# σ -Algebra and σ-Baire in Fuzzy Soft Setting

Authors: Shuker Mahmood Khalil; Mayadah Ulrazaq; Samaher Abdul-Ghani; Abu Firas Al-Musawi
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

We first introduce some new notions of Baireness in fuzzy soft topological space (FSTS). Next, their characterizations and basic properties are investigated in this work. The notions of fuzzy soft dense, fuzzy soft nowhere dense, fuzzy soft meager, fuzzy soft second category, fuzzy soft residual, fuzzy soft Baire, fuzzy softδ-sets, fuzzy soft λσ-sets, fuzzy soft σ-nowhere dense, fuzzy soft σ-meager, fuzzy soft σ-residual, fuzzy soft σ-Baire, fuzzy soft σ-second category, fuzzy soft σ-residual, fuzzy, fuzzy soft submaximal space, fuzzy soft P-space, fuzzy soft almost resolvable space, fuzzy soft hyperconnected space, fuzzy soft A-embedded, fuzzy soft D-Baire, fuzzy soft almost P-spaces, fuzzy soft Borel, and fuzzy soft σ-algebra are introduced. Furthermore, several examples are shown as well.


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## Body

## 1. Introduction

The concepts of Baire spaces have been studied and discussed extensively in general topology in [1–4]. Thangaraj and Balasubramanian [5] studied the notion of somewhat fuzzy continuous functions. Next, Thangaraj and Anjalmose investigated and discussed the notion of Baire space in fuzzy topology [6]. After that, they introduced the notion of fuzzy Baire space [7].Soft sets theory was introduced by Molodtsov [8]. It explains new type of mathematical tool of soft sets and it deals with vagueness when solving problems in practice as in engineering, environment, social science, and economics, which cannot be handled as classical mathematical tools. Also, other authors such as Maji et al. [9–19] have further studied the theory of soft sets and used this theory in pure mathematics to solve some decision making problems. Next, the notion of fuzzy soft set is investigated and discussed [20–22]. Since then much attention has been used to generalize the basic notions of fuzzy topology in soft setting. In other words, the modern theories of fuzzy soft topology have been developed.In recent years, fuzzy soft topology has been found to be very useful in solving many practical problems [23]. Also, rough fuzzy and other applications are studied [24–26]. The main purpose of this work is to introduce new concepts of fuzzy soft Baireness in fuzzy soft topological spaces. In section three, we introduce fuzzy soft dense sets, fuzzy soft nowhere dense sets, fuzzy soft meager sets, fuzzy soft second category sets, fuzzy soft meager spaces, fuzzy soft second category spaces, fuzzy soft residual sets, fuzzy soft Baire spaces, fuzzy soft δ-sets, fuzzy soft λσ-sets, fuzzy soft σ-nowhere dense, fuzzy soft σ-meager, fuzzy soft σ-residual, fuzzy soft σ-Baire, fuzzy soft σ-second category, fuzzy soft σ-residual, fuzzy, fuzzy soft submaximal space, fuzzy soft P-space, fuzzy soft almost resolvable space, fuzzy soft hyperconnected space, fuzzy soft A-embedded, fuzzy soft D-Baire, fuzzy soft almost P-spaces, fuzzy soft Borel, and fuzzy soft σ-algebra. Moreover, several examples are given to illustrate the notions introduced in this work.


## 2. Preliminaries

In this section, we give few definitions and properties regarding fuzzy soft sets.Definition 1 ([20]).
AssumeU is an initial universe set and E is a set of parameters. Let IU refer to the family of all fuzzy soft sets (FSSs) in U and A⊆E. The multivalued map FA:A→IU defined by FA(e)=μFAe is said to be a fuzzy soft set (FSS) over (U,E), where μFAe≠0¯ if e∈A and μFAe=0¯ if e∈E∖A. We refer to family of all (FSSs) over (U,E) by FS(U,E).Definition 2 ([20]).
We sayFϕ∈FS(U,E) is null (FSS) and we refer to it by Φ, if ∀e∈E, F(e) is the null (FSS) 0¯ of U, where 0¯(x)=0∀x∈U.Definition 3 ([20]).
AssumeFE∈FS(U,E) and FE(e)=1¯∀e∈E, where 1¯(x)=1∀x∈U. We say FE is absolute (FSS) and we refer to it by E¯.Definition 4 ([20]).
We sayFA is a fuzzy soft subset of a (FSS) GB over a common universe U if A⊆B and FA(e)⊆GB(e)∀e∈A; i.e., if μFAe(x)≤μGBe(x)∀x∈U and ∀e∈E and denoted by FA⊆~GB.Definition 5 ([20]).
AssumeFA and GB are (FSSs) over a common universe U. We say they are fuzzy soft equal if FA⊆~GB and GB⊆~FA.Definition 6 ([20]).
AssumeFA and GB are (FSSs) over a common universe U. Their union is the (FSS) HC, defined by HC(e)=μHCe=μFAe∪μGBe∀e∈E, where C=A⋃B. For this case, we write HC=FA∪~GB.Definition 7 ([20]).
AssumeFA and GB are (FSSs) over a common universe U. Their intersection is a (FSS) HC, defined by HC(e)=μHCe=μFAe∩μGBe∀e∈E, where C=A⋂B. For this case, we write HC=FA∩~GB.Definition 8 ([27]).
AssumeFA∈FS(U,E) is a (FSS). We refer to its complement by FAc, and it is defined as(1)FAc=1¯-μFAeife∈A,1¯ife∉A.Remark 9.
LetK={FAii∣i∈I} be a family of fuzzy soft sets. The union (intersection) of any number (J⊆I) of family K is defined by HC(e)=μHCe=⋃j∈J{μFAiie}(HC(e)=μHCe=⋂j∈J{μFAiie}), where C=⋃j∈J{Ai}(C=⋂j∈J{Ai}).Definition 10 ([28]).
Assumeψ is the family of (FSSs) over U. We say ψ is a fuzzy soft topology on U if ψ the following axioms hold:
(i)Φ,E¯ belong to ψ.
(ii) The union of any number of (FSSs) inψ belongs to ψ.
(iii) The intersection of any two (FSSs) inψ belongs to ψ.
We say(U,E,ψ) is a fuzzy soft topological space (FSTS) over U. Each member in ψ is called (FSOS) (FSOS) in U and its complement is called fuzzy soft closed set (FSCS) in U, where C=⋃j∈J{Ai}(C=⋂j∈J{Ai}).Definition 11 ([28]).
The union of all fuzzy soft open subsets ofFA over (U,E) is called the interior of FA and is denoted by intfs(FA).Proposition 12 ([28]).
i n t f s ( F A ∩ ~ G B ) = i n t f s ( F A ) ∩ ~ i n t f s ( G B ).Definition 13 ([28]).
