The Net in a Fuzzy Soft Topological Space and Its Applications

In 2001,Maji et al. [1] combined fuzzy sets [2] with soft sets [3] and proposed the concept of fuzzy soft sets. After that, the fuzzy soft set was applied to group theory, decision-making, medical diagnosis, and other fields (see [4–13]). Meanwhile, the theory of fuzzy soft set has been developed rapidly. In particular, the research on fuzzy soft topology hasmade a lot of achievements (see [14–24]). Noting the contribution of the pointed approach in fuzzy topology, Roy and Samanta [21] defined a fuzzy soft point on a fuzzy soft topological space. In 2018, Ibedou and Abbas [17] redefined this concept. Recently, Gao and Wu [7] studied the properties of fuzzy soft points introduced in [17] deeply and pointed that the fuzzy soft point given in [17] was more effective than that given in [21]. *ey also gave the definitions of a fuzzy soft net consisting of fuzzy soft points and its convergence. On these bases, they characterized the continuity of fuzzy soft mappings by the net approach. *is paper aims to further the study of [7]. In Section 2, some preliminaries will be recalled. In Section 3, the characterizations of some important results involving closure, separation, and compactness will be obtained by means of fuzzy soft nets. 2. Preliminaries

Noting the contribution of the pointed approach in fuzzy topology, Roy and Samanta [21] defined a fuzzy soft point on a fuzzy soft topological space. In 2018, Ibedou and Abbas [17] redefined this concept. Recently, Gao and Wu [7] studied the properties of fuzzy soft points introduced in [17] deeply and pointed that the fuzzy soft point given in [17] was more effective than that given in [21]. ey also gave the definitions of a fuzzy soft net consisting of fuzzy soft points and its convergence. On these bases, they characterized the continuity of fuzzy soft mappings by the net approach.
is paper aims to further the study of [7]. In Section 2, some preliminaries will be recalled. In Section 3, the characterizations of some important results involving closure, separation, and compactness will be obtained by means of fuzzy soft nets.

Preliminaries
roughout this paper, U refers to an initial universe and E is the set of all parameters for U. In this case, U is also denoted by (U, E). I U is the set of all fuzzy subsets over U, where I � [0, 1]. e elements 0, 1 ∈ I U , respectively, refer to the functions 0(x) � 0 and 1(x) � 1 for all x ∈ U. For an element A ∈ I U , if there exists an x ∈ U such that A(x) � λ > 0 and A(y) � 0, ∀y ∈ (U/ x { }), then A is called a fuzzy point over U and is denoted by x λ , x and λ are its support and height, respectively. e definitions in this section are all sourced from the existing literature [7,17,21,22].
e set of all fuzzy soft sets over (U, E) is denoted by FS(U, E). e fuzzy soft set F ϕ ∈ FS(U, E) is called the null fuzzy soft set and is denoted by Φ. Here, F ϕ (e) � 0 for every e ∈ E.
For F E ∈ FS(U, E), if F E (e) � 1 for all e ∈ E, then F E is called the absolute fuzzy soft set and is denoted by E.
for all e ∈ E, then F A is said to be a fuzzy soft subset of F B and is denoted by (1) e complement of F A , denoted by F c A , is then defined as (2) e union of F A and F B is also a fuzzy soft set Similarly, the union (intersection) of a family of fuzzy soft sets may be defined as F C α : α ∈ Λ and denoted by where Λ is an arbitrary index set.

