# New Soft Structure: Infra Soft Topological Spaces

Authors: Tareq M. Al-shami
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

It is always convenient to find the weakest conditions that preserve some topologically inspired properties. To this end, we introduce the concept of an infra soft topology which is a collection of subsets that extend the concept of soft topology by dispensing with the postulate that the collection is closed under arbitrary unions. We study the basic concepts of infra soft topological spaces such as infra soft open and infra soft closed sets, infra soft interior and infra soft closure operators, and infra soft limit and infra soft boundary points of a soft set. We reveal the main properties of these concepts with the help of some elucidative examples. Then, we present some methods to generate infra soft topologies such as infra soft neighbourhood systems, basis of infra soft topology, and infra soft relative topology. We also investigate how we initiate an infra soft topology from crisp infra topologies. In the end, we explore the concept of continuity between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces.


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## Body

## 1. Introduction

This paper is at the junction of two disciplines, namely, infra topology and soft set theory. Their hybridization has produced an interesting structure called infra soft topology which is the framework for our contribution. Let us summarize the antecedents and state-of-the-art of the topic.In 1999, Molodtsov [1] proposed the concept of soft sets as a new mathematical approach to cope with problems containing uncertainties, and he explained the potentiality of soft sets to handle many problems in different areas. This theory has gained much attention from researchers and scientists because of its diverse applications. It is possible to see a rapid growth in soft sets’ research in the last few years, see, for example, [2, 3].In 2003, Maji et al. [4] put forward some soft operations such as union and intersection and subset and equality relations between two soft sets. They also defined the null and absolute soft sets as a soft version of the empty and universal crisp sets. Ali et al. [5] showed some shortcoming given in [4], defined certain new operations on soft sets, and explored their main properties. Abbas et al. [6] and Qin and Hong [7] described new types of soft equality, which they applied to introduce new types of algebraic structures. Recently, Al-shami and El-Shafei [8] have defined and discussed new types of operations between soft sets.In 2011, Shabir and Naz [9] introduced a topological structure on soft setting. They defined the fundamental notions of soft topologies such soft open and closed sets, soft subspaces, and belonging and nonbelonging relations which are used to initiate soft separation axioms. Zorlutuna et al. [10] came up with the idea of soft point which helps to study some properties of soft interior points and soft neighbourhood systems. This concept was independently reformulated by Samanta et al. [11, 12], while Das and Samanta [11] applied the new version of the soft point to study the concept of soft metric spaces and Nazmul and Samanta [12] used it to discuss soft neighbourhood systems and reveal some relations of soft limit points of a soft set. Many scholars analyzed the properties of soft topologies and compared their performance with the case of classical topologies, see, for example, [13–23]. Generalizations of open sets were investigated in soft topologies, see [24, 25]. In [26], we corrected some alleged results concerning soft separation axioms, especially those defined using soft points.Some generalizations of a soft topology were given by weakening a soft topology’s conditions. For example, in 2014, El-Sheikh and Abd El-Latif [27] established the concept of supra soft topological spaces by neglecting a finite intersection condition of a soft topology. This path, therefore, attracted a lot of researchers who studied essential notions related to supra soft topologies, see, for example, [28, 29]. Thomas and John [30] formulated the concept of soft generalized topological spaces which are defined as a family of soft sets that satisfy an arbitrary union condition of a soft topology, and Zakari et al. [31] originated the concepts of soft weak structures which are defined as a family of soft sets that contain the null soft set Φ. Ittanagi [32] studied the concept of soft bitopological space which can be regarded as a soft topological space when the two soft topologies are identical. Lately, Al-shami et al. [33] have constructed soft topology on ordered setting as an extension of soft topology. Similarly, Al-shami and El-Shafei [34] studied supra soft topology on ordered setting.We note that many properties of soft topological spaces are still valid on infra soft topological spaces, and initiating examples that show some relationships between certain topological concepts are easier on infra soft topological spaces. Therefore, we aim in this paper to perform an exhaustive analysis of infra soft topological spaces.This paper is structured as follows: after this introduction, Section2 addresses some definitions and properties that help the reader to well understand this manuscript. In Section 3, we introduce the concept of infra soft topological spaces and disclose the main properties of infra soft interior, infra soft closure, infra soft limit, and infra soft boundary points of a soft set. In Section 4, we tackle some techniques of generating infra soft topology such as infra soft neighbourhoods and infra soft subspaces. In Section 5, we formulate the concept of infra continuous maps between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces. We give some conclusions and make a plan for future works in Section 6.


## 2. Preliminaries

In this section, we recall the technical concepts that we need in this paper.The notation2X refers to the set of subsets of X.Definition 1 (see [1]).
A pairG,E is a soft set over a nonempty set X provided that G is a map from the set of parameters E to 2X.
For the sake of brevity and ease, henceforth, a soft set is symbolized byGE instead of G,E. It is identified as GE=e,Ge:e∈EandGe∈2X.
The set of all soft sets overX under a set of parameters E is symbolized by SXE.Definition 2 (see [11]).
A soft setGE is called finite (resp., countable) if Ge is finite (resp., countable) for each e∈E.Definition 3 (see [35]).
The relative complement of a soft setGE is a soft set GEc, where Gc:E⟶2X is a map defined by Gce=X∖Ge for each e∈E.Definition 4 (see [4]).
A soft setGE over X is said to be the null soft set, symbolized by Φ, if Ge=∅ for each e∈E. Its relative complement is said to be the absolute soft set, symbolized by X˜.Definition 5 (see [11, 12]).
A soft pointPE over X is a soft set such that Pe is a singleton set and Pe′ is the empty set for each e′≠e. This soft point will be briefly symbolized by Pex.Since a soft topological space and its generalizations, which are the theme of this manuscript, are defined under a fixed set of parameters, we will mention the definitions and findings given in the previous studies under a fixed set of parameters.Definition 6 (see [4]).
A soft setGE is a soft subset of a soft set FE, symbolized by GE⊑FE, if Ge⊆Fe for all e∈E.
The soft setsGE and FE are called soft equal if each is a soft subset of the other.Definition 7 (see [5]).
The intersection of two soft setsGE and FE over X, symbolized by GE⊓FE, is a soft set HE, where a map H:E⟶2X is given by He=Ge∩Fe for each e∈E.Definition 8 (see [4]).
The union of two soft setsGE and FE over X, symbolized by GE⊔FE, is a soft set HE, where a map H:E⟶2X is given by He=Ge∪Fe for each e∈E.Definition 9 (see [9, 19]).
For a soft setGE over X and x∈X, we say that(i)
x∈GE (it reads as x totally belongs to GE) if x∈Ge for each e∈E(ii)
x∉GE (it reads as x does not partially belong to GE) if x∉Ge for some e∈E(iii)
x⋐GE (it reads as x partially belongs to GE) if x∈Ge for some e∈E(iv)
x⋐GE (it reads as x does not totally belong to GE) if x∉Ge for each e∈ESoft maps are recalled in the next two definitions with some modifications to be convenient for defining the concepts of soft continuous maps.Definition 10 (see [36]).
A soft mapping betweenSXE and SYE is a pair f,φ, denoted also by fφ, of mappings such that f:X⟶Y and φ:E⟶E. Let GE and HE be subsets of SXE and SYE, respectively. Then, the image of GE and preimage of HE are defined as follows:(i)
fφGE=fG,E is a soft set in SYE such that(1)fGe=∪˜a∈φ−1efGe,φ−1e≠∅,∅,φ−1e=∅.for eache∈E.(ii)
fφ−1HE=f−1H,E is a soft set in SXE such that(2)f−1He=f−1Hφefor eache∈E.Remark 1.
Henceforth, a soft mapfφ:SXE⟶SYE implies a map f from the universal set X to the universal set Y and a map φ from the set of parameters E to itself.Definition 11 (see [10]).
A soft mapfφ:SXE⟶SYE is said to be injective (resp. surjective and bijective) if f and φ are injective (resp. surjective and bijective).Definition 12 (see [37]).
An infratopology onX is a collection τ of subsets of X, that is, closed under finite intersections and satisfies ∅∈τ.Definition 13 (see [9]).
The collectionϑ of soft sets over X under a fixed set of parameters E is called a soft topology on X if it satisfies the following axioms:(i)
X˜ and Φ belong to ϑ(ii)
The union of an arbitrary family of soft sets inϑ belongs to ϑ(iii)
The intersection of a finite family of soft sets inϑ belongs to ϑ
The tripleX,ϑ,E is called a soft topological space. The term given to each member of ϑ is a soft open set, and the relative complement of each member of ϑ is a soft closed set.


