# Fuzzy Soft Compact Topological Spaces

Authors: Seema Mishra; Rekha Srivastava
Journal: Journal of Mathematics
 (2016)

---

## Abstract

In this paper, we have studied compactness in fuzzy soft topological spaces which is a generalization of the corresponding concept by R. Lowen in the case of fuzzy topological spaces. Several basic desirable results have been established. In particular, we have proved the counterparts of Alexander’s subbase lemma and Tychonoff theorem for fuzzy soft topological spaces.


---

## Body

## 1. Introduction

Molodtsov introduced the concept of a soft set in 1999 (cf. [1]) as a new approach to model uncertainties. He also applied his theory in several directions, for example, stability and regularization, game theory, and soft analysis (cf. [1]).Maji et al. [2] introduced and studied fuzzy soft sets. Yang et al. [3] introduced interval valued fuzzy soft sets. Majumdar and Samanta [4] defined generalized fuzzy soft set.Algebraic structures of soft sets and fuzzy soft sets have been studied by many researchers. Aktaş and Çağman [5] introduced soft groups. Feng et al. [6] gave the concept of a soft semiring and many related concepts. Aygünoğlu and Aygün [7] introduced fuzzy soft groups and Varol et al. [8] studied fuzzy soft rings.Shabir and Naz [9] defined soft topological spaces and many related basic concepts. Aygünoğlu and Aygün [10] also studied soft topological spaces. Fuzzy soft topology was introduced by Tanay and Kandemir [11] and it was further studied by Varol and Aygün [12], Mahanta and Das [13], Mishra and Srivastava [14], and so forth.In [15], Varol et al. have introduced a soft topology and an L-fuzzy soft topology in a different way, using Šostak approach [16], and studied soft compactness and L-fuzzy soft compactness.In fuzzy topological spaces, compactness was first introduced by Chang [17], but it is well known by now that compactness in the sense of Chang does not satisfy the Tychonoff property. Lowen [18] introduced compactness in a fuzzy topological space, in another way which satisfies the Tychonoff property and has many other desirable properties.Gain et al. [19], Osmanoğlu and Tokat [20], and Sreedevi and Ravi Shankar [21] have studied fuzzy soft compactness in a fuzzy soft topological space introduced by Tanay and Kandemir [11]. These authors have introduced fuzzy soft compactness as a generalization of Chang’s fuzzy compactness.In this paper, we have introduced and studied fuzzy soft compactness as a generalization of Lowen’s fuzzy compactness, in a fuzzy soft topological space introduced by Varol and Aygün [12].Several basic desirable results have been established. In particular, we have proved the counterparts of Alexander’s subbase lemma and Tychonoff theorem for fuzzy soft topological spaces.


