# On Some New Types of Fuzzy Soft Compact Spaces

Authors: S. Saleh; Tareq M. Al-shami; Abdelwaheb Mhemdi
Journal: Journal of Mathematics
 (2023)

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## Abstract

In this article, we apply the concept of fuzzy softδ-open sets to define new types of compactness in fuzzy soft topological spaces, namely, δ-compactness and δ∗-compactness. We explore some of their basic properties and reveal the relationships between these types and that which are defined by other authors. Also, we show that δ∗-compactness is more general than that which is presented in other papers. To clarify the obtained results and relationships, some illustrative examples are given.


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## Body

## 1. Introduction

Conventional mathematical tools are not sufficient to cope with many practical problems that existed in various disciplines such as social science, economics, engineering, medical science, and environment, involving vagueness/uncertainty. Zadeh [1] proposed the remarkable theory of fuzzy set (F-set) for handling these kinds of uncertainties where traditional tools fail. This theory represents a real revolution and grand paradigmatic change in mathematics. Since it is advent, it has contributed to handling numerous types of practical issues and overcoming many real-life problems. On the other hand, this theory has its inherent difficulties, which are possibly attributed to the inadequacy of the parameterization tool, as Molodtsov pointed out in [2]. So, he familiarized the idea of the soft set (S-set) as an efficient instrument for coping with uncertainties that are free from the abovementioned difficulties. Shabir and Naz [3] studied the topological structures of S sets. Then, many contributions have been conducted to soft topologies by many authors; see for example [4–6].In recent times, the fuzzification of the S-set is progressing rapidly combining F sets and S sets, Maji et al. [7] established a novel model known as a fuzzy soft set (FS-set). The important notion of mapping FS classes was studied by Kharal and Ahmad [8]. Subsequently, the topological structure of FS sets was started by Tanay and Kandemir [9]. FS sets and their applications have been studied in different aspects, such as algebraic ([10–12]) and decision-making ([13, 14]). Mukherjee et al. [15] introduced the notions of fuzzy soft δ-open (closed) sets and fuzzy soft δ-closure (interior) with fuzzy soft δ-continuous mappings. The notion of compactness in fuzzy soft topological space (FSTS) was defined and studied in ([16, 17]) as a generalization of Chang’s notion [18]. Taha [19] discussed compactness via the structure of fuzzy soft and r-Minimal spaces. Also, Saleh and Salemi [20] defined and studied a general type of fuzzy soft compactness.In this work, we define and study new types of compactness in FSTS, namely,δ-compactness and δ∗-compactness. Some of their basic properties are studied. The relationships between these notions and that which are defined by other authors are investigated. We show that δ∗-compactness is more general than that which are presented in [16, 17, 20]. Some results and theorems related to these notions are studied with some necessary examples.