## 3. Infra Soft Topological Spaces

In this section, we introduce the concept of infra soft topology as a class of soft sets, that is, closed under finite soft intersection and contains the null soft set. It lies between soft topology and soft weak structure, and it is independent of a supra soft topology. We define the main concepts of infra soft topology and reveal their main properties. One of the merits of infra soft topology is that many results of soft topology are still valid on infra soft topology, especially those are related to the soft interior and closure operators. A number of examples are provided to validate the obtained results.Definition 14.
The collectionϑ of soft sets over X under a fixed set of parameters E is said to be an infra soft topology on X if it is closed under finite soft intersection and the null soft set is a member of ϑ.
The tripleX,ϑ,E is called an infra soft topological space. Every member of ϑ is called an infra soft open set, and its relative complement is called an infra soft closed set.The following examples elucidate the uniqueness of infra soft topology than the other celebrated soft structures.Example 1.
LetE=e1,e2 be a set of parameters and X=a,b be the universal set. Then, ϑa=X˜,FE⊑X˜:a∉FE is an infra soft topology on X; we called ϑa an excluding point infra soft topology. On the contrary, UE=e1,a,e2,∅ and VE=e1,∅,e2,a are infra soft open sets, but their union is not an infra soft open set. Therefore, ϑa is neither supra soft topology nor generalized soft topology. Hence, it is not a soft topology.Example 2.
LetE=e1,e2 be a set of parameters and X=a,b be the universal set. Then, ϑa=Φ,FE⊑X˜:a⋐FE is a supra soft topology on X; we called ϑa a particular point supra soft topology. On the contrary, UE=e1,b,e2,X and VE=e1,X,e2,b are supra soft open sets, but their intersection is not a supra soft open set. Therefore, ϑa is not an infra soft topology. Moreover, it is not a soft topology.Remark 2.
It can be examined that the intersection of any family of infra soft topologies is always infra soft topology. But the union of two infra soft topologies need not be an infra soft topology. We show this fact in the following example.Example 3.
Consider the soft sets below defined onX=a,b,c with a set of parameters E=e1,e2:(3)F1E=e1,a,b,e2,c,F2E=e1,c,e2,b,F3E=e1,a,e2,∅,F4E=e1,b,c,e2,b.
Then,ϑ1=Φ,X˜,F1E,F2E and ϑ2=Φ,X˜,F3E,F4E are two infra soft topologies on X. But ϑ1∪ϑ2=Φ,X˜,F1E,F2E,F3E,F4E is not an infra soft topology on X because F1E⊓F4E∉ϑ1∪ϑ2.The following results present two techniques to originate infra soft topology using soft maps.Proposition 1.
Letfφ:SXE⟶SYE be a soft map. If ϑ is an infra soft topology on Y, then a class θ=fφ−1GE⊑X˜:GE∈ϑ is an infra soft topology on X.Proof.
SinceΦandY˜∈ϑ, then fφ−1Φ=Φandfφ−1Y˜=X˜∈θ. Let F1EandF2E∈θ. Then, there exist H1EandH2E∈ϑ such that fφ−1H1E=F1E and fφ−1H2E=F2E. Therefore, F1E⊓F2E=fφ−1H1E⊓fφ−1H2E=fφ−1H1E⊓H2E. Since H1E⊓H2E∈ϑ, then F1E⊓F2E∈θ. Hence, the proof is complete.In a similar manner, one can prove the following result.Proposition 2.
Letfφ:SXE⟶SYE be a bijective soft map. If ϑ is an infra soft topology on X, then a class θ=fφ−1GE⊑Y˜:GE∈ϑ is an infra soft topology on Y.Definition 15.
We define the infra soft interior points and infra soft closure points of a soft subsetHE of X,ϑ,E which are, respectively, denoted by IntHE and ClHE as follows:(i)
IntHE is the union of all infra soft open sets that are contained in HE(ii)
ClHE is the intersection of all infra soft closed sets containing HEThe following example illustrates that the infra soft interior points and infra soft closure points of a soft set need not be infra soft open and infra soft closed sets, respectively.Example 4.
LetE=e1,e2 be a set of parameters. Then, {ϑ = ℝ,FE⊏ℝ:FE is finite} is an infra soft topology on the set of real numbers ℝ. It is clear that ℝ,ϑ,E is not a soft topological space. Let GE=e1,ℕ,e2,ℕ, where ℕ is the set of natural numbers. We find that IntGE=ClGE equals to GE which is neither an infra soft open set nor an infra soft closed set.Proposition 3.
LetHE be a soft subset of X,ϑ,E. Then, the following properties hold.(i)
IfHE is an infra soft open set, then IntHE=HE(ii)
IfHE is an infra soft closed set, then ClHE=HEProof.
It immediately follows from Definition15.Example4 shows that the converse of the two properties given in the above proposition need not be true in general. These two properties are an example of some soft topological properties that are losing on infra soft topological spaces.Proposition 4.
