A topological view on application of L-fuzzy soft sets: Compactness

Abstract

It is known that compactness occupies a very important place in general topology and also in fuzzy topology. In this paper, a stronger form of a fuzzy soft topology which is called a parameterized L-fuzzy soft topology is presented and compactness of a L-fuzzy soft set is established in the described topological space, where L is a complete DeMorgan algebra. Then the fundamental properties and characterizations of compactness are observed.

Keywords

Fuzzy soft set fuzzy soft topology continuity compactness

1 Introduction

In 1999, Molodtsov [26] proposed a completely new concept called soft set theory to model uncertainty, which associates a set with a set of parameters. Later, Maji et al. [25] introduced the concept of a fuzzy soft set which combines fuzzy sets and soft sets. All over the globe, (fuzzy) soft set theory is a topic of interest for many authors working in diverse areas due to its rich potential for applications in several directions. Nowadays the scholars study the theoretics and also applications of (fuzzy) soft sets in algebra [2 , 40] and topology [6 , 34]. Both of soft sets and fuzzy soft sets have many practical applications in decision making problems [8 , 38]. In addition, soft sets and fuzzy soft sets have been imposed to daily life problems [15, 16].

Compactness is one of the most important concepts in general topology since it is sort of a topological generalization of finiteness. If you have some object, then compactness allows you to extend results that you know are true for all finite subobjects to the object itself. A very closely related example is the compactness theorem in propositional logic: an infinite collection of sentences is consistent if every finite subcollection is consistent. The notion of compactness has been generalized to L-topological space by many authors [7 , 31]. In 1997, Aygün et al. [4] first introduced the concept of smooth compactness in L-fuzzy topological spaces, which is a generalization of strong compactness in [19, 21].

The aim of this work is to describe compactness of a L-fuzzy soft set in a fuzzy soft topological space which satisfy the certain properties of compactness known in general topology. For this reason, we first propose a new approach to topological structure of L-fuzzy soft sets named as parameterized L-fuzzy soft topology. Second, we define the compactness of a L-fuzzy soft set where L is a complete DeMorgan algebra. This definition does not rely on the structure of the basis lattice L and no distributivity in L is required. We show that the union of compact sets is also compact, and the intersection of a compact set f_A and a L-fuzzy soft set g_B, which is $τ_{e} (g_{B}^{c}) = 1,$ for any e ∈ E is compact, too. We then prove that the compactness is preserved under continuous mappings between parameterized L-fuzzy soft topological spaces. Finally, we consider the different characteristics of presented compactness.

2 Preliminaries

Throughout this paper, X refers to an initial universe and E is a nonempty set of all parameters for X. L^X denotes the set of all L-fuzzy sets on X (where, L = (L, ∨ , ∧ , ′) is a complete DeMorgan algebra with the unit element 1 and the zero element 0) and for λ ∈ L, $\bar{λ} (x) = λ$ for all x ∈ X . $\bar{0}$ and $\bar{1}$ denotes the smallest element and the largest element in L^X respectively.

An element a in L is called a prime element if a ≥ b ∧ c implies a ≥ b or a ≥ c . a in L is called a co-prime element if a′ is a prime element [14]. The set of all prime elements in L, different from the unit of the lattice, is denoted by P (L) . The set of all nonzero co-prime elements in L is denoted by M (L) . We say that a is wedge below b in L, denoted by a ◃ b, if for every subset D ⊆ L, ∨D ≥ b implies d ≥ a for some d ∈ D .

According to [35], we know that if L is completely distributive, then each element a in L has the greatest minimal family (the greatest maximal family), denoted by β (a) = {b ∈ L ∣ b ◃ a} (α (a) = {b ∈ L ∣ b′ ◃ a′}) . In this case, β^* (a) = β (a) ∩ M (L) is a minimal family of a and α^* (a) = α (a) ∩ P (L) is a maximal family of a .

Definition 2.1. [29] f_A is called a L-fuzzy soft set on X, where f is a mapping from E into L^X, i.e., f_e : = f (e) : X → L is a L-fuzzy set on X, for each e ∈ A and $f_{e} = \bar{0}$ , if e ∉ A, where $\bar{0}$ is zero function on X.

From now on, the family of all L-fuzzy soft sets on X will be denoted by $\tilde{(X, E)}$ .

For a subfamily $U \subseteq \tilde{(X, E)}, 2^{(U)}$ denotes the set of all finite subfamilies of $U .$

Definition 2.2. [29] (1) For two L-fuzzy soft sets f_A and g_B on X, we say that f_A is a L-fuzzy soft subset of g_B and write f_A ⊑ g_B if f_e ≤ g_e, for each e ∈ E. f_A and g_B are called equal if f_A ⊑ g_B and g_B ⊑ f_A.

(2) Union of two L-fuzzy soft sets f_A and g_B on X is the L-fuzzy soft set h_C = f_A ⊔ g_B, where C = A ∪ B and h_e = f_e ∨ g_e, for each e ∈ E.

(3) Intersection of two L-fuzzy soft sets f_A and g_B on X is the L-fuzzy soft set h_C = f_A ⊓ g_B, where C = A ∩ B and h_e = f_e ∧ g_e, for each e ∈ E.

