# A Net with Applications for Continuity in a Fuzzy Soft Topological Space

Authors: Rui Gao; Jianrong Wu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the concept of a fuzzy soft point is redefined, and the definition of a fuzzy soft net in a fuzzy soft topological space is given. On this basis, the convergence of a fuzzy soft net is defined by using the Q-neighborhood theory, and the continuity of fuzzy soft mappings is characterized by the net. The obtained results demonstrate that the concepts proposed in this paper are very suitable and will provide powerful research tools for further research in this field.


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## Body

## 1. Introduction

In 1965, Zadeh introduced the concept of fuzzy sets in his classic work [1]. In 1999, Molodtsov [2] introduced the theory of soft sets, which have been applied in several fields, including the smoothness of functions, game theory, Riemann integration, and the theory of probability [3]. In 2001, Maji et al. [4] combined fuzzy sets with soft sets and proposed the concept of fuzzy soft sets. Since then, many researchers have applied fuzzy soft sets to group theory [5], decision making, and medical diagnosis [6,7]. In 2009, Kharal and Ahmad [8] studied the properties of fuzzy soft images and fuzzy soft inverse images of fuzzy soft sets.In 2011, Tanay and Kandemir [9] proposed the concept of a fuzzy soft set in a fuzzy soft topological space and explained some of its structural properties. They also claimed that fuzzy soft topological spaces may be used in the theory of information systems. In 2012, Mahanta and Das [10] introduced the definition of a fuzzy soft point and its neighborhood. They also studied the interior and closure of a fuzzy soft set and investigated the separation axioms and connectedness. Varol and Aygün [11] introduced the fuzzy soft continuity of fuzzy soft mappings. In 2013, Gunduz (Aras) and Bayramov [3] presented fuzzy soft continuous mappings, fuzzy soft open and fuzzy soft closed mappings, and fuzzy soft homeomorphism. In 2014, Ping et al. [12] proposed the sum of fuzzy soft topological spaces. In 2016, Mishra and Srivastava [13] studied compactness in fuzzy soft topological spaces. In 2017, Kandil et al. [14] discussed the connectedness of fuzzy soft sets. Riaz and Hashmi [15] proposed the concept of fuzzifying soft sets, called fuzzy parameterized fuzzy soft sets (FPFS-sets). Mahanta and Das [16] studied fuzzy soft closure and the fuzzy soft interior. In 2018, Abbas et al. [17] explored connectedness in fuzzy soft topological spaces. In 2019, Riaz and Tehrim [18] proved some properties of bipolar fuzzy soft topology (BFS-topology) via the use of the concept of the Q-neighborhood.In 2012, Roy and Samanta [19] redefined the concept of fuzzy soft topology and obtained some basic results. In a subsequent work [20], they adopted a new definition of a fuzzy soft point, proposed the concepts of quasi-coincidence and Q-neighborhoods, and demonstrated the relationship between the limit point and the closure of a fuzzy soft set.As pointed out in Example1 in this paper, the existing concept of fuzzy soft points does not satisfy selectivity, which makes it difficult for a fuzzy soft point to play its expected role. Therefore, a more suitable definition of a fuzzy soft point must be given. Moreover, as is commonly known, the net plays an important role in classical topology theory; however, the concept of the net has not been introduced into fuzzy soft topological spaces.In view of these considerations, this paper first redefines the concept of a fuzzy soft point and introduces the net into fuzzy soft topological spaces. The continuity of a mapping in fuzzy soft topological spaces with the use of a net is then studied. The remainder of this article is organized as follows. InSection 2, some necessary concepts of fuzzy soft sets and fuzzy soft topological spaces are recalled. In Section 3, fuzzy soft points are refined and the notion of a fuzzy soft net consisting of fuzzy soft points is introduced. By using the theory of the Q-neighborhood, the concept of the convergence of a fuzzy soft net is introduced. In Section 4, the net is applied to characterize the continuity of fuzzy soft mappings. Section 5 presents a discussion of convergence for a net of fuzzy soft mappings. In Section 6, an example of the application of the fuzzy soft set theory to medical diagnosis is provided. Finally, the conclusion of this paper is given in Section 7.


