Measures of compactness in ( L,M )-fuzzy Q-convergence spaces

Abstract

In this paper, the concept of compactness measures of (L, M)-fuzzy Q-convergence spaces is introduced. It is used to define compactness measures of (L, M)-fuzzy topological spaces and pointwise (L, M)-fuzzy quasi-uniform spaces. It is shown that the concept of compactness measures satisfies the lattice-valued Tychonoff theorem in the framework of (L, M)-fuzzy Q-convergence spaces and (L, M)-fuzzy topological spaces. Also, it is proved that the compactness measures of a pointwise (L, M)-fuzzy quasi-uniform space is equal to the compactness measures of its induced (L, M)-fuzzy topological space.

Keywords

Compactness

1 Introduction

Compactness is an important topological property in topological spaces. The concept of compactness of a topological space is usually defined in terms of open covers [10]. Actually, compactness can be also characterized by ultrafilter convergence. Based on this fact, compactness is generalized to axiomatic filter convergence spaces [22]. Concretely, an axiomatic filter convergence space is called compact provided that every ultrafilter converges.

With the development of fuzzy topology, compactness is defined in different ways in the framework of fuzzy topological spaces. Gantner et al. [3] introduced α-compactness for an L-topological space. For an L-subset, Liu [13] introduced Q-compactness, Zhao [29] introduced N-compactness, and Shi [24] introduced S^∗-compactness. Recently, Shi [25] introduced a new approach to compactness of an L-topological space by an inequality. In 1985, Šostak [26] introduced a definition of compactness degrees for a fuzzy set by means of the level I-topology. Later, Eş and Çoker [1] defined the degrees of almost compactness, near compactness, countable compactness, light compactness and strong compactness for a fuzzy set. Afterwards, Shi [23] introduced new definitions of countable compactness and the Lindelöf property in L-topological spaces, which do not rely on the structure of the basis lattice L. In 2010, Li and Shi [12] introduced the compactness degrees of L-fuzzy topological spaces. Naturally, this is a motivation to consider compactness in fuzzy convergence spaces. For stratified L-generalized convergence spaces [4], Jäger investigated its compactness [5, 7] and compactification [6 , 9]. In the situation of (L, M)-fuzzy convergence spaces [16], Pang and Shi [17] introduced the concept of compactness degrees of an (L, M)-fuzzy convergence space and applied this method to define compactness degrees of an (L, M)-fuzzy topological space. Besides (L, M)-fuzzy convergence structures, there are some other kinds of fuzzy convergence structures in the framework of (L, M)-fuzzy topological spaces, such as (L, M)-fuzzy Q-convergence spaces [14, 15 , 19], enriched (L, M)-fuzzy convergence spaces [18, 21, 18, 21] and L-fuzzy N-convergence spaces [20]. Inspired by this, we consider compactness theory in (L, M)-fuzzy Q-convergence spaces.

Not only in fuzzy topological spaces, but also in fuzzy convergence spaces, topological properties are endowed with some degrees or some measures. This requirement seems to be necessary in the development of fuzzy topology. In this paper, we will endow (L, M)-fuzzy Q-convergence spaces with some compactness measures and use it to define compactness measures of (L, M)-fuzzy topological spaces and pointwise (L, M)-fuzzy quasi-uniform spaces.

2 Preliminaries

Throughout this paper, both L and M denote completely distributive lattices and ^′ is an order-reversing involution on L. The smallest element and the largest element in L (M) are denoted by ⊥_L (⊥_M) and ⊤_L (⊤_M), respectively. For a, b ∈ L, we say that a is wedge below b in L, in symbols a ≺ b, if for every subset D ⊆ L, ⋁D ⩾ b implies d ⩾ a for some d ∈ D. A complete lattice L is completely distributive if and only if b = ⋁ {a ∈ L ∣ a ≺ b} for each b ∈ L. An element a in L is called coprime if a ⩽ b ∨ c implies a ⩽ b or a ⩽ c. The set of nonzero coprime elements in L (M) is denoted by J (L) (J (M)), respectively. We define a residual implication operation → : L × L → L as the right adjoint for the meet operation ∧ by $a \to b = ⋁ {c \in L ∣ a \land c ⩽ b} .$

This operation plays a particular role in the sequel.

For a nonempty set X, L^X denotes the set of all L-subsets on X. The smallest element and the largest element in L^X are denoted by $⊥_{L}^{X}$ and $⊤_{L}^{X}$ , respectively. L^X is also a completely distributive De Morgan algebra when it inherits the structure of the lattice L in a natural way, by defining ⋁, ⋀ , ⩽ and ^′ pointwisely. For each x ∈ X and a ∈ L, the L-subset x_a, defined by x_a (y) = a if y = x, and x_a (y) = ⊥ _L if y ≠ x, is called a fuzzy point. The set of nonzero coprime elements in L^X is denoted by J (L^X). It is easy to see that J (L^X) = {x_λ ∣ x ∈ X, λ ∈ J (L)}. We say that a fuzzy point x_λ ∈ J (L^X) quasi-coincides with A, denoted by $x_{λ} \hat{q} A$ , if λnotleqslantA′ (x) or equivalently x_λnotleqslantA′ . The relation “does not quasi-coincide with” is denoted by $\neg \hat{q} .$ Let φ : X ⟶ Y be a mapping. Define φ^→ : L^X ⟶ L^Y and φ^← : L^Y ⟶ L^X by φ^→ (A) (y) = ⋁ _φ(x)=yA (x) for A ∈ L^X and y ∈ Y, and φ^← (B) = B ∘ φ for B ∈ L^Y, respectively.

Definition 2.1. (Yao [27]) A mapping $F : L^{X} ⟶ M$ is called an (L, M)-fuzzy filter on X if it satisfies:

$F (⊥_{L}^{X}) = ⊥_{M}, F (⊤_{L}^{X}) = ⊤_{M}$ ;

$F (A \land B) = F (A) \land F (B)$ .

The family of all (L, M)-fuzzy filters on X is denoted by $F_{LM} (X) .$ On the set $F_{LM} (X)$ of all (L, M)-fuzzy filters on X, we define an order by $F ⩽ G$ if $F (A) ⩽ G (A)$ for all A ∈ L^X. Then $(F_{LM} (X), ⩽)$ is a partially ordered set.