## 2. Preliminaries

Throughout this paper,X denotes a nonempty set, called the universe, E denotes the set of parameters for the universe X, and A⊆E.Definition 1 (see [22]).
A fuzzy set inX is a function f:X→[0,1]. Now we define some basic fuzzy set operations as follows.
Letf and g be fuzzy sets in X. Then (1)
f=g if f(x)=g(x),∀x∈X.(2)
f⊆g if f(x)⩽g(x),∀x∈X.(3)
(f∪g)(x)=max⁡{f(x),g(x)},∀x∈X.(4)
(f∩g)(x)=min⁡fx,gx,∀x∈X.(5)
fc(x)=1-f(x),∀x∈X (here fc denotes the complement of f).The constant fuzzy set inX, taking value α∈[0,1], will be denoted by αX.Definition 2 (see [23]).
LetΩ be an index set and {fi:i∈Ω} be a family of fuzzy sets in X. Then their union ⋃i∈Ωfi and intersection ⋂i∈Ωfi are defined, respectively, as follows: (1)
⋃i∈Ωfix=sup⁡fix:i∈Ω,∀x∈X.(2)
(⋂i∈Ωfi)(x)=inf⁡fix:i∈Ω,∀x∈X.Definition 3 (see [24]).
A fuzzy pointxλ(0<λ<1) in X is a fuzzy set in X given by (1)xλx′=λ,ifx′=x0,otherwise.Here x and λ are, respectively, called thesupport and thevalue of xλ.Definition 4 (see [18]).
A fuzzy topological space is a pair(X,T), where X is a nonempty set and T is a family of fuzzy sets in X such that the following conditions are satisfied: (1)
αX∈T,∀α∈[0,1].(2)
Iff,g∈T, then f∩g∈T.(3)
If{fi:i∈Ω} is an arbitrary family of fuzzy sets in T, then ⋃i∈Ωfi∈T.
ThenT is called a fuzzy topology on X and members of T are called fuzzy open sets. A fuzzy set in X is called fuzzy closed if fc∈T.Definition 5 (see [18]).
A fuzzy setf in X is said to be fuzzy compact if for any family β⊆T such that ⋃μ∈βμ⊇f and for all ϵ>0, there exists a finite subfamily βo⊆β such that ⋃μ∈βoμ⊇f-ϵX.Definition 6 (see [18]).
A fuzzy topological space(X,T) is said to be fuzzy compact if each constant fuzzy set in X is fuzzy compact.Definition 7 (see [2]).
A pair(f,E) is called a fuzzy soft set over X if f is a mapping from E to IX; that is, f:E→IX, where IX is the collection of all fuzzy sets in X.Definition 8 (see [12]).
A fuzzy soft setfA over X is a mapping from E to IX; that is, fA:E→IX such that fA(e)≠0X, if e∈A⊆E and fA(e)=0X, otherwise, where 0X denotes the constant fuzzy set in X taking value 0.Definition 9 (see [12]).
A constant fuzzy soft setαE over X, where α∈[0,1], is the fuzzy soft set over X such that αE(e)=αX,∀e∈E.From here onwards, we will denote byF(X,E) the collection of all fuzzy soft sets over X, where E is the parameters set for X.Definition 10 (see [12]).
LetfA,gB∈F(X,E). Then (1)
fA is said to be afuzzy soft subset of gB (or that fA is contained in gB), denoted by fA⊑gB, if fA(e)⊆gB(e),∀e∈E(2)
fA and gB are said to beequal, denoted by fA=gB, if fA⊑gB and gB⊑fA(3)
Theunion of fA and gB, denoted by fA⊔gB, is the fuzzy soft set over X defined by (2)fA⊔gBe=fAe∪gBe,∀e∈E(4)
Theintersection of fA and gB, denoted by fA⊓gB, is the fuzzy soft set over X defined by (3)fA⊓gBe=fAe∩gBe,∀e∈ETwo fuzzy soft setsfA and gB over X are said to bedisjoint if fA⊓gB=0E(5)
LetΩ be an index set and {fAi:i∈Ω} be a family of fuzzy soft sets over X. Then theirunion⨆i∈Ω⁡fAi andintersection⨅i∈Ω⁡fAi are defined, respectively, as follows:
(a)
(⨆i∈Ω⁡fAi)(e)=⋃i∈ΩfAi(e),∀e∈E(b)
(⨅i∈Ω⁡fAi)(e)=⋂i∈ΩfAi(e),∀e∈E(6)
Thecomplement of fA, denoted by fAc, is the fuzzy soft set over X, defined by (4)fAce=1X-fAe,∀e∈EDefinition 11 (see [7]).
LetF(X,E) and F(Y,K) be the collections of all fuzzy soft sets over the universe sets X and Y, respectively, where E and K are the parameters sets for X and Y, respectively. Let φ:X→Y and ψ:E→K be two maps. Then the pair (φ,ψ) is called a fuzzy soft mapping from X to Y and is denoted by (5)φ,ψ:FX,E⟶FY,K.(1)
LetfA∈F(X,E). Then theimage of fA under the fuzzy soft mapping (φ,ψ) is the fuzzy soft set over Y, denoted by (φ,ψ)fA, and is defined as follows: (6)φ,ψfAky=supφx=ysupψe=kfAex,ifφ-1y≠ϕ,ψ-1k≠ϕ0,otherwise,∀k∈K,∀y∈Y.(2)
LetgB∈F(Y,K). Then theinverse image of gB under the fuzzy soft mapping (φ,ψ) is the fuzzy soft set over X, denoted by (φ,ψ)-1gB, and is defined as follows: (7)φ,ψ-1gBex=gBψeφx,∀e∈E,∀x∈X.Proposition 12 (see [25]).
