net in X has a subnet that β-converges to some point in X.

Each net in X has an β-adherent point.

Each family of β-closed sets in X that has the finite intersection property has a non-void intersection.

Definition 4.3. Let {(X_s, τ_s) : s ∈ S} be a family of fuzzifying topological spaces, $\prod_{s \in S} X_{s}$ be the cartesian product of {X_s : s ∈ S} and $φ = {p_{s}^{- 1} (U_{s}) : s \in S, U_{s} \in p (X_{s})},$ where $p_{t} : \prod_{s \in S} X_{s} \to X_{t} (t \in S)$ is a projection. For Φ ⊆ φ, S (Φ) stands for the set of indices of elements in Φ. The β-base $ß_{β} \in ℑ (\prod_{s \in S} X_{s})$ of $\prod_{s \in S} (τ_{β})_{s}$ is defined as

V ∈ ß _β : = (∃ Φ) (Φ ⊂ _fφ ∧ (⋂ Φ = V)) → ∀ s (s ∈ S (Φ) → V_s ∈ (τ_β) _s) , i.e.,

$α_{β} (V) = \underset{Φ \subset_{f} φ, \cap Φ = V}{\lor} \underset{s \in S (Φ)}{\land} {(τ_{β})}_{s} (V_{s}) .$

Definition 4.4. Let (X, τ) , (Y, σ) be two fuzzifying topological space. A unary fuzzy predicate O_β ∈ ℑ (Y^X) , is called fuzzifying β-openness, is given as: O_β (f) : = ∀ U (U ∈ τ_β → f (U) ∈ σ_β) . Intuitively, the degree to which f is β-open is $[O_{β} (f)] = \underset{B \subseteq X}{⋀} min (1, 1 - τ_{β} (U) + σ_{β} (f (U))) .$

Lemma 4.5. Let (X, τ) and (Y, σ) be two fuzzifying topological space. For any f ∈ Y^X,

$O_{β} (f) : = \forall B (B \in ß_{β}^{X} \to f (B) \in σ_{β}),$ where $ß_{β}^{X}$ is an β-base of τ_β.

Proof. Clearly, $[O_{β} (f)] \leq [\forall U (U \in ß_{β}^{X} \to f (U) \in σ_{β})]$ . Conversely, for any U ⊆ X, we are going to prove min(1, 1 - τ_β (U)+ σ_β (f (U))) ≥ [∀ V (V ∈ $ß_{β}^{X} \to f (V) \in σ_{β})]$ . If τ_β (U) ≤ σ_β (f (U)) , it is hold clearly. Now assume τ_β (U) > σ_β (f (U)).

If $C \subseteq P (X)$ with $⋃ C = U,$ then $⋃_{V \in C} f (V) = f (⋃ C) = f (U) .$ Therefore $\begin{matrix} τ_{β} (U) - σ_{β} (f (U)) \\ = \underset{C \subseteq P (X), ⋃ C = U}{⋁} \underset{V \in C}{⋀} ß_{β}^{X} (V) \\ - \underset{K \subseteq P (Y), ⋃ K = f (U)}{⋁} \underset{W \in K}{⋀} σ_{β} (W) \\ \leq \underset{C \subseteq P (X), ⋃ C = U}{⋁} \underset{V \in C}{⋀} ß_{β}^{X} (V) \\ - \underset{C \subseteq P (X), ⋃ C = U}{⋁} \underset{V \in C}{⋀} σ_{β} (f (V)) \\ \leq \underset{C \subseteq P (X), ⋃ C = U}{⋁} \underset{V \in C}{⋀} (ß_{β}^{X} (V) - σ_{β} (f (V))) \end{matrix}$ $\begin{matrix} min (1, 1 - τ_{β} (U) + σ_{β} (f (U))) \\ \geq \underset{C \subseteq P (X), ⋃ C = U}{⋁} \underset{V \in C}{⋀} min (1, 1 - ß_{β}^{X} (V) + σ_{β} (f (V))) \\ \geq [\forall V (V \in ß_{β}^{X} \to f (V) \in σ_{β})] . \end{matrix}$

Lemma 4.6. For any family {(X_s, τ_s) : s ∈ S} of fuzzifying topological spaces.

⊨ (∀ s) (s ∈ S → p_s ∈ O_β) ;

⊨ (∀ s) (s ∈ S → p_s ∈ C_β).

Proof. (1) For any t ∈ S, we have $O_{β} (p_{t}) = \underset{U \in P (s \in S \prod X_{s})}{⋀} min (1, 1 - (\prod_{s \in S} (τ_{β})_{s}) (U) + (τ_{β})_{t} (p_{t} (U))) .$ Then it suffices to show that for any $U \in P (\prod_{s \in S} X_{s}),$ we have $(τ_{β})_{t} (p_{t} (U)) \geq (\prod_{s \in S} (τ_{β})_{s}) (U) .$

Assume $\begin{matrix} (\prod_{s \in S} (τ_{β})_{s}) (U) \\ = \underset{λ \in Λ \cup B_{λ} = U}{⋁} \underset{λ \in Λ}{⋀} \underset{Φ_{λ} \subset_{f} φ, \cap Φ_{λ} = B_{λ}}{⋁} \underset{s \in S (Φ_{λ})}{⋀} (τ_{β})_{s} (V_{s}) > μ, \end{matrix}$ where $Φ_{λ} = {p_{s}^{- 1} (V_{s}) : s \in S (Φ_{λ})} (λ \in Λ) .$

Hence there exists ${B_{λ} : λ \in Λ} \subseteq P (\prod_{s \in S} X_{s})$ such that $⋃_{λ \in Λ} B_{λ} = U$ and furthermore, for any λ ∈ Λ, there exists Φ_λ ⊂ _fφ such that ∩Φ_λ = B_λ and $⋂_{s \in S (Φ_{λ})} p_{s}^{- 1} (V_{s}) = B_{λ},$ where for any s ∈ S (Φ_λ) we have (τ_β) _s (V_s) > μ . Thus $p_{t} (U) = p_{t} (⋃_{λ \in Λ} ⋂_{s \in S (Φ_{λ})} p_{s}^{- 1} (V_{s})) .$

(1) If for any λ ∈ Λ, $⋂_{s \in S (Φ_{λ})} p_{s}^{- 1} (V_{s}) = φ,$ then U = φ, p_t (U) = φ and (τ_β) _t (p_t (U)) = 1 . Therefore $(τ_{β})_{t} (p_{t} (U)) \geq (\prod_{s \in S} (τ_{β})_{s}) (U) .$

(2) If there exists λ_∘ ∈ Λ, such that $φ \neq ⋂_{s \in S (Φ_{λ})} p_{s}^{- 1} (V_{s}) = B_{λ_{\circ}},$

(i) If t ∉ S (Φ_{λ
_∘}) , i.e., t ∈ S - S (Φ_{λ
_∘}), p_t (B_{λ
_∘}) = X_t . Therefore (τ_β) _t (p_t (B_{λ
_∘})) = (τ_β) _t (X_t) =1.