LetFA∈FS(U,E) be a (FSS). Then the intersection of all closed sets, each containing FA, is called the closure of FA and is denoted by clfs(FA).Remark 14.
(1) For any (FSS)FA in a (FSTS) (U,E,ψ), it is easy to see that (clfs(FA))c=intfs(FAc) and (intfs(FA))c=clfs(FAc).
(2) For any fuzzy softFA subset of a (FSTS) (U,E,ψ), we define the fuzzy soft subspace topology ψFA on FA by KD∈ψFA if KD=FA∩~GB for some GB∈ψ.
(3) For any fuzzy softHC in FA fuzzy soft subspace of a (FSTS), we denote the interior and closure of HC in FA by intFAfs(HC) and clFAfs(HC), respectively.


## 3. Fuzzy Softσ-Baire Spaces

In this section, we introduce new notions of (FSTSs) using new classes of (FSSs) which are introduced in this section and obtained their properties.Definition 15.
A (FSS)FA in a (FSTS) (U,E,ψ) is called fuzzy soft dense if there exists no (FSCS) GB in (U,E,ψ) such that FA⊂~GB⊂~E¯.Definition 16.
A (FSS)FA in a (FSTS) (U,E,ψ) is called fuzzy soft nowhere dense if there exists no nonempty (FSOS) GB in (U,E,ψ) such that GB⊂~clfs(FA). That is, intfs(clfs(FA))=Φ.Definition 17.
A (FSS)FA in a (FSTS) (U,E,ψ) is called fuzzy soft meager if FA is a countable union of fuzzy soft nowhere dense sets [i.e., if FA=⋃~i∈I{FAii}, where FAii’s are fuzzy soft nowhere dense sets in (U,E,ψ), ∀i∈I⊆N]. Otherwise, FA will be called a fuzzy soft second category set.Definition 18.
A (FSTS)(U,E,ψ) is called fuzzy soft meager or (fuzzy soft first category) space if the (FSS) E¯ is a fuzzy soft meager set in (U,E,ψ). That is, E¯=⋃~i∈I{FAii}, where (FAiic)’s are fuzzy soft nowhere dense sets in (U,E,ψ). Otherwise, (U,E,ψ) will be called a fuzzy soft second category space.Definition 19.
AssumeFA is a fuzzy soft meager set in (U,E,ψ). We say FAc is a fuzzy soft residual set in (U,E,ψ).Definition 20.
Assume(U,E,ψ) is a (FSTS). We say (U,E,ψ) is a fuzzy soft Baire space if each sequence {FA11,FA22,FA33,…} of fuzzy soft nowhere dense sets in (U,E,ψ) such that intfs(⋃~i∈I{FAii})=Φ.Definition 21.
A (FSS)FA in a (FSTS) (U,E,ψ) is called fuzzy soft δ-set in (U,E,ψ) if FA=⋂~i∈I{FAii}, where FAii∈ψ∀i∈I.Remark 22.
Definition20 can be written in other equivalent forms as follows:
( i ) We say (U,E,ψ) is a fuzzy soft Baire space if every countable fuzzy soft closed cover {FAii;i∈I} of E¯, the set ⋃~i∈I{intfs(FAii)} is fuzzy soft dense in E¯.
( i i ) We say (U,E,ψ) is a fuzzy soft Baire space if every sequence {FA11,FA22,FA33,…} of (FSOSs) with the same closure FA, we have FA=clfs(⋂~i∈I{FAii}).
( i i i ) We say (U,E,ψ) is a fuzzy soft Baire space if every fuzzy soft meager and fuzzy soft δ-set in E¯ is fuzzy soft nowhere dense.Remark 23.
We saySS(UE) is a family of all soft sets over a universe set U and the parameter set E. Moreover, the cardinality of SS(UE) is given by n(SS(UA))=2n(U)×n(E). Therefore, in this paper for each (FSS) FE over (U,E) we can define FE by using matrix form as follows:(2)FE=a11a12a13…a1ka21a22a23…a2k⋮⋮⋮⋱⋮am1am2am3…amkm×k
The order of this matrix is given bym×k, where m=n(E),k=n(U), and aij=μFEei(cj), ∀1≤i≤m and 1≤j≤k.Definition 24.
A (FSS)FA in a (FSTS) (U,E,ψ) is called fuzzy soft λσ-set in (U,E,ψ) if FA=⋃~i∈I{FAii}, where FAiic∈ψ∀i∈I.Definition 25.
A (FSS)FA in a (FSTS) (U,E,ψ) is called fuzzy soft σ-nowhere dense set if FA is a fuzzy soft λσ-set in (U,E,ψ) such that intfs(FA)=ϕ.Example 26.
As an illustration, consider the following example. Suppose the (FSSs)FE,GE,HE,JE,KE,LE,TE describe attractiveness of the cars with respect to the given parameters, which my friends are going to buy. U={x1,x2,x3} which is the set of all cars under consideration. Let IU be the collection of all fuzzy subsets of U. Also, let E={e1,e2,e3}, where e1,e2,e3 stand for the attributes of cheap, qualification, and colorful, respectively. Let FE,GE,HE,JE,KE,LE,TE be defined as follows:(3)FE=.2.3.6.15.25.10.3.12.6,GE=.4.7.3.8.75.7.8.10.9,HE=.1.5.3.5.75.4.8.4.1,JE=.2.5.6.5.75.4.8.4.6,KE=.2.3.3.15.25.10.3.10.6,LE=.1.3.3.15.25.10.3.12.1,TE=.4.7.6.8.75.7.8.12.9.
Then,ψ={Φ,E¯,FE,GE,HE,JE,KE,LE,TE} is clearly a fuzzy soft topology on E¯. Now consider the (FSS)(4)α=TEc∪~JEc=.8.5.4.5.25.6.2.88.4in (U,E,ψ). Then α is a fuzzy soft λσ-set in (U,E,ψ) and intfs(α)=Φ and hence α is a fuzzy soft σ-nowhere dense set in (U,E,ψ). The (FSS)(5)β=FEc∪~HEc=.9.7.7.85.75.90.7.88.9is a fuzzy soft λσ-set in (U,E,ψ) and intfs(β)=FE≠Φ and hence β is not a fuzzy soft σ-nowhere dense set in (U,E,ψ).Definition 27.
A (FSS)FA in a (FSTS) (U,E,ψ) is called fuzzy soft σ-meager if FA=⋃~i∈I{FAii}, where (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Any other (FSS) in (U,E,ψ) is said to be of fuzzy soft σ-second category.Definition 28.
A (FSTS)(U,E,ψ) is called fuzzy soft σ-meager space if the (FSS) E¯ is a fuzzy soft σ-meager set in (U,E,ψ). That is, E¯=⋃~i∈I{FAii}, where (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Otherwise, (U,E,ψ) will be called a fuzzy soft σ-second category space.Definition 29.
