# The Net in a Fuzzy Soft Topological Space and Its Applications

Authors: Rui Gao; Jianrong Wu
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, the fuzzy soft net is explored to study the properties of a fuzzy soft topological space. Some important results about the closure, separation, and compactness are obtained. It is demonstrated that the net is a powerful tool for studying fuzzy soft topological spaces.


---

## Body

## 1. Introduction

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 has made 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]. They 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.This 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

Throughout 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. IU is the set of all fuzzy subsets over U, where I=0,1. The elements 0¯,1¯∈IU, respectively, refer to the functions 0¯x=0 and 1¯x=1 for all x∈U. For an element A∈IU, if there exists an x∈U such that Ax=λ>0 and Ay=0, ∀y∈U/x, then A is called a fuzzy point over U and is denoted by xλ, x and λ are its support and height, respectively.The definitions in this section are all sourced from the existing literature [7, 17, 21, 22].Definition 1.
LetA⊆E. A mapping FA:E⟶IU, is called a fuzzy soft set over U,E, where FAe=0¯ if e∈E/A and FAe≠0¯ if e∈A.
The set of all fuzzy soft sets overU,E is denoted by FSU,E.
The fuzzy soft setFϕ∈FSU,E is called the null fuzzy soft set and is denoted by Φ˜. Here, Fϕe=0¯ for every e∈E.
ForFE∈FSU,E, if FEe=1¯ for all e∈E, then FE is called the absolute fuzzy soft set and is denoted by E˜.
LetFA,FB∈FSU,E. If FAe⊆FBe for all e∈E, then FA is said to be a fuzzy soft subset of FB and is denoted by FA⊆˜FB or FB⊇˜FA. In addition, FA⊆˜FB means that FA is not a fuzzy soft subset of FB. If FA⊆˜FB and FB⊆˜FA, then FA and FB are said to be equivalent, which is denoted by FA=FB.Remark 1.
IfFA⊆˜FB, then A⊆B.Definition 2.
LetFA, FB∈FSU,E.(1)
The complement ofFA, denoted by FAc, is then defined as(1)FAce=1¯−FAefore∈A,1¯,otherwise.(2)
The union ofFA and FB is also a fuzzy soft set FC defined by FCe=FAe∪FBe for all e∈E, where C=A∪B, and is denoted by FC=FA∪˜FB.(3)
The intersection ofFA and FB is also a fuzzy soft set FC defined by FCe=FAe∩FBe for all e∈E, where C=A∩B, and is denoted by FC=FA∩˜FB.
Similarly, the union (intersection) of a family of fuzzy soft sets may be defined asFCα:α∈Λ and denoted by ∪˜α∈ΛFCα∩˜α∈ΛFCα, where Λ is an arbitrary index set.Remark 2.
The following is, therefore, clear:(1)
Φ˜c=E˜,E˜c=Φ˜(2)
∪˜α∈ΛFAαc=∩˜α∈ΛFAαc(3)
∩˜α∈ΛFAαc=∪˜α∈ΛFAαcDefinition 3.
A fuzzy soft topologyτ over U,E is a family of fuzzy soft sets over U,E satisfying the following properties:(1)
Φ˜,E˜∈τ(2)
IfFA,FB∈τ,then FA∩˜FB∈τ(3)
IfFAα∈τ for all α∈Λ (an index set), then ∪˜α∈ΛFAα∈τ
Ifτ 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 FAc is an open set, then FA is called a closed set.Definition 4.
LetU,E,τ be a fuzzy soft topological space, FA∈FSU,E.(1)