(4) The complement of a L-fuzzy soft set f_A is denoted by $f_{A}^{c}$ , where f^c : E → L^X is a mapping given by $f_{e}^{c} = (f_{e})^{'}$ , for each e ∈ E. Clearly ${(f_{A}^{c})}^{c} = f_{A}$ .

(5) (Null L-fuzzy soft set) A L-fuzzy soft set f_E on X is called a null L-fuzzy soft set and denoted by Φ, if $f_{e} = \bar{0}$ , for each e ∈ E.

(6) (Absolute L-fuzzy soft set) A L-fuzzy soft set f_E on X is called an absolute L-fuzzy soft set and denoted by $\tilde{E}$ , if $f_{e} = \bar{1}$ , for each e ∈ E. Clearly $(\tilde{E})^{c} = Φ$ and $Φ^{c} = \tilde{E}$ .

Definition 2.3. (λ-absolute L-fuzzy soft set) AL-fuzzy soft set f_E on X is called a λ-absolute L-fuzzy soft set and denoted by ${\tilde{E}}^{λ}$ , if $f_{e} = \bar{λ}$ , for each e ∈ E. Clearly, $({\tilde{E}}^{λ})^{c} = {\tilde{E}}^{λ^{'}}$ .

Proposition 2.1. [1] Let Γ be an index set and f_A, g_B, (f_A) _i, (g_B) _i $\in \tilde{(X, E)}$ , for all i ∈ Γ, then the following properties are satisfied:

f_A ⊓ (⨆ _i∈Γ (g_B) _i) = ⨆ _i∈Γ (f_A ⊓ (g_B) _i).

f_A ⊔ (⊓ _i∈Γ (g_B) _i) = ⊓ _i∈Γ (f_A ⊔ (g_B) _i).

${(⊓_{i \in Γ} (f_{A})_{i})}^{c} = ⨆_{i \in Γ} (f_{A})_{i}^{c}$ and ${(⨆_{i \in Γ} (f_{A})_{i})}^{c} = ⊓_{i \in Γ} (f_{A})_{i}^{c}$ .

If f_A ⊑ g_B, then $g_{B}^{c} ⊑ f_{A}^{c}$ .

Definition 2.4. [5] Let φ : X → Y and ψ : E → F be two mappings, where E and F are parameter sets for the crisp sets X and Y, respectively. Then the pair φ_ψ is called a L-fuzzy soft mapping from X to Y.

Definition 2.5. [5] Let f_A and g_B be two L-fuzzy soft sets on X and Y, respectively and let φ_ψ be a L-fuzzy soft mapping from X to Y.

(1) The image of f_A under the L-fuzzy soft mapping φ_ψ, denoted by φ_ψ (f_A), is a L-fuzzy soft set on Y defined by for all k ∈ K, y ∈ Y, $\begin{matrix} φ_{ψ} (f)_{k} (y) \\ = {\begin{matrix} ⋁_{φ (x) = y} ⋁_{ψ (a) = k} f_{a} (x), & if x \in φ^{- 1} (y), a \in ψ^{- 1} (k); \\ 0, & otherwise . \end{matrix} \end{matrix}$

(2) The pre-image of g_B under the L-fuzzy soft mapping φ_ψ, denoted by $φ_{ψ}^{- 1} (g_{B})$ , is a L-fuzzy soft set on X defined by for all e ∈ E, x ∈ X, $φ_{ψ}^{- 1} (g)_{e} (x) = g_{ψ (e)} (φ (x)) .$

If φ and ψ is injective (surjective), then φ_ψ is said to be injective (surjective).

(3) Let φ_ψ be a L-fuzzy soft mapping from X to Y and $φ_{ψ^{*}}^{*}$ be a L-fuzzy soft mapping from Y to Z. Then the composition of these mappings from X to Z is defined as follows: $φ_{ψ} \circ φ_{ψ^{*}}^{*} = (φ \circ φ^{*})_{ψ \circ ψ^{*}},$ where ψ : E → F and ψ^* : F → G .

Proposition 2.2. [18] Let X and Y be two universes $(f_{A})_{i} \in \tilde{(X, E)}$ and $g_{B}, (g_{B})_{i} \in \tilde{(Y, F)}$ for all i ∈ Γ, where Γ is an index set. Then the followings are valid.

φ_ψ (⨆ _i∈Γ (f_A) _i) = ⨆ _i∈Γφ_ψ ((f_A) _i).

φ_ψ (⊓ _i∈Γ (f_A) _i) ⊑ ⊓ _i∈Γφ_ψ ((f_A) _i), the equality holds if φ_ψ is injective.

$φ_{ψ}^{- 1} (⨆_{i \in Γ} (g_{B})_{i}) = ⨆_{i \in Γ} φ_{ψ}^{- 1} ((g_{B})_{i})$ and $φ_{ψ}^{- 1} (⊓_{i \in Γ} (g_{B})_{i}) = ⊓_{i \in Γ} φ_{ψ}^{- 1} ((g_{B})_{i})$ .