(ii) If t ∈ S (Φ_{λ
_∘}), then p_t (B_{λ
_∘}) = V_t ⊆ X_t. Thus $\begin{matrix} p_{t} (U) & = & p_{t} ((⋃_{t \in S (Φ_{λ_{\circ}})} B_{λ_{\circ}}) \cup (⋃_{t \notin S (Φ_{λ_{\circ}})} B_{λ_{\circ}})) \\ = & (⋃_{t \in S (Φ_{λ_{\circ}})} p_{t} (B_{λ_{\circ}})) \cup (⋃_{t \notin S (Φ_{λ_{\circ}})} p_{t} (B_{λ_{\circ}})) \\ = & V_{t} \cup X_{t} = X_{t} . \end{matrix}$ Hence (τ_β) _t (p_t (U)) = (τ_β) _t (X_t) =1 or (τ_β) _t (p_t (U)) = (τ_β) _t (V_t) > λ.

Therefore $(τ_{β})_{t} (p_{t} (U)) \geq (\prod_{s \in S} (τ_{β})_{s}) (U) .$ Thus O_β (p_t) =1.

(2) From Lemma 3.1 in [15] we have ⊨ (∀ s) (s ∈ S → p_s ∈ C) . Furthermore, for any two fuzzifying topological spaces (X, τ) and (Y, σ) and f ∈ Y^X, we have C (f) ≤ C_β (f) (Theorem 3.3 in [3]). Therefore ⊨ (∀ s) (s ∈ S → p_s ∈ C_β).

Theorem 4.7. Let {(X_s, τ_s) : s ∈ S} be the family of fuzzifying topological spaces, then $⊨ \exists U (U \subseteq \prod_{s \in S} X_{s} \land Γ_{β} (U, τ / U) \land \exists x (x \in {Int}_{β} (U)) \to \exists T (T \subset_{f} S \land \forall t (t \in S - T \land Γ_{β} (X_{t}, τ_{t}))) .$

Proof. It suffices to show that $\begin{matrix} \underset{U \in P (s \in S \prod X_{s})}{⋁} (Γ_{β} (U, τ / U) \land \underset{x \in s \in S \prod X_{s}}{⋁} N_{x}^{β} (U)) \\ \leq \underset{T \subset_{f} S}{⋁} \underset{t \in S - T}{⋀} Γ_{β} (X_{t}, τ_{t}) . \end{matrix}$

Indeed, if $\underset{U \in P (s \in S \prod X_{s})}{⋁} (Γ_{β} (U, τ / U) \land \underset{x \in s \in S \prod X_{s}}{⋁} N_{x}^{β} (U)) > μ > 0,$ then there exists $U \in P (\prod_{s \in S} X_{s})$ such that Γ_β (U, τ/U) > μ and $\underset{x \in s \in S \prod X_{s}}{⋁} N_{x}^{β} (U) > μ,$ where $N_{x}^{β} (U) = \underset{x \in V \subseteq U}{⋁} (\prod_{s \in S} (τ_{β})_{s}) (V)$ . Furthermore, there exists V such that x ∈ V ⊆ U and $(\prod_{s \in S} (τ_{β})_{s}) (V) > μ .$ Since ß_β is an β-base of $\prod_{s \in S} (τ_{β})_{s},$ $\begin{matrix} (\prod_{s \in S} (τ_{β})_{s}) (V) = \underset{λ \in Λ \cup B_{λ} = V}{⋁} \underset{λ \in Λ}{⋀} ß_{β} (B_{λ}) \\ = \underset{λ \in Λ \cup B_{λ} = V}{⋁} \underset{λ \in Λ}{⋀} \underset{Φ_{λ} \subset_{f} φ, \cap Φ_{λ} = B_{λ}}{⋁} \underset{s \in S (Φ_{λ})}{⋀} (τ_{β})_{s} (V_{s}) > μ, \end{matrix}$ where $Φ_{λ} = {p_{s}^{- 1} (V_{s}) : s \in S (Φ_{λ})} (λ \in Λ) .$

Hence there exists ${B_{λ} : λ \in Λ} \subseteq P (\prod_{s \in S} X_{s})$ such that $\underset{λ \in Λ}{\cup} B_{λ} = V .$ Furthermore, for any λ ∈ Λ, there exists Φ_λ ⊂ _fφ such that ∩Φ_λ = B_λ and for any s ∈ S (Φ_λ) , we have (τ_β) _s (V_s) > μ . Since x ∈ V, there exists B_{λ
_x} such that x ∈ B_{λ
_x} ⊆ V ⊆ U . Hence there exists Φ_{λ
_x} ⊂ _fφ such that ∩Φ_{λ
_x} = B_{λ
_x} and $⋂_{s \in S (Φ_{λ})} p_{s}^{- 1} (V_{s}) = B_{λ_{x}} \subseteq \prod_{s \in S} X_{s}$ and for any s ∈ S (Φ_λ) , we have (τ_β) _s (V_s) >1 - μ . By $⋂_{s \in S (Φ_{λ})} p_{s}^{- 1} (V_{s}) = B_{λ_{x}},$ we have p_δ (B_{λ
_x}) = V_δ ⊆ X_δ, if δ∈ S (Φ_{λ
_x}) ; p_δ (B_{λ
_x}) = X_δ, if δ ∈ S - S (Φ_{λ
_x}) . Since B_{λ
_x} ⊆ U, for any δ ∈ S - S (Φ_{λ
_x}) , we have p_δ (U) ⊇ p_δ (B_{λ
_x}) = X_δ and p_δ (U) = X_δ . On the other hand, since for any s ∈ S and $U_{s} \in P (X_{s}), (\prod_{t \in S} (τ_{β})_{t}) (p_{s}^{- 1} (U_{s})) \geq (τ_{β})_{s} (U_{s}),$ wehave, for any $s \in S, I_{β} (p_{s}) = \underset{U_{s} \in P (X_{s})}{⋀} min (1, 1 - (τ_{β})_{s} (U_{s}) + \prod_{t \in S} (τ_{β})_{t} (p_{s}^{- 1} (U_{s}))) = 1 .$ Furthermore, since by Theorem 5.3 in [11], we have ⊨Γ_β (X, τ) ⊗ I_β (f) → Γ_β (f (X)) , then Γ_β (U, τ/U) = Γ_β (U, τ/U) ⊗ I_β (p_s) ≤ Γ_β (p_δ (U) , τ_δ) = Γ_β (X_δ, τ_δ) . Therefore $\underset{T \subset_{f} S}{⋁} \underset{t \in S - T}{⋀} Γ_{β} (X_{t}, τ_{t}) \geq \underset{δ \in S - S (Φ_{λ})}{⋀} Γ_{β} (X_{δ}, τ_{δ}) \geq Γ_{β} (U, τ / U) > μ .$

The above theorem is a generalization of the following corollary.