Example 2.2. (Pang [16]) For each x_λ ∈ J (L^X) , $\hat{q} (x_{λ}) : L^{X} \to M$ we define $\hat{q} (x_{λ}) : L^{X} ⟶ M$ as follows: $\begin{matrix} \forall A \in L^{X}, \hat{q} (x_{λ}) (A) = {\begin{matrix} ⊤_{M}, & x_{λ} \hat{q} A, \\ ⊥_{M}, & x_{λ} \neg \hat{q} A . \end{matrix} \end{matrix}$

Then $\hat{q} (x_{λ})$ is an (L, M)-fuzzy filter.

Definition 2.3. (Yao [27]) Let $F \in F_{LM} (X)$ and φ : X ⟶ Y be a mapping. Then $φ^{\Rightarrow} (F) : L^{Y} ⟶ M, A ⟼ F (φ^{\leftarrow} (A))$ is an (L, M)-fuzzy filter on Y and is called the image of $F$ under φ.

Definition 2.4. (Pang [14] for L = M) An (L, M)-fuzzy Q-convergence structure on X is a mapping $qc : F_{LM} (X) ⟶ L^{X}$ which satisfies:

$x_{λ} ⩽ qc (\hat{q} (x_{λ}))$ ;

$F ⩽ G$ implies $qc (F) ⩽ qc (G)$ .

For an (L, M)-fuzzy Q-convergence structure qc on X, the pair (X, qc) is called an (L, M)-fuzzy Q-convergence space.

A mapping φ : (X, qc_X) ⟶ (Y, qc_Y) between (L, M)-fuzzy Q-convergence spaces is called continuous provided that $x_{λ} ⩽ {qc}_{X} (F) implies φ (x)_{λ} ⩽ {qc}_{Y} (φ^{\Rightarrow} (F))$ for each $F \in F_{LM} (X)$ and x_λ ∈ J (L^X). That is, $φ^{\to} ({qc}_{X} (F)) ⩽ {qc}_{Y} (φ^{\Rightarrow} (F))$ for each $F \in F_{LM} (X)$ . The category of (L, M)-fuzzy Q-convergence spaces with continuous mappings as morphisms will be denoted by (L, M)-FQC.

Definition 2.5. (Kubiak [11] and Šostak [26]) An (L, M)-fuzzy topology on X is a mapping τ : L^X ⟶ M which satisfies:

$τ (⊥_{L}^{X}) = τ (⊤_{L}^{X}) = ⊤_{M}$ ;

τ (A ∧ B) ⩾ τ (A) ∧ τ (B);

τ (⋁ _j∈JA_j) ⩾ ⋀ _j∈Jτ (A_j).

For an (L, M)-fuzzy topology τ on X, the pair (X, τ) is called an (L, M)-fuzzy topological space.

A continuous mapping between (L, M)-fuzzy topological spaces (X, τ_X) and (Y, τ_Y) is a mapping φ : X ⟶ Y such that τ_X (φ^← (A)) ⩾ τ_Y (A) for each A ∈ L^Y. The category of (L, M)-fuzzy topological spaces with continuous mappings as morphisms will be denoted by (L, M)-FTop.

Definition 2.6. (Fang [2]) An (L, M)-fuzzy quasi-coincident neighborhood system on X is defined to be a set $Q = {Q_{x_{λ}} ∣ x_{λ} \in J (L^{X})}$ of mappings $Q_{x_{λ}} : L^{X} ⟶ M$ satisfying the following conditions:

$Q_{x_{λ}} (⊤_{L}^{X}) = ⊤_{M}, Q_{x_{λ}} (⊥_{L}^{X}) = ⊥_{M}$ ;

$Q_{x_{λ}} (A) \neq ⊥_{M} \Rightarrow x_{λ} \hat{q} A$ ;

$Q_{x_{λ}} (A \land B) = Q_{x_{λ}} (A) \land Q_{x_{λ}} (B)$ .

The pair $(X, Q)$ is called an (L, M)-fuzzy quasi-coincident neighborhood space, and it will be called topological if it satisfies moreover,

$Q_{x_{λ}} (A) = ⋁_{x_{λ} \hat{q} B ⩽ A} ⋀_{y_{μ} \hat{q} B} Q_{y_{μ}} (B)$ .

A continuous mapping between two (L, M)-fuzzy quasi-coincident neighborhood spaces $(X, Q^{X})$ and $(Y, Q^{Y})$ is a mapping φ : X ⟶ Y such that $Q_{x_{λ}}^{M} (φ^{\leftarrow} (A)) ⩾ Q_{φ (x)_{λ}}^{Y} (A)$ for each x_λ ∈ J (L^X) and A ∈ L^Y. The category of topological (L, M)-fuzzy quasi-coincident neighborhood spaces with continuous mappings as morphisms will be denoted by (L, M)-FTQN.

Proposition 2.7. (Fang [2]) Let $Q = {Q_{x_{λ}} ∣ x_{λ} \in J (L^{X})}$ be an (L, M)-fuzzy quasi-coincident neighborhood system on X and define $τ^{Q} : L^{X} ⟶ M$ by $\forall A \in L^{X}, τ^{Q} (A) = ⋀_{x_{λ} \hat{q} A} Q_{x_{λ}} (A) .$

Then $τ^{Q}$ is an (L, M)-fuzzy topology on X.

Proposition 2.8. (Fang [2]) Let τ be an (L, M)-fuzzy topology on X and define $Q_{x_{λ}}^{τ} : L^{X} ⟶ M$ by $Q_{x_{λ}}^{τ} (A) = ⋁_{x_{λ} \hat{q} B ⩽ A} τ (B) .$

Then $Q^{τ} = {Q_{x_{λ}}^{τ} ∣ x_{λ} \in J (L^{X})}$ is a topological (L, M)-fuzzy quasi-coincident neighborhood system on X. Moreover, $τ (A) = ⋀_{x_{λ} \hat{q} A} Q_{x_{λ}}^{τ} (A)$ .

Theorem 2.9. (Fang [2]) (L, M)-FTop is isomorphic to (L, M)-FTQN.

3 Measures of compactness in (L, M)-fuzzy Q-convergence spaces

In order to define compactness measures of (L, M)-fuzzy Q-convergence spaces, we first recall some conclusions with respect to (L, M)-fuzzy ultrafilters in [17].

Lemma 3.1. (Pang [17]) The partially ordered set $(F_{LM} (X), ⩽)$ has maximal elements.

Definition 3.2. (Pang [17]) An (L, M)-fuzzy filter $F^{u}$ is called an (L, M)-fuzzy ultrafilter provided that it is a maximal element of $(F_{LM} (X), ⩽)$ . The set of all (L, M)-fuzzy ultrafilters is denoted by $F_{LM}^{u} (X)$ .