## 2. Preliminaries

Throughout this paper,U refers to an initial universe and E is the set of all parameters of 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, and x and λ are called the support and height of xλ, respectively. The set of all fuzzy points over U is denoted by FPU.The following definitions in this section were obtained from the existing literature [19,20].Definition 1.
LetA⊆E. A mapping FA:E⟶IU, defined by FAe=μFAe (a fuzzy subset of U), is called a fuzzy soft set over U,E, where μFAe=0¯ if e∈E\A and μFAe≠0¯ if e∈A.
IfA=E, then FA is shortened to F.
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 defined as(1)FAce=1¯−FAe,fore∈A,1¯,otherwise.(2)
The union ofFA and FB, denoted by FC=FA∪˜FB, is defined as FCe=FAe∪FBe for all e∈E, where C=A∪B.(3)
The intersection ofFA and FB, denoted by FC=FA∩˜FB, is defined as FCe=FAe∩FBe for all e∈E, where C=A∩B.
Similarly, the union (intersection) of a family of fuzzy soft setsFCα:α∈Λ can be defined and is denoted by ∪˜α∈ΛFCα∪˜α∈ΛFCα, where Λ is an arbitrary index set.Remark 2.
It is clear that(1)
Φ˜c=E˜,E˜c=Φ˜(2)
∪˜α∈ΛFAαc=∩˜α∈ΛFAαc, ∩˜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 (any 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 F¯A(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. It is evident that(1)
F¯A is closed and intFA is open(2)
FA is closed if and only if FA = F¯A(3)
FA is open if and only if FA = intFA