Let(ϕ,ψ):(X,τ)→(Y,δ) be a fuzzy soft mapping and fA1 and fA2 be fuzzy soft sets over X such that fA1⊑fA2. Then (ϕ,ψ)fA1⊑(ϕ,ψ)fA2.Proposition 13 (see [25]).
Let(ϕ,ψ):(X,τ)→(Y,δ) be a fuzzy soft mapping and {fAi:i∈Ω} be a family of fuzzy soft sets over X. Then(1)
ϕ,ψ⨆i∈Ω⁡fAi=⨆i∈Ω⁡(ϕ,ψ)fAi.(2)
(ϕ,ψ)(⨅i∈Ω⁡fAi)⊑⨅i∈Ω⁡(ϕ,ψ)fAi.Proposition 14 (see [25]).
Let(ϕ,ψ):(X,τ)→(Y,δ) be a fuzzy soft mapping and {gBi:i∈Ω} be a family of fuzzy soft sets over Y. Then (1)
(ϕ,ψ)-1(⨆i∈Ω⁡gBi)=⨆i∈Ω⁡(ϕ,ψ)-1gBi.(2)
(ϕ,ψ)-1(⨅i∈ΩgBi)=⨅i∈Ω(ϕ,ψ)-1gBi.Definition 15.
LetfA and gB be fuzzy soft sets over X such that fA⊒gB. Then fA-gB is the fuzzy soft set over X given by (8)fA-gBe=fAe-gBe,∀e∈E.Definition 16 (see [12]).
LetfA∈F(X,E) and gB∈F(Y,K). Then the fuzzy soft product of fA and gB, denoted by fA×gB, is the fuzzy soft set over X×Y and is defined by (9)fA×gBe,k=fAe×gBk,∀e,k∈E×K,and, for (x,y)∈X×Y,(10)fAe×gBkx,y=min⁡fAex,gBky.Definition 17 (see [11, 12]).
A fuzzy soft topological space relative to the parameters setE is a pair (X,τ) consisting of a nonempty set X and a family τ of fuzzy soft sets over X satisfying the following conditions: (1)
αE∈τ,∀α∈[0,1].(2)
IffA,gB∈τ, then fA⊓gB∈τ.(3)
IffAj∈τ,∀j∈Ω, where Ω is some index set, then ⨆j∈Ω⁡fAj∈τ.
Thenτ is called afuzzy soft topology over X and members of τ are calledfuzzy soft open sets. A fuzzy soft set gB over X is calledfuzzy soft closed if (gB)c∈τ.We mention here that the fuzzy soft topology defined above has been called “enriched fuzzy soft topology” in [12].Definition 18 (see [12]).
Let(X,τ) be a fuzzy soft topological space. Then a subfamily B of τ is called a base for τ if every member of τ can be written as a union of members of B.Definition 19 (see [12]).
Let(X,τ) be a fuzzy soft topological space. Then a subfamily S of τ is called a subbase for τ if the family of finite intersections of its members forms a base for τ.Definition 20 (see [12]).
A fuzzy soft topologyτ over X is said to be generated by a subfamily S of fuzzy soft sets over X if every member of τ is a union of finite intersections of members of S.Definition 21 (see [12]).
Let{(Xi,τi)}i∈Ω be a family of fuzzy soft topological spaces relative to the parameters sets Ei, respectively, and, for each i∈Ω, let (φ,ψ)i:X→(Xi,τi) be a fuzzy soft mapping. Then the fuzzy soft topology τ over X is said to be initial with respect to the family {(φ,ψ)i}i∈Ω if τ has as subbase the set (11)S=φ,ψi-1fAi:i∈Ω,fAi∈τi;that is, the fuzzy soft topology τ over X is generated by S.Definition 22 (see [12]).
Let{(Xi,τi)}i∈Ω be a family of fuzzy soft topological spaces relative to the parameters sets Ei, respectively. Then their product is defined as the fuzzy soft topological space (X,τ) relative to the parameters set E, where X=∏iXi, E=∏iEi, and τ is the fuzzy soft topology over X which is initial with respect to the family {(pXi,qEi)}i∈Ω, where pXi:∏iXi→Xi and qEi:∏iEi→Ei are the projection maps; that is, τ is generated by (12)pXi,qEi-1fAi:i∈Ω,fAi∈τi.Definition 23 (see [14]).
A fuzzy soft pointexλ over X is a fuzzy soft set over X defined as follows: (13)exλe′=xλ,ife′=e0X,ife′∈E-e,where xλ is the fuzzy point in X with support x and value λ, λ∈(0,1).
A fuzzy soft pointexλ is said tobelong to a fuzzy soft set fA, denoted by exλ∈fA, if λ<fA(e)(x) and two fuzzy soft points exλ and eys′ are said to bedistinct if x≠y or e≠e′.Proposition 24 (see [14]).
Let(X,τ) be a fuzzy soft topological space. Then a fuzzy soft set fA is fuzzy soft open iff ∀exr∈fA; there exists a basic fuzzy soft open set gB such that exr∈gB⊑fA.Definition 25 (see [26]).
A family of sets is said to be of finite character iff each finite subset of a member of the family is also a member, and each set belongs to this family if each of its finite subsets belong to it.Lemma 26 (TUKEY, [26]).
Each nonempty family of sets of finite character has a maximal element.