Lemma 3.3. (Pang [17]) For every (L, M)-fuzzy filter $F$ , there exists an (L, M)-fuzzy ultrafilter $F^{*}$ with $F ⩽ F^{*}$ .

Given a mapping φ : X ⟶ Y and an (L, M)-filter $G$ on Y, the mapping $φ^{\leftarrow} (G) : L^{X} ⟶ M$ defined by $\forall A \in L^{X}, φ^{\leftarrow} (G) (A) = ⋁_{φ^{\leftarrow} (B) ⩽ A} G (B)$ is an (L, M)-fuzzy filter on X if and only if $G (B) = ⊥_{M}$ whenever $φ^{\leftarrow} (B) = ⊥_{L}^{X}$ for all B ∈ L^Y. In the case $φ^{\leftarrow} (G) \in F_{LM} (X)$ , it is called the inverse image of $G$ under φ. In particular, $φ^{\leftarrow} (G) \in F_{LM} (X)$ whenever φ is a surjective mapping.

Lemma 3.4. (Pang [17]) Let φ : X ⟶ Y be a mapping, and let $F \in F_{LM} (X)$ and $G \in F_{LM} (Y)$ . Then

$φ^{\Rightarrow} (φ^{\leftarrow} (G)) ⩾ G .$

$φ^{\Rightarrow} (φ^{\leftarrow} (G)) = G$ provided that φ is surjective.

Lemma 3.5. (Pang [17]) If φ : X ⟶ Y is a mapping and $F^{u} \in F_{LM}^{u} (X)$ , then $φ^{\Rightarrow} (F^{u}) \in F_{LM}^{u} (Y)$ .

Now we can define the compactness measures of an (L, M)-fuzzy Q-convergence space by means of (L, M)-fuzzy ultrafilters.

Definition 3.6. For an (L, M)-fuzzy Q-convergence space (X, qc), define the measure Comp (X, qc) to which (X, qc) is compact by $\begin{matrix} Comp (X, qc) \\ = ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to qc (F) (x)) . \end{matrix}$

Furthermore, (X, qc) is called compact provided that Comp (X, qc) = ⊤ _M.

Whenever L = M = {0, 1}, the equality can be understood as “a convergence space is compact iff every ultrafilter converges”. Hence, this definition can be interpreted as to many-valued logical truth-value of the sentence “a convergence space is compact iff every ultrafilter converges”.

The above notion allows us to talk on compactness of an (L, M)-fuzzy Q-convergence space in some measure even if (X, qc) is not compact. The compactness measure Comp(X,qc) of (X, qc) is a natural characterization for which (X, qc) is a compact (L, M)-fuzzy Q-convergence space. In the sequel, we will show that the compactness measures of an (L, M)-fuzzy Q-convergence space naturally suggests many-valued logical extensions of the properties of compactness in generalized convergence spaces to (L, M)-fuzzy Q-convergence spaces.

Proposition 3.7. If φ : (X, qc_X) ⟶ (Y, qc_Y) is continuous and surjective, then

$C o m p (X, q c_{X}) ⩽ C o m p (Y, q c_{Y}) .$

Proof. It suffices to show that $\begin{matrix} ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}_{X} (F) (x)) \\ ⩽ ⋀_{G^{u} \in F_{LM}^{u} (Y)} ⋀_{μ \in J (L)} ⋁_{y \in Y} (μ \to {qc}_{Y} (G) (y)) . \end{matrix}$

Take any $G^{u} \in F_{LM}^{u} (Y)$ . Since φ : X ⟶ Y is surjective, we know $φ^{\leftarrow} (G^{u}) \in F_{LM} (X) .$ By Lemma 3.3, we choose an (L, M)-fuzzy ultrafilter $F_{0}^{u}$ such that $φ^{\leftarrow} (G^{u}) ⩽ F_{0}^{u}$ . Then by Lemma 3.4, we have $G^{u} = φ^{\Rightarrow} (φ^{\leftarrow} (G^{u})) ⩽ φ^{\Rightarrow} (F_{0}^{u})$ . Since $G^{u}$ is maximal, it follows that $G^{u} = φ^{\Rightarrow} (F_{0}^{u})$ . Hence by the continuity of φ, we have $\begin{matrix} ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}_{X} (F) (x)) \\ ⩽ ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}_{X} (F_{0}^{u}) (x)) \\ ⩽ ⋀_{λ \in J (L)} ⋁_{φ (x) \in Y} (λ \to {qc}_{Y} (φ^{\Rightarrow} (F_{0}^{u})) (φ (x))) \\ ⩽ ⋀_{λ \in J (L)} ⋁_{y \in Y} (λ \to {qc}_{Y} (G^{u}) (y)) \\ = ⋀_{μ \in J (L)} ⋁_{y \in Y} (μ \to {qc}_{Y} (G^{u}) (y)) . \end{matrix}$

By the arbitrariness of $G^{u}$ , we obtain $\begin{matrix} ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}_{X} (F) (x)) \\ ⩽ & ⋀_{G^{u} \in F_{LM}^{u} (Y)} ⋀_{μ \in J (L)} ⋁_{y \in Y} (μ \to {qc}_{Y} (G^{u}) (y)), \end{matrix}$ as desired. □

In [14], Pang and Fang provided the concrete form of initial structures with respect to L-fuzzy Q-convergence spaces. Since (L, M)-fuzzy Q-convergence spaces are generalizations of L-fuzzy Q-convergence spaces, we can easily give the initial structure of (L, M)-fuzzy Q-convergence spaces. As a special case of initial structures, the product of (L, M)-fuzzy Q-convergence structures can be presented as follows.

Definition 3.8. Let {(X_j, qc_j) ∣ j ∈ J} be a family of (L, M)-fuzzy Q-convergence spaces, $X = \prod_{j \in J} X_{j}$ and p_k : X ⟶ X_k be projection mapping. Define $\prod_{j \in J} {qc}_{j} : F_{LM} (X) ⟶ L^{X}$ by $\forall F \in F_{LM} (X), \prod_{j \in J} {qc}_{j} (F) = ⋀_{j \in J} φ_{j}^{\leftarrow} ({qc}_{j} (φ_{j}^{\Rightarrow} (F))) .$

Then $(\prod_{j \in J} X_{j}, \prod_{j \in J} {qc}_{j})$ is called the product space of {(X_j, qc_j) ∣ j ∈ J}.