$φ_{ψ}^{- 1} (g_{B}^{c}) = {(φ_{ψ}^{- 1} (g_{B}))}^{c}$ .

Definition 2.6. (Cartesian product of two L-fuzzy soft sets) Let X₁ and X₂ nonempty crisp sets $f_{A} \in \tilde{(X_{1}, E_{1})}$ and $g_{B} \in \tilde{(X_{2}, E_{2})}$ . The cartesian product f_A × g_B of f_A and g_B is denoted by (f × g) _A×B, where, for each (e, k) ∈ E₁ × E₂,

(f × g) _(e,k) (x, y) = f_e (x) ∧ g_k (y), for all (x, y) ∈ X × Y.

According to this definition the fuzzy soft set (f × g) _A×B is a fuzzy soft set on X₁ × X₂ and the universal parameter set is E₁ × E₂.

Definition 2.7. Let (f_A) ₁ × (f_A) ₂ be a L-fuzzy soft set on X₁ × X₂. The projection mappings (p_q) _i, i ∈ {1, 2}, is defined as follows:

(p_q) _i ((f_A) ₁ × (f_A) ₂) = p_i (f₁ × f₂) _{q_i(A₁×A₂)} = (f_A) _i where p_i : X₁ × X₂ ⟶ X_i and q_i : E₁ × E₂ ⟶ E_i are projection mappings in classical meaning.

3 Parameterized L-fuzzy soft topological spaces

In this section, we present a new approach to topology of L-fuzzy soft sets called parameterized L-fuzzy soft topology which is stronger than the fuzzy soft topology described in [6], is as follows.

Definition 3.1. A mapping $τ : E \to L^{\tilde{(X, E)}}$ is called a parameterized L-fuzzy soft topology on X if it satisfies the following conditions.

(T1) $⋀_{e \in E} τ_{e} (Φ) = ⋀_{e \in E} τ_{e} (\tilde{E}) = 1 .$

(T2) ⋀_e∈Eτ_e (f_A ⊓ g_B) ≥ ⋁ _e∈E (τ_e (f_A) ∧ τ_e (g_B)) for each $f_{A}, g_{B} \in \tilde{(X, E)} .$

(T3) ⋀_e∈Eτ_e (⨆ _i∈Γ (f_A) _i) ≥ ⋁ _e∈E ⋀ _i∈Γτ_e ((f_A) _i) for each $(f_{A})_{i} \in \tilde{(X, E)}, i \in Γ .$

A parameterized L-fuzzy soft topology is called stratified if it provides that

(T1)^′ $⋀_{e \in E} τ_{e} ({\tilde{E}}^{λ}) = 1$ , for each λ ∈ L.

Then the pair (X, τ_E) is called a parameterized L-fuzzy soft topological space. The value τ_e (f_A) is interpreted as the degree of openness of a L-fuzzy soft set f_A with respect to parameter e ∈ E .

Let $τ_{E}^{1}$ and $τ_{E}^{2}$ be parameterized L-fuzzy soft topologies on X . We say that $τ_{E}^{1}$ is finer than $τ_{E}^{2}$ ( $τ_{E}^{2}$ is coarser than $τ_{E}^{1}$ ), denoted by $τ_{E}^{2} ⊑ τ_{E}^{1},$ if $τ_{e}^{2} (f_{A}) \leq τ_{e}^{1} (f_{A})$ for each $e \in E, f_{A} \in \tilde{(X, E)} .$

Example 3.1. Let $T$ be a fuzzy topology on X in Šostak’s sense [33], that is, $T$ is a mapping from [0, 1] ^X to [0, 1]. Take E = [0, 1] and define $\bar{T} : E ⟶ [0, 1]^{X}$ as $\bar{T} (e) (μ) = {\begin{matrix} 1, & if T (μ) \geq e; \\ 0, & otherwise . \end{matrix}$

Clearly $\bar{T} (e)$ is a fuzzy topology in Šostak’s sense and $\bar{T}$ satisfies (T1), (T2) and (T3). Hence $\bar{T}$ is a fuzzy soft topology on X. According to this definition a fuzzy topology can be uniquely represented as a fuzzy soft topology.

Example 3.2. Let L = {(0, 0) , (1, 1)} ∪ {(a, 0) , (0, b) , (a, a) ∣ a, b ∈ (0, 1)} . “≤” is defined as follows:

(m, b) ≤ (n, d) if and only if m ≤ n and b ≤ d.

Define an order reversing involution ^′ : L → Lis as follows: for each x, y ∈ (0, 1) , (x, 0) ′ = (1 - x, 0) , (0, y) ′ = (0, 1 - y) , (x, x) ′ = (1 - x, 1 - x) and (1, 1) ′ = (0, 0) . Then (L, ≤ , ′) is a complete DeMorgan algebra. Let X = {x, y} , E = (0, 0.5] and f_e (x) = f_e (y) = (e, 0), g_e (x) = g_e (y) = (0, e) and h_e (x) = h_e (y) = (e, e) for each e ∈ E. Define a mapping $τ : E \to L^{\tilde{(X, E)}}$ as follows: $τ_{e} (u) = {\begin{matrix} (1, 1), & if u \in {Φ, \tilde{E}, h_{E}}; \\ (e, 0), & if u = f_{E}; \\ (0, e), & if u = g_{E}; \\ (0, 0), & otherwise . \end{matrix}$

Then τ is a parameterized L-fuzzy soft topology on X .

