# Nearly Soft Menger Spaces

Authors: Tareq M. Al-shami; Ljubiša D. R. Kočinac
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we define a weak type of soft Menger spaces, namely, nearly soft Menger spaces. We give their complete description using softs-regular open covers and prove that they coincide with soft Menger spaces in the class of soft regular⋆ spaces. Also, we study the role of enriched and soft regular spaces in preserving nearly soft Mengerness between soft topological spaces and their parametric topological spaces. Finally, we establish some properties of nearly soft Menger spaces with respect to hereditary and topological properties and product spaces.


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## Body

## 1. Introduction

The theory of selection principles is an area of mathematics that studies the possibility of generating a mathematical object of one kind from a sequence of objects of the same or different kind. The beginnings of this theory are going back to Borel, Hurewicz, Menger, Rothberger, and Sierpiński. This theory is one of the important tools of numerous subareas of mathematics such as set theory and general topology, Ramsey theory, game theory, hyperspaces, function spaces, uniform structures, cardinal invariants, and dimension theory.In 1924, Menger [1] studied selection property under the name Menger basis property and Hurewicz [2], in 1925, reformulated it in the present form. Menger’s property is strictly between σ-compactness and Lindelöfness. The papers [3, 4] carried out a systematic study of selection principles in topology and then research in this field expanded immensely and attracted many researchers (see survey papers [5–7] and references therein). Some types of selection principles (so-called weak selection principles) have been formulated by applying the interior and closure operators in the definition of a selection property (see [8–21]) and the other types have been explored by replacing sequences of open covers by sequences of covers by some generalized open sets (see [22–24]). In this paper, we apply the ideas from selection principles theory to soft topological spaces. In fact, we are focused on a weaker form of the classical Menger covering property in the soft topology settings.Soft sets were established by Molotdsov [25], in 1999, as a new technique to approach real-life problems which suffer vague and uncertainties. He investigated merits of soft sets compared with probability theory and fuzzy set theory. Many applications of soft sets have been recently given on the different areas such as decision-making problem, information theory, computer sciences, engineering, and medical sciences. In 2011, Shabir and Naz [26] employed soft sets that are defined over an initial universe set with a fixed set of parameters to introduce the concept of soft topological space. Then, researchers have studied several concepts of classical topological spaces through soft topological spaces. Soft compactness [27] and some weak variants of it [28–32] have been established and investigated. One of divergences between soft topological spaces and classical topological spaces was discussed in [33].This paper is organized as follows. Section2 provides basic definitions and results which are used in this paper. In Section 3, we establish some properties of soft semiopen and soft s-regular open sets which will help us to prove some results in the next sections. Section 4 introduces the concept of nearly soft Menger spaces and investigates its fundamental properties with the help of examples. Section 5 concludes the paper.


## 2. Preliminaries

In what follows, we recall the main definitions and results which shall be used throughout this work.


### 2.1. Selection Principles

Definition 1.
(see [3]). Let A and ℬ be given families of sets. Then, SfinA,ℬ denotes the selection hypothesis: for each sequence An:n∈ℕ of elements of A, there is a sequence Bn:n∈ℕ such that Bn is a finite subset of An for each n and ∪n∈ℕBn∈ℬ.
IfO denotes the collection of all open covers of a topological space X, then spaces satisfying SfinO,O are said to have the Menger (covering) property (or X is a Menger space).Definition 2.
(see [22]). A topological space X,τ is said to be nearly Menger if for each sequence Un:n∈ℕ of open covers of X there is a sequence Vn:n∈ℕ such that, for each n∈ℕ, Vn is a finite subset of Un and ∪n∈ℕ∪IntClV:V∈Vn=X.
Evidently, every nearly compact space [34] (every open cover U has a finite subcover V such that the interiors of closures of members of V cover the space) is nearly Menger, and every nearly Menger space is nearly Lindelöf [35] (every open cover U has a countable subcover V such that the interiors of closures of members of V cover the space).