## 3. Fuzzy Soft Compact Topological Spaces

Definition 27.
Let(X,τ) be a fuzzy soft topological space relative to the parameters set E. Then a fuzzy soft set fA over X is said to be fuzzy soft compact if, for any family β⊆τ such that ⨆gB∈βgB⊒fA and ∀ϵ such that fA⊒ϵE, there exists a finite subfamily βo of β such that ⨆gB∈βogB⊒fA-ϵE.Definition 28.
A fuzzy soft topological space(X,τ) relative to the parameters set E is said to be fuzzy soft compact if each constant fuzzy soft set over X is fuzzy soft compact; that is, for α∈[0,1], if there exists a family β of fuzzy soft open sets over X such that ⨆fA∈βfA⊒αE, then ∀ϵ∈(0,α); there exists a finite subfamily βo of β such that ⨆fA∈βofA⊒(α-ϵ)E.Proposition 29.
Let(X,τ) and (Y,δ) be fuzzy soft topological spaces relative to parameters sets E and K, respectively, (ϕ,ψ):(X,τ)→(Y,δ) be a fuzzy soft continuous mapping, and fA∈F(X,E) be fuzzy soft compact. Then (ϕ,ψ)fA is fuzzy soft compact.Proof.
Letβ⊆δ be such that(14)⨆gB∈β⁡gB⊒ϕ,ψfA⟹⨆gB∈β⁡ϕ,ψ-1gB⊒fA.Since {(ϕ,ψ)-1gB}gB∈β is a family of fuzzy soft open sets over X and fA is fuzzy soft compact, so, ∀ϵ such that fA⊒ϵE, there exists a finite subfamily βo⊆β such that (15)⨆gB∈βo⁡ϕ,ψ-1gB⊒fA-ϵE.Then applying (ϕ,ψ) on both sides, we get (16)⨆gB∈βo⁡gB⊒ϕ,ψfA-ϵE=ϕ,ψfA-ϵK,which implies that (ϕ,ψ)fA is fuzzy soft compact.From the fact that(ϕ,ψ) is surjective if ϕ and ψ both are surjective (cf. [12]), each constant fuzzy soft set αK over Y is the image of constant fuzzy soft set αE over X. Hence we have the following result.Corollary 30.
Let(X,τ) and (Y,δ) be fuzzy soft topological spaces where (X,τ) is fuzzy soft compact and (ϕ,ψ) be a surjective fuzzy soft continuous mapping from (X,τ) to (Y,δ). Then (Y,δ) is fuzzy soft compact.As in the case of soft topological spaces [10], here we have the following.Definition 31.
Let(X,τ) be a fuzzy soft topological space relative to the parameters set E. Then, for e∈E, the e-parameter fuzzy topological spaces are given by (X,τe), where τe={fA(e):fA∈τ}.The following proposition is a counterpart of Theorem  4.1 in [10].Proposition 32.
Let(X,τ) be a fuzzy soft topological space relative to the parameters set E, which is finite. Then (X,τ) is fuzzy soft compact if each e-parameter fuzzy topological space is fuzzy compact.Proof.
Suppose that eache-parameter fuzzy topological space is fuzzy compact. Then to show that αE,α∈[0,1] is fuzzy soft compact, consider a family β of fuzzy soft open sets over X such that (17)αE⊑⨆fA∈β⁡fA⟹αEe⊆⋃fA∈βfAe,∀e∈E⟹αX⊆⋃fA∈βfAe,∀e∈E.Then, for e∈E, by fuzzy compactness of (X,τe), for ϵ∈(0,α), there exists a finite subfamily βoe of β such that α-ϵX⊆⋃fA∈βoefA(e).