## 3. Fuzzy Soft Point and Fuzzy Soft Net

Roy and Samanta [20] defined a fuzzy soft point Fe over U,E as a special fuzzy soft set such that Fea≠0¯ if a=e, and Fea=0¯ if a≠e. For FA∈FSU,E, they stated that a fuzzy soft point Fe belongs in FA, denoted by Fe∈FA, if and only if Feex≤FAex for all x∈U.Example 1.
LetU=x,y and E=e,f. A fuzzy soft point Fe is defined as(2)Feet=35,t=x,23,t=y,Fef=0¯.
Also, fuzzy soft setsFA and FB are defined as(3)FAet=35,t=x,13,t=y,FAft≡12,t=x,y,FBet=25,t=x,23,t=y,FBft≡12t=x,y.
Then,(4)FA∪˜FBet=35,t=x,23,t=y,FA∪˜FBft≡12,t=x,y,
It is clear thatFe∈FA∪˜FB. However, Fe∉FA and Fe∉FB.
This example demonstrates that a fuzzy soft point that belongs in the union of two fuzzy soft sets does not need to belong in one of these two fuzzy soft sets.
To overcome this shortcoming, a fuzzy soft point is redefined as follows.Definition 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.
It is clear thatFe in Example 1 is not a fuzzy soft point in the sense of Definition 5.Example 2.
LetU=x,y and E=e,f. A mapping Fe:E⟶IU is defined as(5)Feet=0.2,t=x,0,t=y,Fef=0¯.
Then,Fe is a fuzzy soft point in the sense of Definition 5 and can be written as Pex0.2.
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,Eα∈Λ:(1)
IfPexλ∈FA∪˜FB, then Pexλ∈FA or Pexλ∈FB(2)
Pexλ∈FAα∀α∈Λ if and only if Pexλ∈∩˜α∈ΛFAαProof.
(1)
BecausePexλ∈FA∪˜FB, then λ≤μFA∪˜FBex; that is, λ ≤ μFAex ∨ μFBex. Therefore, λ≤μFAex or λ≤μFBex. Hence, Pexλ∈FA or Pexλ∈FB.(2)
Pexλ∈FAα∀α∈Λ is equal to λ≤μFAαex∀α∈Λ. Equivalently, λ≤∧α∈ΛμFAαex; that is, Pexλ∈∩˜α∈ΛFAα.
However, it is clear that Theorem1(1) does not hold for the union of infinite fuzzy soft sets. For this reason, the concept of quasi-coincidence is subsequently introduced.Definition 6.
A pointPexλ is said to be quasi-coincident with FA∈FSU,E, which is denoted by Pexλ∈˜FA, if λ+μFAex>1.
ThatPexλ is not quasi-coincident with FA is denoted by Pexλ∉˜FA.Remark 4.
In the work by Roy and Samanta [20], a fuzzy soft point Fe over U,E was said to be quasi-coincident with FA∈FSU,E if μFex +μFAex>1 for some x∈U. If Fe is a fuzzy soft point in the sense of Definition 5, then the definition of quasi-coincidence in this paper is equivalent to that in the work by Roy and Samanta [20].Remark 5.
It is evident 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∈ΛPexλ∈˜FA∩˜FB if and only if Pexλ∈˜FA and Pexλ∈˜FB(3)
Pexλ∈˜∪˜α∈ΛFAα if and only if there exists an α0∈Λ such that Pexλ∈˜FAα0Proof.
(1)
The necessity is evident. To prove the sufficiency, it is supposed thatPexλ∈˜FB for any point Pexλ∈˜FA. If FA⊆˜FB is not true, then there exist an e∈E and x∈U such that μFAex>μFBex. For λ∈0,1 with μFAex>1−λ>μFBex, the point Pexλ∈˜FA and Pexλ∉˜FB, which contradicts the assumption. Thus, FA⊆˜FB.(2)
IfPexλ∈˜∩˜α∈ΛFAα, then ∧α∈ΛμFAαex+λ>1. Hence, μFAαex+λ>1∀α∈Λ; that is, Pexλ∈˜FAα∀α∈Λ.From Definition6, it is clear that Pexλ∈˜FA∩˜FB if and only if μFA∩˜FBex+λ>1. Equivalently, μFAex∧μFBex+λ>1. Thus, μFAex+λ>1 and μFBex+λ>1; that is, Pexλ∈˜FA and Pexλ∈˜FB.(3)
From Definition6, it can be seen that Pexλ∈˜∪˜α∈ΛFAα if and only if ∨α∈ΛμFAαex+λ>1. Equivalently, there exists an α0∈Λ such that μFAα0ex+λ>1; that is, Pexλ∈˜FAα0.Definition 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.Remark 6.
It is clear thatFA∩˜FB≠Φ˜ if and only if FA and FB are quasi-coincident at a point ξ∈FSPU,E.Definition 8.
(see the work by Roy and Samanta [20]). A fuzzy soft set FA is said to be quasi-coincident with FB, which is denoted by FAqFB, if μFAex+μFBex>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˜.Proof.
Suppose thatFAqFB; then, there exist an x∈U and e∈A∩B such that μFAex+μFBex>1. Set λ=maxμFAex,μFBex. Then, Pexλ∈˜FA∩˜FB. Hence, FA∩˜FB≠Φ˜.Remark 7.
The converse of Theorem3 does not hold. Indeed, in Example 1,(6)FA∩˜FBet=25,t=x,13,t=y,FA∩˜FBft≡12,t=x,y.
Because(7)FAex+FBex=35+25=1,FAfx+FBfx=12+12=1,FAey+FBey=13+23=1,FAfy+FBfy=12+12=1,then FAq¯FB. However, FA∩˜FB≠Φ˜.Theorem 4.
LetFA,FB∈FSU,E. If there is a Pexλ∈FSPU,E with λ∈0,1/2 such that Pexλ∈˜FA∩˜FB, then FAqFB.Proof.
BecausePexλ∈˜FA∩˜FB, via Theorem 2(2), μFAex+λ>1 and μFBex+λ>1; that is, μFAex+μFBex>2−2λ. From λ∈0,1/2, 1≤2−2λ≤2. Therefore, μFAex+μFBex>1; that is, FAqFB.Definition 9.
Letξ∈FSPU,E and FA,FB∈FSU,E:(1)
FA is said to be a neighborhood of ξ if there exists an FB∈τ such that ξ∈FB⊆˜FA(2)
FA is called a Q-neighborhood of ξ if there exists an FB∈τ such that ξ∈˜FB⊆˜FA
The set of all Q-neighborhoods ofξ is denoted by Uξ.
In the remainder of this paper,Δ is a directed set with the partial order “≺.”Remark 8.
It is clear thatUξ is a directed set with the partial order “⊆˜.”Definition 10.
The mappingS:Δ⟶FSPU,E is called a fuzzy soft net in U,E and is denoted by Sδ,δ∈Δ, or S for simplicity.
In particular, if there exists anFA∈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 in U,E. If there exists a 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 a 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 in 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.