Example 3.3. Let L = {0, a, b, 1} be a diamond-type lattice with the order reversing involution ^′ : L → L defined by 0′ = 1, 1′ = 0, a′ = a and b′ = b. Then (L, ≤ , ′) is a complete DeMorgan algebra. Let X = {x, y}, E = {1, 2} and f_E, g_E be two L-fuzzy soft sets defined as follows: f₁ (x) = f₂ (y) = g₁ (y) = g₂ (x) = a and f₁ (y) = f₂ (x) = g₁ (x) = g₂ (y) = b. Define a mapping $τ : E \to L^{\tilde{(X, E)}}$ as follows: $τ_{e} (h) = {\begin{matrix} 1, & if h \in {Φ, \tilde{E}}, \forall e \in E; \\ a, & if h = f_{E}, e = 1 or h = g_{E}, e = 2; \\ b, & if h = g_{E}, e = 1 or h = f_{E}, e = 2; \\ 0, & otherwise . \end{matrix}$

Then it is easy to testify that the mapping τ is a parameterized L-fuzzy soft topology on X .

Definition 3.2. Let (X, τ_E) and $(Y, τ_{F}^{*})$ be parameterized L-fuzzy soft topological spaces and S ⊂ E. A fuzzy soft mapping φ_ψ from X to Y is called

(1) a fuzzy soft continuous map if $⋀_{e \in E} τ_{e} (φ_{ψ}^{- 1} (g_{B})) \geq ⋁_{k \in F} τ_{k}^{*} (g_{B})$ for all $g_{B} \in \tilde{(Y, F)}$ .

(2) a fuzzy S-soft continuous map if $⋀_{s \in S} τ_{s} (φ_{ψ}^{- 1} (g_{B})) \geq ⋁_{k \in ψ (S)} τ_{k}^{*} (g_{B})$ for all $g_{B} \in \tilde{(Y, F)}$ .

Proposition 3.1. Let {τ_k} _k∈Γ be a family of parameterized L-fuzzy soft topologies on X . Then τ = ⋀ _k∈Γτ_k is also a parameterized L-fuzzy soft topology on X, where $τ_{e} (f_{A}) = ⋀_{k \in Γ} {(τ_{k})}_{e} (f_{A}), \forall e \in E, f_{A} \in \tilde{(X, E)} .$

Definition 3.3. A mapping $η : E ⟶ L^{\tilde{(X, E)}}$ is called a parameterized L-fuzzy soft cotopology on X if it satisfies the following conditions.

(C1) $⋀_{e \in E} η_{e} (Φ) = ⋀_{e \in E} η_{e} (\tilde{E}) = 1 .$

(C2) ⋀_e∈Eη_e (f_A ⊔ g_B) ≥ ⋁ _e∈E (η_e (f_A) ∧ η_e (g_B))for each $f_{A}, g_{B} \in \tilde{(X, E)} .$

(C3) ⋀_e∈Eη_e (⊓ _i∈Γ (f_A) _i) ≥ ⋁ _e∈E ⋀ _i∈Γη_e ((f_A) _i) for each $(f_{A})_{i} \in \tilde{(X, E)}, i \in Γ .$

The pair (X, η_E) is called a parameterized L-fuzzy soft cotopological space.

Let τ be a parameterized L-fuzzy soft topology on X, then the mapping $η : E \to L^{\tilde{(X, E)}}$ defined by $η_{e} (f_{A}) = τ_{e} (f_{A}^{c}),$ for all e ∈ E is a parameterized L-fuzzy soft cotopology on X . Let η be a parameterized L-fuzzy soft cotopology on X, then the mapping $τ : E \to L^{\tilde{(X, E)}}$ defined by $τ_{e} (f_{A}) = η_{e} (f_{A}^{c}),$ for all e ∈ E is a parameterized L-fuzzy soft topology on X .

4 Compactness of a L-fuzzy soft set

In order to generalize the notion of compactness to L-fuzzy soft topological spaces, first let us research compactness in general topology. Let $(X, T)$ be a topological space and G ⊆ X be a crisp set. G is said to be compact if each open cover $U$ of G has a finite subfamily $V$ which is an open cover of G. By the following fact (where G′ denotes the complement of G): $\begin{matrix} ⋀_{x \in X} (G^{'} (x) \lor ⋁_{A \in U} A (x)) = 1 \\ \Leftrightarrow \forall x \in X, G^{'} (x) \lor ⋁_{A \in U} A (x) = 1 \\ \Leftrightarrow \forall x \in X, G^{'} (x) \neg = 1 implies ⋁_{A \in U} A (x) = 1 \\ \Leftrightarrow \forall x \in X, G (x) = 1 implies ⋁_{A \in U} A (x) = 1 \\ \Leftrightarrow \forall x \in X, G (x) \leq ⋁_{A \in U} A (x) \end{matrix}$ we know that G is compact if and only if $\begin{matrix} ⋀_{x \in X} (G^{'} (x) \lor ⋁_{A \in U} A (x)) = 1 implies \\ ⋁_{V \in 2^{U}} ⋀_{x \in X} (G^{'} (x) \lor ⋁_{A \in V} A (x)) = 1 [32] \end{matrix}$

Definition 4.1. Let (X, τ_E) be a parameterizedL-fuzzy soft topological space and S ⊂ E .