AssumeFA is a fuzzy soft σ-meager set in a (FSTS) (U,E,ψ). We say FAc is a fuzzy soft σ-residual set in (U,E,ψ).Proposition 30.
Let(U,E,ψ) be a (FSTS). Then the following are equivalent:
(1)(U,E,ψ) is a fuzzy soft Baire space.
(2)intfs(FA)=Φ for every fuzzy soft meager set FA in (U,E,ψ).
(3)clfs(GB)=E¯ for every fuzzy soft residual set GB in (U,E,ψ).Proof.
(1)⇒ (2). Let FA be a fuzzy soft meager set in (U,E,ψ). Then FA=(⋃~i∈I{FAii}), where (FAii)’s, i∈I are fuzzy soft nowhere dense sets in (U,E,ψ). Then, we have intfs(FA)=intfs(⋃~i∈I{FAii}). Since (U,E,ψ) is a fuzzy soft Baire space, intfs(⋃~i∈I{FAii})=Φ. Hence, intfs(FA)=Φ for any fuzzy soft meager set FA in (U,E,ψ).
(2)⇒ (3). Let GB be a fuzzy soft residual set in (U,E,ψ). Then GBc is a fuzzy soft σ-meager set in (U,E,ψ). By hypothesis, intfs(GBc)=Φ. Then (clfs(GB))c=Φ. Hence, clfs(GB)=E¯ for any fuzzy soft residual set GB in (U,E,ψ).
(3)⇒ (1). Let FA be a fuzzy soft meager set in (U,E,ψ). Then FA=(⋃~i∈I{FAii}), where (FAii)’s are fuzzy soft nowhere dense sets in (U,E,ψ). Now FA is a fuzzy soft meager set in (U,E,ψ) implying that FAc is a fuzzy soft residual set in (U,E,ψ). By hypothesis, we have clfs(FA)c=E¯. Then (intfs(FA))c=E¯. Hence, intfs(FA)=Φ. That is, intfs(⋃~i∈I{FAii})=Φ, where (FAii)’s are fuzzy soft nowhere dense sets in (U,E,ψ). Hence, (U,E,ψ) is a fuzzy soft Baire space.Proposition 31.
IfFA is a fuzzy soft dense and fuzzy soft δ-set in a (FSTS) (U,E,ψ), then FAc is a fuzzy soft meager set in (U,E,ψ).Proof.
SinceFA is a fuzzy soft δ-set in (U,E,ψ), FA=⋂~i∈I{FAii}, where FAii∈ψ and since FA is a fuzzy soft dense set in (U,E,ψ), clfs(FA)=E¯. Then clfs(⋂~i∈I{FAii})=E¯. But clfs(⋂~i∈I{FAii})⊆~⋂~i∈I{clfs(FAii)}. Hence, E¯⊆~⋂~i∈I{clfs(FAii)}. That is, ⋂~i∈I{clfs(FAii)}=E¯. Then we have clfs(FAii)=E¯ for each FAii∈ψ and hence clfs(intfs(FAii))=E¯ which implies that (clfs(intfs(FAii)))c=Φ and hence intfs(clfs(FAiic))=Φ. Therefore, (FAiic) is a fuzzy soft nowhere dense set in (U,E,ψ). Now FAc=(⋂~i∈I{FAii})c=⋃~i∈I{FAiic}. Therefore, FAc=⋃~i∈I{FAiic}, where (FAiic)’s are fuzzy soft nowhere dense sets in (U,E,ψ). Hence, FAc is a fuzzy soft meager set in (U,E,ψ).Lemma 32.
IfFA is a fuzzy soft dense and fuzzy soft δ-set in a (FSTS) (U,E,ψ), then FA is a fuzzy soft residual set in (U,E,ψ).Proof.
SinceFA is a fuzzy soft dense and fuzzy soft δ-set in (U,E,ψ), by Proposition 31, we have that FAc is a fuzzy soft meager set in (U,E,ψ) and hence FA is a fuzzy soft residual set in (U,E,ψ).Proposition 33.
In a (FSTS)(U,E,ψ), a (FSS) FA is a fuzzy soft σ-nowhere dense set in (U,E,ψ) if and only if FAc is a fuzzy soft dense and fuzzy soft δ-set in (U,E,ψ).Proof.
LetFA be a fuzzy soft σ-nowhere dense set in (U,E,ψ). Then FA=(⋃~i∈I{FAii}), where FAiic∈ψ, ∀i∈I and intfs(FA)=Φ. Then (intfs(FA))c=Φc=E¯ implies that clfs(FAc)=E¯. Also FAc=(⋃~i∈I{FAii})c=⋂~i∈I{FAiic}, where FAiic∈ψ, for i∈I. Hence, we have FAiic is a fuzzy soft dense and fuzzy soft δ-set in (U,E,ψ).
Conversely, letFA be a fuzzy soft dense and fuzzy soft δ-set in (U,E,ψ). Then FA=⋂~i∈I{FAii}, where FAii∈ψ for i∈I. Now FAc=(⋂~i∈I{FAii})c=⋃~i∈I{FAiic}. Hence, FAc is a λσ-set in (U,E,ψ) and intfs(FAc)=(clfs(FA))c=(E¯)c=Φ [since FA is a fuzzy soft dense]. Therefore, FAc is a fuzzy soft σ-nowhere dense set in (U,E,ψ).Definition 34.
Assume(U,E,ψ) is a (FSTS). We say (U,E,ψ) is a fuzzy soft σ-Baire space if each sequence {FA11,FA22,FA33,…} of fuzzy soft σ-nowhere dense sets in (U,E,ψ) such that intfs(⋃~i∈I{FAii})=Φ.Example 35.
LetU={c1,c2,c3} be the set of three flats and E=costly (e1), modern (e2), security services (e3) be the set of parameters. Then, we consider that ψ={Φ,E¯,FE,GE,HE,}∪~{TEi∣i=1,2,3,…,14} is a fuzzy soft topology on E¯ defined as follows:(6)FE=.7.5.6.6.4.5.5.3.4,GE=.5.8.7.4.7.6.3.6.5,HE=.6.4.8.5.3.7.4.2.6.TE1=FE∩~GE, TE2=FE∩~HE, TE3=GE∩~HE, TE4=FE∪~GE, TE5=FE∪~HE, TE6=GE∪~HE, TE7=FE∩~(GE∪~HE), TE8=FE∪~(GE∩~HE), TE9=GE∩~(FE∪~HE), TE10=GE∪~(FE∩~HE), TE11=HE∩~FE∪~GE, TE12=HE∪~(FE∩~GE), TE13=FE∪~GE∪~HE, and TE14=FE∩~GE∩~HE. Now consider the (FSSs)(7)α=FEc∪~TE8c∪~TE13c=.3.5.4.4.6.5.5.7.6,β=GEc∪~TE5c∪~TE10c=.5.5.3.6.6.4.7.7.5,and(8)θ=HEc∪~TE1c∪~TE4c∪~TE6c=.5.6.4.6.7.5.7.8.6,in (U,E,ψ). Then α,β and θ are fuzzy soft λσ-sets in (U,E,ψ) and intfs(α)=Φ, intfs(β)=Φ, and intfs(θ)=Φ.Then α,β and θ are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Moreover, (TE2)c∪~(TE3)c∪~(TE7)c∪~(TE9)c∪~(TE11)c∪~(TE12)c∪~(TE14)c=θ and also intfs(α∪~β∪~θ)=intfs(θ)=Φ and therefore (U,E,ψ) is a fuzzy soft σ-Baire space.Remark 36.