## 4. Fuzzy Soft Continuous Mapping

In this section, the definition of fuzzy soft continuous mapping is first recalled, and the net is then applied to characterize the continuity.Definition 13.
(see the work of Aygünoğlu and Aygün [5]). Let FSU1,E1 and FSU2,E2 be the families of all fuzzy soft sets over U1 and U2, respectively. Let φ:U1⟶U2 and ψ:E1⟶E2 be two functions. Then, the pair φ,ψ is called a fuzzy soft mapping from U1 to U2 and is denoted by φ,ψ: FSU1,E1⟶FSU2,E2:(1)
LetFA∈FSU1,E1. Then, the image of FA under the fuzzy soft mapping φ,ψ is the fuzzy soft set over U2 defined by φ,ψFA, where φ,ψFAky(8)=∨φx=y∨ψe=kFAex,ifx∈φ−1y,0¯,otherwise,∀k∈ψE1,∀y∈U2.(2)
LetFB∈FSU2,E2. Then, the pre-image of FB under the fuzzy soft mapping φ,ψ is the fuzzy soft set over U1 defined by φ,ψ−1FB, where(9)φ,ψ−1FBex=FBψeφx,∀e∈ψ−1E2∀x∈U1.
If bothφ and ψ are injective (surjective), then the fuzzy soft mapping φ,ψ is said to be injective (surjective).
The composition of two fuzzy soft mappingsφ,ψ from U1,E1 to U2,E2 and φ′,ψ′ from U2,E2 to U3,E3 is defined as φ′∘φ,ψ′∘ψ from U1,E1 to U3,E3.Lemma 1.
(see the work of Kharal and Ahmad [8]). Let FA, FAi∈FSU1,E1, and FB, FBi∈FSU2,E2, i=1,2. The following is then obtained:(1)
IfFA1⊆˜FA2, then φ,ψFA1⊆˜φ,ψFA2(2)
IfFB1⊆˜FB2, then φ,ψ−1FB1⊆˜φ,ψ−1FB2(3)
FA⊆˜φ,ψ−1φ,ψFA; the equality holds if φ,ψ is injective(4)
φ,ψφ,ψ−1FB⊆˜FB; the equality holds if φ,ψ is surjective
It is simple to verify the following lemma.Lemma 2.
Letφ,ψ be a fuzzy soft mapping, Pexλ∈FSPU1,E1 and FA∈FSU1,E1; then(1)
φ,ψPexλ=Pψeφxλ(2)
IfPexλ∈˜FA, then φ,ψPexλ∈˜φ,ψFA(3)
IfPexλ∈FA, then φ,ψPexλ∈φ,ψFATheorem 5.
LetFA, FB∈FSU1,E1, and φ,ψ: U1,E1⟶U2,E2 be a fuzzy soft mapping. If FAqFB, then φ,ψFAqφ,ψFB.Proof.
Suppose thatFAqFB. Then, there are an x∈U1 and e∈E1 such that FAex+FBex>1. Let FAex=λ. Then, Pexλ∈FA and Pexλ∈˜FB. From Lemma 2, it can be found that φ,ψPexλ∈φ,ψFA, φ,ψPexλ∈˜φ,ψFB. Thus,(10)φ,ψFBψeφx+λ>1,φ,ψFAψeφx≥λ.
Therefore,(11)φ,ψFBψeφx+φ,ψFAψeφx>1.
Equivalently,(12)μφ,ψFBψeφx+μφ,ψFAψeφx>1.which implies that φ,ψFAqφ,ψFB.Definition 14.
(see the work by Varol and Aygün [11]). Let U1,E1,τ1 and U2,E2,τ2 be two fuzzy soft topological spaces. A fuzzy soft mapping φ,ψ: U1,E1,τ1⟶U2,E2,τ2 is said to be fuzzy soft continuous if φ,ψ−1FB∈τ1, ∀FB∈τ2.
If bothφ,ψ: U1,E1,τ1⟶U2,E2,τ2 and φ′,ψ′: U2,E2,τ2⟶U3,E3,τ3 are fuzzy soft continuous, then it is clear that φ′,ψ′∘φ,ψ is also fuzzy soft continuous. In fact, for a fuzzy soft set FA on U3,(13)φ′,ψ′∘φ,ψ−1FAe=φ′∘φ,ψ′∘ψ−1FAe=φ′∘φ−1FAψ′ψe=φ−1φ′−1FAψ′ψe=φ,ψ−1φ′,ψ′−1FAe.