## 2. Basic Definitions and Results

Here, we recall the basic definitions which will be needed in this paper.X refers to the universal set, T is the set of all parameters for X, FS-refers to fuzzy soft, X,ϑ,T means an FSTS, and I=0,1.Definition 1 (see [1, 18]).
An F-setA of X is a mapping A:X⟶I and A can be represented by the pairs A=x,Ax:x∈X,Ax∈I. IX refers to the set of all F sets on X. An F-pointxλ,λ∈0,1, is an F-set in X given by xλy=λ at x=y and xλy=0 otherwise for all y∈X/x. For α∈I, α¯∈IX refers to the F-constant function where α¯x=α for all x∈X.Definition 2 (see [21]).
An FS-setf,T over X symbolized by fT is the set of ordered pairs fT=t,ft:t∈T,ft∈IX. FSSXT refers to the set of all FS sets over X. The complement of fT symbolized by fTc, where fc:T⟶IX defined by ftc=1¯−ft∀t∈T. Clearly, fTcc=fT. α˜T∈FSSXT, α˜T=tα¯:t∈T,α¯∈IX, α¯:t∈T,α¯∈IX is called an FS-constant set.Definition 3 (see [7, 21]).
LetfTandgT be two FS sets on X. Then,(1)
fT is called a null (resp universal) FS-set, symbolized by 0˜Tresp.1˜T if ft=0¯resp.ft=1¯ for all t∈T(2)
fT is a subset of gT if ft≤gt∀t∈T, symbolized by f⊑g(3)
fT and gT are equal if fT⊑gT and gT⊑fT. It is symbolized by fT=gT(4)
The union offT and gT is an FS-sethT defined by ht=ft∨gt for all t∈T. hT is symbolized by fT⊔gT.(5)
The intersection offT and gT is an FS-setlT defined by lt=ft∧gt for all t∈T. lT is symbolized by fT⊓gTDefinition 4 (see [22–24]).
An FS-pointxαz over X is an FS-set over X defined by, xαzt′=xα if t′=t and xαzt′=0¯ if t′∈T−t, where xα is the F-point in X. FSPXT refers to the set of all FS points in X. An FS-pointxαz is said to belong to an FS-setfT, symbolized by xαz∈˜fT if α≤ftx. Moreover, every non-nullFS-setfT can be expressed as the union of all the FS points belonging to fT.Definition 5 (see [9, 22]).
A tripletX,ϑ,T is FSTS, where ϑ is a family of FS sets on X which satisfies the following conditions:(1)
0˜T and 1˜T belong to ϑ(2)
ϑ is closed under arbitrary union and finite intersectionDefinition 6 (see [22, 24, 25]).
The FS setsfT and gT are called quasi-coincident, symbolized by fTqgT if there exist t∈T, x∈X such that, ftx+gtx>1. If fT is not quasi-coincident with gT, then we write fTq˜gT.Definition 7 (see [15, 24]).
An FS-setfT of X,ϑ,T is called Q-neighborhood (briefly, q-nbd) of xαz if there exists a fuzzy soft open set (FSO-set) gT such that xαzqgT⊑fT.Definition 8 (see [8]).
LetFSSXT and FSSYK be two families of all FS sets over X and Y, respectively. Let u:X⟶Y and p:T⟶K be two maps. Then, fup:FSSXT⟶FSSYK is called a fuzzy soft map (FS-map), for which(i)
IffT∈FSSXT, then the image of fT symbolized by fupfT is an FS-set on Y given by fupfTk=supuft:t∈p−1kifp−1k≠∅, and fupfTk=0˜Y otherwise, for all k∈K(ii)
IfgK∈FSSYK, then the preimage of gK symbolized by fup−1gK is an FS-set on X defined by, fup−1gKt=u−1gpt for all t∈T
An FS-mapfup is called one-to-one(onto), if u and p are one-to-one(onto).Theorem 1 (see [8]).
LetfiT∈FSSXTandgiK∈FSSYKfor alli∈J, whereJis an index set. Then, for an FS-mapfup:FSSXT⟶FSSYYK, we have(1)
iff1T⊑f2T, thenfupf1T⊑fupf2T(2)
ifg1K⊑g2K, thenfup−1g1K⊑fup−1g2K(3)
fup⊔i∈JfiT=⊔i∈JfupfiT(4)