Corollary 4.8. If there exists a coordinate β-neighborhood β-compact subset U of some point x ∈ X of the product space, then all except a finite number of coordinate spaces are β-compact.

Lemma 4.9. For any fuzzifying topological space (X, τ) , A ⊆ X, $⊨ T_{2}^{β} (X, τ) \to T_{2}^{β} (A, τ / A) .$

Proof. $\begin{matrix} [T_{2}^{β} (X, τ)] \\ = \underset{x, y \in X, x \neq y}{⋀} \underset{U, V \in P (X), U \cap V = φ}{⋁} (N_{x}^{β} (U), N_{y}^{β} (V)) \\ \leq \underset{x, y \in X, x \neq y}{⋀} \underset{(U \cap A) \cap (V \cap A) = φ}{⋁} (N_{x}^{β^{A}} (U \cap A), N_{y}^{β^{A}} (V \cap A)) \\ \leq \underset{x, y \in A, x \neq y}{⋀} \underset{U^{'} \cap V^{'} = φ, U^{'}, V^{'} \in P (A)}{⋁} (N_{x}^{β^{A}} (U^{'}), N_{y}^{β^{A}} (V^{'})) \\ = T_{2}^{β} (A, τ / A), \end{matrix}$

where $N_{x}^{β^{A}} (U) = \underset{x \in C \subseteq U}{⋁} τ_{β} / A (C)$ and $τ_{β} / A (B) = \underset{B = V \cap A}{⋁} τ_{β} (V) .$

Lemma 4.10. For any fuzzifying β-topological space (X, τ), $⊨ T_{2}^{β} (X, τ) \otimes Γ_{β} (X, τ) \to T_{4}^{β} (X, τ) .$

For the definition of $T_{4}^{β} (X, τ)$ see [10, Definition 2.1].

Proof. If $[T_{2}^{β} (X, τ) \otimes Γ_{β} (X, τ)] = 0$ , then the result holds. Now, suppose that $[T_{2}^{β} (X, τ) \otimes Γ_{β} (X, τ)] > λ > 0 .$ Then $T_{2}^{β} (X, τ) + Γ_{β} (X, τ) - 1 > λ > 0$ . Therefore from Theorem 5.4 in [11] $T_{2}^{β} (X, τ) \otimes (Γ_{β} (A) \land Γ_{β} (B)) \land (A \cap B = φ) ⊨^{ws} T_{2}^{β} (X, τ) \to$ (∃U) (∃ V) ((U ∈ τ_β) ∧ (V ∈ τ_β) ∧ (A ⊆ U) ∧ (B ⊆ V) ∧ (A ∩ B = φ)) . Then for any A, B ⊆ X, A ∩ B = φ, $T_{2}^{β} (X, τ) \otimes (Γ_{β} (A) \land Γ_{β} (B)) \leq \underset{U \cap V = φ, A \subseteq U, B \subseteq V}{⋁} min (τ_{β} (U), τ_{β} (V))$ or equivalently $T_{2}^{β} (X, τ) \leq Γ_{β} (A) \land Γ_{β} (B) \to \underset{U \cap V = φ, A \subseteq U, B \subseteq V}{⋁} min (τ_{β} (U), τ_{β} (V))$ .

Hence for any $A, B \subseteq X, A \cap B = φ, 1 - [Γ_{β} (A) \land Γ_{β} (B)] + \underset{U \cap V = φ, A \subseteq U, B \subseteq V}{⋁} min (τ_{β} (U), τ_{β} (V))$ +Γ_β (X, τ) -1 > λ.

From Theorem 5.1 in [11] we have ⊨Γ_β (X, τ) ⊗ A ∈ Γ_β → Γ_β (A) . Then Γ_β (X, τ)+ [τ_β (A^c) ∧ τ_β (B^c)] -1 = (Γ_β (X, τ) + τ_β (A^c) -1) ∧ (Γ_β (X, τ)+ τ_β (B^c) -1) ≤ (Γ_β (X, τ) ⊗ τ_β (A^c)) ∧ (Γ_β (X, τ)⊗ τ_β (B^c)) ≤ [Γ_β (A) ∧ Γ_β (B)] . ThusΓ_β (X, τ) - [Γ_β (A) ∧ Γ_β (B)] -1 ≤ - [τ_β (A^c) ∧ τ_β (B^c)] . So, $1 - [τ_{β} (A^{c}) \land τ_{β} (B^{c})] + \underset{U \cap V = φ, A \subseteq U, B \subseteq V}{⋁} min (τ_{β} (U), τ_{β} (V)) > λ,$ i.e.,

$T_{4}^{β} (X, τ) = \underset{A \cap B = φ}{⋀} min (1, 1 - [τ_{β} (A^{c}) \land τ_{β} (B^{c})] +$ $\underset{U \cap V = φ, A \subseteq U, B \subseteq V}{⋁} min (τ_{β} (U), τ_{β} (V))) > λ$ .

The above lemma is a generalization of the following corollary.

Corollary 4.11. Every β-compact β-Hausdorff topological space is β-normal.

Lemma 4.12. For any fuzzifying β-topological space (X, τ) , $⊨ T_{2}^{β} (X, τ) \otimes Γ_{β} (X, τ) \to T_{3}^{β} (X, τ) .$ For the definition of $T_{3}^{β} (X, τ)$ see [10, Definition 2.1].

Proof. Immediate, set A = {x} in the above lemma.

The above lemma is a generalization of the following corollary.

Corollary 4.13. Every β-compact β-Hausdorff topological space is β-regular.