LetHE be a soft subset of X,ϑ,E. Then, the following properties hold.(i)
Pex∈IntHE if and only if there exists an infra soft open set GE such that Pex∈GE⊑HE(ii)
Pex∈ClHE if and only if HE⊓GE≠Φ for every infra soft open set GE containing PexProof.
(i)
Straightforward.(ii)
Necessity: letPex∈ClHE. Then, Pex belongs to every infra soft closed set containing HE. Suppose that there exists an infra soft open set GE containing Pex such that HE⊓GE=Φ, so HE⊑GEc. This is a contradiction. Thus, the necessary part holds.
Sufficiency: letPex∉ClHE. Then, ClHEc is an infra soft open set containing Pex such that HE⊓ClHEc=Φ. Thus, the proof is complete.Proposition 5.
LetHE be a soft subset of X,ϑ,E. Then,(i)
IntHEc=ClHEc(ii)
ClHEc=IntHEcProof.
(i)
IntHEc = {⊔i∈IGiE:GiE is an infra soft open set included in HE}c = ⊓i∈I{GEc:GEc is an infra soft closed set including HEc}=ClHEc
In a similar manner, we prove (ii).Proposition 6.
LetHE be a soft subset of X,ϑ,E. Then,(i)
IfUE is an infra soft open set, then UE⊓ClHE⊑ClUE⊓GE(ii)
IfFE is an infra soft closed set, then IntFE⊔HE⊑FE⊔IntHEProof.
(i)
LetPex∈UE⊓ClHE. Then, Pex∈UE and Pex∈ClHE. This means that, for each infra soft open set VE containing Pex, we have VE⊓HE≠Φ. Since ϑ is infra soft topology, then UE⊓VE is an infra soft open set containing Pex; consequently, VE⊓UE⊓HE≠Φ⟺VE⊓UE⊓HE≠Φ. Thus, Pex∈ClUE⊓HE. Hence, UE⊓ClHE⊑ClUE⊓HE.(ii)
LetPex∉FE⊔IntHE. Then, Pex∉FE and Pex∉IntHE. Therefore, there is an infra soft open set UE such that Pex∈UE⊑FEc. Suppose that Pex∈IntFE⊔HE. Then, there is an infra soft open set VE such that Pex∈VE⊑FE⊔HE. Now, we have UE⊓VE which is an infra soft open set containing Pex such that UE⊓VE⊑FEc and UE⊓VE⊑FE⊔HE. This means that Pex∈IntHE. But this contradicts Pex∉IntHE. Thus, Pex∉IntFE⊔HE. Hence, IntUE⊔GE⊑FE⊔IntGE.Theorem 1.
LetFEandGE be soft subsets of X,ϑ,E. Then, the following properties hold:(i)
IntX˜=X˜(ii)
IntFE⊑FE(iii)
IfGE⊑FE, then IntGE⊑IntFE(iv)
IntIntFE=IntFE(v)
IntFE⊓GE=IntFE⊓IntGEProof.
The proofs of (i), (ii), and (iii) are obvious.(iv)
It follows from (ii) thatIntIntFE⊑IntFE. Conversely, let Pex∈IntFE. Then, there exists an infra soft open set GE such that Pex∈GE⊑FE. By (iii), we have Pex∈IntGE=GE⊑IntFE. Therefore, Pex∈IntIntFE. This ends the proof of IntIntFE=IntFE.(v)
It is clear thatIntFE⊓GE⊑IntFE⊓IntGE. Conversely, let Pex∈IntFE⊓IntGE. Then, there exist two infra soft open sets UEandVE such that Pex∈UE⊑FE and Pex∈VE⊑GE. Now, UE⊓VE is an infra soft open set containing Pex such that Pex∈UE⊓VE⊑FE⊓GE. Therefore, Pex∈IntFE⊓GE. Thus, IntFE⊓IntGE⊑IntFE⊓GE. Hence, the proof is complete.One can prove the following result similarly.Theorem 2.
LetFEandGE be soft subsets of X,ϑ,E. Then, the following properties hold:(i)
ClΦ=Φ(ii)
FE⊑ClFE(iii)
IfGE⊑FE, then ClGE⊑ClFE(iv)
ClClFE=ClFE(v)
ClFE⊔GE=ClFE⊔ClGEThe following example clarifies that the following two equalities are not always true.(i)
Int⊓n=1∞GnE=⊓n=1∞IntGnE(ii)
Cl⊔n=1∞GnE=⊔n=1∞ClGnEExample 5.
LetE=e1,e2 be a set of parameters and ϑ = {Φ,FE⊑ℝ˜:FEc is finite} be a family of soft sets on the set of real numbers ℝ. Then, ℝ,ϑ,E is an infra soft topological space. Let HnE=e1,ℝ∖n,e2,ℝ∖n. Then, IntHnE=HnE. Therefore, ⊓n=1∞IntGnE=e1,ℝ∖ℕ,e2,ℝ∖ℕ. But Int⊓n=1∞GnE=Φ. Also, let GnE=e1,n,e2,n. Then, ClGnE=GnE for each n. Therefore, ⊔n=1∞ClGnE=e1,ℕ,e2,ℕ. But Cl⊔n=1∞GnE=e1,ℝ,e2,ℝ.Proposition 7.
IfFEandGE are soft subsets of X,ϑ,E such that ClFE⊓ClGE=Φ, then IntFE⊔GE=IntFE⊔IntGE.Proof.
It follows from (iii) of Theorem1 that IntFE⊔IntGE⊑IntFE⊔GE. To prove that IntFE⊔GE⊑IntFE⊔IntGE, let Pex∈IntFE⊔GE. Then, there exists an infra soft open set UE containing Pex such that Pex∈UE⊑FE⊔GE. Now, we have three cases:Case 1.
UE⊑FE. Then, Pex∈IntFE.Case 2.
UE⊑GE. Then, Pex∈IntGE.Case 3.
UE⊑FE and UE⊑GE. Then, UE⊓FE≠Φ and UE⊓GE≠Φ. This implies that, for each infra soft open set VE containing Pex, we obtain Pex∈ClFE and Pex∈ClGE. But this contradicts ClFE⊓ClGE=Φ. Therefore, this case is impossible.
Thus, Cases1 and 2 are only valid, and they imply that IntFE⊔GE⊑IntFE⊔IntGE. Hence, the proof is complete.The following result explains two methods of generating crisp infra topologies from an infra soft topology.Proposition 8.
LetX,ϑ,E be an infra soft topological space. Then,(i)
X,ϑe is an infra topological space for each e∈E, where ϑe=Fe:FE∈ϑ(ii)
E,ϑx is an infra topological space for each x∈X, where ϑx=e:x∈Fe:FE∈ϑProof.
(i)
SinceΦandX˜∈ϑ, then ∅andX∈ϑe. To prove that ϑe is closed under finite intersection, let UandV∈ϑe. Then, there exists two infra soft open subsets FEandGE of X,ϑ,E such that Fe=U and Ge=V. Owing to the fact that FE⊓GE∈ϑ, we obtain U∩V∈ϑe, as required.