Hence,(14)φ′,ψ′∘φ,ψ−1FA=φ,ψ−1φ′,ψ′−1FA.Theorem 6.
LetU1,E1,τ1 and U2,E2,τ2 be two fuzzy soft topological spaces and φ,ψ: U1,E1,τ1⟶U2,E2,τ2 be a fuzzy soft mapping. The following are then equivalent:(1)
φ,ψ: U1,E1,τ1⟶U2,E2,τ2 is fuzzy soft continuous(2)
For eachξ∈FSPU1,E1 and each neighborhood FB of φ,ψξ, there exists a neighborhood FA of ξ such that φ,ψFA⊆˜FB(3)
For eachPexλ∈FSPU1,E1 and each Q-neighborhood FB of φ,ψPexλ, there exists a Q-neighborhood FA of Pexλ such that φ,ψFA⊆˜FB(4)
For each netS=Sδ,δ∈Δ in U1,E1, if S converges to Pexλ∈FSPU1,E1, then φ,ψ∘S=φ,ψSδ,δ∈Δ is a net in U2,E2 and converges to φ,ψPexλProof.
1⇒2: Suppose that ξ∈FSPU1,E1 and FB is a neighborhood of φ,ψξ. Then, there exists an FC∈τ2 such that φ,ψξ∈FC⊆˜FB. Because φ,ψ is fuzzy soft continuous, FA=φ,ψ−1FC∈τ1. Then, FA is a neighborhood of ξ and φ,ψFA⊆˜FC⊆˜FB follows from Lemma 1(4).2⇒3: Suppose that Pexλ∈FSPU1,E1 and FB is a Q-neighborhood of φ,ψPexλ. Then, there is an open fuzzy soft set FB′∈Uφ,ψPexλ such that FB′⊆˜FB. From Remark 5 and Lemma 2(1), it can be determined that there exists 0<μ<λ such that φ,ψPex1−μ∈FB′. Thus, FB′ is a neighborhood of φ,ψPex1−μ. Under condition (2), there exists an open neighborhood FA of Pex1−μ such that φ,ψFA⊆˜FB′. Because 0<μ<λ, μFAex≥1−μ>1−λ. Therefore, FA is a Q-neighborhood of Pexλ, and φ,ψFA⊆˜FB.3⇒4: Suppose a net S=Sδ,δ∈Δ in U1,E1 converges to Pexλ∈FSPU1,E1. For any FB∈Uφ,ψPexλ, under condition (3), there exists an FA∈UPexλ such that φ,ψFA⊆˜FB. From the supposition, there is a δ0∈Δ such that Sδ∈˜FA whenever δ0≺δ. Lemma 2 implies that φ,ψSδ∈˜φ,ψFA⊆˜FB. Therefore, φ,ψ∘S=φ,ψSδ,δ∈Δ converges to φ,ψPexλ.4⇒1: This is proven by contradiction. If φ,ψ is not fuzzy soft continuous, then there is an FB∈τ2 such that φ,ψ−1FB is not open. From Remark 3(3), it can be found that there exists a Pexλ∈FSPU1,E1 such that Pexλ∈˜φ,ψ−1FB, Pexλ∈˜intφ,ψ−1FB. By Definition 4(2), there is an FA⊆˜φ,ψ−1FB for any FA∈UPexλ. It follows from Theorem 2(1) that there exists a ξFA∈FSPU1,E1 such that ξFA∈˜FA and ξFA∈˜φ,ψ−1FB. Recall from Remark 8 that UPexλ is a directed set with the partial order “⊆˜”; thus, a net S=ξFA:FA∈UPexλ in U1,E1 is obtained. It is easy to verify that S converges to Pexλ. Under condition (4), the net φ,ψ∘S converges to φ,ψPexλ. Recall that Pexλ∈˜φ,ψ−1FB, or φ,ψPexλ∈˜FB∈τ2; it is known that FB is a Q-neighborhood of φ,ψPexλ. Thus, φ,ψ∘S is eventually quasi-coincident with FB, which conflicts with the fact that ξFA∈˜φ,ψ−1FB, or φ,ψξFA∈˜FB, for any FA∈UPexλ. Therefore, φ,ψ is fuzzy soft continuous.