fup⊓i∈JfiT⊑⊓i∈JfupfiT, the equality holds if fup is one-to-one(5)
fup−1⊔i∈JgiK=⊔i∈Jfup−1giK(6)
fup−1⊓i∈JgiK=⊓i∈Jfup−1giK(7)
fup1˜T=1˜K if fup is onto(8)
fup−11˜K=1˜Tandfup−10˜K=0˜TDefinition 9 (see [15, 26]).
For an FS-setfT in X,ϑ,T, we have(i)
An FS-closure offT is the intersection of all FSC sets containing fT, symbolized by clfT, and an FS-interior of fT is the union of all FSO sets contained in fT, symbolized by intfT(ii)
fT is said to be a fuzzy soft regular open set (FSRO-set) if fT=intclfT and the complement of an FSRO-set is called a fuzzy soft-regular closed set (FSRC-set). FSROXT denotes the set of all FSRO sets, and FSRCXT denotes the set of all FSRC sets(iii)
fT is said to be an FS semiopen set if fT⊑clintfT and the complement of an FS semiopen set is called an FS semiclosed set(iv)
fT is said to be an FSδ-neighborhood (briefly, δ-nbd) of xαz if and only if there exists FSRO q-nbd (FSRO q-nbd) gT of xαz, such that gT⊑fTDefinition 10 (see [15]).
LetfT be an FS-set in X,ϑ,T. Then,(i)
An FS-pointxαz is called an FSδ-cluster point of fT if and only if every FSRO q-nbdhT of xαz, hTqfT. The set of all FS δ-cluster points of fT is called the FS δ-closure of fT, symbolized by clδfT, that is, clδfT=⊓gT∈FSRCXT:fT⊑gT.(ii)
An FS-setfT is called an FS δ-closed set (FS δC-set, briefly) if and only if fT=clδfT and the complement of an FS δ-closed set is called an FS δ-open set(FS δO-set, briefly). FSδCXT denotes to the set of all FSδC sets, and the set of all FSδOsets is symbolized by FSδOXT.(iii)
TheFSδ-interior of fT is symbolized by intδfT and defined by intδfT=1˜T−clδfTc, that is, intδfT=⊔gT∈FSROXT:gT⊑fT. Consequently, fT is FSδO-set if and only if fT=intδfT.Result 1.
[15] Every FSRO-set is an FSδO-set, and every FSδO-set is an FSO-set. Moreover, if fT is an FS semiopen set in X,ϑ,T, then clfT=clδfT.Result 2.
[15] If fT is an FSO-set in X,ϑ,T, then clfT is an FSRC-set, that is, clgT:gT∈ϑ=hT:hT∈FSRCXT, and for any FS-setfT in X,ϑ,T, clδfT=⊓clgT:fT⊑gT,gT∈ϑ.Theorem 2 (see [15]).
For FS setsfTandgTinX,ϑ,Twe have(1)
clδ0˜T=0˜T and clδ1˜T=1˜T(2)
IffT⊑gT, thenclfT⊑clgT(3)
clδfT is an FSδC-set, that is, clδclδfT=clδfT(4)
IffT,gTareFSδCsets, thenfT⊔gTis anFSδC-setResult 3.
(Remark 4.15, [15]) The FS δ-closure operator on X,ϑ,T satisfies the Kuratowski closure axioms. So that, there exists one topology over X, this topology is defined as follows:
The set of allFSδO sets of X,ϑ,T forms FST, symbolized by ϑδ. It is called FSδ-topology over X, and the triplet X,ϑδ,T is called an FSδ-topological space. Moreover, ϑδ⊑ϑ.Definition 11 (see [15]).
Letfup:X,ϑ,T⟶Y,η,K be an FS-map. Then, fup is FSδ-open (FSδ-closed) if fupfT is FSδO-set(FSδC-set) in Y,η,K for every FSδO-set(FSδC-set) fT in X,ϑ,T.Theorem 3 (see [15]).
Letfup:X,ϑ,T⟶Y,η,Kbe an FS-map. Then, the following items are equivalence:(1)
fup is FSδ-continuous(2)
fup−1gK is an FSδO-set in X,ϑ,T for every FSδO-set gK in Y,η,K(3)
fup−1gK is an FSδC-set in X,ϑ,T for every FSδC-set gK in Y,η,KDefinition 12 (see [22, 27]).
(i)
LetX,σ be an FTS. Then, the collection ϑσ=fT:ft=Afor allt∈TandA∈σ defines FST on X(ii)
LetX,ϑ,T be an FSTS. Then, the collection ϑt=ft:fT∈ϑ, for every t∈T defines FT on X