Following similar arguments, one can prove (ii).We calledX,ϑe given in the above proposition “a parametric infra topological space.”The converse of the above proposition need not be true, as shown in the following example.Example 6.
Consider the soft sets below defined onX=a,b,c with a set of parameters E=e1,e2:(4)F1E=e1,a,e2,b,c,F2E=e1,a,e2,a.
Then,ϑ=Φ,X˜,F1E,F2E is not an infra soft topology on X. On the contrary, ϑe1=∅,X,a and ϑe2=∅,X,a,b,c are two infra soft topologies on X. Also, ϑa=∅,E,e1, ϑb=∅,E,e2, and ϑc=∅,E are three infra soft topologies on E.Definition 16.
LetHE be a soft subset of X,ϑ,E. Then,(i)
ClHE is a soft set given by ClHe=ClHe, where ClHe is the closure of He in X,ϑe for each e∈E(ii)
IntHE is a soft set given by IntHe=IntHe, where IntHe is the interior of He in X,ϑe for each e∈EProposition 9.
LetHE be a soft subset of X,ϑ,E. Then,(i)
IntHE⊑IntHE(ii)
ClHE⊑ClHEProof.
(i)
LetPex∈IntHE. Then, there exists an infra soft open set UE containing Pex such that Pex∈UE⊑HE. Now, Ue is an infra open subset of X,ϑe for every e∈E such that Ue⊆He. Therefore, Pex∈IntHe; thus, Pex∈IntHE, as required.
Following similar arguments to those given in the proof of (i), one can prove (ii).The converse of properties (i) and (ii) in the above theorem is not always true as explained in the following example.Example 7.
Assume thatX,ϑ1,E is the infra soft topological space given in Example 3. Let UE=e1,a,c,e2,b,c and VE=e1,a,c,e2,a,b. By calculating, we find IntUE=e1,c,e2,b and ClVE=X˜. On the contrary, IntHE=e1,c,e2,b,c and ClHE=e1,X,e2,a,b. Hence, IntHE⊑IntHE and ClHE⊑ClHE.Definition 17.
LetHE be a soft subset of an infra soft topological space X,ϑ,E. A soft point Pex is said to be an infra soft limit point of HE if UE\Pex⊓HE≠Φ for each infra soft open set UE containing Pex.
All infra soft limit points ofHE are said to be an infra-derived soft set of HE and denoted by HEi′.Proposition 10.
IfGE and HE are soft subsets of X,ϑ,E, then the following properties hold:(i)
Φi′=Φ and X˜i′⊑X˜(ii)
IfGE⊑HE, then GEi′⊑HEi′(iii)
IfPex∈GEi′, then x∈GE∖Pexi′(iv)
GEi′⊔HEi′=GE⊔HEi′Proof.
The proofs of (i), (ii), and (ii) are obvious.(iv)
It follows from (ii) thatGEi′⊔HEi′⊑GE⊔HEi′. Conversely, let Pex∉GEi′⊔HEi′. Then, there exist infra soft open sets UEandVE containing Pex such that UE\Pex⊓GE=Φ and VE\Pex⊓HE=Φ. It is clear that UE⊓VE is an infra soft open set containing Pex such that UE⊓VE\Pex⊓GE⊔HE=Φ. This means that Pex∉GE⊔HEi′. Thus, GE⊔HEi′⊑GEi′⊔HEi′. Hence, the proof is complete.The next result investigates the role of infra soft limit points in studying the infra soft closure points of a soft set.Theorem 3.
LetFE be a soft subset of X,ϑ,E. Then,(i)
IfFE is infra soft closed, then FEi′⊆FE.(ii)
FE⊔FEi′i′⊑FE⊔FEi′.(iii)
ClFE=FE⊔FEi′.Proof.
(i)
LetFE be an infra soft closed set and Pex∉FE. Then, Pex∈FEc. Now, FEc is an infra soft open set such that FEc⊓FE=Φ. Therefore, Pex∉FEi′. Thus, FEi′⊑FE, as required.(ii)
LetPex∉FE⊔FEi′. Then, Pex∉FE and Pex∉FEi′. Therefore, there exists an infra soft open set GE such that(5)GE⊓FE=Φ.On the contrary, for eachPex∈GE, we have Pex∉FE. Then, GE∖Pex⊓FE=Φ. Therefore, Pex∉FEi′, and this implies that(6)GE⊓FEi′=Φ.From (1) and (2), we get GE⊓FE⊔FEi′=Φ. Thus, pex∉FE⊔FEi′i′. Hence, FE⊔FEi′i′⊑FE⊔FEi′, as required.(iii)
It follows from (ii) of Proposition4 that FEi′⊑ClFE; also, it is clear that FE⊑ClFE. Then, FE⊔FEi′⊑ClFE. On the contrary, let Pex∈ClFE. Then, FE⊓GE≠Φ for every infra soft open set containing Pex. Without loss of generality, suppose that Pex∉FE. Then, FE\Pex⊓GE≠Φ. Therefore, Pex∈FEi′. Hence, we obtain the desired result.The converse of property (i) in the above theorem is not always true as explained in the following example.Example 8.
From Example4, we demonstrate that ClGE=GE. This directly means that GEi′⊑GE. But GE is not an infra soft closed set.Definition 18.
LetHE be a soft subset of X,ϑ,E. The infra soft boundary points of HE, denoted by BHE, are all soft points which belong to the relative complement of IntHE⊔IntHEc.Proposition 11.
LetHE be a soft subset of X,ϑ,E. Then,(i)
BHE=ClHE⊓ClHEc(ii)
BHE=ClHE∖IntHEProof.
(i)
BHE = {Pex∈X˜:Pex∉IntHE and Pex∉IntHEc}= {Pex∈X˜:Pex∉ClHEcc and Pex∉ClHEc}= {Pex∈X˜:Pex∈ClHEc and Pex∈ClHE}=ClHE⊓ClHEc(ii)
BHE=ClHE⊓ClHEc=ClHE⊓IntHEc=ClHE∖IntHECorollary 1.
LetHE be a soft subset of X,ϑ,E. Then,(i)
BHE=BHEc(ii)
ClHE=IntHE⊔BHE(iii)
IfHE is infra soft open, then BHE⊓HE=Φ(iv)
IfHE is infra soft closed, then BHE⊑HE(v)
IfHE is both infra soft open and infra soft closed, then BHE=Φ