A fuzzy softσ-Baire space need not be a fuzzy soft Baire space. For, consider the following example.Example 37.
AssumeX={c1,c2,c3} is a set of soldiers under consideration and E=courageous (e1); strong (e2); smart(e3) is a set of parameters framed to choose the best soldier. Then, we consider that ψ={Φ,E¯,FE,GE,HE}∪~{TEi∣i=1,2,3,…,9} is a fuzzy soft topology on E¯ defined as follows:(9)HE=.8.5.1.7.4.1.9.6.1,FE=.1.3.8.1.2.7.1.4.9,GE=.4.1.3.3.1.2.5.1.4.TE1=FE∩~GE, TE2=FE∩~HE, TE3=GE∩~HE, TE4=FE∪~GE, TE5=FE∪~HE, TE6=GE∪~HE, TE7=GE∪~(FE∩~HE), TE8=HE∪~(GE∩~FE), and TE9=HE∩~(FE∪~GE). Now {FEc,GEc,HEc,(TE1)c,(TE2)c,(TE3)c,(TE5)c,(TE7)c,(TE8)c,(TE9)c} are fuzzy soft nowhere dense sets in (U,E,ψ). (TE1)c=FEc∪~GEc∪~HEc∪~(TE2)c∪~(TE3)c∪~(TE5)c∪~(TE7)c∪~(TE8)c∪~(TE9)c. Therefore, (TE1)c is a fuzzy soft meager set in (U,E,ψ). intfs((TE1)c)=TE1≠Φ. Hence, (U,E,ψ) is not a fuzzy soft Baire space. Now consider the (FSSs) α=(HE)c∪~(TE5)c∪~(TE6)c, and β=(FE)c∪~(TE2)c∪~(TE4)c∪~(TE7)c∪~(TE8)c in (U,E,ψ) and also intfs(α)=Φ, intfs(β)=Φ. Then α and β are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Now the (FSS) (α∪~β) is a fuzzy soft σ-meager set in (U,E,ψ) and intfs(α∪~β)=Φ. Hence, (U,E,ψ) is a fuzzy soft σ-Baire space.Proposition 38.
Let(U,E,ψ)be a (FSTS). Then the following are equivalent:
(1)(U,E,ψ) is a fuzzy soft σ-Baire space.
(2)intfs(FA)=Φ for every fuzzy soft σ-meager set FA in (U,E,ψ).
(3)clfs(GB)=E¯ for every fuzzy soft σ-residual set GB in (U,E,ψ).Proof.
(1)⇒ (2). Let FA be a fuzzy soft σ-meager set in (U,E,ψ). Then FA=(⋃~i∈I{FAii}), where ((FAii)’s, i∈I are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Then, we have intfs(FA)=intfs(⋃~i∈I{FAii}). Since (U,E,ψ) is a fuzzy soft σ-Baire space, intfs(⋃~i∈I{FAii})=Φ. Hence, intfs(FA)=Φ for any fuzzy soft σ-meager set FA in (U,E,ψ).
(2)⇒ (3). Let GB be a fuzzy soft σ-residual set in (U,E,ψ). Then GBc is a fuzzy soft σ-meager set in (U,E,ψ). By hypothesis, intfs(GBc)=Φ. Then (clfs(GBc))c=Φ. Hence, clfs(GB)=E¯ for any fuzzy soft σ-residual set GB in (U,E,ψ).
(3)⇒ (1). Let FA be a fuzzy soft σ-meager set in (U,E,ψ). Then FA=(⋃~i∈I{FAii}), where (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Now FA is a fuzzy soft σ-meager set in (U,E,ψ) implying that FAc is a fuzzy soft σ-residual set in (U,E,ψ). By hypothesis, we have clfs(FA)c=E¯. Then (intfs(FA))c=E¯. Hence, intfs(FA)=Φ. That is, intfs(⋃~i∈I{FAii})=Φ, where (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Hence, (U,E,ψ) is a fuzzy soft σ-Baire space.Proposition 39.
If the (FSTS)(U,E,ψ) is a fuzzy soft σ-Baire space, then clfs(⋃~i∈I{FAii})=E¯, where the (FSSs) (FAii)’s∀i∈I are fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ).Proof.
Let(FAii)’s, i∈I be fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ). By Proposition 33, (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Then the (FSS) FA=⋃~i∈I{FAiic} is a fuzzy soft σ-meager set in (U,E,ψ). Now intfs(FA)=intfs(⋃~i∈I{FAiic})=intfs(⋂~i∈I{FAii})c=(clfs(⋂~i∈I{FAii}))c. Since (U,E,ψ) is a fuzzy soft σ-Baire space, by Proposition 38, we have intfs(FA)=Φ. Then (clfs(⋂~i∈I{FAii}))c=Φ. This implies that clfs(⋂~i∈I{FAii})=E¯.Proposition 40.
If the (FSTS)(U,E,ψ) is a fuzzy soft σ-Baire space, then intfs(⋃~i∈I{FAiic})=Φ, where the (FSSs) sets (FAii)’s, i∈I are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft δ-sets FAii in (U,E,ψ).Proof.
Let the (FSTS)(U,E,ψ) be a fuzzy soft σ-Baire space and the (FSSs) (FAii)’s, i∈I be fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ). By Proposition 39, clfs(⋂~i∈I{FAii})=E¯. Then (clfs(⋂~i∈I{FAii}))c=Φ. This implies that intfs(⋃~i∈I{FAiic})=Φ. Also by Proposition 31, (FAii)’s are fuzzy soft meager sets in (U,E,ψ). Hence, intfs(⋃~i∈I{FAiic})=Φ, where the (FSSs) (FAii)’s, i∈I are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft δ-sets FAii in (U,E,ψ).Proposition 41.
If the fuzzy soft meager sets are formed from the fuzzy soft dense and fuzzy softδ-sets in a fuzzy soft σ-Baire space (U,E,ψ), then (U,E,ψ) is a fuzzy soft Baire space.Proof.