Based on the definition of product space of (L, M)-fuzzy Q-convergence spaces, we give the lattice-valued Tychonoff theorem in the framework of (L, M)-fuzzy Q-convergence spaces.

Theorem 3.9. Let {(X_j, qc_j) ∣ j ∈ J} be a family of (L, M)-fuzzy Q-convergence spaces. Then

$C o m p (\prod_{j \in J} X_{j}, \prod_{j \in J} q c_{j}) = \underset{j \in J}{\land} C o m p (X_{j}, q c_{j}) .$

Proof. For convenience, put $X = \prod_{j \in J} X_{j}$ and ${qc}_{X} = \prod_{j \in J} {qc}_{j}$ . By the definition of product of (L, M)-fuzzy Q-convergence spaces, we know that the projection mapping p_k : (X, qc_X) ⟶ (X_k, qc_k) is surjective and continuous for all k ∈ J. Then by Proposition 3.6, it follows that $C o m p (X, q c_{X}) ⩽ C o m p (Y, q c_{Y}) .$ for all j ∈ J. This implies that

$C o m p (\prod_{j \in J} X_{j}, q c_{X}) ⩽ \underset{j \in J}{\land} C o m p (X_{j}, q c_{j}) .$

Next we prove the inverse, i.e.,

$C o m p (\prod_{j \in J} X_{j}, \prod_{j \in J} q c_{j}) \geq \underset{j \in J}{\land} C o m p (X_{j}, q c_{j}) .$

Take any α ∈ J (M) such that $\begin{matrix} α & ≺ & ⋀_{j \in J} Comp (X_{j}, {qc}_{j}) \\ = & ⋀_{j \in J} ⋀_{F_{j}^{u} \in F_{LM}^{u} (X_{j})} ⋀_{λ \in J (L)} ⋁_{x^{j} \in X_{j}} (λ \to {qc}_{j} (F_{j}^{u}) (x^{j})) . \end{matrix}$

Then for each j ∈ J, $F_{j}^{u} \in F_{LM}^{u} (X_{j})$ and λ ∈ J (L), there exists x^j ∈ X_j such that $α ⩽ λ \to {qc}_{j} (F_{j}^{u}) (x^{j}) .$

Next we will show that $α ⩽ ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}_{X} (F^{u}) (x)) .$ By Lemma 3.5, for each $F^{u} \in F_{LM}^{u} (X)$ , we have $p_{j}^{\Rightarrow} (F^{u}) \in F_{LM}^{u} (X_{j})$ for all j ∈ J. Further for each λ ∈ J (L), there exists x^j ∈ X_j such that $α ⩽ λ \to {qc}_{j} (p_{j}^{\Rightarrow} (F^{u})) (x^{j}) .$ Put x = (x^j) ∈ X. Then it follows that $\begin{matrix} α & ⩽ & ⋀_{j \in J} (λ \to {qc}_{j} (p_{j}^{\Rightarrow} (F^{u})) (x^{j})) \\ ⩽ & ⋁_{x \in X} ⋀_{j \in J} (λ \to {qc}_{j} (p_{j}^{\Rightarrow} (F^{u})) (p_{j} (x)) \\ = & ⋁_{x \in X} (λ \to ⋀_{j \in J} {qc}_{j} (p_{j}^{\Rightarrow} (F^{u})) (p_{j} (x)) \\ = & ⋁_{x \in X} (λ \to {qc}_{X} (F^{u}) (x)) . \end{matrix}$

By the arbitrariness of λ and $F^{u}$ , we obtain $α ⩽ ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}_{X} (F^{u}) (x)) .$

This proves that

$C o m p (\prod_{j \in J} X_{j}, \prod_{j \in J} q c_{j}) ⩾ \underset{j \in J}{\land} C o m p (X_{j}, q c_{j}),$

as desired. □

4 Measures of compactness in (L, M)-fuzzy topological spaces

In this section, we define the compactness measures of an (L, M)-fuzzy topological space by convergence of (L, M)-fuzzy ultrafilters. Also, we give the lattice-valued representations of the Tychonoff theorem in (L, M)-fuzzy topological spaces.

Firstly, we study the relations between (L, M)-fuzzy Q-convergence spaces and (L, M)-fuzzy topological spaces from a categorical aspect.

Proposition 4.1. (1) Let (X, qc) be an (L, M)-fuzzy Q-convergence space and define τ^qc : L^X ⟶ M by $\forall A \in L^{X}, τ^{qc} (A) = ⋀_{x_{λ} \hat{q} A} F_{x_{λ}}^{qc} (A),$ where $F_{x_{λ}}^{qc} (A) = ⋀_{x_{λ} ⩽ qc (F)} F (A)$ for each A ∈ L^X. Then τ^qc is an (L, M)-fuzzy topology on X.

(2) If φ : (X, qc_X) ⟶ (Y, qc_Y) is continuous, then so is φ : (X, τ^{qc
_X}) ⟶ (Y, τ^{qc
_Y}).

Proof. (1) It suffices to verify that τ^qc satisfies (LFT1)–(LFT3). In fact,

(LFT1) By the definition of τ^qc, we obatin $τ^{qc} (⊥_{L}^{X}) = ⋀_{x_{λ} \hat{q} ⊥_{L}^{X}} F_{x_{λ}}^{qc} (⊥_{L}^{X}) = ⋀ \emptyset = ⊤_{M}$ and $τ^{qc} (⊤_{L}^{X}) = ⋀_{x_{λ} \hat{q} ⊤_{L}^{X}} F_{x_{λ}}^{qc} (⊤_{L}^{X}) = ⋀ ⊤_{M} = ⊤_{M} .$

(LFT2) Take any A, B ∈ L^X. Then

\begin{matrix} τ^{qc} (A) \land τ^{qc} (B) \\ = ⋀_{x_{λ} \hat{q} A} F_{x_{λ}}^{qc} (A) \land ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B) \\ = ⋀_{x_{λ} \hat{q} A} ⋀_{y_{μ} \hat{q} B} (F_{x_{λ}}^{qc} (A) \land F_{y_{μ}}^{qc} (B)) \\ ⩽ ⋀_{x_{λ} \hat{q} (A \land B)} ⋀_{y_{μ} \hat{q} (A \land B)} (F_{x_{λ}}^{qc} (A) \land F_{y_{μ}}^{qc} (B)) \\ ⩽ ⋀_{x_{λ} \hat{q} (A \land B)} (F_{x_{λ}}^{qc} (A) \land F_{x_{λ}}^{qc} (B)) \\ = ⋀_{x_{λ} \hat{q} (A \land B)} F_{x_{λ}}^{qc} (A \land B) \\ = τ^{qc} (A \land B) . \end{matrix}