The intersection of all closed setsFB⊇˜FA is called the closure of FA and is denoted by FA¯(2)
The union of all open subsets ofFA over U,E,τ is called the interior of FA and is denoted by intFARemark 3.
LetFA∈FSU,E. The following is evident:(1)
FA¯ is closed and intFA is open(2)
FA is closed if and only if FA = FA¯(3)
FA is open if and only if and FA = intFADefinition 5.
A mappingξ:E⟶IU is called a fuzzy soft point over U,E if there is an e∈E such that ξe=xλ∈FPU, and ξa=0¯ when a∈E/e.
In this case,ξ is also denoted by Pexλ, and e is called its parameter support. The set of all fuzzy soft points over U,E is denoted by FSPU,E.
In the remainder of this paper, a fuzzy soft point is always referred to as given by Definition5 and is called a point for short.
ForPexλ, Pfyβ∈FSPU,E, if x≠y or λ≠β or e≠f, then it is said that Pexλ and Pfyβ are different, which is written as Pexλ≠Pfyβ.Theorem 1.
LetPexλ∈FSPU,E, FA,FB∈FSU,E, and FAα∈FSU,E∀α∈Λ.(1)
Pexλ∈FA∪˜FB if and only if Pexλ∈FA or Pexλ∈FB(2)
Pexλ∈FAα∀α∈Λ if and only if Pexλ∈∩˜α∈ΛFAαDefinition 6.
A pointPexλ is said to be quasi-coincident with FA∈FSU,E, which is denoted by Pexλ∈˜FA, if λ+FAex>1.
ThatPexλ is not quasi-coincident with FA is denoted by Pexλ∉˜FA.Remark 4.
It is obvious thatPexλ∈˜FA if and only if Pexλ∉FAc. Additionally, if Pexλ∈˜FA, then there exists 0<μ<λ such that Pex1−μ∈FA.Theorem 2.
LetFA, FB, FAα∈FSU,E∀α∈Λ, and Pexλ∈FSPU,E.(1)
FA⊆˜FB if and only if Pexλ∈˜FB for any Pexλ∈˜FA(2)
Pexλ∈˜∩˜α∈ΛFAα implies Pexλ∈˜FAα∀α∈Λ(3)
Pexλ∈˜FA∩˜FB if and only if Pexλ∈˜FA and Pexλ∈˜FB(4)
Pexλ∈˜∪˜α∈ΛFAα if and only if there exists an α0∈Λ such that Pexλ∈˜FAα0Definition 7.
LetFA,FB∈FSU,E and ξ∈FSPU,E. If ξ∈˜FA∩˜FB, then it is said that FA and FB are quasi-coincident at ξ.Definition 8.
A fuzzy soft setFA is said to be quasi-coincident with FB, which is denoted by FAqFB, if FAex+FBex>1 for some x∈U and e∈A∩B.
ThatFA is not quasi-coincident with FB is denoted by FAq¯FB.Theorem 3.
LetFA,FB∈FSU,E. If FAqFB, then FA∩˜FB≠Φ˜.Definition 9.
Letξ∈FSPU,E and FA,FB∈FSU,E.(1)
FA is said to be a neighborhood of ξ if there exists FB∈τ such that ξ∈FB⊆˜FA(2)
FA is called a Q-neighborhood of ξ if there exists FB∈τ such that ξ∈˜FB⊆˜FA
The set of all Q-neighborhoods ofξ is denoted by Aξ.
In the remainder of this paper,Δ is a directed set with the partial order “≺“.Definition 10.
The mappingS:Δ⟶FSPU,E is called a fuzzy soft net over U,E and is denoted by Sδ,δ∈Δ or S for simplicity.
In particular, if there existsFA∈FSU,E such that Sδ∈FA for any δ∈Δ, then S is said to be a fuzzy soft net in FA, or a net for simplicity.Definition 11.
LetFA∈FSU,E and S=Sδ,δ∈Δ be a net over U,E. If there exists δ0∈Δ such that Sδ∈˜FA whenever δ0≺δ, then S is said to be eventually quasi-coincident with FA. If for each δ∈Δ there exists δ0∈Δ with δ≺δ0 such that Sδ0∈˜FA, then S is said to be frequently quasi-coincident with FA.Definition 12.
A netSδ,δ∈Δ over U,E,τ 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 limSδ.