### 2.2. Soft Sets

In this paper,X will be a nonempty set (called the initial universal set), 2X its power set, E a fixed set (called the set of parameters), and A,B,… subsets of E.Definition 3.
(see [25]). A pair G,E is said to be a soft set over X provided that G is a mapping of a set of parameters E into 2X.
A soft set is identified with the set of ordered pairsG,E=e,Ge:e∈EandGe∈2X. The collection of all soft sets defined over X under a parameter set E is denoted by SSXE.Definition 4.
(see [36]). A soft set G,E over X is said to be finite (resp. countable) if Ge is finite (resp. countable) for each e∈E. Otherwise, it is infinite (resp. uncountable).Definition 5.
(see [36]). A soft set G,E over X is called a soft point if there exist e∈E and x∈X such that Ge=x and Gb=∅ for each b∈E\e.
Throughout this study such soft point is briefly denoted byPex.
In a similar way, we definePex,y as a soft set G,E such that Ge=x,y and Gb=∅ for each b∈E\e.Definition 6.
(see [37]). The relative complement of a soft set G,E, denoted by G,Ec, is given by G,Ec=Gc,E, where Gc:E⟶2X is the mapping defined by Gce=X\Ge for each e∈E.Definition 7.
(see [38]). A soft set G,E over X is said to be a null soft set, denoted by ∅˜, if Ge=∅ for each e∈E. G,E is said to the absolute soft set, denoted by X˜, if Ge=X for each e∈E.Definition 8.
(see [38]). The union of two soft sets G,A and F,B over X, denoted by G,A∪˜F,B, is a soft set H,D, where D=A∪B and the mapping H:D⟶2X is given as follows:(1)Hd=Gd,d∈A\B,Fd,d∈B\A,Gd∪Fd,d∈A∩B.If Gi,E, i∈I, is an indexed family of soft sets over X, then ∪˜Gi,EL:i∈I=H,E, where He=∪i∈IGie.Definition 9.
(see [38]). The intersection of two soft sets G,A and F,B over X, denoted by G,A∩˜F,B, is the soft set H,D, where D=A∩B≠∅, and the mapping H:D⟶2X is given by Hd=Gd∩Fd for every d∈D.
For a familyGi,E, i∈I, of soft sets over X, one defines ∩˜Gi,EL:i∈I=H,E, where He=∩i∈IGie for every e∈E.Definition 10.
A soft setG,A is a soft subset of a soft set F,B, denoted by G,A⊆˜F,B, if A⊆B and for all a∈A, we have Ga⊆Fa.The soft sets G,A and F,B are soft equal if each one of them is a soft subset of the other.Definition 11.
(see [39]). A soft mapping between SSXA and SSYB is a pair f,φ, denoted also by fφ, of mappings such that f:X⟶Y and φ:A⟶B and the image of G,A∈SSXA and preimage of H,B∈SSYB are defined by(i)
fφG,Ab=fφG,B=f∪a∈φ−1bGa, b∈B(ii)
fφ−1H,Ba=fφ−1H,A=fφ−1Hφa. a∈AA soft mapping fφ:SSXA⟶SSYB is said to be injective (resp. surjective and bijective) if f and φ are injective (resp. surjective and bijective).


### 2.3. Soft Topological Spaces

Definition 12.
(see [26]). A triple X,τ,E is said to be a soft topological space if τ is a collection of soft sets over X satisfying the following axioms:(ST.1)∅˜ and X˜ belong to τ(ST.2) IfG,E∈τ and F,E∈τ, then G,E∩˜F,E∈τ(ST.3) IfGi,E:i∈I is any subset of τ, then ∪˜Gi,E:i∈I∈τElements of τ are called soft open sets, and their relative complements are called soft closed.
If (ST.1) is replaced by (ST.1′)G,E∈τ, whenever Ge=X or Ge=∅ for each e∈E, then τ is called the enriched soft topology (see [27]).
Throughout this paper, the ordered tripleX,τ,E indicates a soft topological space.Proposition 1. (see [26]).
LetX,τ,E be a soft topological space. Then, the collection τe=Ge:G,E∈τ defines a topology on X for each e∈E. This collection is called a parametric topology on X.The family,(2)τ⋆=G,E:Ge∈τefor eache∈E,is a soft topology on X finer than τ.The soft topologyτ⋆ given in the proposition above is called an extended soft topology.The equivalence between the enriched and extended soft topologieswas proved in [40].Definition 13.
(see [29, 41]). Let G,E be a soft set in X,τ,E. Then,(i)
The interior ofG,E, denoted by IntG,E, is the union of all soft open sets in X contained in G,E; the closure of G,E, denoted by ClG,E, is the intersection of all soft closed sets containing G,E(ii)
G,E is soft semiopen in X if G,E⊆˜ClIntG,E; the relative complement of a semi open set is called semiclosed(iii)
The semi interior ofG,E, denoted by sIntG,E, is the union of all soft semiopen sets contained in G,E; the semi closure of G,E, denoted by sClG,E, is the intersection of all soft semi closed sets containing G,EDefinition 14.
(see [26]). For a subset Y≠∅˜ of X,τ,E, the family τY=Y˜∩˜G,E:G,E∈τ is called a soft relative topology on Y and the triple Y,τY,E is called a soft subspace of X,τ,E.Definition 15.
(see [26, 42]). Let G,E be a soft set over X and x∈X. We write(i)
x⋐G,E if x∈Ge for some e∈E; y⋐G,E if x∉Ge for every e∈E.(ii)
x∈G,E if x∈Ge for every e∈E; y∉G,E if x∉Ge for some e∈E.Definition 16.
(see [26, 43]). X,τ,E is said to be(i)
Soft regular if for every soft closed setH,E and x∈X such that x∉H,E, there are disjoint soft open sets U,E and V,E containing H,E and x, respectively(ii)
Soft regular⋆ if for every soft closed set H,E and Pex∈X˜ such that Pex∉H,E, there are disjoint soft open sets U,E and V,E, containing H,E and Pex, respectively.Definition 17.
(see [42]). A soft set G,E over X is called stable provided that there is S⊆X such that Ge=S for each e∈E; X,τ,E is called stable provided that all proper nonnull soft open sets are stable.
A familyGi,E:i∈I of soft sets in X,τ,E is said to be a soft cover of X,τ,E (or a soft cover of X˜) if ∪˜i∈IGi,E=X˜. A soft cover Gi,E:i∈I is said to be locally finite if for each soft point Pex has a soft neighbourhood intersecting only finitely many Gi,E. A soft cover Gi,E:i∈I is a soft refinement of a soft cover Hj,E:j∈J if for each Gi,E there is Hj,E such that Gi,E⊂˜Hj,E.Definition 18.
A soft spaceX,τ,E is said to be(i)
Soft compact (resp. soft Lindelöf) [27] provided that every soft open cover of X has a finite (resp. countable) subcover(ii)
Soft paracompact [30, 44] if every soft open cover has a soft open, locally finite refinementDefinition 19.
(see [36]). A soft map fφ:X,τ,A⟶Y,θ,B is said to be(i)
Soft continuous if the inverse image of each soft open set is soft open(ii)
Soft open (resp. soft closed) if the image of each soft open (resp. soft closed) set is soft open (resp. soft closed)(iii)
Soft homeomorphism if it is bijective, soft continuous, and openDefinition 20.
LetG,A and F,B be soft sets over X and Y, respectively. The cartesian product of G,A and F,B is a soft set G×F,A×B over X×Y such that G×Fa,b=Ga×Fb for each a,b∈A×B.Theorem 1. (see [40]).
X,τ,E is extended if and only if IntH,E=IntH,E and ClH,E=ClH,E for any soft subset H,E of X,τ,E.Theorem 2. (see [40]).
IfX,τ,E is a soft regular space, then IntH,E=IntH,E and ClH,E=ClH,E for any stable soft subset H,E of X,τ,E.