Now setβo=⋃e∈Eβoe. Then βo is a finite subfamily of β such that α-ϵX⊆⋃fA∈βofA(e),∀e∈E. Hence (α-ϵ)E⊑⨆fA∈βofA, which shows that (X,τ) is fuzzy soft compact.Now we consider the mappings (cf. [27]) h:F(X,E)→IX×E, where IX×E is the set of all fuzzy sets in X×E, defined as follows: (18)hfAx,e=fAex,∀fA∈FX,E,and g:IX×E→F(X,E) as follows: (19)gU=fEU,∀U∈IX×E,where∀e∈E,∀x∈X,fEUex=Ux,e.In view of the above, we state the following theorem proved in [27].Theorem 33 (see [27]).
( F X , E , ⊔ , ⊓ , )   c is isomorphic to (IX×E∪,∩,)c, where IX×E denotes the set of all fuzzy sets in X×E.Then it is easy to verify that if(X,τ) is a fuzzy soft topological space relative to the parameters set E, then (X×E,h(τ)) is a fuzzy topological space and also if (X×E,T) is a fuzzy topological space, then (X,g(T)) is a fuzzy soft topological space relative to the parameters set E, where h(τ)={h(fA):fA∈τ} and g(T)={g(U):U∈T} (cf. [27]).Proposition 34.
A fuzzy soft topological space(X,τ) relative to the parameters set E is fuzzy soft compact iff (X×E,h(τ)) is fuzzy compact.Proof.
First, suppose that(X,τ) is fuzzy soft compact. Then to show that (X×E,h(τ)) is fuzzy compact, consider a family β⊆h(τ) of fuzzy open sets in X×E such that (20)αX×E⊆⋃ν∈βν⟹gαX×E⊑g⋃ν∈βν⟹αE⊑⨆ν∈β⁡gν.Note that if ν∈β⊆h(τ), then g(ν)∈τ. So by fuzzy soft compactness of (X,τ), for ϵ∈(0,α), there exists a finite subfamily βo of β such that (21)α-ϵE⊑⨆ν∈βo⁡gν⟹hα-ϵE⊆h⨆ν∈βo⁡gν⟹α-ϵX×E⊆⋃ν∈βohogν⟹α-ϵX×E⊆⋃ν∈βoν,which proves the fuzzy compactness of (X×E,h(τ)).
Conversely, assume that(X×E,h(τ)) is fuzzy compact. To show that (X,τ) is fuzzy soft compact, we have to show that αE,α∈[0,1] is fuzzy soft compact. Let β⊆τ such that(22)αE⊑⨆fA∈β⁡fA⟹hαE⊆h⨆fA∈β⁡fA⟹αX×E⊆⋃fA∈βhfA.Then by the fuzzy compactness of (X×E,h(τ)), for ϵ∈(0,α), there exists a finite subfamily βo of β such that (23)α-ϵX×E⊆⋃fA∈βohfA⟹gα-ϵX×E⊑g⋃fA∈βohfA⟹α-ϵE⊑⨆fA∈βo⁡gohfA⟹α-ϵE⊑⨆fA∈βo⁡fA,which proves that the fuzzy soft topological space (X,τ) is fuzzy soft compact.Now we prove the counterparts of the well known Alexander’s subbase lemma and the Tychonoff theorem for fuzzy soft topological spaces, the proofs of which are based on the proofs of the corresponding results given in [18] and [28], respectively.Theorem 35.
Let(X,τ) be a fuzzy soft topological space relative to the parameters set E. Then (X,τ) is fuzzy soft compact iff, for any subbase σ for τ, if there is a family β⊆σ such that ⨆fA∈βfA⊒αE, then, for ϵ∈(0,α), there exists a finite subfamily βo of β such that ⨆fA∈βofA⊒(α-ϵ)E.Proof.