## 3. New Types of Fuzzy Soft Compact Spaces

In this section, we are going to give the definitions of two types of FS-compactness, namely,FSδ-compactness and FSδ∗-compactness via FSδO sets, and study some properties of them.Definition 13.
Letℱ be a family of FSδO sets of X,ϑ,T and gT be an FS-set over X. Then,(i)
ℱ is said to be a fuzzy soft δ-open cover (FSδ-open cover, briefly) of gT if gT⊑⊔fiT:fiT∈ℱ. A finite subcover is a finite subfamily of ℱ, which is again an FSδ-open cover(ii)
ℱ is said to be a fuzzy soft δ∗-open cover (FSδ∗-open cover, briefly) of gT if for every xαz∈˜gT there is fT∈ℱ satisfying xαz∈˜fTRemark 1.
Clearly, everyFSδ-open cover is an FS-open cover. The converse is not necessarily true. This follows from the fact that an FSO-set does not imply an FS δ-open set.Remark 2.
Clearly, everyFSδ∗-open cover is an FSδ-open cover. The converse is not necessarily true, the next Example 1 shows it.Definition 14.
LetfT be an FS-set in X,ϑ,T. Then,(i)
fT is said to be an FS δ-compact if every FSδ-open cover of fT has a finite subcover. X,ϑ,T is said to be fuzzy soft δ-compact (FSδ-compact, briefly) if 1˜T itself is FSδ-compact(ii)
X,ϑ,T is said to be an FS δ∗-compact if every FSδ∗-open cover of 1˜T has a finite subcoverExample 1.
LetX=0,1 and T=t. Then, the family ϑ=0˜T,1˜T∪fiT:i∈N is FST ϑ on X, where(1)fiTtx=1¯,ifx=0,1−1i,if0<x≤1i,1¯,if1i<x≤1.
It is clear that,clhT:hT∈ϑ=gT:gT∈FSRCXT, and for any lT in X,ϑ,T, clδlT=⊓gT∈FSSXT:lT⊑gT,gT∈FSRCXT. Since clfiT=1˜T∀i∈N, cl1˜T=1˜T, and cl0˜T=0˜T, then FSRCXT=0˜T,1˜T. So, FSδCXT=0˜T,1˜T. Therefore, the only FSδ-open cover of 1˜T is 1˜T. So, X,ϑ,T is an FSδ-compact, and also is FSδ∗-compact.Proposition 1.
EveryFSδ-compact space is FSδ∗-compact.Proof.
It follows directly from Remark2.
The next example shows that the converse is not true in general.Example 2.
LetX be an infinite set, T=t and uiT:i∈N,viT:i∈N be two families of FS sets on X, given byuitx=1−1/i and vitx=1/i for all x∈X, i∈N. We consider FST ϑ on X which is generated by the subbase uiT,viT:i∈N. Then, intcluiT=uiT for all i∈N. So, every uiT is FS δ-open. Similarly, every viT is FS δ-open. Since the family ℱ=uiT:i∈N is a FSδ-open cover of 1˜T which has no finite subcover, then X,ϑ,T is not FSδ-compact.
On the other hand,ℱ=uiT:i∈N is not FSδ∗-open cover of 1˜T. Indeed, for x1z∈˜1˜T, there is no FSδ-open set uT in ℱ such that x1z∈˜uT, so the only FSδ∗-open cover of 1˜T is the family viT:i∈N which has 1˜T as a finite subcover. Therefore, X,ϑ,T is FSδ∗-compact.Proposition 2.
IfX and T are finite sets, then every FSTS X,ϑ,T is FSδ-compact (FSδ∗-compact).
The converse is not necessarily true, as the next example shows.Example 3.