## 2.1. Selection Principles

Definition 1.
(see [3]). Let A and ℬ be given families of sets. Then, SfinA,ℬ denotes the selection hypothesis: for each sequence An:n∈ℕ of elements of A, there is a sequence Bn:n∈ℕ such that Bn is a finite subset of An for each n and ∪n∈ℕBn∈ℬ.
IfO denotes the collection of all open covers of a topological space X, then spaces satisfying SfinO,O are said to have the Menger (covering) property (or X is a Menger space).Definition 2.
(see [22]). A topological space X,τ is said to be nearly Menger if for each sequence Un:n∈ℕ of open covers of X there is a sequence Vn:n∈ℕ such that, for each n∈ℕ, Vn is a finite subset of Un and ∪n∈ℕ∪IntClV:V∈Vn=X.
Evidently, every nearly compact space [34] (every open cover U has a finite subcover V such that the interiors of closures of members of V cover the space) is nearly Menger, and every nearly Menger space is nearly Lindelöf [35] (every open cover U has a countable subcover V such that the interiors of closures of members of V cover the space).


## 2.2. Soft Sets

In this paper,X will be a nonempty set (called the initial universal set), 2X its power set, E a fixed set (called the set of parameters), and A,B,… subsets of E.Definition 3.
(see [25]). A pair G,E is said to be a soft set over X provided that G is a mapping of a set of parameters E into 2X.
A soft set is identified with the set of ordered pairsG,E=e,Ge:e∈EandGe∈2X. The collection of all soft sets defined over X under a parameter set E is denoted by SSXE.Definition 4.
(see [36]). A soft set G,E over X is said to be finite (resp. countable) if Ge is finite (resp. countable) for each e∈E. Otherwise, it is infinite (resp. uncountable).Definition 5.
(see [36]). A soft set G,E over X is called a soft point if there exist e∈E and x∈X such that Ge=x and Gb=∅ for each b∈E\e.
Throughout this study such soft point is briefly denoted byPex.
In a similar way, we definePex,y as a soft set G,E such that Ge=x,y and Gb=∅ for each b∈E\e.Definition 6.
(see [37]). The relative complement of a soft set G,E, denoted by G,Ec, is given by G,Ec=Gc,E, where Gc:E⟶2X is the mapping defined by Gce=X\Ge for each e∈E.Definition 7.
(see [38]). A soft set G,E over X is said to be a null soft set, denoted by ∅˜, if Ge=∅ for each e∈E. G,E is said to the absolute soft set, denoted by X˜, if Ge=X for each e∈E.Definition 8.
(see [38]). The union of two soft sets G,A and F,B over X, denoted by G,A∪˜F,B, is a soft set H,D, where D=A∪B and the mapping H:D⟶2X is given as follows:(1)Hd=Gd,d∈A\B,Fd,d∈B\A,Gd∪Fd,d∈A∩B.If Gi,E, i∈I, is an indexed family of soft sets over X, then ∪˜Gi,EL:i∈I=H,E, where He=∪i∈IGie.Definition 9.
(see [38]). The intersection of two soft sets G,A and F,B over X, denoted by G,A∩˜F,B, is the soft set H,D, where D=A∩B≠∅, and the mapping H:D⟶2X is given by Hd=Gd∩Fd for every d∈D.
For a familyGi,E, i∈I, of soft sets over X, one defines ∩˜Gi,EL:i∈I=H,E, where He=∩i∈IGie for every e∈E.Definition 10.
A soft setG,A is a soft subset of a soft set F,B, denoted by G,A⊆˜F,B, if A⊆B and for all a∈A, we have Ga⊆Fa.The soft sets G,A and F,B are soft equal if each one of them is a soft subset of the other.Definition 11.
(see [39]). A soft mapping between SSXA and SSYB is a pair f,φ, denoted also by fφ, of mappings such that f:X⟶Y and φ:A⟶B and the image of G,A∈SSXA and preimage of H,B∈SSYB are defined by(i)
fφG,Ab=fφG,B=f∪a∈φ−1bGa, b∈B(ii)
fφ−1H,Ba=fφ−1H,A=fφ−1Hφa. a∈AA soft mapping fφ:SSXA⟶SSYB is said to be injective (resp. surjective and bijective) if f and φ are injective (resp. surjective and bijective).