## 4. Methods of Generating Infra Soft Topologies

In this section, we present some methods of generating infra soft topologies different from those given in Propositions1 and 2. These methods are infra soft neighbourhood systems, infra soft basis, infra soft subspace, and crisp infra topologies. We research these methods with the help of elucidative examples.


### 4.1. Infra Soft Neighbourhood and Infra Soft Neighbourhood Systems

Definition 19.
A soft subsetWE of X,ϑ,E is said to be an infra soft neighbourhood of a soft point Pex∈X˜ if there exists an infra soft open set GE such that Pex∈GE⊑WE.Definition 20.
The infra soft neighbourhood system of a soft pointPex∈X˜, denoted by ℐNPex, is the class of all infra soft neighbourhoods of Pex. In other words, ℐNPex = {WE⊑X˜:WE is an infra soft neighbourhood of a soft point Pex}.Proposition 12.
IfGE is an infra soft open subset of X,ϑ,E, then it is an infra soft neighbourhood of its all soft points.Proof.
Straightforward.To demonstrate that the converse of the above proposition fails, consider Example5. It is clear that every infinite infra soft set is an infra soft neighbourhood of its all soft points, but it is not an infra soft open set.Theorem 4.
The infra soft neighbourhood system of a soft pointPex∈X,ϑ,E satisfies the following properties.(IN1):ℐNPex≠∅(IN2): ifWE∈ℐNPex and WE⊑NE, then NE∈ℐNPex(IN3): ifVE,WE∈ℐNPex, then VE⊓WE∈ℐNPex(IN4): for eachWE∈ℐNPex, there is an infra soft neighbourhood VE of a soft point Pex contained in WE such that VE∈ℐNPay for each Pay∈VEProof.
(IN1): sinceX˜ is an infra soft open set containing every soft point Pex, then it is an infra soft neighbourhood of its all soft points. Therefore, X˜∈ℐNPex for every Pex. Thus, ℐNPex≠∅.(IN2): letWE∈ℐNPex. Then, there exists an infra soft open set GE such that Pex∈GE⊑WE. If WE⊑NE, then Pex∈GE⊑NE. Therefore, NE∈ℐNPex.(IN3): letVEandWE∈ℐNPex. Then, there exist two infra soft open sets FEandGE such that Pex∈FE⊑VE and Pex∈GE⊑WE. Since ϑ is an infra soft topology, then FE⊓GE is an infra soft open set. Obviously, Pex∈FE⊓GE⊑VE⊓WE. Thus, ℐNPex≠∅.(IN4): letWE∈ℐNPex. Then, there exists an infra soft open set GE such that Pex∈GE⊑WE. Putting VE=GE, we obtain the desired result.Theorem 5.
LetℐNPex is the class of all families satisfying the four properties given in Theorem 4. Then, ℳ = {UE⊑X˜: for each Pex∈UE⇒UE∈ℐNPex} forms an infra soft topology on the universal set X. Moreover, it forms a soft topology.Proof.
It is well known thatX˜ is an infra soft neighbourhood of all its soft points; then, X˜∈ℳ; also, Φ∈ℳ. Let UEandVE∈ℳ. For each Pex∈UE⊓VE, we have Pex∈UE and Pex∈VE. Then, UE∈ℐNPex and VE∈ℐNPex. It follows from (IN3) that UE⊓VE∈ℐNPex. Thus, UE⊓VE∈ℳ. Hence, ℳ is an infra soft topology.
To prove thatℳ is a soft topology, let UiE∈ℳ for each i∈I. Suppose that Pex∈⊔i∈IUiE. Then, there exists j∈I such that Pex∈UjE. Therefore, UjE∈ℐNPex. Since UjE⊑⊔i∈IUiE, then, by (IN2), we obtain ⊔i∈IUiE∈ℐNPex; hence, ⊔i∈IUiE∈ℳ, as required.


### 4.2. Basis of Infra Soft Topology

Definition 21.
LetX,ϑ,E be an infra soft topological space. A class ℬ⊆ϑ is said to be a basis for ϑ if the finite soft intersection of members of ℬ forms ϑ.It is clear thatϑ is a basis for itself.Proposition 13.
Every subclassℬ of SXE containing Φ (or two disjoint infra soft open sets) is a basis for a unique infra soft topology on X.Proof.
Letϑ be a family of soft sets generated by ℬ. Since the empty soft intersection is the absolute soft set, then X˜∈ϑ. By hypothesis, Φ∈ℬ, so Φ∈ϑ. It follows from the definition of ϑ, which is generated by ℬ, that ϑ is closed under finite soft intersection. Hence, we obtain the desired result.
To prove the uniqueness, letϑ1 be another infra soft topology generated by ℬ. Let FE∈ϑ1. Then, FE can be expressed as a finite soft intersection of members of ℬ. Therefore, FE∈ϑ. Thus, ϑ1⊆ϑ. Similarly, one can prove that ϑ⊆ϑ1. Hence, ϑ=ϑ1, as required.Example 9.
Consider the classℬ=FiE:i=1,2,3 defined on X=a,b,c with a set of parameters E=e1,e2, where(7)F1E=e1,X,e2,c,F2E=e1,a,b,e2,a,c,F3E=e1,c,e2,b.
By taking the finite soft intersection of the members ofℬ, we obtain θ=Φ,X˜,F1E,F2E,F3E,H1E,H2E, where(8)H1E=e1,a,b,e2,c,H2E=e1,c,e2,∅.
Obviously,θ is an infra soft topology on X.Remark 3.
The basis for an infra soft topology need not be unique. In other words, there are more than one basis for an infra soft topology in general.


### 4.3. Subspace

Definition 22.
LetX,ϑ,E be an infra soft topological space and Y be a nonempty subset of X. A class ϑY=Y˜⊓GE:GE∈ϑ is called an infra soft relative topology on Y, and Y,ϑY,E is called an infra soft subspace of X,ϑ,E.It can be easily checked thatϑY given in the above definition is an infra soft topology on Y.Theorem 6.
LetY,ϑY,E be an infra soft subspace of X,ϑ,E. Then, HE is an infra soft closed subset of Y,ϑY,E if and only if there exists an infra soft closed subset FE of X,ϑ,E such that HE=Y˜⊓FE.Proof.
Necessity: letHE be an infra soft closed subset of Y,ϑY,E. Then, there exists an infra soft open set UE in Y,ϑY,E such that UE=Y˜∖HE. This means that there exists an infra soft open set VE in X,ϑ,E such that UE=Y˜⊓VE. Therefore, HE=Y˜∖Y˜⊓VE=Y˜⊓VEc. Putting FE=VEc ends the proof of the necessary part.
Sufficiency: letHE=Y˜⊓FE such that FE is an infra soft closed set in X,ϑ,E. Then, Y˜∖HE=Y˜∖Y˜⊓FE=Y˜⊓X˜∖Y˜⊓FE=Y˜⊓X˜∖FE. Since X˜∖FE is an infra soft open set in X,ϑ,E, then Y˜∖HE is an infra soft open set in Y,ϑY,E. Therefore, HE is an infra soft closed set in Y,ϑY,E. Hence, the proof is complete.The proofs of the following two propositions are straightforward, and thus, they are omitted.Proposition 14.
LetY˜ be an infra soft open subset of X,ϑ,E. Then, UE is an infra soft open subset of Y,ϑY,E if and only if it is an infra soft open subset of X,ϑ,E.Proposition 15.
LetY,ϑY,E be an infra soft subspace of X,ϑ,E. Then, VE is an infra soft neighbourhood of Pey in Y,ϑY,E if and only if there exists an infra soft neighbourhood WE of Pey in X,ϑ,E such that VE=Y˜⊓WE.Theorem 7.
LetY,ϑY,E be an infra soft subspace of X,ϑ,E such that HE⊑Y˜. Let IntY, ClY, and yi′ be, respectively, the infra soft interior, infra soft closure, and infra soft limit points of a soft set in Y,ϑY,E, and let Int, Cl, and i′ be, respectively, the infra soft interior, infra soft closure, and infra soft limit points of a soft set in X,ϑ,E. Then,(i)
IntHE=IntYHE⊓IntY˜(ii)
ClYHE=ClHE⊓Y˜(iii)
HEyi′=HEi′⊓Y˜Proof.
We only prove (i). One can prove the other cases using similar techniques.
LetPex∈IntHE. Then, there exists an infra soft open subset UE of X,ϑ,E such that Pex∈UE⊑HE⊑Y˜. Therefore, Pex∈UE⊓Y˜⊑HE. Thus, Pex∈IntYHE and Pex∈IntY˜. Hence, IntHE⊑IntYHE⊓IntY˜. Conversely, let Pex∈IntYHE⊓IntY˜. Then, there exist two infra soft open subsets UEandVE of X,ϑ,E such that Pex∈UE⊑Y˜ and Pex∈VE⊓Y˜⊑HE. Since ϑ is an infra soft topology, then UE⊓VE is an infra soft open set containing Pex such that UE⊓VE⊑FE. Consequently, Pex∈IntHE. Thus, IntYHE⊓IntY˜⊑IntHE. Hence, the proof is complete.