LetX be an infinite set and ϑ=α˜T:α∈I. Clearly, ϑ is an FST on X. One can check that X,ϑ,T is FSδ-compact, while X is an infinite set.Remark 3.
For any infinite FS-setfT in the discrete FSTS, fT is not FSδ∗-compact (not FSδ-compact). Indeed, let X,ϑ,T be the discrete FS-space. Then, A=xαz:xαz∈FSPXT is a family of FSO sets. Clearly, intclxαz=xαz for all xαz∈A. So, every xαz is FS δ-open. Since fT=⊔xαz:xαz∈˜fT, then the family xαz:xαz∈fT is an FSδ∗-open cover of fT which has no finite subcover. So, fT is not FSδ∗-compact. When fT=1˜T, then X,ϑ,T is not FSδ∗-compact.Proposition 3.
IfX,ϑ1,T is FSδ-compact (FSδ∗-compact) and ϑ2⊆ϑ1, then X,ϑ2,T is FSδ-compact (FSδ∗-compact).Proof.
Obvious.Theorem 4.
IfX,ϑ,T is FSδ-compact and fT is an FSδC-set of X,ϑ,T, then fT is FSδ-compact.Proof.
LetfT be an FS δC-set in X,ϑ,T and ℱ=fiT:i∈J be an arbitrary FSδ-open cover of fT. Since fTc is FS δO-set, then ℱ⊔fTc is FSδ-open cover of 1˜T and X,ϑ,T is FSδ-compact, then ℱ⊔fTc has a finite subcover, say fiT:i=1,2,3,…,n⊔fTc, that is, 1˜T⊑⊔fiT:i=1,2,3,…,n⊔fTc. However, fT,fTc are disjoint. Thus, fT⊑⊔fiT:i=1,2,3,…,n and so, the result holds.Corollary 1.
EveryFSδ-closed subspace of FSδ-compact is FSδ-compact.Theorem 5.
LetfT be an FSδ-compact and gT be an FSδC-set in X,ϑ,T. Then, fT⊓gT is an FSδ-compact.Proof.
A similar technique given in Theorem4 is followed.Definition 15 (see [20]).
LetA=fiT:i∈J be a family of FS sets and gT be an FS-set. Then,(i)
A is said to behave fuzzy soft Q-intersection (FSQ-intersection, briefly) with respect to gT if there exists xαz∈˜gT with xαzqfiT∀i∈J(ii)
A is called has a fuzzy soft finite intersection property FSFIP brieflyw.r.togT if every finite subfamily of A has FSQ-intersection w.r.togT
The next theorem gives a nice characterization ofFSδ∗-compact space.Theorem 6.
X,ϑ,T is an FSδ∗-compact if and only if every family of FSδC sets on X, having the FSFIPw.r.to1˜T has FSQ-intersection w.r.to1˜T.Proof.
LetX,ϑ,T be FSδ∗-compact. Then 1˜T is FSδ∗-compact and let A=fiT:i∈J be the family of FSδC sets over X, which has the FSFIPw.r.to1˜T. Suppose that, A has no FSQ-intersection w.r.to1˜T. Then, for all xαz∈˜1˜T there is i∈J such that xαzq˜fiT. Thus, Ac=fiTc:i∈J is an FSδ∗-open cover of 1˜T. Since 1˜T is FSδ∗-compact, there is a finite subcover of Ac say, fiTc:i∈I0andI0is a finite subset ofJ. Hence fiTc:i∈I0andI0is.a.finite.subset.ofJ has no FSQ-intersection w.r.to1˜T. This contradicts that A has FSFIPw.r.to1˜T. Then, the result holds.
Conversely, letA=Oxαzi:i∈I be an FSδ∗-open cover of 1˜T. Then, Ac=Oxαzic:i∈I has no FSQ-intersection w.r.to1˜T. Thus, Ac has no FSFIPw.r.to1˜T. So, there are i1,i2,…,in⊂J, such that Oxαzijc:j=1,2,…,n has no FSQ-intersection w.r.to1˜T. Then, Oxαzij:j=1,2,…,n is a finite subcover of 1˜T. Then, 1˜T is FSδ∗-compact. The result holds.Definition 16 (see [16, 17]).