## 3. Main Results

In this section, nets are applied to characterize the closure of a fuzzy soft set,T2 separation, and compactness of a fuzzy soft topological space.Obviously, ifPexλ∈FSPU,E, then APexλ forms a directed set under the relation ⊇˜.Theorem 4.
LetU,E,τ be a fuzzy soft topological space. A point ξ∈˜intFA if and only if there exists an FB∈Aξ such that FB⊆˜FA.Proof.
(Necessity). Letξ∈˜intFA. From Definition 4 and Theorem 2 (4), there exists an open set FB⊆˜FA such that ξ∈˜FB. Because FB is open, FB∈Aξ.
(Sufficiency) LetFB∈Aξ and FB⊆˜FA. From Definition 9, there is FC∈τ such that ξ∈˜FC⊆˜FB. It is obvious that FC⊆˜intFB⊆˜intFA. Then, ξ∈˜intFA.Theorem 5.
A pointPexλ∈FA¯ if and only if each Q-neighborhood of Pexλ is quasi-coincident with FA.Proof:.
(Necessity). LetPexλ∈FA¯, FB∈APexλ. Without loss of generality, it is supposed that FB is open. To complete the proof of necessity, it is sufficient to show FBqFA. In fact, if FBq¯FA, then FAdy+FBdy≤1 for any y∈U and d∈E. Thus, FA⊆˜FBc. Note that FBc is closed and FA¯⊆˜FBc. So, Pexλ∈FBc, which together with Remark 4 conflicts with Pexλ∈˜FB. Hence, FBqFA.
(Sufficiency) Suppose each Q-neighborhood ofPexλ is quasi-coincident with FA. If Pexλ∉FA¯, then Pexλ∈˜FA¯c. Thus, FA¯c∈APexλ. It follows from the supposition that FA¯cqFA. So, there exist y∈U and d∈E such that(2)FA¯cdy+FAdy>1.
Hence,FA¯dy<FAdy, which is a contradiction.Theorem 6.
LetFA∈FSU,E. Then, a point Pexλ∈FA¯ if and only if there is a net S=Sδ:δ∈Δ in FA such that limSδ=Pexλ.Proof:.
(Necessity). LetPexλ∈FA¯. Take APexλ,⊇˜ as a directed set. For any FB∈APexλ, by Theorem 5, FBqFA. Then, there exist dB∈E and yB∈U such that(3)FBdByB+FAdByB>1.
LetλB=FAdByB. Then, the points PdByBλB∈FA and PdByBλB∈˜FB from (1). Thus PdByBλB:FB∈APexλ is a net in FA.
It is now proven thatPdByBλB:FB∈APexλ converges to Pexλ. In fact, for any FC∈APexλ, let FB0=FC. When FB0⊇˜FB∈APexλ, by (3), the following is obtained:(4)1<FBdByB+FAdByB≤FB0dByB+λB.
So,PdByBλB∈˜FB0, that is, PdByBλB∈˜FC. From the arbitrariness of FC, PdByBλB:FB∈APexλ converges to Pexλ.
Sufficiency: it is supposed thatlimSδ=Pexλ, where S=Sδ:δ∈Δ is in FA. Then, for any FB∈APexλ, there is a δ0∈Δ such that Sδ0∈FA and Sδ0∈˜FB. Sδ0 is written as PdyB, where d∈E, y∈U, and β∈0,1. Thus FAdy≥β and β+FBdy>1. So, FAdy+FBdy>1, which means that FBqFA. From the arbitrariness of FB and Theorem 5, Pexλ∈FA¯ is obtained.Definition 13.
LetU,E,τ be a fuzzy soft topological space. If for any two different points Pexλ and Pfyβ there exist FA∈APexλ and FB∈APfyβ such that FA∩˜FB=Φ˜, then U,E,τ is said to be T2 separated.Theorem 7.