## 2.3. Soft Topological Spaces

Definition 12.
(see [26]). A triple X,τ,E is said to be a soft topological space if τ is a collection of soft sets over X satisfying the following axioms:(ST.1)∅˜ and X˜ belong to τ(ST.2) IfG,E∈τ and F,E∈τ, then G,E∩˜F,E∈τ(ST.3) IfGi,E:i∈I is any subset of τ, then ∪˜Gi,E:i∈I∈τElements of τ are called soft open sets, and their relative complements are called soft closed.
If (ST.1) is replaced by (ST.1′)G,E∈τ, whenever Ge=X or Ge=∅ for each e∈E, then τ is called the enriched soft topology (see [27]).
Throughout this paper, the ordered tripleX,τ,E indicates a soft topological space.Proposition 1. (see [26]).
LetX,τ,E be a soft topological space. Then, the collection τe=Ge:G,E∈τ defines a topology on X for each e∈E. This collection is called a parametric topology on X.The family,(2)τ⋆=G,E:Ge∈τefor eache∈E,is a soft topology on X finer than τ.The soft topologyτ⋆ given in the proposition above is called an extended soft topology.The equivalence between the enriched and extended soft topologieswas proved in [40].Definition 13.
(see [29, 41]). Let G,E be a soft set in X,τ,E. Then,(i)
The interior ofG,E, denoted by IntG,E, is the union of all soft open sets in X contained in G,E; the closure of G,E, denoted by ClG,E, is the intersection of all soft closed sets containing G,E(ii)
G,E is soft semiopen in X if G,E⊆˜ClIntG,E; the relative complement of a semi open set is called semiclosed(iii)
The semi interior ofG,E, denoted by sIntG,E, is the union of all soft semiopen sets contained in G,E; the semi closure of G,E, denoted by sClG,E, is the intersection of all soft semi closed sets containing G,EDefinition 14.
(see [26]). For a subset Y≠∅˜ of X,τ,E, the family τY=Y˜∩˜G,E:G,E∈τ is called a soft relative topology on Y and the triple Y,τY,E is called a soft subspace of X,τ,E.Definition 15.
(see [26, 42]). Let G,E be a soft set over X and x∈X. We write(i)
x⋐G,E if x∈Ge for some e∈E; y⋐G,E if x∉Ge for every e∈E.(ii)
x∈G,E if x∈Ge for every e∈E; y∉G,E if x∉Ge for some e∈E.Definition 16.
(see [26, 43]). X,τ,E is said to be(i)
Soft regular if for every soft closed setH,E and x∈X such that x∉H,E, there are disjoint soft open sets U,E and V,E containing H,E and x, respectively(ii)
Soft regular⋆ if for every soft closed set H,E and Pex∈X˜ such that Pex∉H,E, there are disjoint soft open sets U,E and V,E, containing H,E and Pex, respectively.Definition 17.
(see [42]). A soft set G,E over X is called stable provided that there is S⊆X such that Ge=S for each e∈E; X,τ,E is called stable provided that all proper nonnull soft open sets are stable.
A familyGi,E:i∈I of soft sets in X,τ,E is said to be a soft cover of X,τ,E (or a soft cover of X˜) if ∪˜i∈IGi,E=X˜. A soft cover Gi,E:i∈I is said to be locally finite if for each soft point Pex has a soft neighbourhood intersecting only finitely many Gi,E. A soft cover Gi,E:i∈I is a soft refinement of a soft cover Hj,E:j∈J if for each Gi,E there is Hj,E such that Gi,E⊂˜Hj,E.Definition 18.
A soft spaceX,τ,E is said to be(i)
Soft compact (resp. soft Lindelöf) [27] provided that every soft open cover of X has a finite (resp. countable) subcover(ii)
Soft paracompact [30, 44] if every soft open cover has a soft open, locally finite refinementDefinition 19.
(see [36]). A soft map fφ:X,τ,A⟶Y,θ,B is said to be(i)
Soft continuous if the inverse image of each soft open set is soft open(ii)
Soft open (resp. soft closed) if the image of each soft open (resp. soft closed) set is soft open (resp. soft closed)(iii)
Soft homeomorphism if it is bijective, soft continuous, and openDefinition 20.
LetG,A and F,B be soft sets over X and Y, respectively. The cartesian product of G,A and F,B is a soft set G×F,A×B over X×Y such that G×Fa,b=Ga×Fb for each a,b∈A×B.Theorem 1. (see [40]).
X,τ,E is extended if and only if IntH,E=IntH,E and ClH,E=ClH,E for any soft subset H,E of X,τ,E.Theorem 2. (see [40]).
IfX,τ,E is a soft regular space, then IntH,E=IntH,E and ClH,E=ClH,E for any stable soft subset H,E of X,τ,E.