First assume that(X,τ) is fuzzy soft compact. Choose β⊆σ such that ⨆fA∈βfA⊒αE,α∈[0,1]. Now since σ⊆τ and (X,τ) is fuzzy soft compact, there exists a finite subfamily βo of β such that ⨆fA∈βofA⊒(α-ϵ)E.
Conversely, to show that(X,τ) is fuzzy soft compact, we have to show that if, for β⊆τ, there exist α and ϵ∈(0,α) such that there does not exist any finite subfamily βo of β, such that ⨆fA∈βofA⊒(α-ϵ)E, then it must follow that ⨆fA∈βfA does not contain αE.
Consider the family(24)C=β⊆τ:  there  does  not  exist  any  finite  subfamilyβoofβsuch  that⨆fA∈βo⁡fA⊒α-ϵE.Then C is of finite character. Now it follows from Tukey’s lemma that, ∀β∈C, there exists a maximal element β′∈C containing β.
Next, we show that iffA∈β′ and fA1,fA2,…,fAn∈τ such that fA⊒fA1⊓fA2⊓⋯⊓fAn, then there exists some k, k=1,2,…,n such that fAk∈β′. For this we proceed as follows.
If we takefA1∈τ such that fA1∉β′, then since β′ is maximal, for the family {fA1}∪β′, there exists a finite subfamily {fA1,gB1,gB2,…,gBp}, where gBi∈β′,∀i=1,2,…,p, such that fA1⊔gB1⊔gB2⊔⋯⊔gBp⊒(α-ϵ)E, which implies that if, for e∈E and x∈X, (25)sup⁡gB1ex,gB2ex,…,gBpex<α-ϵ,then(26)fA1ex≥α-ϵ.Similarly, if we take another fA2∈τ such that fA2∉β′, then again since β′ is maximal, for the family {fA2}∪β′, there exists a finite subfamily {fA2,gB1′,gB2′,…,gBq′}, where gBi′∈β′,∀i=1,2,…,q, such that fA2⊔gB1′⊔gB2′⊔⋯⊔gBq′⊒(α-ϵ)E, which implies that if, for e′∈E and x′∈X, (27)sup⁡gB1e′x′,gB2e′x′,…,gBqe′x′<α-ϵ,then(28)fA2e′x′≥α-ϵ.Now we show that (29)fA1⊓fA2⊔gB1⊔gB2⊔⋯⊔gBp⊔gB1′⊔gB2′⊔⋯⊔gBq′⊒α-ϵE,as follows.
If fork∈E and y∈X, (gB1⊔gB2⊔⋯⊔gBp⊔gB1′⊔gB2′⊔⋯⊔gBq′)(k)(y)<α-ϵ which implies that supj=1,2,…,pgBj(k)(y)<α-ϵ and supi=1,2,…,qgBi′(k)(y)<α-ϵ, then, from (26) and (28), we get fA1(k)(y)≥α-ϵ and fA2(k)(y)≥α-ϵ. This implies that (fA1⊓fA2)(k)(y)≥α-ϵ and hence fA1⊓fA2∉β′. Thus, in general, if fA1,fA2,…,fAn do not belong to β′, then fA1⊓fA2⊓⋯⊓fAn does not belong to β′ implying that there is no fuzzy soft open set containing fA1⊓fA2⊓⋯⊓fAn which belong to β′. Thus we have shown that if fAi,i=1,2,…,n do not belong to β′, then no fuzzy soft open set fA such that fA⊒fA1⊓fA2⊓⋯⊓fAn belongs to β′. Equivalently, if fA∈β′ such that fA⊒fA1⊓fA2⊓⋯⊓fAn, then there exists some k, k=1,2,…,n such that fAk∈β′.
Next, considerβ′∩σ. Then, from the assumption of the theorem, we have that ⨆fA∈β′∩σfA does not contain αE. Now we show ⨆fA∈β′fA⊑⨆gB∈β′∩σgB. Since, ∀fA∈β′,∀e∈E and ∀x∈X such that fA(e)(x)>0 and ∀a<fA(e)(x), then efA(e)(x)-a∈fA and hence, using Proposition 24, there exist fA1a,fA2a,…,fAna∈σ such that fA1a⊓fA2a⊓⋯⊓fAna⊑fA and fA1a(e)(x)∧fA2a(e)(x)∧⋯∧fAna(e)(x)>fA(e)(x)-a. Since fA∈β′ and β′ is maximal, it follows that there exists some k,k=1,2,…,n such that