(LFT3) For {A_j ∣ j ∈ J} ⊆ L^X, it follows that $\begin{matrix} τ^{qc} (⋁_{j \in J} A_{j}) & = & ⋀_{x_{λ} \hat{q} ⋁_{j \in J} A_{j}} F_{x_{λ}}^{qc} (⋁_{j \in J} A_{j}) \\ = & ⋀_{j \in J} ⋀_{x_{λ} \hat{q} A_{j}} F_{x_{λ}}^{qc} (⋁_{j \in J} A_{j}) \\ ⩾ & ⋀_{j \in J} ⋀_{x_{λ} \hat{q} A_{j}} F_{x_{λ}}^{qc} (A_{j}) \\ = & ⋀_{j \in J} τ^{qc} (A_{j}) . \end{matrix}$

This shows that τ^qc is an (L, M)-fuzzy topology on X.

(2) Since φ : (X, qc_X) ⟶ (X, qc_X) is continuous, it follows that $\forall F \in F_{LM} (X), φ^{\to} ({qc}_{X} (F)) ⩽ {qc}_{Y} (φ^{\Rightarrow} (F)) .$

Then for each B ∈ L^Y, we have $\begin{matrix} τ^{{qc}_{X}} (φ^{\leftarrow} (B)) \\ = ⋀_{x_{λ} \hat{q} φ^{\leftarrow} (B)} F_{x_{λ}}^{{qc}_{X}} (φ^{\leftarrow} (B)) \\ = ⋀_{x_{λ} \hat{q} φ^{\leftarrow} (B)} φ^{\Rightarrow} (F_{x_{λ}}^{{qc}_{X}}) (B) \\ = ⋀_{φ (x)_{λ} \hat{q} B} ⋀_{x_{λ} ⩽ {qc}_{X} (F)} φ^{\Rightarrow} (F) (B) \\ ⩾ ⋀_{φ (x)_{λ} \hat{q} B} ⋀_{φ (x)_{λ} ⩽ {qc}_{Y} (φ^{\Rightarrow} (F))} φ^{\Rightarrow} (F) (B) \\ ⩾ ⋀_{y_{μ} \hat{q} B} ⋀_{y_{μ} ⩽ {qc}_{Y} (G)} G (B) \\ = ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{{qc}_{Y}} (B) \\ = τ^{{qc}_{Y}} (B) . \end{matrix}$

Thus, φ : (X, τ^{qc
_X}) ⟶ (Y, τ^{qc
_Y}) is continuous. □

Hence, we obtain a functor as follows: $\begin{matrix} 𝕋_{𝔽} : {\begin{matrix} (L, M) - FQC & ⟶ & (L, M) - FTop, \\ (X, qc) & ⟼ & (X, τ^{qc}), \\ φ & ⟼ & φ . \end{matrix} \end{matrix}$

Lemma 4.2. Let (X, τ) be an (L, M)-fuzzy topological space and define ${qc}^{τ} : F_{LM} (X) ⟶ L^{X}$ as follows: $\forall F \in F_{LM} (X), {qc}^{τ} (F) = ⋁_{Q_{y_{μ}}^{τ} ⩽ F} y_{μ} .$

Then for each x_λ ∈ J (L^X) and $F \in F_{LM} (X)$ , $x_{λ} ⩽ {qc}^{τ} (F) if and only if Q_{x_{λ}}^{τ} ⩽ F .$

Proof. The sufficiency is obvious. It suffices to show the necessity. Suppose that $x_{λ} ⩽ {qc}^{τ} (F)$ . By Proposition 2.8, take any A ∈ L^X and α ∈ J (M) such that $α ≺ Q_{x_{λ}}^{τ} (A) = ⋁_{x_{λ} \hat{q} B ⩽ A} ⋀_{y_{μ} \hat{q} B} Q_{y_{μ}}^{τ} (B) .$

Then there exists B_α ∈ L^X such that $x_{λ} \hat{q} B_{α} ⩽ A$ and for each $y_{μ} \hat{q} B_{α}$ , it follows that $α ⩽ Q_{y_{μ}}^{τ} (B_{α})$ . Then we have ${qc}^{τ} (F) \geq_{λ} ≰ B_{α}^{'}$ . By the definition of qc^τ, there exists z_ν ∈ J (L^X) such that $Q_{z_{ν}}^{τ} ⩽ F$ and $z_{ν} ≰ B_{α}^{'}$ . Thus, $α ⩽ Q_{z_{ν}}^{τ} (B_{α}) ⩽ F (B_{α}) ⩽ F (A) .$

By the arbitrariness of α, we get $F (A) ⩾ {α \in J (M) ∣ α ≺ Q_{x_{λ}}^{τ} (A)} = Q_{x_{λ}}^{τ} (A),$ as desired. □

Proposition 4.3. (1) Let (X, τ) be an (L, M)-fuzzy topological space. Then qc^τ is an (L, M)-fuzzy Q-convergence structure on X.

(2) If φ : (X, τ_X) ⟶ (Y, τ_Y) is continuous, then so is φ : (X, qc^{τ
_X}) ⟶ (Y, qc^{τ
_Y}).

Proof. (1) It is enough to show that qc^τ satisfies (LFQC1) and (LFQC2).

(LFQC1) Take any x_λ ∈ J (L^X). Then $Q_{x_{λ}}^{τ} \in F_{LM} (X)$ and $Q_{x_{λ}}^{τ} ⩽ \hat{q} (x_{λ})$ . Further, it follows that ${qc}^{τ} (\hat{q} (x_{λ})) = ⋁_{Q_{y_{μ}}^{τ} ⩽ \hat{q} (x_{λ})} y_{μ} ⩾ x_{λ} .$

(LFQC2) Obviously.