## 3. Further Properties of Soft Semiopen Sets

This section is devoted to investigation of some properties of soft semiopen and soft s-regular open sets, which we need to prove some results in Section4.Proposition 2.. (see [45]).
IfG,E is a soft open set, then(3)G,E∩˜ClH,E⊆˜ClG,E∪˜H,E,for every soft subset H,E of X,τ,E.Proposition 3.
IfF,E is a soft closed set, then(4)IntF,E∪˜H,E⊆˜F,E∪˜IntH,E,for every soft subset F,E of X,τ,E.Proof.
LetPex∉F,E∪˜IntH,E. Then, Pex∉F,E and Pex∉IntH,E. Therefore, there is a soft open set U,E such that Pex∈U,E⊆˜Fc,E. Suppose that Pex∈IntF,E∪˜H,E. Then, there is a soft open set V,E such that Pex∈V,E⊆˜F,E∪˜H,E. Now, U,E∩˜V,E is a soft open set containing Pex such that U,E∩˜V,E⊆˜Fc,E and U,E∩˜V,E⊆˜F,E∩˜H,E. This implies that Pex∈IntH,E. This is a contradiction. Thus, Pex∉IntH,E∪˜F,E, as required.Proposition 4.
G,E∪˜IntClG,E=sClG,E for every soft subset G,E of X,τ,E.Proof.
SincesClG,E is soft semiclosed, then IntClsClG,E⊆˜sClG,E. Obviously,(5)IntClG,E⊆˜IntClsClG,E,G,E⊆˜sClG,E.
Therefore,(6)G,E∪˜IntClG,E⊆˜sClG,E.
Conversely, it can be observed that (using Proposition3)(7)IntClG,E∪˜IntClG,E=IntClG,E∪˜ClIntClG,E⊆˜ClG,E∪˜IntClIntClG,E⊆˜ClG,E∪˜ClIntG,E=ClG,E.
Now, we have(8)IntClG,E∪˜IntClG,E⊆˜IntClG,E⊆˜G,E∪˜IntClG,E.
Thus,G,E∪˜IntClG,E is a soft semiclosed set. Hence,(9)sClG,E⊆˜G,E∪˜IntClG,E,as required.Corollary 1.
IntClG,E=sClG,E for every soft open subset G,E of X,τ,E.Definition 21.
A soft subsetH,E of X,τ,E is said to be soft s-regular open if H,E=IntsClH,E.Proposition 5.
Every softs-regular open set is soft open and soft semiclosed.Proof.
LetH,E be a soft s-regular open set. Then, H,E=IntsClH,E. Obviously, H,E is soft regular open. It follows, from Corollary 1, that IntsClH,E=IntIntClH,E=IntClH,E. Thus, H,E=IntClH,E and IntClH,E⊆˜H,E. Hence, the desired result is proved.Proposition 6.
IntsClH,E is a soft s-regular open set for every soft subset H,E of X,τ,E.Proof.
SinceIntsClH,E⊆˜sClH,E, then(10)IntsClIntsClH,E⊆˜IntsClH,E.
Conversely,IntsClH,E⊆˜sClIntsClH,E, then(11)IntsClH,E⊆˜IntsClIntsClH,E.
Thus,IntsClH,E=IntsClIntsClH,E, as required.