Theorem 4.14. For any fuzzifying topological space (X, τ) and A ⊆ X, $⊨ T_{2}^{β} (X, τ) \otimes Γ_{β} (A) \to A \in Γ_{β} .$

Proof. For any {x} ⊂ A^c, we have {x} ∩ A = φ and Γ_β ({x}) =1 . By Theorem 5.4 in [11] $[T_{2}^{β} (X, τ) \otimes (Γ_{β} (A) \land Γ_{β} ({x}))] \leq \underset{G \cap H_{x} = φ, A \subseteq G, x \in H_{x}}{⋁} min (τ_{β} (G), τ_{β} (H_{x}))) .$ Assume γ_x = {H_x : A ∩ H_x = φ, x ∈ H_x} , $⋃_{x \in A^{c}} f (x) \supseteq A^{c}$ and $⋃_{x \in A^{c}} f (x) \cap A = ⋃_{x \in A^{c}} (f (x) \cap A) = φ .$ So, $⋃_{x \in A^{c}} f (x) = A^{c} .$ Therefore $[T_{2}^{β} (X, τ) \otimes Γ_{β} (A)] \leq \underset{G \cap H_{x} = φ, A \subseteq G, x \in H_{x}}{⋁} τ_{β} (H_{x}) \leq \underset{x \in A^{c}}{⋀} \underset{A \cap H_{x} = φ, x \in H_{x}}{⋁} τ_{β} (H_{x}) = \underset{f \in x \in A^{c} \prod γ_{x}}{⋁} \underset{x \in A^{c}}{⋀} τ_{β} (f (x)) \leq \underset{f \in x \in A^{c} \prod γ_{x}}{⋁} τ_{β} (⋃_{x \in A^{c}} f (x)) = \underset{f \in x \in A^{c} \prod γ_{x}}{⋁} τ_{β} (A^{c}) = Γ_{β} (A) .$

The above theorem is a generalization of the following corollary.

Corollary 4.15. β-compact subspace of β-Hausdorff topological space is β-closed.

5 Fuzzifying locally β-compactness

Definition 5.1. Let Ω be a class of fuzzifying topological spaces. A unary fuzzy predicate L_βC ∈ ℑ (Ω) , called fuzzifying locally β-compactness, is given as follows:

(X, τ) ∈ L_βC : = (∀ x) (∃ B) ((x ∈ Int_β (B) ⊗ Γ_β (B, τ/B)) . Since $[x \in {Int}_{β} (X)] = N_{x}^{β} (X) = 1,$ then L_βC (X, τ) ≥ Γ_β (X, τ) . Therefore, ⊨ (X, τ) ∈ Γ_β → (X, τ) ∈ L_βC . Also, since ⊨ (X, τ) ∈ Γ → (X, τ) ∈ LC [12] and ⊨ (X, τ) ∈ Γ_β → (X, τ) ∈ Γ [11], ⊨ (X, τ) ∈ Γ_β → (X, τ) ∈ LC.

Theorem 5.2. For any fuzzifying topological space (X, τ) and A ⊆ X, ⊨ (X, τ) ∈ L_βC ⊗ A ∈ Γ_β → (A, τ/A) ∈ L_βC .

Proof. We have $\begin{matrix} L_{β} C (X, τ) \\ = \underset{x \in X}{⋀} \underset{B \subseteq X}{⋁} max (0, N_{x}^{β^{X}} (B) + Γ_{β} (B, τ / B) - 1) \\ L_{β} C (A, τ / A) \\ = \underset{x \in A}{⋀} \underset{G \subseteq A}{⋁} max (0, N_{x}^{β^{A}} (G) + Γ_{β} (G, (τ / A) / G) - 1) . \end{matrix}$ Now, suppose that [(X, τ) ∈ L_βC ⊗ A ∈ Γ_β] > λ > 0 . Then for any x ∈ A, there exists B ⊆ X such that $N_{x}^{β^{X}} (B) + Γ_{β} (B, τ / B) + τ_{β} (X - A) - 2 > λ$ . (*)

Set E = A ∩ B ∈ P (A) . Then $N_{x}^{β^{A}} (E) = \underset{E = C \cap B}{⋁} N_{x}^{β^{X}} (C) \geq N_{x}^{β^{X}} (B)$ and for any U ∈ P (E), we have $\begin{matrix} (τ_{β} / A)_{β} / E (U) & = & \underset{U = C \cap E}{⋁} τ_{β} / A (C) \\ = & \underset{U = C \cap E}{⋁} \underset{C = D \cap A}{⋁} τ_{β} (D) \\ = & \underset{U = D \cap A \cap E}{⋁} τ_{β} (D) = \underset{U = D \cap E}{⋁} τ_{β} (D) . \end{matrix}$ Similarly, $(τ_{β} / B)_{β} / E (U) = \underset{U = D \cap E}{⋁} τ_{β} (D) .$ Thus,(τ_β/B) _β/E = (τ_β/A) _β/E and Γ_β (E, (τ/A)/E) = Γ_β (E, (τ/B)/E) . Furthermore, $[E \in Γ_{β} / B] = τ_{β} / B (B - E) = τ_{β} / B (B \cap E^{c}) = \underset{B \cap E^{c} = B \cap D}{⋁} τ_{β} (D) \geq$ τ_β (X - A) = Γ_β (A) . Since ⊨ (X, τ) ∈ Γ_β ⊗ A ∈ Γ_β → (A, τ/A) ∈ Γ_β, from (*) we have for any x ∈ A that $\begin{matrix} \underset{G \subseteq A}{⋁} max (0, N_{x}^{β^{A}} (G) + Γ_{β} (G, (τ / A) / G) - 1) \\ \geq N_{x}^{β^{A}} (E) + Γ_{β} (E, (τ / A) / E) - 1 \\ = N_{x}^{β^{A}} (E) + Γ_{β} (E, (τ / B) / E) - 1 \\ \geq N_{x}^{β^{X}} (B) + [Γ_{β} (B, τ / B) \otimes E \in Γ_{β} / B] - 1 \\ \geq N_{x}^{β^{X}} (B) + Γ_{β} (B, τ / B) + [E \in Γ_{β} / B] - 2 \\ \geq N_{x}^{β^{X}} (B) + Γ_{β} (B, τ / B) + [A \in Γ_{β}] - 2 > λ . \end{matrix}$ Therefore $L_{β} C (A, τ / A) = \underset{x \in A}{⋀} \underset{G \subseteq A}{⋁} max (0, N_{x}^{β^{A}} (G) + Γ_{β} (G, (τ / A) / G) - 1) > λ .$

Hence [(X, τ) ∈ L_βC ⊗ A ∈ Γ_β] ≤ L_βC (A, τ/A).