## 5. Convergence for a Net of Fuzzy Soft Mappings

LetU1,E1,τ1 and U2,E2,τ2 be two fuzzy soft topological spaces. FSMU1,U2 denotes the set of all mappings of U1,E1,τ1 into U2,E2,τ2.Definition 15.
A netφ,ψμ,μ∈M in FSMU1,U2 is said to be convergent to φ,ψ∈FSMU1,U2 if, for every point ξ∈FSPU1,E1 and for every FB∈Uφ,ψξ, there exist a μ0∈M and FA∈Uξ such that φ,ψμFA⊆˜FB for every μ∈M with μ≥μ0.Theorem 7.
A netφ,ψμ,μ∈M in FSMU1,U2 converges to φ,ψ∈FSMU1,U2 if and only if, for every net Sδ,δ∈Δ in U1,E1,τ1 which converges to a point ξ∈FSMU1,E1, the net φ,ψμSδ,δ,μ∈Δ×M converges to the point φ,ψξ∈FSPU2,E2, where Δ×M is the product of Δ and M.Proof.
(Necessity) LetSδ,δ∈Δ be a net in FSMU1,U2 that converges to ξ∈FSPU1,E1 and let FB∈Uφ,ψξ arbitrarily. By assumption, there exist an FA∈Uξ and μ0∈M such that φ,ψμFA⊆˜FB for any μ∈M with μ0≺μ. Because Sδ,δ∈Δ converges to ξ, there is a δ0∈Δ such that Sδ∈˜FA for every δ∈Δ with δ0≺δ. Then, for every δ,μ∈Δ×M with δ0,μ0≺δ,μ, it follows from Lemma 2(2) that φ,ψμSδ∈˜φ,ψμFA⊆˜FB, which means φ,ψμ,μ∈M converges to φ,ψ.
(Sufficiency) Proceeding by contradiction, it is supposed that there are aξ∈FSPU1,E1, FB∈Uφ,ψξ such that, for every μ∈M and for every FA∈Uξ, there exists a μ′≥μ such that φ,ψμ′FA⊆˜FB. From Theorem 2(1) and Lemma 2(1), a point ξFA∈FSPU1,E1 is obtained such that ξFA∈˜FA and φ,ψμ′ξFA∉˜FB. As can be determined from Remark 8, the net S=ξFA:FA∈Uξ converges to ξ. However, the net φ,ψμSFA,FA,μ∈Uξ×M does not converge to φ,ψξ. This conflicts with the condition that φ,ψμ,μ∈M converges to φ,ψ.


## 6. Application of Fuzzy Soft Set Theory to Medical Diagnosis

In a hospital, some doctors usually decide what disease a patient is suffering from by observing the patient’s symptoms. However, due to the complexity of symptoms, it is difficult to find the precise relationship between diseases and symptoms. The concept of fuzzy soft sets partially resolves this difficulty.Suppose that the initial universeU=x1,x2,…,xm is the set of all the disease objects that the patient may be infected with, and the set of parameters E=e1,e2,…,en is all of the patient’s symptoms. Generally speaking, from a symptom e∈E, one cannot completely determine the corresponding disease x∈U; however, one can determine the membership degree in which object x∈U holds parameter e∈E, which is denoted by μFex; that is, for every e∈E, there is a fuzzy subset Fe of U. Obviously, the mapping F:E⟶IU is a fuzzy soft set over U,E.Let(15)μxi=∑k=1nμFekxi.Ifμxi0=max1≤i≤mμxi, then it may be claimed that the patient has disease xi0.


## 7. Conclusions

In this paper, the new concepts of fuzzy soft points and fuzzy soft nets were introduced to fuzzy soft topological spaces. On these bases, the fuzzy soft net was used to accurately describe the convergence, which was used to characterize the continuity. Moreover, the convergence for a net of fuzzy soft mappings was investigated. The obtained results demonstrate that the concepts proposed in this paper are very useful and will provide powerful research tools for further research in this field. Particularly, the convergence of fuzzy soft nets may be used to characterize some important properties of fuzzy soft topological spaces, such as closure, separation, compactness, etc.



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*Source: 9098410-2020-06-30.xml*