(1) $f_{A} \in \tilde{(X, E)}$ is said to be compact if for every family $U \subseteq \tilde{(X, E)}$ , it follows that $\begin{matrix} ⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \\ \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) \\ \leq ⋁_{V \in 2^{(U)}} ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) \end{matrix}$

(2) $f_{A} \in \tilde{(X, E)}$ is said to be S-compact if for every family $U \subseteq \tilde{(X, E)}$ , it follows that $\begin{matrix} ⋀_{g_{B} \in U} ⋁_{s \in S} τ_{s} (g_{B}) \land (⋀_{s \in S} ⋀_{x \in X} (f_{s}^{c} (x) \lor ⋁_{g_{B} \in U} g_{s} (x))) \\ \leq ⋁_{V \in 2^{(U)}} ⋀_{s \in S} ⋀_{x \in X} (f_{s}^{c} (x) \lor ⋁_{g_{B} \in V} g_{s} (x)) . \end{matrix}$

Example 4.1. Let (X, τ_E) be the parameterizedL-fuzzy soft topology defined in Example 3.2. Then the L-fuzzy soft sets f_E, g_E and h_E which are defined in Example 3.2 are compact in the sense of Definition 4.1.

Theorem 4.1. Let (X, τ_E) be a parameterized L-fuzzy soft topological space and S ⊂ E.

(1) $f_{A} \in \tilde{(X, E)}$ is compact if and only if for every family $P \subseteq \tilde{(X, E)}$ we have that $\begin{matrix} ⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \\ \lor (⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} h_{e} (x))) \\ \geq ⋀_{H \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in H} h_{e} (x)) . \end{matrix}$

(2) $f_{A} \in \tilde{(X, E)}$ is S-compact if and only if for every family $P \subseteq \tilde{(X, E)}$ we have that

$\begin{matrix} ⋁_{h_{C} \in P} ⋀_{s \in S} τ_{s}^{'} (h_{C}^{c}) \\ \lor (⋁_{s \in S} ⋁_{x \in X} (f_{s} (x) \land ⋀_{h_{C} \in P} h_{s} (x))) \\ \geq ⋀_{H \in 2^{(P)}} ⋁_{s \in S} ⋁_{x \in X} (f_{s} (x) \land ⋀_{h_{C} \in H} h_{s} (x)) . \end{matrix}$

Proof. From definition we easily obtain by using quasi-complement “^′”.

Theorem 4.2. Let (X, τ_E) be a parameterized L-fuzzy soft topological space. If both $f_{A}, g_{B} \in \tilde{(X, E)}$ are compact (respectively, S-compact) in (X, τ_E), then $f_{A} ⊔ g_{B} \in \tilde{(X, E)}$ is compact (respectively, S-compact) in (X, τ_E).

Proof. $\begin{matrix} ⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \lor (⋁_{e \in E} ⋁_{x \in X} ((f_{e} \lor g_{e}) (x) \land ⋀_{h_{C} \in P} h_{e} (x))) \\ = ⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \\ \lor (⋁_{e \in E} ({⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} h_{e} (x))} \\ \lor {⋁_{x \in X} (g_{e} (x) \land ⋀_{h_{C} \in P} h_{e} (x))})) \\ = {⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \lor (⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} h_{e} (x)))} \\ \lor {⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \lor (⋁_{e \in E} ⋁_{x \in X} (g_{e} (x) \land ⋀_{h_{C} \in P} h_{e} (x)))} \\ \geq {⋀_{H \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in H} h_{e} (x))} \\ \lor {⋀_{H \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} (g_{e} (x) \land ⋀_{h_{C} \in H} h_{e} (x))} \\ = ⋀_{H \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} ((f_{e} \lor g_{e}) (x) \land ⋀_{h_{C} \in H} h_{e} (x)) . \end{matrix}$

Theorem 4.3. Let (X, τ_E) be a parameterized L-fuzzy soft topological space. If $f_{A} \in \tilde{(X, E)}$ is compact, and $τ_{e} (g_{B}^{c}) = 1,$ for each e ∈ E (respectively, $f_{A} \in \tilde{(X, E)}$ is S-compact and $τ_{s} (g_{B}^{c}) = 1, \forall s \in S$ ) then $f_{A} ⊓ g_{B} \in \tilde{(X, E)}$ is compact (respectively, S-compact).