As a crisp result of the above theorem we have the following corollary.

Corollary 5.3. Let A be an β-closed subset of locally β-compact space (X, τ) . Then A with the relative topology τ/A is locally β-compact.

The following theorem is a generalization of the statement “If X is an β-Hausdorff topological space and A is an β-dense locally β-compact subspace, then A is β-open”, where A is an β-dense in a topological space X if and only if the β-closure of A is X.

Theorem 5.4. For any fuzzifying β-topological space (X, τ) and A ⊆ X, $⊨ T_{2}^{β} (X, τ) \otimes L_{β} C (A, τ / A) \otimes ({Cl}_{β} (A) \equiv X) \to A \in τ_{β} .$

Proof. Assume $[T_{2}^{β} (X, τ) \otimes L_{β} C (A, τ / A) \otimes ({Cl}_{β} (A) \equiv X)] > λ > 0 .$ Then $L_{β} C (A, τ / A) > λ - [T_{2}^{β} (X, τ) \otimes ({Cl}_{β} (A) \equiv X)] + 1 = λ^{'} > λ$ , i.e., $\underset{x \in A}{⋀} \underset{B \subseteq A}{⋁} max (0, N_{x}^{β^{A}} (B) + Γ_{β} (B, (τ / A) / B) - 1) > λ^{'} .$ Thus for any x ∈ A, there exists B_x ⊆ A such that $N_{x}^{β^{A}} (B_{x}) + Γ_{β} (B_{x}, (τ / A) / B_{x}) - 1 > λ^{'} .$ i.e., $\underset{H \cap A = B_{x}}{⋁} \underset{x \in K \subseteq H}{⋁} τ_{β} (K)$ +Γ_β (B_x, (τ/A)/B_x) -1 > λ′ . Hence there exists K_x such that K_x ∩ A = B_x, τ_β (K_x) +Γ_β (B_x, (τ/A)/B_x) -1 > λ′ . Therefore τ_β (K_x) > λ′.

(1) If for any x ∈ A there exists K_x such that x ∈ K_x ⊆ B_x ⊆ A, then $⋃_{x \in A} K_{x} = A$ and $τ_{β} (A) = τ_{β} (⋃_{x \in A} K_{x}) \geq \underset{x \in A}{⋀} τ_{β} (K_{x}) \geq λ^{'} > λ .$

(2) If there exists x_∘ ∈ A such that $K_{x_{\circ}} \cap (B_{x_{\circ}}^{c}) \neq φ,$ τ_β (K_{x
_∘}) +Γ_β (B_{x
_∘}, (τ/A)/B_{x
_∘}) -1 > λ′ . From the hypothesis $[T_{2}^{β} (X, τ) \otimes L_{β} C (A, τ / A) \otimes ({Cl}_{β} (A) \equiv X)] > λ > 0$ , we have $[T_{2}^{β} (X, τ) \otimes ({Cl}_{β} (A) \equiv X)] \neq 0$ . So τ_β (K_{x
_∘}) + Γ_β (B_{x
_∘}, (τ/A)/B_{x
_∘}) -1 $+ [T_{2}^{β} (X, τ) \otimes ({Cl}_{β} (A) \equiv X)] - 1 > 0 .$ Therefore τ_β (K_{x
_∘}) $+ Γ_{β} (B_{x_{\circ}}, (τ / A) / B_{x_{\circ}}) - 1 + T_{2}^{β} (X, τ) + [({Cl}_{β} (A) \equiv X)] - 1 - 1 > λ .$ Since $\begin{matrix} (τ_{β} / A)_{β} / B_{x_{\circ}} (U) = \underset{U = C \cap B_{x_{\circ}}}{⋁} τ_{β} / A (C) \\ = \underset{U = C \cap B_{x_{\circ}}}{⋁} \underset{C = D \cap A}{⋁} τ_{β} (D) \\ = \underset{U = D \cap B_{x_{\circ}}}{⋁} τ_{β} (D) = τ_{β} / B_{x_{\circ}} (U), \end{matrix}$ Γ_β (B_{x
_∘}, (τ/A)/B_{x
_∘}) = Γ_β (B_{x
_∘}, τ/B_{x
_∘}) . From Theorem 4.3, we have $\begin{matrix} τ_{β} (B_{x_{\circ}}^{c}) \geq T_{2}^{β} (X, τ) \otimes Γ_{β} (B_{x_{\circ}}, τ / B_{x_{\circ}}) \\ \geq T_{2}^{β} (X, τ) + Γ_{β} (B_{x_{\circ}}, τ / B_{x_{\circ}}) - 1 . \end{matrix}$ Hence $τ_{β} (K_{x_{\circ}}) + τ_{β} (B_{x_{\circ}}^{c}) + [{Cl}_{β} (A) \equiv X] - 2 > λ .$ Now, for any y ∈ A^c we have $[{Cl}_{β} (A) \equiv X] = \underset{x \in X}{⋀} (1 - N_{x}^{β^{X}} (A^{c})) \leq 1 - N_{y}^{β^{X}} (A^{c})$ . Since (X, τ) is a fuzzifying β-topological space, $\begin{matrix} τ_{β} (K_{x_{\circ}}) + τ_{β} (B_{x_{\circ}}^{c}) - 1 \leq τ_{β} (K_{x_{\circ}}) \otimes τ_{β} (B_{x_{\circ}}^{c}) \\ \leq τ_{β} (K_{x_{\circ}}) \land τ_{β} (B_{x_{\circ}}^{c}) \leq τ_{β} (K_{x_{\circ}} \cap B_{x_{\circ}}^{c}) \\ \leq N_{y}^{β^{X}} (K_{x_{\circ}} \cap B_{x_{\circ}}^{c}) \leq N_{y}^{β^{X}} (A^{c}), \end{matrix}$ where $y \in K_{x_{\circ}} \cap B_{x_{\circ}}^{c} \subseteq H_{x_{\circ}} \cap (H_{x_{\circ}} \cap A)^{c} =$ $H_{x_{\circ}} \cap (H_{x_{\circ}}^{c} \cup A^{c}) = H_{x_{\circ}} \cap A^{c} \subseteq A^{c} .$ Therefore $0 < λ < τ_{β} (K_{x_{\circ}}) + τ_{β} (B_{x_{\circ}}^{c}) + [{Cl}_{β} (A) \equiv X] - 2 =$ $τ_{β} (K_{x_{\circ}}) + τ_{β} (B_{x_{\circ}}^{c}) - 1 +$ [Cl_β (A) ≡ X] -1 $\leq N_{y}^{β^{X}} (A^{c}) + 1 - N_{y}^{β^{X}} (A^{c}) - 1 = 0,$ a contradiction. So, case (2) does not hold. We complete the proof.