A familyA=fiT:i∈J of FS sets is called has the FS-finite intersection property if the intersection of the members of each finite subfamily of A is not the null FS-set.
In the next result, we characterize anFSδ-compact space using FS δC sets.Theorem 7.
X,ϑ,T is FSδ-compact if and only if every family of FSδC sets on X, which has the FS-finite intersection property has a non-null intersection.Proof.
LetX,ϑ,T be FSδ-compact and let A=fiT:i∈I be the family of FSδCsets in X,ϑ,T which has the FS-finite intersection property. To prove that fiT:i∈I is non-null, suppose otherwise, i.e., ⊓fiT:i∈I=0˜T, then ⊔fiTc:i∈I=1˜T, that is, fiTc:i∈I is an FSδ-open cover of 1˜T. By compactness, fiTc:i∈I has a finite subcover of 1˜T say, fiTc:i=i1,i2,…,in. This implies that ⊓fiT:i=i1,i2,…,in=0˜T. This contradicts that the family A has the FS-finite intersection property. So the result holds.
Conversely, letgiT:i∈I be an FSδ-open cover of 1˜T. Then, giTc:i∈I is the family of FS δ-closed sets in X,ϑ,T. Since giT:i∈I is an FSδ-open cover of 1˜T, then ⊓giTc:i∈I=0˜T. Thus, giTc:i∈I has no finite intersection property, that is, there is a finite subset i1,i2,…,in of I such that ⊓giTc:i=i1,i2,…,in=0˜T. This implies that giTc:i=i1,i2,…,in is a finite subcover of 1˜T. So the result holds.
We close this section by discussing the image and preimage ofFSδ-compactness under FSδ-continuous maps.Theorem 8.
IfX,ϑ1,T is FSδ-compact and fup:X,ϑ1,T⟶Y,ϑ2,K is FSδ-continuous and onto, then, Y,ϑ2,K is FSδ-compact.Proof.
LetgiT:i∈J be an FSδ-open cover of 1˜K in Y,ϑ2,K. Since fup is FSδ-continuous. Then, the family ℱ=fup−1giT:i∈J is an FSδ-open cover of 1˜T in X,ϑ1,T and X,ϑ1,T is FSδ-compact, there is a finite subfamily of ℱ say, fup−1giT:i=1,2,3,…,n covers 1˜T, i.e., 1˜T⊑⊔fup−1giT:i=1,2,…,n. Thus,(2)fup1˜T⊑fup⊔fup−1giT:i=1,2,…,n=⊔fupfup−1giT:i=1,2,…,n=⊔giT:i=1,2,…,n.
Sincefup is onto, then 1˜K=fup1˜T⊑⊔giT:i=1,2,…,n. Hence, Y,ϑ2,K is FSδ-compact.Theorem 9.
Letfup:X,ϑ1,T⟶Y,ϑ2,K be FSδ-continuous. If gT is FSδ-compact in X,ϑ1,T, then fupgT is FSδ-compact in Y,ϑ2,K.Proof.
LetfiT:i∈J be an FSδ-open cover of fupgT. Since fup is δ-continuous, fup−1fiT is FSδ-open in X,ϑ1,T for all i∈J. Thus, the family ℱ=fup−1fiT:i∈J is an FSδ-open cover of gT in X,ϑ1,T. Since gT is FSδ-compact, there is a finite subfamily of ℱ say fup−1fiT:i=1,2,3,…,n covers gT, i.e., gT⊑⊔fup−1fiT:i=1,2,…,n. Thus,(3)fupgT⊑fup⊔fup−1fiT:i=1,2,…,n=⊔fupfup−1fiT:i=1,2,…,n=⊔fiT:i=1,2,…,n.
Hence,fupgT is FSδ-compact.Theorem 10.
Letfup: X,ϑ1,T⟶Y,ϑ2,K be an FS δ-continuous and one-to-one map. If gT is FSδ-compact in Y,ϑ2,K, then fup−1gT is FSδ-compact in X,ϑ1,T.Proof.
The proof is similar to that in the above theorem.