## 4. Nearly Soft Menger Spaces

Definition 22.
(see [46]). A soft topological space X,τ,E is said to be soft Menger if for each sequence Un:n∈ℕ of soft open covers of X˜ there is a sequence Vn:n∈ℕ such that Vn is a finite subset of Un for each n and ∪n∈ℕVn is a soft open cover of X˜.
In this section, we formulate the concept of nearly soft Menger spaces and show its relationships with soft compact, soft Lindelöf, and soft Menger spaces with the help of examples. Also, we characterize this concept in terms of soft s-regular open covers and study under what conditions the concepts of soft Menger and nearly soft Menger spaces are equivalent.Definition 23.
A soft topological spaceX,τ,E is said to be nearly soft Menger if for each sequence Un:n∈ℕ of soft open covers of X˜ there is a sequence Vn:n∈ℕ such that, for every n, Vn is a finite subset of Un and ∪˜n∈ℕ∪˜IntClV,E:V,E∈Vn=X˜.Definition 24.
A soft spaceX,τ,E is said to be nearly soft compact (resp. nearly soft Lindelöf) if every soft open cover of X˜ has a finite (resp. countable) subcover such that the soft interiors of soft closures of whose members cover X˜.
Clearly, every nearly soft compact space and every soft Menger space are nearly soft Menger, and every nearly soft Menger space is nearly soft Lindelöf.
The three examples below show that the above implications are not reversible.Example 1.
LetE=e1,e2 be a set of parameters. Consider the soft topological space ℤ,τ,E, where ℤ is the set of integers, and τ is the discrete soft topology on ℤ. Then, ℤ,τ,E is a nearly soft Menger space, but it is not nearly soft compact.Example 2.
LetE=e1,e2 be a parameters set and τ=G,E⊆ℝ˜:1∈G,EorG,E=∅˜ be a soft topology on the set of real numbers ℝ. Then, ℝ,τ,E is not a soft Lindelöf space, so it is not soft Menger. To show that ℝ,τ,E is a nearly soft Menger space, let G,E be a nonnull soft open set. Then, ℝ˜ is the only soft closed set containing G,E. This implies that IntClG,E=ℝ˜. Therefore, for any sequence Un:n∈ℕ of soft open covers of ℝ˜, we choose only one element Gn,E of Un. Set Vn=Gn,E, n∈ℕ. We have the sequence of finite subsets of Un, n∈ℕ, such that ∪˜n∈ℕ∪˜IntClGn,E:Gn,E∈Vn=ℝ˜. Hence, ℝ,τ,E is a nearly soft Menger space.Example 3.
It is well known that a soft topological spaceℝ,τ,E is the classical topological space if E=e is a singleton. Then, we can consider the Sorgenfrey line as an example of a soft Lindelöf, hence a soft nearly Lindelöf space. Since for any open subset G,E of ℝ˜ we have IntClG,E=G,E, one concludes that it is not a Menger space.
An immediate consequence of Corollary1 gives a characterization of nearly soft Menger spaces in terms of soft semiclosed sets.Proposition 7.
A soft spaceX,τ,E is nearly soft Menger if for each sequence Un:n∈ℕ of soft open covers of X˜ there is a sequence Vn:n∈ℕ such that Vn is a finite subset of Un for each n, and ∪˜n∈ℕ∪˜sClV,E:V,E∈Vn=X˜.The following result is a consequence of Corollary1 and Definition 23 and characterizes a nearly soft Menger space in terms of soft s-regular open covers.Corollary 2.
A soft spaceX,τ,E is a nearly soft Menger space if and only if it satisfies SfinsℛR,sℛR, where sℛR denotes the family of all soft s-regular open covers of X˜.We investigate in the following result under what condition the concepts of soft Menger and nearly soft Menger spaces are equivalent.Theorem 3.
The following two properties are equivalent ifX,τ,E is a soft regular⋆ space:(i)
X,τ,E is a soft Menger space(ii)
X,τ,E is a nearly soft Menger spaceProof.
(i)
⇒ (ii) It follows from Proposition 7.(ii)
⇒ (i) Let Un:n∈ℕ be a sequence of soft open covers of X,τ,E. Then, for every n∈ℕ, there is a sequence Vn:n∈ℕ of soft open covers such that Vn′=sClV,E:V,E∈Vn forms a soft open refinement of Un (because X is soft regular∗). Since X,τ,E is a nearly soft Menger space, there is a sequence Wn:n∈ℕ such that Wn is a finite subset of Vn′ for each n, and ∪n∈ℕ∪sClW,E:W,E∈Wn=X˜. For every n∈ℕ and every W,E∈Wn, there is UW,E∈Un such that W,E⊆˜UW,E. Let Un′=UW,E:W,E∈Wn. To prove that ∪n∈ℕUn′ covers X˜, let Pex∈X˜. Then, there are n∈ℕ and W,E∈Wn such that Pex∈W,E. This means that there is UW,E∈Un′ such that W,E⊆˜UW,E. Thus, Pex∈UW,E. Hence, X˜=∪˜n∈ℕ∪˜UW,E:UW,E∈Un.Corollary 3.
The following two properties are equivalent ifX,τ,E is a soft paracompact:(i)
X,τ,E is a soft Menger space(ii)
X,τ,E is a nearly soft Menger space
One of topological results states that Mengerness and near Mengerness coincide in spacesX in which IntClA is a finite set for any A⊆X. This result does not valid for soft topological spaces as the following example shows.Example 4.
Let the set of parametersE be the set of real numbers ℝ and τ=G,E⊆X˜:1∈G,E or G,E is the null soft set } be a soft topology on X=x,y. Following the similar arguments given in Example 2, we infer that X,τ,E is a nearly soft Menger space, but it is not soft Menger.Definition 25.