Let the (FSTS)(U,E,ψ) be a fuzzy soft σ-Baire space. By Proposition 40, intfs(⋃~i∈I{FAiic})=Φ, where the (FSSs) (FAii)’s, i∈I are fuzzy soft meager sets formed from the fuzzy soft dense and fuzzy soft δ-sets FAii in (U,E,ψ). Now ⋃~i∈I(intfs{FAiic})⊆~intfs(⋃~i∈I{FAiic}). Then we have ⋃~i∈I{intfs(FAiic)}=Φ. This implies that intfs(FAiic)=Φ, where (FAiic) is a fuzzy soft meager set in (U,E,ψ). By Proposition 30, (U,E,ψ) is a fuzzy soft Baire space.Proposition 42.
If the (FSTS)(U,E,ψ) is a fuzzy soft σ-meager space, then (U,E,ψ) is not a fuzzy soft σ-Baire space.Proof.
Let the (FSTS)(U,E,ψ) be a fuzzy soft σ-meager space. Then ⋃~i∈I{FAii}=E¯, where (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Now intfs(⋃~i∈I{FAii})=intfs(E¯)≠Φ. Hence, by definition, (U,E,ψ) is not a fuzzy soft σ-Baire space.Remark 43.
By Proposition42, we consider that if the (FSTS) (U,E,ψ) is a fuzzy soft σ-Baire space, then (U,E,ψ) is a fuzzy soft σ-second category space. Moreover, the converse is not true in general. That means a fuzzy soft σ-second category space need not be a fuzzy soft σ-Baire space.Example 44.
Let there be three houses in the universeU given by U={s1,s2,s3} and let E=stone (e1); steel (e2); and brick (e3) be the set of parameters framed to choose one house to rent, where (brick) means the brick built houses, (steel) means the steel built houses, and (stone) means the stone built houses. Then, we consider ψ={Φ,E¯,FE,GE,HE}∪~{TEi∣i=1,2,3,…,10} is a fuzzy soft topology defined as follows:(10)FE=.7.3.4.5.1.2.8.4.5,GE=.4.5.6.2.3.4.5.6.7,HE=.6.4.7.4.2.5.7.5.8.TE1=FE∩~GE, TE2=GE∩~HE, TE3=FE∩~HE, TE4=FE∪~GE, TE5=GE∪~HE, TE6=FE∪~HE, TE7=GE∪~(FE∩~HE), TE8=FE∪~(GE∩~HE), TE9=HE∩~(FE∪~GE), TE10=FE∪~GE∪~HE. Now consider the (FSSs) α=(GE)c∪~(TE4)c∪~(TE5)c, and β=(FE)c∪~(HE)c in (U,E,ψ). Then α and β are fuzzy soft λσ-sets in (U,E,ψ) and intfs(α)=Φ,intfs(β)=Φ. Then α and β are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Now (α∪~β)≠E¯. Therefore, (U,E,ψ) is a fuzzy soft σ-second category space. But intfs(α∪~β)≠Φ and therefore (U,E,ψ) is not a fuzzy soft σ-Baire space.Proposition 45.
If(⋂~i∈I{FAii})≠Φ, where the (FSSs) (FAii)’s are fuzzy soft dense and fuzzy soft δ-sets in a (FSTS) (U,E,ψ), then (U,E,ψ) is a fuzzy soft σ-second category space.Proof.
Let(FAii)’s, i∈I be fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ). By Proposition 33, (FAiic)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Now (⋂~i∈I{FAii})≠Φ implies that (⋂~i∈I{FAii})c≠E¯. Then ⋃~i∈I{(FAii)c}≠E¯. Hence, (U,E,ψ) is not a fuzzy soft σ-meager space and therefore (U,E,ψ) is a fuzzy soft σ-second category space.Proposition 46.
IfFA is a fuzzy soft σ-meager set in (U,E,ψ), then there is a fuzzy soft λσ-set GB in (U,E,ψ) such that FA⊆~GB.Proof.
LetFA be a fuzzy soft σ-meager set in (U,E,ψ). Thus, FA=⋃~i∈I{FAii}, where (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Now [(clfsFAii)c]’s are (FSOSs) in (U,E,ψ). Then T=⋂~i∈I{(clfs(FAii))c}is a fuzzy soft δ-set in (U,E,ψ) and Tc=(⋂~i∈I{(clfs(FAii))c})c=⋃~i∈I{clfs(FAii)}. Now FA=⋃~i∈I{FAii}⊆~⋃~i∈I{clfs(FAii)}=Tc. That is, FA⊆~Tc and Tc is a fuzzy soft λσ-set in (U,E,ψ). Let GBc⊆~T. Hence, if FA is a fuzzy soft σ-meager set in (U,E,ψ), then there is a fuzzy soft λσ-set GB in (U,E,ψ) such that FA⊆~GB.Proposition 47.
IfGB is a fuzzy soft σ-residual set in a (FSTS) (U,E,ψ) such that GB in (U,E,ψ) such that FA⊆~GB, where FA is a fuzzy soft dense and fuzzy soft δ-set in (U,E,ψ), then (U,E,ψ) is a fuzzy soft σ-Baire space.Proof.
LetGB be a fuzzy soft σ-residual set in a (FSTS) (U,E,ψ). Thus, GBc is a fuzzy soft σ-meager set in (U,E,ψ). Now by Proposition 46, there is a fuzzy soft λσ-set T in (U,E,ψ) such that GBc⊆~T. This implies that Tc⊆~GB. Let FA=Tc. Then FA is a fuzzy soft δ-set in (U,E,ψ) and FA⊆~GB implies that clfs(FA)⊆~clfs(GB). If clfs(FA)=E¯, then we have clfs(GB)=E¯. Hence, by Proposition 30, (U,E,ψ) is a fuzzy soft σ-Baire space.Proposition 48.
If the (FSTS)(U,E,ψ) is a fuzzy soft σ-Baire space and if ⋃~i∈I{FAii}=E¯, then there exists at least one λσ-set FAii such that intfs(FAii)≠Φ.Proof.
Suppose thatintfs(FA)=Φ, ∀ (i∈I), where (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Then ⋃~i∈I{FAii}=E¯, implying that intfs(⋃~i∈I{FAii})=intfs(E¯)=E¯≠Φ, a contradiction to (U,E,ψ) being a fuzzy soft σ-Baire space. Hence, int(FAii)≠Φ, for at least one λσ-set FAii in (U,E,ψ).Proposition 49.
If the (FSTS)(U,E,ψ) is a fuzzy soft σ-Baire space, then no nonempty (FSOS) is a fuzzy soft σ-meager set in (U,E,ψ).Proof.
LetFA be nonempty (FSOS) in a fuzzy soft σ-Baire space (U,E,ψ). Suppose that FA=⋃~i∈I{FAii}, where the (FSSs) (FAii)’s are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Then intfs(FA)=intfs(⋃~i∈I{FAii}). Since (U,E,ψ) is a fuzzy soft σ-Baire space, intfs(⋃~i∈I{FAii})=Φ. This implies that intfs(FA)=Φ. Then we will have FA=Φ, which is a contradiction, since FA∈ψ implies that intfs(FA)=FA≠Φ. Hence, no nonempty (FSOS) is a fuzzy soft σ-meager set in (U,E,ψ).Definition 50.