Proof. $\begin{matrix} ⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \lor (⋁_{e \in E} ⋁_{x \in X} ((f_{e} \land g_{e}) (x) \land ⋀_{h_{C} \in P} h_{e} (x))) \\ = ⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \lor (⋁_{e \in E} ⋁_{x \in X} ((f_{e} (x) \land g_{e} (x)) \land ⋀_{h_{C} \in P} h_{e} (x))) \\ = ⋁_{h_{C} \in P \cup {g_{B}}} ⋀_{e \in E} τ_{e}^{'} (h_{C}^{c}) \lor (⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P \cup {g_{B}}} h_{e} (x))) \\ \geq ⋀_{F \in 2^{(P) \cup {g_{B}}}} ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in F} h_{e} (x)) \\ = {⋀_{F \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in F} h_{e} (x))} \\ \land {⋀_{F \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land (g_{e} (x) \land ⋀_{h_{C} \in F} h_{e} (x)))} \\ = ⋀_{F \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} ((f_{e} \land g_{e}) (x) \land ⋀_{h_{C} \in F} h_{e} (x)) . \end{matrix}$

Lemma 4.1. Let L be a complete DeMorgan frame, then the following equalities are satisfied. $\begin{matrix} (1) & ⋁_{k \in F} ⋁_{y \in Y} (φ_{ψ} (f)_{k} (y) \land ⋀_{h_{C} \in P} h_{k} (y)) \\ = ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} φ_{ψ}^{- 1} (h)_{e} (x)) . \\ (2) & ⋁_{k \in ψ (S)} ⋁_{y \in Y} (φ_{ψ} (f)_{k} (y) \land ⋀_{h_{C} \in P} h_{k} (y)) \\ = ⋁_{s \in S} ⋁_{x \in X} (f_{s} (x) \land ⋀_{h_{C} \in P} φ_{ψ}^{- 1} (h)_{s} (x)) . \end{matrix}$

Proof. (1) This is proved from the following facts. $\begin{matrix} ⋁_{k \in F} ⋁_{y \in Y} (φ_{ψ} (f)_{k} (y) \land ⋀_{h_{C} \in P} h_{k} (y)) \\ = ⋁_{k \in F} ⋁_{y \in Y} ((⋁_{e \in ψ^{- 1} (k)} ⋁_{x \in φ^{- 1} (y)} f_{e} (x)) \land ⋀_{h_{C} \in P} h_{k} (y)) \\ = ⋁_{k \in F} ⋁_{y \in Y} (⋁_{e \in ψ^{- 1} (k)} ⋁_{x \in φ^{- 1} (y)} (f_{e} (x) \land ⋀_{h_{C} \in P} h_{ψ (e)} (φ (x)))) \\ = ⋁_{k \in F} ⋁_{e \in ψ^{- 1} (k)} ⋁_{y \in Y} ⋁_{x \in φ^{- 1} (y)} (f_{e} (x) \land ⋀_{h_{C} \in P} h_{ψ (e)} (φ (x))) \\ = ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} φ_{ψ}^{- 1} (h)_{e} (x)) . \end{matrix}$

(2) One can prove this equality similarly to (1).

Theorem 4.4. Let (X, τ_E) and $(Y, τ_{F}^{*})$ be parameterized L-fuzzy soft topological spaces and S ⊂ E . Then the following properties are satisfied.

(1) If φ_ψ is a fuzzy soft continuous mapping from X to Y and $f_{A} \in \tilde{(X, E)}$ is compact in (X, τ_E) , then $φ_{ψ} (f_{A}) \in \tilde{(Y, F)}$ is compact in $(Y, τ_{F}^{*})$ .

(2) If φ_ψ is a fuzzy S-soft continuous mapping from X to Y and $f_{A} \in \tilde{(X, E)}$ is S-compact in (X, τ_E) , then $φ_{ψ} (f_{A}) \in \tilde{(Y, F)}$ is ψ (S)-compact in $(Y, τ_{F}^{*})$ .

Proof. (1) For any family $P \subset \tilde{(Y, F)}$ , by Lemma 4.1 and compactness of f_A we have that $\begin{matrix} ⋁_{h_{C} \in P} ⋀_{k \in F} τ_{k}^{*^{'}} (h_{C}^{c}) \\ \lor (⋁_{k \in F} ⋁_{y \in Y} (φ_{ψ} (f)_{k} (y) \land ⋀_{h_{C} \in P} h_{k} (y))) \\ = ⋁_{h_{C} \in P} ⋀_{k \in F} τ_{k}^{*^{'}} (h_{C}^{c}) \\ \lor (⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} φ_{ψ}^{- 1} (h)_{e} (x))) \\ \geq ⋁_{h_{C} \in P} ⋁_{e \in E} τ_{e}^{'} (φ_{ψ}^{- 1} (h_{C}^{c})) \\ \lor (⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} φ_{ψ}^{- 1} (h)_{e} (x))) \\ \geq ⋁_{h_{C} \in P} ⋀_{e \in E} τ_{e}^{'} (φ_{ψ}^{- 1} (h_{C}^{c})) \\ \lor (⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} φ_{ψ}^{- 1} (h)_{e} (x))) \\ \geq ⋀_{H \in 2^{(P)}} ⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in H} φ_{ψ}^{- 1} (h)_{e} (x)) \\ = ⋀_{H \in 2^{(P)}} ⋁_{k \in F} ⋁_{y \in Y} (φ_{ψ} (f)_{k} (y) \land ⋀_{h_{C} \in H} h_{k} (y)) . \end{matrix}$

(2) One can prove this similarly to (1).