fAka∈β′. Thus, ∀a>0, there exists fAka such that fAka(e)(x)>fA(e)(x)-a,∀e∈E,∀x∈X and fAka∈β′∩σ. Now fix e and x both. Then ∀fA∈β′ such that fA(e)(x)>0 and a<fA(e)(x), where a>0, there exists fAa∈β′∩σ such that(30)fAaex>fAex-a⟹⨆gB∈β′∩σ⁡gBex≥fAex⟹⨆fA∈β′⁡fA⊑⨆gB∈β′∩σ⁡gB.This implies that ⨆fA∈β′fA does not contain αE and hence ⨆fA∈βfA does not contain αE. Thus, (X,τ) is fuzzy soft compact.Theorem 36.
If{(Xi,τi):i∈Ω} is a family of fuzzy soft topological spaces relative to the parameters sets Ei, respectively, then the product of fuzzy soft topological space,(31)X,τ=∏i∈ΩXi,τi,is fuzzy soft compact if and only if each coordinate fuzzy soft topological space (Xi,τi) is fuzzy soft compact.Proof.
Let us first assume that each coordinate space(Xi,τi),i∈Ω, is fuzzy soft compact. From the previous theorem, to show that (X,τ) is fuzzy soft compact, it is sufficient to show that for any family β⊆σ={(pXi,qEi)-1(fAi):i∈Ω,fAi∈τi}, if there exist α,ϵ,α>ϵ>0 such that ∀ finite subfamily βo of β, ⨆fA∈βofA does not contain (α-ϵ)E, then it must follow that ⨆fA∈βfA does not contain αE.
Letβ be such a family. Then, ∀j∈Ω, put βj={fAj∈τj:(pXj,qEj)-1(fAj)∈β}. Then, ∀ finite subfamily (βj)o of βj, {(pXj,qEj)-1(fAj):fAj∈(βj)o} is a finite subfamily of β. Hence, from our assumption there exist some e∈E and x∈X such that (32)⨆fAj∈βjo⁡pXj,qEj-1fAjex<α-ϵ⟹supfAj∈βjo⁡fAjejxj<α-ϵ2-ϵ2.Since the above inequality holds ∀ finite subfamily (βj)o of βj, so, from the fuzzy soft compactness of (Xj,τj), there exist ej′∈Ej and xj′∈Xj such that (33)supfAj∈βj⁡fAjej′xj′<α-ϵ2.The same inequality holds for all j∈Ω. Finally, if we set e=(ej′)j∈Ω and x=(xj′)j∈Ω, then(34)⨆fA∈β⁡fAex=supfA∈β⁡fAex=supj∈Ω⁡supfA∈β∩pXj,qEj-1τj⁡fAex=supj∈Ω⁡supfAj∈βj⁡pXj,qEj-1fAjex=supj∈Ω⁡supfAj∈βj⁡fAjej′xj′≤α-ϵ2<α⟹⨆fA∈β⁡fAdoes  not  containαE.
The converse part follows using Corollary30 as well as the fact that (pXj,qEj) are fuzzy soft continuous maps, ∀j∈Ω.


## 4. Conclusion

Soft sets were introduced by Molodtsov in 1999 [1]. Maji et al. [2] introduced and studied fuzzy soft sets. The theory of fuzzy soft topology was initiated by Tanay and Kandemir [11] and further studied by Varol and Aygün [12], Mahanta and Das [13], and so forth. In this paper we have introduced fuzzy soft compactness in a fuzzy soft topological space, which is an extension of Lowen’s concept of fuzzy compactness in the case of fuzzy topological spaces. Several basic desirable results have been obtained. In particular the counterparts of Alexander’s subbase lemma and the Tychonoff theorem for fuzzy soft topological spaces have been proved.



---

*Source: 2480842-2016-12-25.xml*