A (FSTS)(U,E,ψ) is called a fuzzy soft submaximal space if for each (FSS) FA in (U,E,ψ) such that clfs(FA)=E¯; then FA∈ψ in (U,E,ψ).Proposition 51.
If the (FSTS)(U,E,ψ) is a fuzzy soft submaximal space and if FA is a fuzzy soft σ-meager set in (U,E,ψ), then FA is a fuzzy soft meager set in (U,E,ψ).Proof.
LetFA=⋃~i∈I{FAii} be a fuzzy soft σ-meager set in (U,E,ψ), where the (FSSs) (FAii)’s, i∈I are fuzzy soft σ-nowhere dense sets in (U,E,ψ). Then we have intfs(FAii)=Φ and (FAii)’s, i∈I are fuzzy soft λσ-sets in (U,E,ψ). Now intfs(FAii)=Φ, implying that (intfs(FAii))c=Φc=E¯ and hence clfs(FAiic)=E¯. Since (U,E,ψ) is a fuzzy soft submaximal space, the fuzzy soft dense sets (FAiic)’s are (FSOSs) in (U,E,ψ) and hence (FAii)’s are (FSCSs) in (U,E,ψ). Then clfs(FAii)=(FAii) and intfs(FAii)=Φ imply that intfs(clfs(FAii))=intfs(FAii)=Φ. That is, (FAii)’s are fuzzy soft nowhere dense sets in (U,E,ψ). Therefore, FA=⋃~i∈I{FAii} is a fuzzy soft meager set in (U,E,ψ).Proposition 52.
If the (FSTS)(U,E,ψ) is a fuzzy soft σ-Baire space and fuzzy soft submaximal space, then (U,E,ψ) is a fuzzy soft Baire space.Proof.
LetFA be a fuzzy soft σ-meager set in (U,E,ψ). Since (U,E,ψ) is a fuzzy soft submaximal space, by Proposition 51, FA is a fuzzy soft meager set in (U,E,ψ). Since (U,E,ψ) is a fuzzy soft σ-Baire space, by Proposition 38, intfs(FA)=Φ. Hence, for the fuzzy soft meager set FA in (U,E,ψ), we have intfs(FA)=Φ. Therefore, by Proposition 30, (U,E,ψ) is a fuzzy soft Baire space.Definition 53.
A fuzzy softP-space is a (FSTS)(U,E,ψ) with the property that states that if countable intersection of fuzzy soft open sets in (U,E,ψ) is fuzzy soft open. That is, every non−empty fuzzy soft δ-set in (U,E,ψ) is fuzzy soft open in (U,E,ψ).Proposition 54.
If the (FSTS)(U,E,ψ) is a fuzzy soft σ-Baire space and fuzzy soft P-space, then (U,E,ψ) is a fuzzy soft Baire space.Proof.
Let the (FSTS)(U,E,ψ) be a fuzzy soft σ-Baire space. Then, by Proposition 39, clfs(⋂~i∈I{FAii})=E¯, where the (FSSs) (FAii)’s, i∈I are fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ). Now from clfs(⋂~i∈I{FAii})=E¯, we have (clfs(⋂~i∈I{FAii}))c=Φ. This implies that intfs(⋃~i∈I(FAii)c)=Φ. Since the (FSSs) (FAii)’s are fuzzy soft dense in (U,E,ψ), clfs(FAii)=E¯. Then we have (clfs(FAii))c=Φ. This implies that intfs(FAiic)=Φ. Also, since (U,E,ψ) is a fuzzy soft P-space, the non−empty fuzzy soft δ-sets (FAii)’s in (U,E,ψ) are fuzzy soft open in (U,E,ψ). Then (FAiic)’s are (FSCSs) in (U,E,ψ). Then clfs(FAiic)=FAiic and intfs(FAiic)=Φ imply that intfs(clfs(FAiic))=intfs(FAiic)=Φ. That is, (FAiic)’s are fuzzy soft nowhere dense sets in (U,E,ψ). Therefore, we have intfs(⋃~i∈I{FAiic})=Φ, where (FAiic)’s are fuzzy soft nowhere dense sets in (U,E,ψ). Hence, by Proposition 30, (U,E,ψ) is a fuzzy soft Baire space.Definition 55.
A (FSTS)(U,E,ψ) is called a fuzzy soft almost resolvable space if ⋃~i∈I{FAii}=E¯, where the (FSSs) (FAii)’s in (U,E,ψ) are such that intfs(FAii)=Φ. Otherwise, (U,E,ψ) is called a fuzzy soft almost irresolvable space.Proposition 56.
If the (FSTS)(U,E,ψ) is a fuzzy soft almost irresolvable space, then (U,E,ψ) is a fuzzy soft σ-second category space.Proof.
Let(FAii)’s, i∈I be the fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ). Now clfs(FAii)=E¯ implies that (clfs(FAii))c=Φ. That is, intfs(FAiic)=Φ. Since (U,E,ψ) is a fuzzy soft almost irresolvable space, ⋃~i∈I{FAiic}≠E¯, where the (FSSs) (FAii)’s in (U,E,ψ) are such that intfs(FAiic)=Φ. Now ⋃~i∈I{FAiic}≠E¯ implies that (⋃~i∈I{(FAiic})c≠Φ. Hence, we have ⋂~i∈I{FAii}≠Φ, where the (FSSs) (FAii)’s are fuzzy soft dense and fuzzy soft δ-sets in a (FSTS) (U,E,ψ). Thus, by Proposition 45, (U,E,ψ) is a fuzzy soft σ-second category space.Definition 57.
A (FSTS)(U,E,ψ) is called a fuzzy soft hyperconnected space if every (FSOS) FA is fuzzy soft dense in (U,E,ψ). That is, clfs(FA)=E¯∀Φ≠FA∈ψ.Proposition 58.
Ifclfs(⋂~i∈I{FAii})=E¯, where (FAii)’s are fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ), then (U,E,ψ) is a fuzzy soft σ-Baire space.Proof.
The proof is obvious.Proposition 59.
Ifclfs(⋂~i∈I{FAii})=E¯, where the (FSSs) (FAii)’s are fuzzy soft δ-sets in a fuzzy soft hyperconnected and fuzzy soft P-space (U,E,ψ), then (U,E,ψ) is a fuzzy soft σ-Baire space.Proof.