### 4.4. Producing Infra Soft Topology from Crisp Infra Topologies

Proposition 16.
Suppose thatΨ=Ωee∈E is a family of crisp infra topologies on X. Then,(9)ϑΨ=e,Fe:e∈E∈SXEsuch thatFe∈Ωefor eache∈E,which defines an infra soft topology on X.Proof.
Let the given assumptions be satisfied. Since∅andX∈Ωe for each e∈E, then ΦandX˜∈ϑΨ. To prove that ϑΨ is closed under finite soft intersection, let UEandVE∈ϑΨ. According to the structure of ϑΨ, we have UeandVe∈Ωe for each e∈E. Since Ωe is infra topology, then Ue∩Ve∈Ωe. This implies that UE⊓VE∈ϑΨ, as required. Hence, X,ϑΨ,E is an infra soft topological space.Definition 23.
The infra soft topological space given in the above proposition is called the infra soft topology onX generated by Ψ.
We writeϑΨ=ϑΩ if Ωe=Ωe′=Ω for each eande′∈E.Remark 4.
The largest infra soft topology whose parametric infra topologies areΨ=Ωee∈E is ϑΨ.We explain in the following example how we can apply Proposition16 to construct an infra soft topology from crisp infra topologies.Example 10.
LetΩe1=∅,X,a,b,a,c and Ωe2=∅,X,a,c be two crisp infra topologies on X=a,b,c. To produce an infra soft topology from Ωe1 and Ωe2, we construct infra soft open sets HiE by choosing any set in Ωe1 as an image of e1, say, ∅. Then, we can choose the image of e2 by four different ways because the number of infra open sets in Ωe2 is four. Therefore, we obtain the following four infra soft sets:(10)H1E=e1,∅,e2,∅,H2E=e1,∅,e2,X,H3E=e1,∅,e2,a,H4E=e1,∅,e2,c.
We repeat this manner with each set inΩe1. Then, we obtain the following infra soft open sets:(11)H5E=e1,X,e2,∅,H6E=e1,X,e2,X,H7E=e1,X,e2,a,H8E=e1,X,e2,c,H9E=e1,a,e2,∅,H10E=e1,a,e2,X,H11E=e1,a,e2,a,H12E=e1,a,e2,c,H13E=e1,b,e2,∅,H14E=e1,b,e2,X,H15E=e1,b,e2,a,H17E=e1,a,c,e2,∅,H17E=e1,a,c,e2,∅,H18E=e1,a,c,e2,X,H19E=e1,a,c,e2,a,H20E=e1,a,c,e2,c.
Hence,ϑ=Φ,X˜,HiE:i=1,2,…,20 is an infra soft topology on X=a,b,c.Proposition 17.
Every infra soft topology generated by infra topologiesΩee∈E contains all soft sets, in which their e-components are X or ∅.Proof.
Since∅andX∈Ωe for each e∈E, then any soft set GE defined as Ge is X or ∅, which is a member of ϑΨ.The converse of the above proposition fails as illustrated in the following example.Example 11.
LetE=e1,e2 and ϑ=Φ,X˜,FiE:i=1,2,3 be an infra soft topology on X=a,b,c, where(12)F1E=e1,X,e2,∅,F2E=e1,∅,e2,X,F2E=e1,a,e2,∅.
It is clear thatϑ contains all soft sets, in which their e-components are X or ∅. But ϑ does not generate from crisp infra topologies because a∈ϑe1 and X∈ϑe2; however, e1,a,e2,X∉ϑ.In the following two examples, we show how we can examine whether the infra soft topology is generated by crisp infra topologies or not?Example 12.
LetE=e1,e2 and ϑ=Φ,X˜,FE,HE be an infra soft topology on X=a,b,c, where(13)FE=e1,a,e2,c,HE=e1,b,e2,b.
Then,ϑe1=∅,X,a,b and ϑe2=∅,X,b,c are the parametric (crisp) infra topologies of an infra soft topology ϑ. Consider Ωe1=ϑe1 and Ωe2=ϑe2. Now, a∈Ωe1 and b∈Ωe2; however, e1,a,e2,b∉ϑ. Thus, ϑ is not generated from crisp infra topologies.Example 13.
Consider the soft sets below defined onX=a,b,c with a set of parameters E=e1,e2:(14)H1E=e1,a,e2,X,H2E=e1,b,e2,X,H3E=e1,a,e2,∅,H4E=e1,b,e2,∅,H5E=e1,X,e2,∅,H6E=e1,∅,e2,X.
Then,ϑ=Φ,X˜,HiE:i=1,2,…,6 is an infra soft topology on X=a,b,c. It is clear that ϑe1=∅,X,a,b and ϑe2=∅,X are the parametric (crisp) infra topologies of an infra soft topology ϑ. Consider Ωe1=ϑe1 and Ωe2=ϑe2. It can be seen that ϑ is generated from the crisp infra topologies Ωe1 and Ωe2.