## 4. The Relations among Different Types of Fuzzy Soft Compactness

In this section, we investigate the relations between our notions of FS-compactness in this paper and those in other papers; we make a comparison between our types of FS-compactness and that given in ([16, 17, 20]). First, we mention that Osmanoglu and Tokat [17] and Hussain [16] gave the same definition of FS-compactness as a generalization of Chang’s notion [18]. Their definitions are as follows.Definition 17 (see [16, 17]).
(i)
A familyA of FS sets in X,ϑ,T is a cover of FS-setgT if gT⊑⊔fiT:fiT∈Aandi∈J. It is open cover if each member of A is an FSO-set. A subcover of A is a subfamily of A which is also a cover.(ii)
An FS-setfTinX,ϑ,T is called a fuzzy soft compact (briefly, FS-compact) if each FS-open cover of fT has a finite subcover. X,ϑ,T is called an FS-compact if every FS-open cover of 1˜T has a finite subcover.
Saleh and Salemi [20] defined another type of FS-compactness which is more general than that of the above definition. His definition is as follows.Definition 18.
(i)
A familyγ of FSO sets is called a fuzzy soft P-open cover (briefly, FSP-open cover) of an FS-setfT if for every xαz∈˜fT there is gT∈γ such that xαz∈˜gT. A finite p-subcover of γ is a finite subfamily of γ which is also an FSP-open cover.(ii)
An FS-setfTinX,ϑ,T is called a fuzzy soft P-compact (briefly, FSP-compact) if every FSP-open cover of fT has a finite subcover. X,ϑ,T is called an FSP-compact if every FSP-open cover of 1˜T has a finite p-subcover.Remark 4.
Clearly, everyFSP-cover is an FS-cover, but the converse may not be true, as shown by Example (3.3) in [20].Theorem 11.
ForT finite, FS-compact ⟹FSδ-compact.Proof.
LetX,ϑ,T be an FS-compact and fiT:i∈J be an FS δ-open cover of X,ϑ,T. Since every FSδO-set is FS-open, thus, fiT:i∈J is an FS-open cover of X,ϑ,T. Since 1˜T is FS-compact, there is a finite subset I0 of J such that 1˜T⊑⊔fiT:i∈I0. Hence, X,ϑ,T is FSδ-compact.Corollary 2.
FSδ-compact ⟹FS-compact.Proof.
The proof follows by using the next example.
LetX=0,1 and T=t. Then, the family ϑ=0˜T,1˜T∪fiT:i∈N is an FST on X, where fiT is defined as follows:(4)fiTtx=1¯,ifx=0,ix,if0<x≤1i,1¯,if1i<x≤1.
Since the onlyFSδ-open cover of 1˜T is 1˜T, then X,ϑ,T is FSδ-compact.
On the other hand,X,ϑ,T is not FS-compact. Indeed, the family fiT:i∈N is an FS-open cover of 1˜T which does not have a finite subcover.
From Theorem11 and Proposition 1, we obtain the next result.Result 4.
ForT finite, FS-compact ⟹FSδ-compact ⟹FSδ∗-compact.Remark 5.
The converse in the implications of the above result may not be true. Corollary2 and Example 2 show it.Theorem 12.
ForT finite, FSP-compact ⟹FSδ∗-compact.Proof.
LetX,ϑ,T be an FSP-compact and fiT:i∈J be a family of FSδO sets ofX,ϑ,T which is an FSδ∗-open cover of 1˜T. It is clear that fiT:i∈J is also, a family of FSO sets, which is an FSP-open cover of 1˜T. Since X,ϑ,T is an FSP-compact, then there is a finite subset I0 of J such that 1˜T⊑⊔fiT:i∈I0. The result holds.Corollary 3.
FSδ∗-compact ⟹FSP-compact.Proof.
It follows directly from the fact thatFSP-open cover need not imply FSδ∗-open cover, indeed FSO sets need not FSδO sets.
In the next, we show that the notions ofFSδ-compact and FSP-compact are independent.Corollary 4.
FSδ-compact⟹FSP-compactProof.
Example1, shows that X,ϑ,T is an FSδ-compact, but it is not FSP-compact, indeed the family fiT:i∈N is an FSP-open cover of 1˜T which has no finite p-subcover.Corollary 5.
FSP-compact ⟹FSδ-compactProof.
Example2, shows that X,ϑ,T is an FSP-compact, but it is not FSδ-compact, indeed the family uiT:i∈N is an FSδ-open cover of 1˜T which has no finite subcover.Proposition 4.
ForT finite, FS-compact ⟹FSP-compact.Proof.
It follows directly from Remark4.Corollary 6.