Let(FAii)’s, i∈I be the fuzzy soft δ-sets in (U,E,ψ) such that clfs(⋂~i∈I{FAii})=E¯. Since (U,E,ψ) is a fuzzy soft P-space, the fuzzy soft δ-sets (FAii)’s in (U,E,ψ) are fuzzy soft open in (U,E,ψ). Also since (U,E,ψ) is a fuzzy soft hyperconnected space, the (FSOSs) (FAii)’s in (U,E,ψ) are fuzzy soft dense sets in (U,E,ψ). Hence, the (FSSs) (FAii)’s, i∈I are fuzzy soft dense and fuzzy soft δ-sets in (U,E,ψ) and clfs(⋂~i∈I{FAii})=E¯. Hence, by Proposition 58, (U,E,ψ) is a fuzzy soft σ-Baire space.Definition 60.
LetFA be a fuzzy soft subset of a fuzzy soft space E¯. Then FA is said to be fuzzy soft A-embedded in E¯ if each fuzzy soft δ-subset GB of E¯ which is contained in FA is fuzzy soft nowhere dense in FA (i.e., intFAfs(clFAfs(HC))=Φ).Proposition 61.
LetFA be a fuzzy soft dense subspace of a fuzzy soft Baire space E¯. If E¯∖FA is fuzzy soft A-embedded in E¯, then FA is a fuzzy soft Baire space.Proof.
Observe that ifFA is not a fuzzy soft Baire space; then there is a sequence GB⊇~FA11⊇~FA22⊇~FA33⊇~… of fuzzy soft open subsets of FA such that each FAii is fuzzy soft dense in GB and yet ⋂~i∈I{FAii}=Φ. Then there is a sequence HC⊇~KD11⊇~KD22⊇~KD33⊇~… of fuzzy soft open subsets of E¯ such that GB=HC∩~FA and FAii=KDii∩~FA. Each KDii is fuzzy soft dense in HC and HC is a fuzzy soft Baire space. Hence, ⋂~i∈I{KDii} is fuzzy soft dense in HC and therefore in HC∩~E¯∖FA. Since ⋂~i∈I{KDii}⊆~E¯∖FA,E¯∖FA is not fuzzy soft A-embedded in E¯.Proposition 62.
LetFA be a fuzzy soft dense subspace of a fuzzy soft Baire space E¯. If E¯∖FA is dense in E¯, then FA is a fuzzy soft Baire space if and only if E¯∖FA is fuzzy soft A-embedded in E¯.Proof.
Assume thatE¯∖FA is not fuzzy soft A-embedded in E¯. Let GB be a fuzzy soft δ-subset of E¯ which is contained in E¯∖FA and which is fuzzy soft dense in some relatively (FSOS) HC of E¯∖FA. Let KD be an fuzzy soft open subset of E¯ with KD∩~(E¯∖FA)=HC. Then TM=KD∩~GB is a δ-subset of E¯ which is fuzzy soft dense in KD and which is contained in HC. Let TM=⋂~i∈I{TMii}, where each TMii is open in E¯ and TMii⊆~KD. The (FSSs) {TMii∩~FA} are fuzzy soft open and fuzzy soft dense subsets of KD∩~FA and yet ⋂~i∈I{TMii∩~FA}. It follows that KD∩~FA is not a fuzzy soft Baire space. Consequently, FA is not a fuzzy soft Baire space. Conversely, assume that E¯∖FA is fuzzy soft A-embedded in E¯ and, by Proposition 61, then FA is a fuzzy soft Baire space.Lemma 63.
LetE¯ be a fuzzy soft Baire space. If GB=⋂~i∈I{TMii} is a nonempty fuzzy soft nowhere dens δ-set of E¯, where TMii is a fuzzy soft open subset of E¯∀i∈I, then for every nonempty fuzzy soft open subset TM of E¯ there is i∈I such that intfs((E¯∖TMii)∩~(TM∖clfs(GB)))≠Φ.Proof.
LetTM be a nonempty fuzzy soft open subset of E¯. Then, TM∖clfs(GB) is a nonempty fuzzy soft open subset of E¯; hence, TM∖clfs(GB) is also fuzzy soft Baire. Since TM∖clfs(GB)⊆~⋃~i∈I(E¯∖TMii) and each E¯∖TMii is a fuzzy soft closed subset of E¯, by Remark 22, then there is i∈I such that intfs((E¯∖TMii)∩~(TM∖clfs(GB)))≠Φ.Proposition 64.
LetE¯ be a fuzzy soft Baire space and let FA⊆~E¯ be fuzzy soft dense. Then FA is a fuzzy soft Baire space if and only if every fuzzy soft δ-set in E¯ contained in E¯∖FA is fuzzy soft nowhere dense.Proof.
Necessity. Let GB=⋂~i∈I{TMii}, where TMii is a fuzzy soft open subset of E¯ for each i∈I, which is contained in E¯∖FA. Then, FA⊆~⋃~i∈I(E¯∖TMii). In virtue of Remark 22, ⋃~i∈IintFAfs(FA∩~(E¯∖TMii)) is fuzzy soft dense in FA. Suppose that intfs(clfs(GB))≠Φ. Then, there is n∈I such that Φ≠intfs(clfs(GB))∩~intFAfs(FA∩~(E¯∖TMnn)). On the other hand, we know that clfs(GB)⊆~clfs(TMnn)=clfs(TMnn∩~FA). Hence, Φ≠intfs(clfs(GB))∩~intFAfs(FA∩~(E¯∖TMnn))⊆~clfs(TMnn∩~FA)∩~FA=clFAfs(TMnn∩~FA) which implies that intfs(clfs(GB))∩~intFAfs(FA∩~(E¯∖TMnn))∩~clfs(TMnn∩~FA)≠Φ, but this is impossible.
Sufficiency. Assume that FA is no fuzzy soft Baire. According to Remark 22, there is a countable fuzzy soft closed cover {FAii;i∈I} of FA such that ⋃~i∈I{intfs(FAii)} is not dense in FA. For each i∈I, choose a fuzzy soft closed subset TMii of E¯ such that FAii=TMii∩~FA, for each i∈I. Let TM=⋂~i∈I(E¯∖TMii) which is a δ-set of E¯ contained in E¯∖FA. If TM=Φ, then {TMii;i∈I} would be a fuzzy soft closed cover of E¯ and, by Remark 22, then ⋃~i∈I{intfs(TMii)} would be fuzzy soft dense in E¯ which is not possible. So, TM≠Φ. Choose a nonempty fuzzy soft open subset KD of E¯ such that KD∩~FA∩~intFAfs(FAii)=Φ, ∀i∈I. By Lemma 63, we can find r∈I such that intfs(TMrr)∩~(KD∖intfs(TM))≠Φ. Hence, Φ≠intfs(TMrr)∩~(KD∖intfs(TM))∩~FA⊆~intFAfs(TMrr∩~FA)∩~KD∩~FA⊆~intFAfs(FAii)∩~KD∩~FA, but this is a contradiction. Thus, FA is fuzzy soft Baire.