5 Other characterizations of compactness of L-fuzzy soft sets

Definition 5.1. Let (X, τ_E) be a parameterizedL-fuzzy soft topological space, a ∈ L - {1} and $f_{A} \in \tilde{(X, E)} .$ A subfamily $U$ in $\tilde{(X, E)}$ is said to be

(1) an “a-shading of f_A” if for any x ∈ X and e ∈ E, it follows that $f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x) ≰ a .$

(2) a “strong a-shading of f_A” if

$⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) ≰ a .$

It is obvious that a strong a- shading of f_A is an a- shading of f_A.

Definition 5.2. Let (X, τ_E) be a parameterizedL-fuzzy soft topological space, a ∈ L - {0} and $f_{A} \in \tilde{(X, E)} .$ A subfamily $P$ in $\tilde{(X, E)}$ is called

(1) an “a-remote family of f_A” if for any x ∈ X and e ∈ E, it follows that $f_{e} (x) \land ⋁_{h_{C} \in P} h_{e} (x) ≱ a .$

(2) a “strong a-remote family of f_A” if $⋁_{e \in E} ⋁_{x \in X} (f_{e} (x) \land ⋀_{h_{C} \in P} h_{e} (x)) ≱ a .$

It is obvious that a strong a-remote family of f_A is an a- remote family of f_A and $P$ is a strong a- remote family of f_A if and only if $P^{'}$ is a strong a′-shading of f_A.

Definition 5.3. Let (X, τ_E) be a parameterizedL-fuzzy soft topological space, a ∈ L - {0} and $f_{A} \in \tilde{(X, E)} .$ A subfamily $U$ in $\tilde{(X, E)}$ is said to be

(1) a “β_a-cover of f_A” if for any x ∈ X and e ∈ E, it follows that $a \in β (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) .$

(2) a “strong β_a-cover of f_A” if $a \in β (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) .$

It is obvious that a strong β_a- cover of f_A is a β_a- cover of f_A.

Definition 5.4. Let (X, τ_E) be a parameterizedL-fuzzy soft topological space, a ∈ L - {0} and $f_{A} \in \tilde{(X, E)} .$ A subfamily $U$ in $\tilde{(X, E)}$ is called a Q_a-cover of f_A if for any x ∈ X and e ∈ E with f_e (x) ≰a′, it follows that $⋁_{g_{B} \in U} g_{e} (x) \geq a .$

It is obvious that for a ∈ M (L) , $U$ is a Q_a-cover of f_A if and only if $a \leq ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) .$

Theorem 5.1. Let (X, τ_E) be a parameterized L-fuzzy soft topological space and $f_{A} \in \tilde{(X, E)} .$ Then the following conditions are equivalent.

f_A is compact.

For any a ∈ M (L) , each strong a-remote family $P$ of f_A with $⋀_{h_{C} \in P} ⋁_{e \in E} τ_{e}^{'} (h_{C}) ≰ a^{'}$ has a finite subfamily $H$ which is a (strong) a-remote family of f_A .

For any a ∈ M (L) and any strong α-remote family $P$ of f_A with $⋀_{h_{C} \in P} ⋁_{e \in E} τ_{e}^{'} (h_{C}) ≰ a^{'},$ there exists a finite subfamily $H$ of $P$ and b ∈ β^* (a) such that $H$ is a (strong) b-remote family of f_A .

For any a ∈ P (L) , each strong a-shading $U$ of f_A with $⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) ≰ a$ has a finite subfamily $V$ which is a (strong) a-shading of f_A .

For any a ∈ P (L) and any strong a-shading $U$ of f_A with $⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) ≰ a,$ there exists a finite subfamily $V$ of $U$ and b ∈ α^* (a) such that $V$ is a (strong) b-shading of f_A .

For any a ∈ M (L) and any b ∈ β^* (a) , each Q_a-cover $U$ of f_A with ⋀_e∈Eτ_e (g_B) ≥ a, $(\forall g_{B} \in U)$ has a finite subfamily $V$ which is a Q_b-cover of f_A .

For any a ∈ M (L) and any b ∈ β^* (a) , each Q_a-cover $U$ of f_A with ⋀_e∈Eτ_e (g_B) ≥ a, $(\forall g_{B} \in U)$ has a finite subfamily $V$ which is a (strong) β_b-cover of f_A .

Proof. (1)⇒(4): Let f_A be compact, a ∈ P (L) and $U$ be a strong a-shading of f_A with $⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) ≰ a .$ Suppose that $⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) \leq a$ , for every $V \in 2^{(U)} .$

Since f_A is compact, then $\begin{matrix} ⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) \\ \leq ⋁_{V \in 2^{(U)}} ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) \leq a \end{matrix}$

Since a ∈ P (L) , then $⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \leq a$ or $⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) \leq a .$

But it contradicts with the hypothesis. Then, there exists $V \in 2^{(U)}$ such that $V$ is a strong a-shading of f_A .