A fuzzy soft topological spaceU,E,τ is T2 separated if and only if the limit of any net is unique.Proof:.
(Necessity). Suppose that a netS=Sλ:λ∈Λ converges to two different points Pexα and Pfyβ. Because U,E,τ is T2 separated, there exist FA∈APexα and FB∈APfyβ such that FA∩˜FB=Φ˜. By Definition 12, there exists Sλ such that Sλ∈˜FA and Sλ∈˜FB. Then, Sλ∈˜FA∩˜FB. There is a contradiction with the fact that FA∩˜FB=Φ˜.
Sufficiency: suppose thatU,E,τ is not T2 separated, then there are two different points Pexα and Pfyβ, and for any FA∈APexα and FB∈APfyβ, FA and FB are quasi-coincident at some points. Thus, SA,B∈FSPU,E such that SA,B∈˜FA and SA,B∈˜FB. Let Δ=FA,FB:FA∈APexα,FB∈APfyβ. Then, Δ forms a directed set under the order relation “≺“ defined as FA,FB≺FC,FD if and only if FA⊇˜FC,FB⊇˜FD. For any δ=FA,FB∈Δ, let Sδ=SA,B. So, a net S=Sδ:δ∈Δ is obtained.
It is now shown thatS converges to Pexα and Pfyβ. In fact, for any FA∈APexα and FB∈APfyβ, FA,FB∈Δ. When FA,FB≺FC,FD∈Δ, SC,D∈˜FC, SC,D∈˜FD, FA⊇˜FC, and FB⊇˜FD. Then, SC,D∈˜FA and SC,D∈˜FB. From the arbitrariness of FA and FB, limSδ=Pexα and limSδ=Pfyβ are obtained, which is a contradiction with the supposition.
A fuzzy soft netTω:ω∈Ω is called a subnet of a fuzzy soft net Sλ:λ∈Λ if there exists a mapping φ:Ω⟶Λ such that(i)
∀p∈Λ, there exists ω∈Ω such that p≺φω(ii)
For anyω1, ω2∈Ω with ω1≺ω2, then φω1≺φω2(iii)
Tω=Sφω
LetΣ be a family of fuzzy soft sets, and FB∈FSU,E. If FB⊂˜∪˜FA∈ΣFA, then Σ is called a cover of FB. If a cover contains finite elements, then it is called a finite cover. If a subset Σ1 of Σ is also a cover of FB, then Σ1 is called a subcover of Σ. If Σ is a cover of FB, every element of Σ being open (closed), then Σ is said to be an open (closed) cover of FB.Definition 14.
LetFA∈FSU,E. If every open cover of FA contains a finite subcover, then FA is said to be compact.Lemma 1.
Letξ∈FSPU,E, FA∈FSU,E, and S=Sλ:λ∈Λ be a net. If any subnet of S does not take ξ as its limit, then there exists Oξ∈Aξ and λξ∈Λ such that Sλ∉˜Oξ whenever λξ≺λ∈Λ.Proof.
Conversely, it is supposed that, for anyO∈Aξ and any λ∈Λ, there exists λO∈Λ with λ≺λO such that SλO∈˜O. Let Λ¯ be a set of the corresponding λO with respect to all O∈Aξ and all λ∈Λ, that is, Λ¯=λO:λ≺λO and SλO∈˜O, O∈Aξ, λ∈Λ. Then, 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 ofS converging to ξ. Set Ω=Aξ. Under the relation ⊇˜, Ω is a directed set. Let Ω×Λ¯ be the usual product directed set of Ω and Λ¯. The mapping φ from Ω×Λ¯ to Λ is defined as φO,λ0=λ0, ∀O,λO∈Ω×Λ¯. Then, for any λ′∈Λ, there always exists O′,λO′∈Ω×Λ¯ such that λ′≺λO′ and SλO′∈˜O′. When O′,λO′≺O,λO∈Ω×Λ¯, O′⊇˜O, λO′≺λO. Thus λ′≺λO′≺λO=φO,λO. So, SλO:O,λO∈Ω×Λ¯ is a subnet of S=Sλ:λ∈Λ. Let O′∈Aξ arbitrarily. Then, O,λO′∈Ω×Λ¯. When O′,λO′≺O,λO∈Ω×Λ¯, O′⊇˜O, λO′≺λO, and SλO∈˜O. Thus, SλO∈˜O′. So, SλO:O,λO∈Ω×Λ¯ converges to ξ. That is, there is a subnet of S converging to ξ.Theorem 8.