Theorem 5.5. For any fuzzifying β-topological space (X, τ) , $⊨ T_{2}^{β} (X, τ) \otimes (L_{β} C (X, τ))^{2} \to \forall x \forall U (U \in N_{x}^{β^{X}} \to \exists V (V \in N_{x}^{β^{X}} \land$ Cl_β (V) ⊆ U ∧ Γ_β (V))) , where (L_βC (X, τ)) ² : = L_βC (X, τ) ⊗ L_βC (X, τ) .

Proof. We need to show that for any x and U, x ∈ U, $T_{2}^{β} (X, τ) \otimes (L_{β} C (X, τ))^{2} \otimes N_{x}^{β^{X}} (U) \leq$ $\underset{V \subseteq X}{⋁} (N_{x}^{β^{X}} (V) \land \underset{y \in U^{c}}{⋀} N_{x}^{β^{X}} (V^{c}) \land Γ_{β} (V, τ / V)) .$ Assu-me that $T_{2}^{β} (X, τ) \otimes (L_{β} C (X, τ))^{2} \otimes N_{x}^{β^{X}} (U) > λ > 0 .$ Then for any x ∈ X there exists C such that

$T_{2}^{β} (X, τ) + N_{x}^{β^{X}} (C) + (L_{β} C (X, τ))^{2} + N_{x}^{β^{X}} (U) - 3 > λ .$ (*)

Since (X, τ) is fuzzifying β-topological space, $N_{x}^{β^{X}} (C) + N_{x}^{β^{X}} (U) - 1 \leq N_{x}^{β^{X}} (C) \otimes N_{x}^{β^{X}} (U) \leq$ $N_{x}^{β^{X}} (C) \land N_{x}^{β^{X}} (U) \leq$ $N_{x}^{β^{X}} (C \cap U) = \underset{x \in W \subseteq C \cap U}{⋁} τ_{β} (W) .$ Therefore there exists W such that x ∈ W ⊆ C ∩ U, and $T_{2}^{β} (X, τ) + (L_{β} C (X, τ))^{2} + τ_{β} (W) - 2 > λ .$ By Lemmas 4.3 and 4.5 we have $T_{2}^{β} (X, τ) \leq T_{2}^{β} (C, τ / C)$ and $T_{2}^{β} (C, τ / C) + Γ_{β} (C, τ / C) - 1 \leq$ $T_{2}^{β} (C, τ / C) \otimes Γ_{β} (C, τ / C) \leq T_{3}^{β} (C, τ / C) .$ Thus $T_{3}^{β} (X, τ) + Γ_{β} (C, τ / C) + τ_{β} (W) - 2 > λ .$ Since for any x ∈ W ⊆ U, we have $T_{3}^{β} (C, τ / C) \leq 1 - τ_{β} / C (W) + \underset{G \subseteq C}{⋁} ((N_{x}^{β^{C}} (G) \land \underset{y \in C - W}{⋀} N_{y}^{β^{C}} (C - G))),$ so there exists G, x ∈ G ⊆ W such that $\begin{matrix} ((N_{x}^{β^{C}} (G) \land \underset{y \in C - W}{⋀} N_{y}^{β^{C}} (C - G))) \\ \geq T_{3}^{β} (C, τ / C) + τ_{β} / C (W) - 1 \\ \geq T_{3}^{β} (C, τ / C) + τ_{β} (W) - 1, \\ ((N_{x}^{β^{C}} (G) \land \underset{y \in C - W}{⋀} N_{y}^{β^{C}} (C - G))) \\ + Γ_{β} (C, τ / C) - 1 > λ . \end{matrix}$ Thus $N_{x}^{β^{C}} (G) = \underset{D \cap C = G}{⋁} N_{x}^{β^{X}} (D) = N_{x}^{β^{X}} (G \cup C^{c}) > λ^{'} = λ + 1 - Γ_{β} (C, τ / C) \geq λ$ . Furthermore, forany y ∈ C - W, $N_{y}^{β^{C}} (C - G) = \underset{D \cap C = C \cap G^{c}}{⋁} N_{y}^{β^{X}} (G^{c} \cup C^{c}) = N_{y}^{β^{X}} (G^{c}) > λ^{'}$ and $N_{x}^{β^{X}} (G) = N_{x}^{β^{X}} ((G \cup C^{c}) \cap C) \geq N_{x}^{β^{X}} (G \cup C^{c}) \land N_{x}^{β^{X}} (C) > λ^{'} .$ Since $N_{y}^{β^{X}} (G^{c}) = \underset{x \in B^{c} \subseteq G^{c}}{⋁} τ_{β} (B^{c}) > λ^{'},$ for any y ∈ C - W, there exists $B_{y}^{c}$ such that $y \in B_{y}^{c} \subseteq G^{c}$ and $τ_{β} (B_{y}^{c}) > λ^{'} .$ Set $B^{c} = ⋃_{y \in C - W} B_{y}^{c} .$ Then C - W ⊆ B^c ⊆ G^c and $τ_{β} (B^{c}) \geq \underset{y \in C - W}{⋀} τ_{β} (B_{y}^{c}) \geq λ^{'} .$ Again, set V = B ∩ C, then V ⊆ (C - W) ^c ∩ C = (C^c ∪ W) ∩ C = C ∩ W = W ⊆ U ∩ C and V^c = B^c ∪ C^c . Since (X, τ) is fuzzifying β-topological space,

$\begin{matrix} N_{x}^{β^{X}} (V) & = & N_{x}^{β^{X}} (B \cap C) \geq N_{x}^{β^{X}} (B) \land N_{x}^{β^{X}} (C) \\ \geq & N_{x}^{β^{X}} (G) \land N_{x}^{β^{X}} (C) > λ . \end{matrix}$ (1) By (*) and Theorem 4.3, $τ_{β} (C^{c}) \geq T_{2}^{β} (X, τ) \otimes Γ_{β} (C, τ / C)$ $\geq T_{2}^{β} (X, τ) + Γ_{β} (C, τ / C) - 1 \geq λ^{'} .$ So τ_β (V^c) = τ_β (B^c∪ C^c) ≥ τ_β (B^c) ∧ τ_β (C^c) ≥ λ′, i.e., τ_β (V^c) + Γ_β (C, τ/C) -1 ≥ λ and