Remark 2.
e following is, therefore, clear: Definition 3. A fuzzy soft topology τ over (U, E) is a family of fuzzy soft sets over (U, E) satisfying the following properties: If τ is a fuzzy soft topology over (U, E), the triple (U, E, τ) is said to be a fuzzy soft topological space. Each element of τ is called an open set. If F c A is an open set, then F A is called a closed set.
. In this case, ξ is also denoted by P x λ e , and e is called its parameter support. e set of all fuzzy soft points over In the remainder of this paper, a fuzzy soft point is always referred to as given by Definition 5 and is called a point for short.
For P e set of all Q-neighborhoods of ξ is denoted by A(ξ). In the remainder of this paper, Δ is a directed set with the partial order "≺".
In particular, if there exists F A ∈ FS(U, E) such that S(δ) ∈ F A for any δ ∈ Δ, then S is said to be a fuzzy soft net in F A , or a net for simplicity.
If there exists δ 0 ∈ Δ such that S(δ)∈ F A whenever δ 0 ≺δ, then S is said to be eventually quasi-coincident with F A . If for each δ ∈ Δ there exists δ 0 ∈ Δ with δ≺δ 0 such that S(δ 0 )∈ F A , then S is said to be frequently quasi-coincident with F A .
is said to be convergent to a point ξ if S is eventually quasi-coincident with each Q-neighborhood of ξ. In this case, ξ is called the limit of S and is denoted by limS(δ).

Main Results
In this section, nets are applied to characterize the closure of a fuzzy soft set, T 2 separation, and compactness of a fuzzy soft topological space.
Obviously, if P x λ e ∈ FSP(U, E), then A(P It is now proven that P (3), the following is obtained: en, S(λ)∈ F A ∩ F B . ere is a contradiction with the fact that Sufficiency: suppose that (U, E, τ) is not T 2 separated, then there are two different points P en, Δ forms a directed set under the order relation "≺" defined as S(A, B). So, a net S � S(δ): δ ∈ Δ { } is obtained. It is now shown that S converges to P Let Σ be a family of fuzzy soft sets, and F B ∈ FS(U, E). If F B ⊂ ∪ F A ∈Σ F A , then Σ is called a cover of F B . If a cover contains finite elements, then it is called a finite cover. If a subset Σ 1 of Σ is also a cover of F B , then Σ 1 is called a subcover of Σ. If Σ is a cover of { } be a net. If any subnet of S does not take ξ as its limit, then there exists O ξ ∈ A(ξ) and λ ξ ∈ Λ such that S(λ) ∉ O ξ whenever λ ξ ≺λ ∈ Λ.
Proof. Conversely, it is supposed that, for any O ∈ A(ξ) and any λ ∈ Λ, there exists λ O ∈ Λ with λ≺λ O such that S(λ O )∈ O. Let Λ be a set of the corresponding λ O with respect to all O ∈ A(ξ) and all λ ∈ Λ, that is, en, Λ is a directed set under the same order relation "≺" as in Λ.
To complete the proof, it must only be shown that there is a subnet of S converging to ξ. Set Ω � A(ξ). Under the relation ⊇ , Ω is a directed set. Let Ω × Λ be the usual product directed set of Ω and Λ. e mapping φ from Ω × Λ to Λ is defined as en, for any λ ′ ∈ Λ, there always exists at is, there is a subnet of S converging to ξ. □ Theorem 8. Let F A ∈ FS(U, E). en, F A is compact if and only if any net whose element is quasi-coincident with F A has a subnet whose limit is quasi-coincident with F A .
Proof (Necessity). Let F A be a compact fuzzy soft set and S � S(λ): λ ∈ Λ { } a net whose element is quasi-coincident with F A . To prove the Necessity, it must only be shown that S has a subnet whose limit is quasi-coincident with F A . Suppose it is not true, then for any point ξ which is quasicoincident with F A , any subnet of S does not take ξ as its limit. erefore, by Lemma 1, there exists O ξ ∈ A(ξ) and λ ξ ∈ Λ such that when λ ξ ≺λ ∈ Λ, and By eorem 2 (1), is is in contradiction with the supposition that S � S(λ): λ ∈ Λ { } is a net whose element is quasi-coincident with F A . us, S has a subnet whose limit is quasi-coincident with F A .

Conclusions
In this paper, we have obtained some important theorems by means of the convergence of a fuzzy soft net. e obtained results demonstrate that the existing net is a powerful tool for studying fuzzy soft topological spaces. It is prospective that fuzzy soft nets will play important roles in characterizing some other properties of fuzzy soft topological spaces.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.