(4)⇒(1): For arbitrary $U \subset \tilde{(X, E)}$ and a ∈ P (L) , let $⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) ≰ a$

Then, $⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) ≰ a .$ $(and ⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) ≰ a)$

By the hypothesis, there exists $V \in 2^{(U)}$ such that $⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) ≰ a .$

Then, $⋁_{V \in 2^{(U)}} ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) ≰ a$ .

Hence, the following is true $\begin{matrix} ⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) \\ \leq ⋁_{V \in 2^{(U)}} ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) . \end{matrix}$

(1)⇔(2) is similarly proved.

(4)⇒(5): Suppose that a ∈ P (L) and $U$ is a strong a-shading of f_A with $⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) ≰ a,$ then $⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) ≰ a .$

Take c ∈ α^* (a) = α (a) ∩ P (L) with $\begin{matrix} ⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) ≰ c such that \\ ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)) ≰ c . \end{matrix}$

Obviously $U$ is a strong c-shading of f_A with $⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) ≰ c,$ by (4) there exists $V \in 2^{(U)}$ which is a c-shading of f_A . Take b ∈ α^* (a) such that c ∈ α^* (b) , then $V \in 2^{(U)}$ is a strong b-shading of f_A .

(5)⇒(4): This implication is trivial.

Dually one can prove the equality of the conditions (2) and (3).

Theorem 5.2. Let (X, τ_E) be a parameterized L-fuzzy soft topological space and f_A be a L-fuzzy soft set on X . If β (c ∧ d) = β (c) ∩ β (d) , (∀ c, d ∈ L) , then the followings are equivalent.

(1) f_A is compact.

(2) For any a ∈ M (L) , each strong β_a-cover $U$ of f_A with $a \in β (⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}))$ has a finite subfamily $V$ which is a (strong) β_a-cover of f_A .

(3) For any a ∈ M (L) and any strong β_a-cover $U$ of f_A with $a \in β (⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B})),$ there exists $V \in 2^{(U)}$ and b ∈ M (L) with a ∈ β^* (b) such that $V$ is a (strong) β_b-cover of f_A .

Proof. (1)⇒ (2): Let f_A be compact, a ∈ M (L) and $U \subset \tilde{(X, E)}$ be a strong β_a-cover of f_A with $a \in β (⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B})) .$

Let $V \in 2^{(U)}$ and let for D ⊆ L, $⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) \leq ⋁ D .$

Since f_A is compact, $\begin{matrix} ⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) \\ \leq ⋁_{V \in 2^{(U)}} ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) \leq ⋁ D . \end{matrix}$

Since $\begin{matrix} a \in β (⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B})) \\ \cap β (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) . \end{matrix}$

By the hypothesis, $\begin{matrix} a \in β (⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \\ \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)))) . \end{matrix}$

Therefore, $\begin{matrix} a ◃ {⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \\ \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)))} . \end{matrix}$

Then there exists d ∈ D such that a ≤ d . By the definition, $a \in β (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x))) .$

Hence, $V$ is a finite subfamily of $U$ which is a strong β_a-cover of f_A .

(2)⇒ (1): Let a ∈ M (L) and $\begin{matrix} a ◃ {⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \\ \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x)))} . \end{matrix}$

By the hypothesis, $\begin{matrix} a ◃ (⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B})) and \\ a ◃ (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))), i . e ., \\ a \in β (⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B})) and \\ a \in β (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) . \end{matrix}$

By (2), there exists finite subfamily $V$ which is a strong β_a-cover of f_A, that is $a \in β (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x))) .$

Hence, $a \leq ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)),$ i.e., $a \leq ⋁_{V \in 2^{(U)}} ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) .$

So, the following is obtained: $\begin{matrix} ⋀_{g_{B} \in U} ⋁_{e \in E} τ_{e} (g_{B}) \\ \land (⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in U} g_{e} (x))) \\ \leq ⋁_{V \in 2^{(U)}} ⋀_{e \in E} ⋀_{x \in X} (f_{e}^{c} (x) \lor ⋁_{g_{B} \in V} g_{e} (x)) . \end{matrix}$

(2) ⇔ (3): One can prove this equality similarly to the equality of (4) ⇔ (5) of Theorem 5.1.

6 Conclusion

It is known that compactness and its stronger and weaker forms play a crucial role in (fuzzy) topology. In addition, El-Naschie [11, 12] has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and e^∞ theory. The fuzzy soft set theory can be used as a newly mathematical tool dealing with uncertainty and also have many applications not only in theoretical areas but also in daily life problems. Consequently, we decided to combine all these theories. We first gave a new definition of a L-fuzzy soft topology which is stronger than the definition given in [6]. Later we proposed and studied the compactness of a L-fuzzy soft set. Since one can characterize the behavior of infinite objects by using finite of them due to the concept of compactness, we hope that the concept of fuzzy soft compactness proposed in this paper will be helpful for the solutions of the daily life problems which consists a large amount of data.

Footnotes

Acknowledgments

The authors are thankful to the associate editor Prof. Jianming Zhan and the anonymous reviewers for their valuable comments.

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