$\begin{matrix} Γ_{β} (V, τ / V) = Γ_{β} (V, (τ / C) / V) \\ \geq τ_{β} / C (C - V) + Γ_{β} (C, τ / C) - 1 \\ \geq τ_{β} (V^{c}) + Γ_{β} (C, τ / C) - 1 \geq λ . \end{matrix}$ (2) $\underset{y \in U^{c}}{⋀} N_{y}^{β^{X}} (V^{c}) \geq \underset{y \in V^{c}}{⋀} N_{y}^{β^{X}} (V^{c}) = τ_{β} (V^{c}) \geq λ .$ (3) Thus by (1), (2) and (3), for any x ∈ U, there exists V ⊆ U such that $N_{x}^{β^{X}} (V) > λ,$ $\underset{y \in U^{c}}{⋀} N_{y}^{β^{X}} (V^{c}) \geq λ$ and Γ_β (V, τ/V) ≥ λ . So $\underset{V \subseteq X}{⋁} (N_{x}^{β^{X}} (V) \land \underset{y \in U^{c}}{⋀} N_{y}^{β^{X}} (V^{c}) \land Γ_{β} (V, τ / V)) \geq λ .$

Theorem 5.6. For any fuzzifying β-topological space (X, τ) , $⊨ T_{2}^{β} (X, τ) \otimes (L_{β} C (X, τ))^{2} \to T_{3}^{β} (X, τ)$

Proof. By Theorem 5.5, for any x ∈ U, we have $\underset{x \in V \subseteq U}{⋁} (N_{x}^{β^{X}} (V) \land \underset{y \in U^{c}}{⋀} N_{y}^{β^{X}} (V^{c}) \geq [T_{2}^{β} (X, τ) \otimes (Γ_{β} (C, τ / C))^{2} \otimes N_{x}^{β^{X}} (U)] .$

Thus $1 - N_{x}^{β^{X}} (U) + \underset{x \in V \subseteq U}{⋁} (N_{x}^{β^{X}} (V) \land \underset{y \in U^{c}}{⋀} N_{y}^{β^{X}} (V^{c}) \geq [T_{2}^{β} (X, τ) \otimes (Γ_{β} (C, τ / C))^{2}],$ i.e., $[T_{3}^{β} (X, τ)] \geq [T_{2}^{β} (X, τ) \otimes (Γ_{β} (C, τ / C))^{2}] .$

Theorem 5.7. For any fuzzifying β-topological space (X, τ), $⊨ T_{3}^{β} (X, τ) \otimes L_{β} C (X, τ) \to \forall A \forall U (U \in N_{A}^{β^{X}} \otimes Γ_{β} (A, τ / A) \to \exists V (V \subseteq U \land U \in N_{A}^{β^{X}} \land τ_{β} (V^{c}) \land Γ_{β} (V, τ / V))),$

where $U \in N_{A}^{β^{X}} : = (\forall x) (x \in A \land U \in N_{x}^{β^{X}}) .$

Proof. It is similar to the proof of Theorem 3.4 [12].

Theorem 5.8. Let (X, τ) and (Y, σ) be two fuzzifying topological space and f ∈ Y^X be surjective. Then ⊨L_βC (X, τ) ⊗ C_β (f) ⊗ O (f) → LC (Y, σ) , where O (f) : = (∀ U) ((U ∈ τ) → (f (U) ∈ σ) .

Proof. If [L_βC (X, τ) ⊗ C_β (f) ⊗ O (f)] > λ > 0, then for any x ∈ X, there exists U ⊆ X, such that $[N_{x}^{β^{X}} (U) \otimes Γ_{β} (U, τ / U) \otimes C_{β} (f) \otimes O (f)] > λ .$ Since $N_{x}^{β^{X}} (U) = \underset{x \in V \subseteq U}{⋁} τ_{β} (V),$ so there exists V′ ⊆ X such that x ∈ V′ ⊆ U and [τ_β (V′) ⊗ Γ_β (U, τ/U) ⊗ C_β (f) ⊗ O (f)] > λ . By Theorem 5.2 in [11], [Γ_β (U, τ/U) ⊗ C_β (f)] ≤ [Γ (f (U) , σ/f (U))] and $\begin{matrix} [τ (V^{'}) \otimes O (f)] = max (0, τ (V^{'}) + O (f) - 1) \\ = max (0, τ (V^{'}) \\ + \underset{V \subseteq X}{⋀} min (1, 1 - τ (V^{'}) + σ (f (V))) - 1) \\ \leq max (0, τ (V^{'}) + 1 - τ (V^{'}) + σ (f (V)) - 1) \\ = σ (f (V)) \leq N_{f (x)}^{Y} (f (V^{'})) \leq N_{f (x)}^{Y} (f (U)) . \end{matrix}$ Since f is surjective, $\begin{matrix} LC (Y, σ) = LC (f (X), σ) \\ = \underset{y \in f (x) \subseteq f (X)}{⋀} \underset{U^{'} = f (U) \subseteq f (X)}{⋁} [N_{y}^{Y} (U^{'}) \otimes [Γ (U^{'}, σ / U^{'})] \\ \geq \underset{y \in f (x) \subseteq f (X)}{⋀} [N_{f (x)}^{Y} (f (U)) \otimes [Γ (f (U), σ / f (U))] \\ \geq \underset{y \in f (x) \subseteq f (X)}{⋀} [τ (V^{'}) \otimes O (f) \otimes Γ_{β} (U, τ / U) \otimes C_{β} (f)] \\ \geq λ . \end{matrix}$

Theorem 5.9. Let (X, τ) and (Y, σ) be two fuzzifying topological space and f ∈ Y^X be surjective. Then ⊨L_βC (X, τ) ⊗ I_β (f) ⊗ O_β (f) → L_βC (Y, σ) .

Proof. By Theorem 5.3 in [11], the proof is similar to the proof of Theorem 5.8.

Theorems 5.8 and 5.9 are a generalization of the following corollary.

Corollary 5.10. Let (X, τ) and (Y, σ) be two topological space and f : (X, τ) → (Y, σ) be surjective mapping. If f is an β-continuous (resp. β-irresolute), open (resp. β-open) and X is locally β-compact, then Y is locally compact (resp. locally β-compact) space.

Theorem 5.11. Let {(X_s, τ_s) : s ∈ S} be a familyof fuzzifying topological spaces, then $⊨ L_{β} C (\prod_{s \in S} X_{s}, \prod_{s \in S} (τ_{β})_{s}) \to \forall s (s \in S \land L_{β} C (X_{s}, (τ_{β})_{s}) \land$ ∃T (T ⊂ _fS ∧ ∀ t (t ∈ S - T ∧ Γ_β (X_t, τ_t))) .