(2) Since φ : (X, τ_X) ⟶ (Y, τ_Y) is continuous, by Theorem 2.9, it follows that $φ : (X, Q^{τ_{X}}) ⟶ (Y, Q^{τ_{Y}})$ is continuous, that is, $\forall x_{λ} \in J (L^{X}), φ^{\Rightarrow} (Q_{x_{λ}}^{τ_{X}}) ⩾ Q_{φ (x)_{λ}}^{τ_{Y}} .$

Then for each $F \in F_{LM} (X)$ , we have $\begin{matrix} x_{λ} ⩽ {qc}^{τ_{X}} (F) \\ \Leftrightarrow & Q_{x_{λ}}^{τ_{X}} ⩽ F (by Lemma 4.2) \\ \Rightarrow & φ^{\Rightarrow} (Q_{x_{λ}}^{τ_{X}}) ⩽ φ^{\Rightarrow} (F) \\ \Rightarrow & Q_{φ (x)_{λ}}^{τ_{Y}} ⩽ φ^{\Rightarrow} (F) \\ \Leftrightarrow & φ (x)_{λ} ⩽ {qc}^{τ_{Y}} (φ^{\Rightarrow} (F)) (by Lemma 4.2) \\ \Leftrightarrow & x_{λ} ⩽ φ^{\leftarrow} ({qc}^{τ_{Y}} (φ^{\Rightarrow} (F))) . \end{matrix}$

By the arbitrariness of x_λ, we obtain ${qc}^{τ_{X}} (F) ⩽ φ^{\leftarrow} ({qc}^{τ_{Y}} (φ^{\Rightarrow} (F)))$ . This shows φ : (X, qc^{τ
_X}) ⟶ (Y, qc^{τ
_Y}) is continuous. □

Thus, we obtain another functor as follows: $\begin{matrix} 𝔽_{𝕋} : {\begin{matrix} (L, M) - FTop & ⟶ & (L, M) - FC, \\ (X, τ) & ⟼ & (X, c^{τ}), \\ φ & ⟼ & φ . \end{matrix} \end{matrix}$

Theorem 4.4. $(𝕋_{𝔽}, 𝔽_{𝕋})$ is a pair of adjoint functors between (L, M)-FQC and (L, M)-FTop.

Proof. It suffices to prove that $𝕋_{𝔽} \circ 𝔽_{𝕋} = 𝕀_{(L, M) - FTop}$ and $𝔽_{𝕋} \circ 𝕋_{𝔽} ⩾ 𝕀_{(L, M) - FC}$ . That is to say, for an (L, M)-fuzzy Q-convergence structure qc on X and an (L, M)-fuzzy topology τ on X, it follows that qc^{τ
^qc} ⩾ qc and τ^{qc
^τ} = τ.

On one hand, take any x_λ ∈ J (L^X) and A ∈ L^X. It follows that $Q_{x_{λ}}^{τ^{qc}} (A) = ⋁_{x_{λ} \hat{q} B ⩽ A} ⋀_{y_{μ} \hat{q} B} F_{y_{μ}}^{qc} (B) ⩽ F_{x_{λ}}^{qc} (A) .$

Then $Q_{x_{λ}}^{τ^{qc}} ⩽ F_{x_{λ}}^{qc}$ . Further, take any $F \in F_{LM} (X)$ . Then $\begin{matrix} x_{λ} ⩽ qc (F) & \Rightarrow & F_{x_{λ}}^{qc} ⩽ F \\ \Rightarrow & Q_{x_{λ}}^{τ^{qc}} ⩽ F \\ \Leftrightarrow & x_{λ} ⩽ {qc}^{τ^{qc}} (F) . (by Lemma 4.2) \end{matrix}$

By the arbitrariness of x_λ, we obtain qc ⩽ qc^{τ
^qc}.

On the other hand, in order to prove τ^{qc
^τ} = τ, by Propositions 2.8 and 4.1, it suffices to show that $Q_{x_{λ}}^{τ} = F_{x_{λ}}^{{qc}^{τ}}$ for each x_λ ∈ J (L^X). By Lemma 4.2, for each A ∈ L^X, we have $F_{x_{λ}}^{{qc}^{τ}} (A) = ⋀_{x_{λ} ⩽ {qc}^{τ} (F)} F (A) = ⋀_{Q_{x_{λ}}^{τ} ⩽ F} F (A) = Q_{x_{λ}}^{τ} (A),$ as desired. □

Next we give the main concept of this section and study its properties.

Definition 4.5. For an (L, M)-fuzzy topological space (X, τ), define the measure Comp(X, τ) to which (X, τ) is compact by

\begin{matrix} Comp (X, τ) \\ = ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}^{τ} (F^{u}) (x)) . \end{matrix}

Proposition 4.6. Let (X, qc) be an (L, M)-fuzzy Q-convergence space. Then

$C o m p (X, q c) ⩽ C o m p (X, τ^{q c}) .$

Proof. By Theorem 4.4, we know that $𝔽_{𝕋} \circ 𝕋_{𝔽} ⩾ 𝕀_{(L, M) - FC}$ . Hence for each $F \in F_{LM} (X)$ , it follows that ${qc}^{τ^{qc}} (F) ⩾ qc (F)$ . Then by Definition 4.5, we have $\begin{matrix} Comp (X, τ^{qc}) \\ = ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}^{τ^{qc}} (F^{u}) (x)) \\ ⩾ ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to qc (F^{u}) (x)) \\ = Comp (X, qc) . □ \end{matrix}$

In general topology, for a surjective and continuous mapping φ : (X, T_X) ⟶ (Y, T_Y) w.r.t. two topological spaces (X, T_X) and (Y, T_Y), the compactness of (X, T_X) implies the compactness of (Y, T_Y). Next we give the lattice-valued characterizations of this conclusion.

Proposition 4.7. If φ : (X, τ_X) ⟶ (Y, τ_Y) is continuous and surjective, then

$C o m p (X, τ_{X}) ⩽ C o m p (Y, τ_{Y}) .$

Proof. By Proposition 4.3, we know the continuity of φ : (X, τ_X) ⟶ (Y, τ_Y) implies the continuity of φ : (X, qc^{τ
_X}) ⟶ (Y, qc^{τ
_Y}). Furthermore, since φ is surjective, it follows that $\begin{matrix} Comp (X, τ_{X}) \\ = ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}^{τ_{X}} (F^{u}) (x)) \\ = Comp (X, {qc}^{τ_{X}}) \\ ⩽ Comp (X, {qc}^{τ_{Y}}) (by Proposition 3.6) \\ = ⋀_{G^{u} \in F_{LM}^{u} (Y)} ⋀_{μ \in J (L)} ⋁_{y \in Y} (μ \to {qc}^{τ_{Y}} (G^{u}) (y)) \\ = Comp (X, τ_{Y}) . □ \end{matrix}$

In order to show the Tychonoff theorem in (L, M)-fuzzy topological spaces, we first give a lemma in category theory.

Lemma 4.8. [22] Let $(𝔽, 𝔾)$ be a pair of adjoint functors between categories A and B. Then the right adjoint $𝔾$ preserves products in B.