## 4.1. Infra Soft Neighbourhood and Infra Soft Neighbourhood Systems

Definition 19.
A soft subsetWE of X,ϑ,E is said to be an infra soft neighbourhood of a soft point Pex∈X˜ if there exists an infra soft open set GE such that Pex∈GE⊑WE.Definition 20.
The infra soft neighbourhood system of a soft pointPex∈X˜, denoted by ℐNPex, is the class of all infra soft neighbourhoods of Pex. In other words, ℐNPex = {WE⊑X˜:WE is an infra soft neighbourhood of a soft point Pex}.Proposition 12.
IfGE is an infra soft open subset of X,ϑ,E, then it is an infra soft neighbourhood of its all soft points.Proof.
Straightforward.To demonstrate that the converse of the above proposition fails, consider Example5. It is clear that every infinite infra soft set is an infra soft neighbourhood of its all soft points, but it is not an infra soft open set.Theorem 4.
The infra soft neighbourhood system of a soft pointPex∈X,ϑ,E satisfies the following properties.(IN1):ℐNPex≠∅(IN2): ifWE∈ℐNPex and WE⊑NE, then NE∈ℐNPex(IN3): ifVE,WE∈ℐNPex, then VE⊓WE∈ℐNPex(IN4): for eachWE∈ℐNPex, there is an infra soft neighbourhood VE of a soft point Pex contained in WE such that VE∈ℐNPay for each Pay∈VEProof.
(IN1): sinceX˜ is an infra soft open set containing every soft point Pex, then it is an infra soft neighbourhood of its all soft points. Therefore, X˜∈ℐNPex for every Pex. Thus, ℐNPex≠∅.(IN2): letWE∈ℐNPex. Then, there exists an infra soft open set GE such that Pex∈GE⊑WE. If WE⊑NE, then Pex∈GE⊑NE. Therefore, NE∈ℐNPex.(IN3): letVEandWE∈ℐNPex. Then, there exist two infra soft open sets FEandGE such that Pex∈FE⊑VE and Pex∈GE⊑WE. Since ϑ is an infra soft topology, then FE⊓GE is an infra soft open set. Obviously, Pex∈FE⊓GE⊑VE⊓WE. Thus, ℐNPex≠∅.(IN4): letWE∈ℐNPex. Then, there exists an infra soft open set GE such that Pex∈GE⊑WE. Putting VE=GE, we obtain the desired result.Theorem 5.
LetℐNPex is the class of all families satisfying the four properties given in Theorem 4. Then, ℳ = {UE⊑X˜: for each Pex∈UE⇒UE∈ℐNPex} forms an infra soft topology on the universal set X. Moreover, it forms a soft topology.Proof.
It is well known thatX˜ is an infra soft neighbourhood of all its soft points; then, X˜∈ℳ; also, Φ∈ℳ. Let UEandVE∈ℳ. For each Pex∈UE⊓VE, we have Pex∈UE and Pex∈VE. Then, UE∈ℐNPex and VE∈ℐNPex. It follows from (IN3) that UE⊓VE∈ℐNPex. Thus, UE⊓VE∈ℳ. Hence, ℳ is an infra soft topology.
To prove thatℳ is a soft topology, let UiE∈ℳ for each i∈I. Suppose that Pex∈⊔i∈IUiE. Then, there exists j∈I such that Pex∈UjE. Therefore, UjE∈ℐNPex. Since UjE⊑⊔i∈IUiE, then, by (IN2), we obtain ⊔i∈IUiE∈ℐNPex; hence, ⊔i∈IUiE∈ℳ, as required.


## 4.2. Basis of Infra Soft Topology

Definition 21.
LetX,ϑ,E be an infra soft topological space. A class ℬ⊆ϑ is said to be a basis for ϑ if the finite soft intersection of members of ℬ forms ϑ.It is clear thatϑ is a basis for itself.Proposition 13.
Every subclassℬ of SXE containing Φ (or two disjoint infra soft open sets) is a basis for a unique infra soft topology on X.Proof.
Letϑ be a family of soft sets generated by ℬ. Since the empty soft intersection is the absolute soft set, then X˜∈ϑ. By hypothesis, Φ∈ℬ, so Φ∈ϑ. It follows from the definition of ϑ, which is generated by ℬ, that ϑ is closed under finite soft intersection. Hence, we obtain the desired result.
To prove the uniqueness, letϑ1 be another infra soft topology generated by ℬ. Let FE∈ϑ1. Then, FE can be expressed as a finite soft intersection of members of ℬ. Therefore, FE∈ϑ. Thus, ϑ1⊆ϑ. Similarly, one can prove that ϑ⊆ϑ1. Hence, ϑ=ϑ1, as required.Example 9.
Consider the classℬ=FiE:i=1,2,3 defined on X=a,b,c with a set of parameters E=e1,e2, where(7)F1E=e1,X,e2,c,F2E=e1,a,b,e2,a,c,F3E=e1,c,e2,b.
By taking the finite soft intersection of the members ofℬ, we obtain θ=Φ,X˜,F1E,F2E,F3E,H1E,H2E, where(8)H1E=e1,a,b,e2,c,H2E=e1,c,e2,∅.
Obviously,θ is an infra soft topology on X.Remark 3.
The basis for an infra soft topology need not be unique. In other words, there are more than one basis for an infra soft topology in general.


## 4.3. Subspace

Definition 22.
LetX,ϑ,E be an infra soft topological space and Y be a nonempty subset of X. A class ϑY=Y˜⊓GE:GE∈ϑ is called an infra soft relative topology on Y, and Y,ϑY,E is called an infra soft subspace of X,ϑ,E.It can be easily checked thatϑY given in the above definition is an infra soft topology on Y.Theorem 6.
LetY,ϑY,E be an infra soft subspace of X,ϑ,E. Then, HE is an infra soft closed subset of Y,ϑY,E if and only if there exists an infra soft closed subset FE of X,ϑ,E such that HE=Y˜⊓FE.Proof.
Necessity: letHE be an infra soft closed subset of Y,ϑY,E. Then, there exists an infra soft open set UE in Y,ϑY,E such that UE=Y˜∖HE. This means that there exists an infra soft open set VE in X,ϑ,E such that UE=Y˜⊓VE. Therefore, HE=Y˜∖Y˜⊓VE=Y˜⊓VEc. Putting FE=VEc ends the proof of the necessary part.
Sufficiency: letHE=Y˜⊓FE such that FE is an infra soft closed set in X,ϑ,E. Then, Y˜∖HE=Y˜∖Y˜⊓FE=Y˜⊓X˜∖Y˜⊓FE=Y˜⊓X˜∖FE. Since X˜∖FE is an infra soft open set in X,ϑ,E, then Y˜∖HE is an infra soft open set in Y,ϑY,E. Therefore, HE is an infra soft closed set in Y,ϑY,E. Hence, the proof is complete.The proofs of the following two propositions are straightforward, and thus, they are omitted.Proposition 14.
LetY˜ be an infra soft open subset of X,ϑ,E. Then, UE is an infra soft open subset of Y,ϑY,E if and only if it is an infra soft open subset of X,ϑ,E.Proposition 15.
LetY,ϑY,E be an infra soft subspace of X,ϑ,E. Then, VE is an infra soft neighbourhood of Pey in Y,ϑY,E if and only if there exists an infra soft neighbourhood WE of Pey in X,ϑ,E such that VE=Y˜⊓WE.Theorem 7.
LetY,ϑY,E be an infra soft subspace of X,ϑ,E such that HE⊑Y˜. Let IntY, ClY, and yi′ be, respectively, the infra soft interior, infra soft closure, and infra soft limit points of a soft set in Y,ϑY,E, and let Int, Cl, and i′ be, respectively, the infra soft interior, infra soft closure, and infra soft limit points of a soft set in X,ϑ,E. Then,(i)
IntHE=IntYHE⊓IntY˜(ii)
ClYHE=ClHE⊓Y˜(iii)
HEyi′=HEi′⊓Y˜Proof.
We only prove (i). One can prove the other cases using similar techniques.
LetPex∈IntHE. Then, there exists an infra soft open subset UE of X,ϑ,E such that Pex∈UE⊑HE⊑Y˜. Therefore, Pex∈UE⊓Y˜⊑HE. Thus, Pex∈IntYHE and Pex∈IntY˜. Hence, IntHE⊑IntYHE⊓IntY˜. Conversely, let Pex∈IntYHE⊓IntY˜. Then, there exist two infra soft open subsets UEandVE of X,ϑ,E such that Pex∈UE⊑Y˜ and Pex∈VE⊓Y˜⊑HE. Since ϑ is an infra soft topology, then UE⊓VE is an infra soft open set containing Pex such that UE⊓VE⊑FE. Consequently, Pex∈IntHE. Thus, IntYHE⊓IntY˜⊑IntHE. Hence, the proof is complete.