A soft subsetG,E of X,τ,E is said to be nearly soft Menger in X (or nearly soft Menger relative to X,τ,E) if for every sequence Un:n∈ℕ of covers by soft open subsets of X,τ,E there is a sequence Vn:n∈ℕ such that Vn is a finite subset of Un for each n and ∪n∈ℕ∪IntClV,E:V,E∈Vn⊃G,E.Proposition 8.
The soft union of two nearly soft Menger sets in a soft spaceX,τ,E is also nearly soft Menger in X.Proof.
LetF1,E and F2,E be two nearly soft Menger sets in X and let Un:n∈ℕ be a sequence of soft open covers of F1,E∪˜F2,E. Then, there is a sequence Vn:n∈ℕ of soft open covers of F1,E and a sequence Vn′:n∈ℕ of soft open covers of F2,E such that Vn and Vn′ are finite subsets of Un for each n, and ∪n∈ℕ∪IntClV,E:V,E∈Vn and ∪n∈ℕ∪IntClV′,E:V′,E∈Vn′ are soft covers of F1,E and F2,E, respectively. Thus, the sequence Wn=Vn∪Vn′:n∈ℕ witnesses for Un:n∈ℕ that F1,E∪˜F2,E is nearly soft Menger in X.Proposition 9.
The property of being a nearly soft Menger is hereditary by soft clopen subspaces.Proof.
LetY,τY,E be a soft clopen subspace of a nearly soft Menger space X,τ,E. Suppose that Un:n∈ℕ is a sequence of soft open covers of Y,τY,E. Then, for each n and each U,E∈Un, there is a soft open subset HU,E of X,τ,E such that U,E=Y˜∩˜HU,E. Set ℋn=HU,E:U,E∈Un∪Yc˜. Now, ℋn:n∈ℕ is a sequence of soft open covers of X,τ,E. By hypothesis, there is a sequence Wn:n∈ℕ such that Wn is a finite subset of ℋn for each n and ∪n∈ℕ∪IntClW,E:W,E∈Wn is a soft cover of X˜. By taking Vn=U,E:HU,E∈Wn for each n, we find that Vn is a finite subset of Un for each n, and ∪n∈ℕ∪IntClU,E:U,E∈Vn is a soft cover of Y˜. Hence, Y,τY,E is a nearly soft Menger space.
Example2 illustrates that the nearly soft Menger property is not soft closed hereditary. The following example illustrates that the nearly soft Menger property is not soft open hereditary as well.Example 5.
LetE=e1,e2 be a set of parameters and τ=G,E⊆ℝ˜:1⋐G,E be a soft topology on the set of real numbers ℝ. Since ℝ,τ,E is soft compact, it is a nearly soft Menger space. On the contrary, let Y=ℝ\1. Then, Y˜ is an uncountable soft open set and τY is the discrete soft topology. Hence, Y,τY,E is not a nearly soft Menger space.Proposition 10.
IfX,τ,E is an enriched nearly soft Menger space, then E is a countable set.Proof.
LetX,τ,E be enriched. Then, U=U,E:Ue=X and Ue′=∅ for each e′≠e is a soft open cover of X˜. We construct a sequence of soft open covers of X˜ as follows: Un=U for each n∈ℕ. By nearly soft Mengerness, there is a sequence Vn:n∈ℕ such that Vn is a finite subset of Un for each n and ∪n∈ℕ∪IntClU,E:U,E∈Vn is a soft cover of X˜. Since U,E is a soft clopen set, then ∪n∈ℕ∪U,E:U,E∈Vn is also a soft cover of X˜. Hence, E must be countable.
Example below shows that the converse of the abovementioned proposition fails.Example 6.
Letτ be the discrete soft topology on the real numbers set ℝ and let E=e1,e2,e3 be a set of parameters. Then, X,τ,E is enriched and E is countable. However, it is not a nearly soft Menger space.
The following results investigate nearly soft Mengerness between a soft topological space and its parametric topological spaces.Proposition 11.
IfX,τ,E is an enriched nearly soft Menger space, then X,τe is a nearly Menger space for each e∈E.Proof.
LetUne:n∈ℕ be a sequence of open covers of X,τe, and let for each n, Une=Uie:i∈I. Set V=Vi,E:i∈I, where Vie=Uie and Vie′=X for each e′≠e, and consider the sequence Vn=V:n∈ℕ of soft open covers of X,τ,E. By near soft Mengerness of X,τ,E, there is a sequence Wn:n∈ℕ such that, for each n, Wn is a finite subset of Vn and ∪n∈ℕ∪∪IntClW,E:W,E∈Wn is a soft cover of X˜. It follows from Theorem 1 that ∪n∈ℕ∪IntClW,E:W,E∈Wn is also a soft cover of X˜. Denote ℋn=IntClWne=IntClUne:Wn,E∈Wn. Then, the sequence ℋn:n∈ℕ witnesses for Une:n∈ℕ that X,τe is a nearly Menger space.Proposition 12.
If allX,τe, e∈E, are nearly Menger spaces, then X,τ,E is a nearly soft Menger space provided that X,τ,E is enriched and E is countable.Proof.
LetUn:n∈ℕ be a sequence of soft open covers of X,τ,E, where Un=Ui,E:i∈In. Fix e and let Vne=Uie:i∈In. Then, Vne:n∈ℕ is a sequence of open covers of X,τe. By near Mengerness of X,τe, there is a sequence Wne:n∈ℕ such that, for each n, Wne is a finite subset of Vne and ∪n∈ℕIntClUie:Uie∈Wne is an open cover of X. Taking a sequence ℋn:n∈ℕ such that ℋn=IntClW,E:We∈Wne. It follows from Theorem 1 that ℋn=IntClW,E:We∈Wne. Since Wne is finite, then ℋn is a finite set and since E is countable, then the sequence ℋn:n∈ℕ witnesses for Un:n∈ℕ that X,τ,E is a nearly soft Menger space.
The following example elucidates that the condition of countability ofE given in the abovementioned proposition is essential.Example 7.
Letτ=G,E⊆˜X˜:Ge=Xor∅ be a soft topology on X=1,2, and E be an uncountable set of parameters. Then, all X,τe are the indiscrete topologies on X. Therefore, all X,τe are Menger spaces. On the contrary, it follows from Proposition 10 that X,τ,E is not a soft Menger space.Proposition 13.
LetX,τ,E be a soft regular space. Then, it is nearly soft Menger if and only if X,τe is nearly Menger for all e∈E.Proof (necessity).