Next we give the lattice-valued Tychonoff theorem in (L, M)-fuzzy topological spaces.

Theorem 4.9. (Tychonoff theorem) Let {(X_j, τ_j) ∣ j ∈ J} be a family of (L, M)-fuzzy topological spaces. Then

$C o m p (\prod_{j \in J} X_{j}, \prod_{j \in J} τ_{j}) = \underset{j \in J}{\land} C o m p (X_{j}, τ_{j}) .$

Proof. By Theorem 4.4, we know $(𝕋_{𝔽}, 𝔽_{𝕋})$ is a pair of adjoint functors between (L, M)-FQC and (L, M)-FTop, i.e., $𝔽_{𝕋}$ is a right adjoint. Then by Lemma 4.8, we know $𝔽_{𝕋}$ preserves products in (L, M)-FTop. Therefore, it follows that $\begin{matrix} Comp (\prod_{j \in J} X_{j}, \prod_{j \in J} τ_{j}) \\ = Comp (\prod_{j \in J} X_{j}, {qc}^{\prod_{j \in J} τ_{j}}) \\ = Comp (𝔽_{𝕋} (\prod_{j \in J} X_{j}, \prod_{j \in J} τ_{j})) \\ = Comp (𝔽_{𝕋} (\prod_{j \in J} (X_{j}, τ_{j}))) \\ = Comp (\prod_{j \in J} 𝔽_{𝕋} (X_{j}, τ_{j})) \\ = Comp (\prod_{j \in J} (X_{j}, {qc}^{τ_{j}})) \\ = Comp (\prod_{j \in J} X_{j}, \prod_{j \in J} {qc}^{τ_{j}}) \\ = ⋀_{j \in J} Comp (X_{j}, {qc}^{τ_{j}}) (by Theorem 3.9) \\ = ⋀_{j \in J} Comp (X_{j}, τ_{j}) . □ \end{matrix}$

5 Measures of compactness in pointwise (L, M)-fuzzy quasi-uniform spaces

Let $D (L^{X})$ be the set of all mappings from J (L^X) to L^X such that x_λnotleqslantd (x_λ) for all x_λ ∈ J (L^X). For any $f, g \in D (L^{X})$ , we define:

f ⩽ g if and only if ∀x_λ ∈ J (L^X) , f (x_λ) ⩽ g (x_λ) .

(f ∨ g) (x_λ) = f (x_λ) ∨ g (x_λ) .

(f ⋄ g) (x_λ) = ⋀ {f (y_μ) ∣ y_μnotleqslantg (x_λ)} .

Then d₀ defined by $d_{0} (x_{λ}) = ⊥_{L}^{X}$ for all x_λ ∈ J (L^X) is the smallest element of $D (L^{X})$ . We can also prove that $f \lor g, f ⋄ g \in D (L^{X}), f ⋄ g ⩽ f, f ⋄ g ⩽ g$ and the operations “∨” and “⋄” satisfy the associate law.

Definition 5.1. (Yue and Fang [28]) A pointwise (L, M)-fuzzy quasi-uniformity on X is a mapping $U : D (L^{X}) ⟶ M$ such that

$U (d_{0}) = ⊤_{M}$ ;

$U (f \lor g) = U (f) \land U (g)$ ;

$U (f) = ⋁_{g ⋄ g ⩾ f} U (g)$ .

For a pointwise (L, M)-fuzzy quasi-uniformity $U$ on X, the pair $(X, U)$ is called a pointwise (L, M)-fuzzy quasi-uniform space. A mapping $φ : (X, U_{X}) ⟶ (Y, U_{Y})$ is called uniformly continuous if $U_{X} (φ^{\leftarrow} (f)) ⩾ U_{Y} (f)$ , where φ^← (f) = φ^← ∘ f ∘ φ^→.

Theorem 5.2. (Pang [17]) Let $(X, U)$ be a pointwise (L, M)-fuzzy quasi-uniform space and define $Q_{x_{λ}}^{U} : L^{X} ⟶ M$ as follows: $Q_{x_{λ}}^{U} (A) = ⋁_{A^{'} ⩽ f (x_{λ})} U (f) .$

Then $Q^{U} = {Q_{x_{λ}}^{U} ∣ x_{λ} \in J (L^{X})}$ is an (L, M)-fuzzy quasi-coincident neighborhood system on X.

Obviously, $Q_{x_{λ}}^{U}$ is an (L, M)-fuzzy filter. Thus we have the following results.

Theorem 5.3. Let $(X, U)$ be a pointwise (L, M)-fuzzy quasi-uniform space and define ${qc}^{U} : F_{LM} (X) ⟶ L^{X}$ by $\forall F \in F_{LM} (X), {qc}^{U} (F) = ⋁_{Q_{y_{μ}}^{U} ⩽ F} y_{μ} .$

Then ${qc}^{U}$ is an (L, M)-fuzzy Q-convergence structure on X.s

Proof. We need only prove (LFQC1) and (LFQC2).

(LFQC1) Since $Q^{U}$ is an (L, M)-fuzzy quasi-coincident neighborhood system on X, we obtain $Q_{x_{λ}}^{U} ⩽ \hat{q} (x_{λ})$ . Then it follows that ${qc}^{U} (\hat{q} (x_{λ})) = ⋁_{Q_{y_{μ}}^{U} ⩽ \hat{q} (x_{λ})} y_{μ} \geq_{λ} .$

(LFQC2) Take $F, G \in F_{LM} (X)$ such that $F ⩽ G$ . Then ${qc}^{U} (F) = ⋁_{Q_{y_{μ}}^{U} ⩽ F} y_{μ} ⩽ ⋁_{Q_{y_{μ}}^{U} ⩽ G} y_{μ} = {qc}^{U} (G) . □$

Now we give the definition of compactness measures of a pointwise (L, M)-fuzzy quasi-uniform space as follows.

Definition 5.4. For a pointwise (L, M)-fuzzy quasi-uniform space $(X, U)$ , define the measure Comp(X, 𝒰) to which $(X, U)$ is compact by $\begin{matrix} Comp (X, U) \\ = ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}^{U} (F^{u}) (x)) . \end{matrix}$

For a quasi-uniform space (X, U) in the classical case, if (X, U) is compact, then its continuous and surjective image is compact. Next we give the lattice-valued characterization of this conclusion. For this, we give the following two lemmas.

Lemma 5.5. (Pang [17]) If $φ : (X, U_{X}) ⟶ (Y, U_{Y})$ is uniformly continuous, then $Q_{φ (x)_{λ}}^{U_{Y}} ⩽ φ^{\Rightarrow} (Q_{x_{λ}}^{U_{X}})$ .