## 4. Baireness in Fuzzy Soft Setting

In this section, we shall study the new class of fuzzy soft Baire spaces.Definition 65.
We say a spaceE¯ is fuzzy soft D-Baire if every fuzzy soft dense subspace of E¯ is fuzzy soft Baire.An immediate consequence of Proposition64 is the following.Corollary 66.
Suppose thatE¯ is a fuzzy soft Baire space. Then, every fuzzy soft δ-set in E¯ with empty interior is fuzzy soft nowhere dense iff E¯ is fuzzy soft D-BaireProof.
It follows from Proposition64 and Definition 65.Definition 67.
We say that a (FSTS)E¯ is a fuzzy soft almost P-space if every non−empty fuzzy soft δ-set in E¯ has a nonempty interior.Corollary 68.
Every fuzzy soft Baire and fuzzy soft almostP-space is fuzzy soft D-Baire.Proof.
This is a consequence of Proposition64 and Definition 67.Definition 69.
A fuzzy soft Borel set is any (FSS) in a (FSTS) that can be formed from (FSOSs) (or, equivalently, from (FSCSs)) through the operations of countable union, countable intersection, and relative complement.Definition 70.
LetE¯ be a (FSTS). Then, the class FSPB(E¯) is the fuzzy soft σ-algebra in E¯ generated by all (FSOSs) and all fuzzy soft nowhere dense sets.Remark 71.
(1) For a (FSTS)(U,E,ψ), the collection of all fuzzy soft Borel sets on E¯ forms a fuzzy soft σ-algebra.
(2) The fuzzy softσ-algebra of fuzzy soft Borel sets is contained in the class FSPB(E¯).
(3) It is clear to show thatFA⊆~E¯ belongs to the class FSPB(E¯) if and only if FA may be expressed in the form FA=GB∪~HC, where GB is a fuzzy soft δ-set and HC is fuzzy soft meager.Theorem 72.
The following seven conditions on a space(U,E,ψ) are equivalent.
(1)(U,E,ψ) is fuzzy soft D-Baire.
(2)(U,E,ψ) is fuzzy soft Baire and every fuzzy soft δ-set with empty interior is fuzzy soft nowhere dense.
(3) Every fuzzy soft meager subsetFA⊆~E¯ is fuzzy soft nowhere dense.
(4)(U,E,ψ) is fuzzy soft Baire and every fuzzy soft dense δ-set has fuzzy soft dense interior.
(5)(U,E,ψ) is fuzzy soft Baire and every set in the class FSPB(E¯) with empty interior is fuzzy soft nowhere dense.
(6)(U,E,ψ) is fuzzy soft Baire and every fuzzy soft Borel set with empty interior is fuzzy soft nowhere dense.
(7)(U,E,ψ) is fuzzy soft Baire and the union of a fuzzy soft δ-set with empty interior and a fuzzy soft meager set of E¯ is fuzzy soft nowhere dense.Proof.
(1)⇔(2). This is Corollary 66.
(2)⇒ (3). Let FA⊆~E¯ be a fuzzy soft meager set. Assume FA=⋃~i∈I{FAii}, where FAii is fuzzy soft nowhere dense ∀i∈I. Therefore, LM=E¯∖⋃~i∈I{clfs(FAii)}=⋂~i∈I{E¯∖clfs(FAii)} is a δ-set in E¯ and LM is fuzzy soft dense in E¯ because its complement is a fuzzy soft meager set and E¯ is fuzzy soft Baire. Let VD=intfs(LM). The (FSS) LM-VD clearly has empty interior. Hence, LM-clfs(VD) is a fuzzy soft δ-set with empty interior; by hypothesis, LM-clfs(VD) is fuzzy soft nowhere dense. Also LM∩~Frfs(VD) is a fuzzy soft nowhere dense set. Therefore, LM-VD=(LM-clfs(VD))∪~(LM∩~Frfs(VD)) is a fuzzy soft nowhere dense set as well. On the other hand, E¯∖VD=(LM∖VD)∪~(E¯∖LM)=(LM∖VD)∪~(⋃~i∈I{FAii}) is a fuzzy soft meager set. Since E¯ is fuzzy soft Baire, Φ=intfs(E¯∖VD)=E¯∖clfs(VD). Therefore, clfs(VD)=E¯ and FA⊆~E¯∖VD=Frfs(VD) is fuzzy soft nowhere dense.
(3)⇒ (4). It follows from Remark 22 that E¯ is a fuzzy soft Baire space. Let LM⊆~E¯ be a fuzzy soft dense δ-set of E¯. Since E¯∖LM is a fuzzy soft meager set, the hypothesis implies that E¯∖LM is fuzzy soft nowhere dense; i.e., clfs(E¯∖LM) has empty interior. Therefore, VD=E¯∖clfs(E¯∖LM)=intfs(LM) is a fuzzy soft open dense subspace of E¯.
(4)⇒ (2). Let GB be a δ-set with empty. First observe that intfs(clfs(GB))⊆~clfs(clfs(GB)∖GB). Since clfs(GB)∖GB is an λσ-set with empty interior, E¯∖(clfs(GB)∖GB) is a fuzzy soft dense δ-set of E¯ By assumption, intfs(E¯∖(clfs(GB)∖GB)) is also fuzzy soft dense in E¯. That is, E¯∖clfs(clfs(GB)∖GB) is fuzzy soft dense in E¯. Hence, intfs(clfs(clfs(GB)∖GB))=Φ and so intfs(clfs(GB))=Φ.
(4)⇒ (5). We have already established above the equivalence among clauses (1), (2), (3), and (4). The fifth clause follows directly from the properties of the class FSPB(E¯) and clauses (2) and (3).
(5)⇒ (6). This implication is obvious because the fuzzy soft δ-algebra of fuzzy soft Borel sets is contained in the class FSPB(E¯).
(6)⇒ (1). It is enough to observe that (6) ⇒(2)⇒ (1).
(1)⇒(7). We know the first six statements are equivalent to each other. Thus, clause (7) follows directly from clauses (2) and (3). (7) ⇒ (1). This is a consequence of Corollary 66.


## 5. Conclusion

In the present paper, we have introduced and discussed new notions of Baireness in fuzzy soft topological spaces. Furthermore, there are many problems and applications in algebra that deal with group theory and spaces. So, future work in this regard would be required to study some applications using the properties ofψ in our new fuzzy soft spaces and new operations depend on fuzzy soft operations ∪~ and ∩~ to consider new fuzzy soft groups and fuzzy soft commutative rings. Also, let us say (U,E,ψ) is fuzzy soft N-Baire if every fuzzy soft set in (U,E,ψ) with empty interior is fuzzy soft nowhere dense. The question we are concerned with is as follows: what are the possible relationships considered between fuzzy soft N-Baire and each concept of our notions that are given in this work?



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