LetUne:n∈ℕ be a sequence of open covers of X,τe such that Une=Uie:i∈In. Then, Vn=Vi,E:i∈I such that Vie=Uie for each e∈E is a soft cover of X,τ,E. Since X,τ,E is soft regular, then every soft open set is stable. Therefore, Vn:n∈ℕ is a sequence of soft open covers of X,τ,E. By nearly soft Mengerness of X,τ,E, there is a sequence Wn:n∈ℕ such that for each n, Wn is a finite subset of Vn and ∪n∈ℕ∪IntClVi,E:Vi,E∈Wn is a soft open cover of X˜. From Theorem 2, it follows that ∪n∈ℕIntClVi,E:Vi,E∈Wn is also a soft open cover of X˜. Set ℋn=IntClWne=IntClUne:Wn,E∈Wn. Then, the sequence ℋn:n∈ℕ testifies for Une:n∈ℕ that X,τe is a nearly Menger space.
Sufficiency: letUn:n∈ℕ be a sequence of soft open covers of X,τ,E such that Un=Ui,E:i∈In. Since Ui,E is stable, then Vn=Uie:i∈In is an open covers of X,τe for each e∈E and each n∈ℕ. By the nearly Menger property of X,τe, there is a sequence Wn:n∈ℕ such that, for each n, Wn is a finite subset of Vn and ∪n∈ℕIntClUie:Uie∈Wn is an open cover of X. Consider the sequence ℋn:n∈ℕ such that ℋn=IntClW,E:We∈Wn for each e∈E, of soft open covers of X,τ,E. Since ℋn=IntClW,E:We∈Wn for each e∈E, we conclude that X,τ,E is a nearly soft Menger space.
In what follows, we examined some features of a nearly soft Menger space under some types of soft mappings.Proposition 14.
The soft continuous open surjective image of a nearly soft Menger space is a nearly soft Menger space.Proof.
Letgφ be a soft continuous open mapping of a nearly soft Menger space X,τ,A onto Y,θ,B and Un:n∈ℕ be a sequence of soft open covers of Y,θ,B. From the soft continuity of gφ, we obtain that Vn=gφ−1Ui,B:Ui,B∈Un is a sequence of soft open covers of X,τ,A. Therefore, there is a sequence ℋn:n∈ℕ such that, for each n, ℋn is a finite subset of Vn and ∪˜n∈ℕ∪˜IntClgφ−1Ui,B:gφ−1Ui,B∈ℋn=X˜. Since gφ is surjective, then Y˜=gφX˜=gφ∪˜n∈ℕ∪˜Intgφ−1ClUi,B:gφ−1Ui,B∈ℋn and since gφ is soft open, then Y˜=∪˜n∈ℕ∪˜Intgφgφ−1ClUi,B:gφ−1Ui,B∈ℋn=∪˜n∈ℕ∪˜IntClUi,B:gφ−1Ui,B∈ℋn. Thus, IntClUi,B:gφ−1Ui,B∈ℋn is a soft cover of Y,θ,B. Hence, the sequence ℋn:n∈ℕ witnesses for Un:n∈ℕ that Y,θ,B is a nearly soft Menger space.Corollary 4.
The property of being a nearly soft Menger is a soft topological property.Example 2.15 in [9] shows that the product of two nearly Menger spaces is not always a nearly Menger space. Since a classical topological space is a special case of a soft topological space when a parameters set is a singleton, then this result is still valid on soft topological spaces.In the rest of this section, we prove that the product of a nearly soft Menger space and a nearly soft compact space is a nearly soft Menger.Lemma 1.
LetG,A and H,B be two subsets of X,τ,A and Y,θ,B, respectively. Then,(i)
sClG,A×sClH,B=sClG,A×H,B(ii)
sIntG,A×sIntH,B=sIntG,A×H,BProof.
(i)
Suppose thatPa,bx,y∉sClG,A×H,B. Then, there exists a soft semiopen subset U,A×V,B of X×Y,τ×θ,A×B containing Pa,bx,y such that U,A×V,B∩˜G,A×H,B=∅˜. So. U,A∩˜G,A=∅˜ or V,B∩˜H,B=∅˜. This means that Pax∉sClG,A or Pby∉sClH,B. Thus, Pa,bx,y∉sClG,A×sClH,B. Hence, sClG,A×sClH,B⊆˜sClG,A×H,B.Suppose now thatPax∉sClG,A or Pby∉sClH,B. Without loss of generality, let Pax∉sClG,A. Then, there exists a soft semiopen subset U,A of X,τ,A containing Pax such that U,A∩˜G,A=∅˜. Now, G,A×Y˜ is a soft semiopen subset of X×Y,τ×θ,A×B containing Pa,bx,y satisfies that G,A×Y˜∩˜G,A×H,B=∅˜. This means Pa,bx,y∉sClG,A×H,B. Hence, sClG,A×H,B⊆˜sClG,A×sClH,B.(ii)
By using a similar argumentation, one can prove item (ii).Theorem 4.
The product of a nearly soft Menger space and a nearly soft compact space is a nearly soft Menger.Proof.
LetX,τ,A be a nearly soft Menger space and Y,θ,B be a nearly soft compact space, and let Wn:n∈ℕ is a sequence of soft open covers of X×Y,τ×θ,A×B. One may assume that, for every n∈ℕ, Wn=Un×Vn, where all Un are soft open covers of X,τ,A and all Vn are soft open covers of Y,θ,B. Since Y,θ,B is nearly soft compact there is a sequence ℋn:n∈ℕ such that, for each n, ℋn is a finite subset of Vn and Y˜=∪˜sClH,B:H,B∈ℋn. As X,τ,A is nearly soft Menger there are finite sets Gn⊂Un such that ∪˜n∈ℕ∪˜sClG,A:G,A∈Gn=X˜. Let ℱn=Gn×ℋn for each n∈ℕ. Then, for every n, ℱn is a finite subset of Wn. By using the previous lemma, it is easy now to see that ∪˜n∈ℕ∪˜sClF,A×B:F,A×B∈ℱn=X×Y˜ which means that X×Y,τ×θ,A×B is nearly soft Menger.


## 5. Conclusion

This study is devoted to introducing and investigating the concept of nearly soft Menger spaces. We provide several examples to discuss its relationships with soft Menger and soft Lindelöf spaces, and to show the interchangeability of nearly Mengerness between a soft space and its parametric spaces. In general, we have initiated many properties of it parallel to their corresponding properties from classical topology. Our hope is that the introduced concepts will be beneficial for the researchers to further promote and advance the study selection principles in soft topologies.



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