LetFA∈FSU,E. Then, FA is compact if and only if any net whose element is quasi-coincident with FA has a subnet whose limit is quasi-coincident with FA.Proof:.
(Necessity). LetFA be a compact fuzzy soft set and S=Sλ:λ∈Λ a net whose element is quasi-coincident with FA. To prove the Necessity, it must only be shown that S has a subnet whose limit is quasi-coincident with FA. Suppose it is not true, then for any point ξ which is quasi-coincident with FA, any subnet of S does not take ξ as its limit. Therefore, by Lemma 1, there exists Oξ∈Aξ and λξ∈Λ such that when λξ≺λ∈Λ, and(5)Sλ∉˜Oξ.
By Theorem2 (1), FA⊆˜∪˜Oξ:Oξ∈Aξ, ξ∈˜FA. Because FA is compact, there exists Ψ=Oξi:i=1, 2…,k⊆Ω such that FA⊆˜∪˜i=1kOξi. For each i=1,2,…,k, by (5), there exists λξi∈Λ, Sλ∉˜Oξi whenever λξi≺λ∈Λ. Because Λ is a directed set, there exists λ0∈Λ such that λξi≺λ0 (i=1,2,…,k). It follows that Sλ0∉˜Oξi(i=1,2,…,k). Then, Sλ0∉˜FA. This is in contradiction with the supposition that S=Sλ:λ∈Λ is a net whose element is quasi-coincident with FA. Thus, S has a subnet whose limit is quasi-coincident with FA.
(Sufficiency) LetΘ=Oi:i∈Γbe an open cover of FA, Λ=λ:λ is a finite subset of Γ. Under the usual inclusion “⊆“ relation, Λ forms a directed set. Suppose that FA is not compact, then for any λ∈Λ, FA⊆˜∪i∈λOi. Then, by Theorem 2 (1), there is a point Sλ∈˜FA; however,(6)Sλ∉˜∪i∈λOi,∀λ∈Λ.
Thus, a netS=Sλ:λ∈Λ is obtained whose element is always quasi-coincident with FA; however, Sλ∉˜∪i∈λOi, ∀λ∈Λ.
Under the assumption, there is a subnetT=Tω:ω∈Ω of S such that limTω=ξ0,∈˜FA. Because FA⊆˜∪i∈ΓOi and due to Theorem 2 (3), there exists i0∈Γ such that ξ0∈˜Oi0. So, there exists ω1∈Ω when ω1≺ω∈Ω:(7)Tω∈˜Oi0.
BecauseT=Tω:ω∈Ω is a subnet of S=Sλ:λ∈Λ, there exists a mapping φ from Ω to Λ. For the above i0, i0∈Λ. Hence, there exists ω2∈Ω such that i0≺φω2. Subsequently, when ω2≺ω∈Ω, the following is obtained: i0≺φω, that is, i0∈φω. From (6), Sφω∉˜Oi0, that is,(8)Tω∉˜Oi0.
Letω∈Ω with ω1≺ω and ω2≺ω; then, both (7) and (8) have a contradiction. Therefore, FA is compact.


## 4. Conclusions

In this paper, we have obtained some important theorems by means of the convergence of a fuzzy soft net. The 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.



---

*Source: 6673976-2021-04-07.xml*