Proof. It suffices to show that $L_{β} C (\prod_{s \in S} X_{s}, \prod_{s \in S} (τ_{β})_{s}) \leq \underset{s \in S}{⋀} [L_{β} C (X_{s}, (τ_{β})_{s}) \land \underset{T \subset_{f} S}{⋁} \underset{t \in S - T}{⋀} Γ_{β} (X_{t}, τ_{t})] .$

From Theorem 5.8 and Lemma 4.5 we have for any t ∈ S, $L_{β} C (\prod_{s \in S} X_{s}, \prod_{s \in S} (τ_{β})_{s}) = [L_{β} C (\prod_{s \in S} X_{s}, \prod_{s \in S} (τ_{β})_{s}) \otimes C_{β} (p_{t}) \otimes O_{β} (p_{t})] \leq$ L_βC (X_t, τ_t).

So, $\underset{t \in S - T}{⋀} L_{β} C (X_{t}, τ_{t}) \geq L_{β} C (\prod_{s \in S} X_{s}, \prod_{s \in S} (τ_{β})_{s}) .$

By Theorem 4.7 we have $\begin{matrix} \underset{T \subset_{f} S}{⋁} \underset{t \in S - T}{⋀} Γ_{β} (X_{t}, τ_{t}) \\ \geq [\underset{U \subseteq s \in S \prod X_{s}}{⋁} Γ_{β} (U, \prod_{s \in S} (τ_{β})_{s} / U) \otimes \underset{X \subseteq s \in S \prod X_{s}}{⋁} N_{x}^{β^{X}} (U))] \\ \geq \underset{U \subseteq s \in S \prod X_{s}}{⋁} \underset{X \subseteq s \in S \prod X_{s}}{⋁} [Γ_{β} (U, \prod_{s \in S} (τ_{β})_{s} / U) \otimes N_{x}^{β^{X}} (U))] \\ \geq \underset{X \subseteq s \in S \prod X_{s}}{⋀} \underset{U \subseteq s \in S \prod X_{s}}{⋁} [Γ_{β} (U, \prod_{s \in S} (τ_{β})_{s} / U) \otimes N_{x}^{β^{X}} (U))] \\ = L_{β} C (\prod_{s \in S} X_{s}, \prod_{s \in S} (τ_{β})_{s}) . \end{matrix}$ Therefore $\begin{matrix} L_{β} C (\prod_{s \in S} X_{s}, \prod_{s \in S} (τ_{β})_{s}) \\ \leq [\underset{t \in S - T}{⋀} L_{β} C (X_{t}, τ_{t}) \land \underset{T \subset_{f} S}{⋁} \underset{t \in S - T}{⋀} Γ_{β} (X_{t}, τ_{t})] . \end{matrix}$

We can obtain the following corollary in crisp setting.

Corollary 5.12. Let {X_λ : λ ∈ Λ} be a family of nonempty topological spaces. If $\prod_{λ \in Λ} X_{λ}$ is locallyβ-compact, then each X_λ is locally β-compact and all but finitely many X_λ are β-compact.

6 Conclusion

The present paper investigates topological notions when these are planted into the framework of Ying’s fuzzifying topological spaces (in semantic method of continuous valued-logic). The main contributions of the present paper are to give characterizations of fuzzifying β-compactness. Also, we define the concept of locally β-compactness of fuzzifying topological spaces and obtain some basic properties of such spaces. There are some open questions for further study:

One obvious problem is: our results are derived in the Łukasiewicz continuous logic. It is possible to generalize them to more general logic setting, like residuated lattice-valued logic considered in [17, 18].

What is the justification for fuzzifying locally β-compactness in the setting of (2, L) topologies?

What is the justification for fuzzifying locally strong compactness in (M, L)-topologiesetc?

References

Abd El-Monsef

M.E.

, El-Deeb

S.N.

and Mahmoud

R.A.

, β-Open sets and β-continuous mappings, Bull Fac Sci Assiut Univ 12 (1983), 77–90.

Abd El-Monsef

M.E.

and Klzae

A.M.

, Some generalized forms of compactness and closedness, Delta J Sci 9 (1985), 257–269.

Abd El-Hakeim

K.M.

, Zeyada

F.M.

and Sayed

O.R.

, β-continuity and D(c, β)-continuity in fuzzifying topology, J Fuzzy Math 7(3) (1999), 547–558.

Allam

A.A.

and Abd El-Hakeim

K.M.

, On β-compact spaces, Bull Calcutta Math Soc 81 (1989), 179–182.

Andrijevic

, Semi-preopen sets, Mat Vesnik 38 (1986), 24–32.

Dontchev

and Przemiski

, On the various decompostions of continuous and some weakly continuous functions, Acta Math Hungar 71(1-2) (1996), 109–120.

Kelley

J.L.

, General Topology, Van Nostrand, New York, 1955.

Rosser

J.B.

and Turquette

A.R.

, Many-Valued Logics, North-Holland, Amsterdam, 1952.

Sayed

O.R.

, On Fuzzifying Topological Spaces, Ph.D. Thesis, Assiut University, Egypt, 2002.

10.

Sayed

O.R.

, β-separation axioms based on lukasiewicz logic, Engineering Science Letters 1(1) (2012), 1–24.

11.

Sayed

O.R.

and Abd-Allah

M.A.

, Fuzzy β-irresolute functions and fuzzy β-compact spaces in fuzzifying topology, Iran J Sci Technol A 30(3) (2006), 297–314.

12.

Shen

, Locally compactness in fuzzifying topology, J Fuzzy Math 2(4) (1994), 695–711.

13.

Ying

M.S.

, A new approach for fuzzy topology (I), Fuzzy Sets and Systems 39 (1991), 303–321.

14.

Ying

M.S.

, A new approach for fuzzy topology (II), Fuzzy Sets and Systems 47 (1992), 221–223.

15.

Ying

M.S.

, A new approach for fuzzy topology (III), Fuzzy Sets and Systems 55 (1993), 193–207.

16.

Ying

M.S.

, Compactness in fuzzifying topology, Fuzzy Sets and Systems 55 (1993), 79–92.

17.

Ying

M.S.

, Fuzzifying topology based on complete residuated lattice-valued logic (I), Fuzzy Sets and Systems 56 (1993), 337–373.

18.

Ying

M.S.

, Fuzzy topology based on residuated lattice-valued logic, Acta Mathematica Sinica 17 (2001), 89–102.