FSP-compact ⟹FS-compact.Proof.
It follows by using the next example.
LetX be an infinite set and T=t. Then, the family ϑ=1˜T∪fiT:i∈N is an FST on X, where fiT given by, fiTtx=1−1/i∀x∈X. Since the only FSP-open cover of 1˜T is 1˜T, then X,ϑ,T is an FSP-compact. However, X,ϑ,T is not FS-compact, indeed for the family fiT:i∈N is FS-open cover of 1˜T which has no finite subcover.Remark 6.
From the previous results, we can summarize the relationships among different types of FS-compactness as in the next diagram.
In the following, we study the relation betweenFSδ-compact spaces and some induced fuzzy compact spaces.Theorem 13.
For finiteT and an FTS X,σ, we have, X,ϑσ,T is FSδ-compact ⇔X,σ is fuzzy δ-compact.Proof.
LetX,ϑσ,T be an FSδ-compact and T=t1,t2,t3,…,tn. Let γ=Ai:i∈J is an arbitrary fuzzy δ-open cover of 1¯. Then, ℱ=fiT:fit=Ai∀t∈TandAi∈σ,i∈J is an FSδ-open cover of 1˜T. Since 1˜T is an FSδ-compact and so, ℱ has a finite subcover say, fiT:fit=Ai∀t∈TandAi∈σ,i=1,2,…,n. Thus, the family Ai∈σ:i=1,2,…,n is a finite subcover of 1¯. Hence, X,σ is a fuzzy δ-compact.
Conversely, letX,σ be a fuzzy δ-compact and ℱ=fiT:fit=Ai∀t∈TandAi∈σ,i∈J be an FSδ-open cover of 1˜T in X,ϑσ,T, then the family γ=Ai∈σ:i∈J is a fuzzy δ-open cover of 1¯. Since X,σ is a fuzzy δ-compact, then γ has a finite subcover say, Ai∈σ:i=1,2,…,n. Thus, ℱ has a finite subcover say, fiT:fit=Ai∀t∈TandAi∈σ,i=1,2,…,n. Hence, X,ϑσ,T is an FSδ-compact.Theorem 14.
For a finite setT with X,ϑ,T. If X,ϑt is fuzzy δ-compact ∀t∈T, then X,ϑ,T is FSδ-compact.Proof.
LetX,ϑt be fuzzy δ-compact ∀t∈T and T=t1,t2,t3,…,tn. Let ℱ=fiT:i∈J be an arbitrary FSδ-open cover of 1˜T in X,ϑ,T. Then, γ=fit:fiT∈ϑ,i∈J∀t∈T is a fuzzy δ-open cover of 1¯. Since X,ϑt is fuzzy δ-compact ∀t∈T, then 1¯ is fuzzy δ-compact. Thus, γ has a finite subcover, say fit:fiT∈ϑ,i=1,2,3,…n. Therefore, the subfamily, fiT:i=1,2,3,…n of ℱ is a finite subcover. The proof is complete.
The converse of the above theorem may not be true in general. This fact is demonstrated by the next example.Example 4.
LetX be an infinite set, T=t1,t2 and let the family ϑ=1˜T∪fiT:fiT=t1,Ai,Ai∈IXandi∈N. Then, one can check that ϑ is an FST on X. Since the only FSδ-open cover of 1˜T is 1˜T, then X,ϑ,T is an FSδ-compact. On the other hand, X,ϑt1, where ϑt1=IX is the discrete FT on X which is not fuzzy δ-compact, indeed the family xαi:x∈X,α∈Iandi∈N is a fuzzy open cover of 1¯T which has no finite subcover.Theorem 15.
X,ϑδ,T is FSδ-compact if and only if every family of FSδC sets on X, which has the FS-finite intersection property has a non-null intersection.Proof.
In a similar way to that in Theorem7, from the above theorem and Theorem 7, we obtain the next result.Corollary 7.
X,ϑ,TisFSδ-compact⟺X,ϑδ,TisFS-compact.


## 5. Conclusion

In this work, we have defined and studied new types of compactness in fuzzy soft topological spaces, namely,δ-compactness and δ∗-compactness. Some of their basic properties have been explored, and the relationships between these notions and that which were defined by other authors have been investigated. We have shown that δ∗-compactness is more general than that which are presented in [16, 17, 20]. Some results and theorems related to these notions have been discussed with some necessary examples. In the future work, we will study the relationships between the concepts given herein and separation axioms via fuzzy soft δ-open sets. Also, we will construct a general frame of fuzzy soft topology inspired by the class of a,b-Fuzzy soft sets [28]. Through this frame, it can be reintroduced all concepts of fuzzy soft topological spaces and explored their properties and characterizations.



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