Lemma 5.6. If $φ : (X, U_{X}) ⟶ (Y, U_{Y})$ is uniformly continuous, then $φ : (X, {qc}^{U_{X}}) ⟶ (Y, {qc}^{U_{Y}})$ is continuous.

Proof. Since $φ : (X, U_{X}) ⟶ (Y, U_{Y})$ is uniformly continuous, by Lemma 5.5, we have $Q_{φ (x)_{λ}}^{U_{Y}} ⩽ φ^{\Rightarrow} (Q_{x_{λ}}^{U_{X}})$ for each x_λ ∈ J (L^X). Then for each $F \in F_{LM} (X)$ , it follows that $\begin{matrix} φ^{\to} ({qc}^{U_{X}} (F)) & = & φ^{\to} (⋁_{Q_{x_{λ}}^{U_{X}} ⩽ F} x_{λ}) \\ ⩽ & ⋁_{φ^{\Rightarrow} (Q_{x_{λ}}^{U_{X}}) ⩽ φ^{\Rightarrow} (F)} φ (x)_{λ} \\ ⩽ & ⋁_{Q_{φ (x)_{λ}}^{U_{Y}} ⩽ φ^{\Rightarrow} (F)} φ (x)_{λ} \\ ⩽ & ⋁_{Q_{y_{μ}}^{U_{Y}} ⩽ φ^{\Rightarrow} (F)} y_{μ} = {qc}^{U_{Y}} (φ^{\Rightarrow} (F)) . \end{matrix}$

This shows the continuity of $φ : (X, {qc}^{U_{X}}) ⟶ (Y, {qc}^{U_{Y}})$ . □

Proposition 5.7. If $φ : (X, U_{X}) ⟶ (Y, U_{Y})$ is uniformly continuous and surjective, then

$C o m p (X, U_{X}) ⩽ C o m p (Y, U_{Y}) .$

Proof. Since $φ : (X, U_{X}) ⟶ (Y, U_{Y})$ is uniformly continuous and surjective, by Lemma 5.6, we know $φ : (X, {qc}^{U_{X}}) ⟶ (Y, {qc}^{U_{Y}})$ is continuous and surjective. Then $\begin{matrix} Comp (X, U_{X}) \\ = ⋀_{F^{u} \in F_{LM}^{u} (X)} ⋀_{λ \in J (L)} ⋁_{x \in X} (λ \to {qc}_{X}^{U} (F^{u}) (x)) \\ = Comp (X, {qc}^{U_{X}}) \\ ⩽ Comp (Y, {qc}^{U_{Y}}) (by Proposition 3.6) \\ = ⋀_{G^{u} \in F_{LM}^{u} (Y)} ⋀_{μ \in J (L)} ⋁_{y \in Y} (μ \to {qc}_{Y}^{U} (G^{u}) (y)) \\ = Comp (Y, U_{Y}) . \end{matrix}$

Theorem 5.8. (Pang [17]) Let $(X, U)$ be a pointwise (L, M)-fuzzy quasi-uniform space and define $τ^{U} : L^{X} ⟶ M$ by $\forall A \in L^{X}, τ^{U} (A) = ⋀_{x_{λ} \hat{q} A} Q_{x_{λ}}^{U} (A) .$

Then $τ^{U}$ is an (L, M)-fuzzy topology on X.

In general topology, a quasi-uniform space is compact if and only if its induced topological space is compact. Now Theorem 5.8 shows a pointwise (L, M)-fuzzy quasi-uniform space $(X, U)$ can induce an (L, M)-fuzzy topological space $(X, τ^{U})$ . Thus, we will show $(X, U)$ and $(X, τ^{U})$ have the same measure of compactness. For this, the following lemma is necessary.

Lemma 5.9. (Pang [17]) Let $(X, U)$ be a pointwise (L, M)-fuzzy quasi-uniform space. Then $Q^{U} = {Q_{x_{λ}}^{U} ∣ x_{λ} \in J (L^{X})}$ is a topological (L, M)-fuzzy quasi-coincident neighborhood system on X.

Theorem 5.10. Let $(X, U)$ be a pointwise (L, M)-fuzzy quasi-uniform space. Then

$C o m p (X, U) = C o m p (X, τ^{U}) .$

Proof. By Definitions 4.5 and 5.4, we have

$C o m p (X, τ^{U}) = \underset{ℱ^{u} \in ℱ_{L M}^{u} (X)}{\land} \underset{λ \in J (L)}{\land} \underset{x \in X}{\lor} (λ \to q c^{τ^{U}} (x))$ and $C o m p (X, U) = \underset{ℱ^{u} \in ℱ_{L M}^{u} (X)}{\land} \underset{λ \in J (L)}{\land} \underset{x \in X}{\lor} (λ \to q c^{U} (x)) .$

In order to prove that $C o m p (X, U) = C o m p (X, τ^{U})$ , by Lemmas 4.2 and 5.3, it suffices to show that $Q_{x_{λ}}^{U} = Q_{x_{λ}}^{τ^{U}}$ for all x_λ ∈ J (L^X). For each A ∈ L^X, we have $\begin{matrix} Q_{x_{λ}}^{U} (A) & = & ⋁_{x_{λ} \hat{q} B ⩽ A} ⋀_{y_{μ} \hat{q} B} Q_{y_{μ}}^{U} (B) (by Lemma 5.9) \\ = & ⋁_{x_{λ} \hat{q} B ⩽ A} τ^{U} (B) (by Theorem 5.8) \\ = & Q_{x_{λ}}^{τ^{U}} (A), \end{matrix}$ as desired. □

6 Conclusion

In this paper, we introduce the compactness measures of an (L, M)-fuzzy Q-convergence space, which has nice properties, such as the Tychonoff theorem. We also use this definition to study the compactness of (L, M)-fuzzy topological spaces and pointwise (L, M)-fuzzy quasi-uniform spaces. This paper provides a new way to define the compactness in the framework of (L, M)-fuzzy topological spaces. In the future, we will consider other topological properties of (L, M)-fuzzy topological spaces by using (L, M)-fuzzy Q-convergence structures.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers and the area editor for their careful reading and constructive comments. This work is supported by the National Natural Science Foundation of China (No. 11501435), the Science and Technology Project of Yulin City (No. 2014CXY-08), the Science and Technology Program of Department of Education in Shaanxi Province (No. 12JK0890).

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