## 4.4. Producing Infra Soft Topology from Crisp Infra Topologies

Proposition 16.
Suppose thatΨ=Ωee∈E is a family of crisp infra topologies on X. Then,(9)ϑΨ=e,Fe:e∈E∈SXEsuch thatFe∈Ωefor eache∈E,which defines an infra soft topology on X.Proof.
Let the given assumptions be satisfied. Since∅andX∈Ωe for each e∈E, then ΦandX˜∈ϑΨ. To prove that ϑΨ is closed under finite soft intersection, let UEandVE∈ϑΨ. According to the structure of ϑΨ, we have UeandVe∈Ωe for each e∈E. Since Ωe is infra topology, then Ue∩Ve∈Ωe. This implies that UE⊓VE∈ϑΨ, as required. Hence, X,ϑΨ,E is an infra soft topological space.Definition 23.
The infra soft topological space given in the above proposition is called the infra soft topology onX generated by Ψ.
We writeϑΨ=ϑΩ if Ωe=Ωe′=Ω for each eande′∈E.Remark 4.
The largest infra soft topology whose parametric infra topologies areΨ=Ωee∈E is ϑΨ.We explain in the following example how we can apply Proposition16 to construct an infra soft topology from crisp infra topologies.Example 10.
LetΩe1=∅,X,a,b,a,c and Ωe2=∅,X,a,c be two crisp infra topologies on X=a,b,c. To produce an infra soft topology from Ωe1 and Ωe2, we construct infra soft open sets HiE by choosing any set in Ωe1 as an image of e1, say, ∅. Then, we can choose the image of e2 by four different ways because the number of infra open sets in Ωe2 is four. Therefore, we obtain the following four infra soft sets:(10)H1E=e1,∅,e2,∅,H2E=e1,∅,e2,X,H3E=e1,∅,e2,a,H4E=e1,∅,e2,c.
We repeat this manner with each set inΩe1. Then, we obtain the following infra soft open sets:(11)H5E=e1,X,e2,∅,H6E=e1,X,e2,X,H7E=e1,X,e2,a,H8E=e1,X,e2,c,H9E=e1,a,e2,∅,H10E=e1,a,e2,X,H11E=e1,a,e2,a,H12E=e1,a,e2,c,H13E=e1,b,e2,∅,H14E=e1,b,e2,X,H15E=e1,b,e2,a,H17E=e1,a,c,e2,∅,H17E=e1,a,c,e2,∅,H18E=e1,a,c,e2,X,H19E=e1,a,c,e2,a,H20E=e1,a,c,e2,c.
Hence,ϑ=Φ,X˜,HiE:i=1,2,…,20 is an infra soft topology on X=a,b,c.Proposition 17.
Every infra soft topology generated by infra topologiesΩee∈E contains all soft sets, in which their e-components are X or ∅.Proof.
Since∅andX∈Ωe for each e∈E, then any soft set GE defined as Ge is X or ∅, which is a member of ϑΨ.The converse of the above proposition fails as illustrated in the following example.Example 11.
LetE=e1,e2 and ϑ=Φ,X˜,FiE:i=1,2,3 be an infra soft topology on X=a,b,c, where(12)F1E=e1,X,e2,∅,F2E=e1,∅,e2,X,F2E=e1,a,e2,∅.
It is clear thatϑ contains all soft sets, in which their e-components are X or ∅. But ϑ does not generate from crisp infra topologies because a∈ϑe1 and X∈ϑe2; however, e1,a,e2,X∉ϑ.In the following two examples, we show how we can examine whether the infra soft topology is generated by crisp infra topologies or not?Example 12.
LetE=e1,e2 and ϑ=Φ,X˜,FE,HE be an infra soft topology on X=a,b,c, where(13)FE=e1,a,e2,c,HE=e1,b,e2,b.
Then,ϑe1=∅,X,a,b and ϑe2=∅,X,b,c are the parametric (crisp) infra topologies of an infra soft topology ϑ. Consider Ωe1=ϑe1 and Ωe2=ϑe2. Now, a∈Ωe1 and b∈Ωe2; however, e1,a,e2,b∉ϑ. Thus, ϑ is not generated from crisp infra topologies.Example 13.
Consider the soft sets below defined onX=a,b,c with a set of parameters E=e1,e2:(14)H1E=e1,a,e2,X,H2E=e1,b,e2,X,H3E=e1,a,e2,∅,H4E=e1,b,e2,∅,H5E=e1,X,e2,∅,H6E=e1,∅,e2,X.
Then,ϑ=Φ,X˜,HiE:i=1,2,…,6 is an infra soft topology on X=a,b,c. It is clear that ϑe1=∅,X,a,b and ϑe2=∅,X are the parametric (crisp) infra topologies of an infra soft topology ϑ. Consider Ωe1=ϑe1 and Ωe2=ϑe2. It can be seen that ϑ is generated from the crisp infra topologies Ωe1 and Ωe2.


## 5. Continuity between Infra Soft Topological Spaces

In this section, we define the concept of continuity between infra soft topological spaces and then give its equivalent conditions using infra soft open and infra soft closed sets. Also, we discuss losing some equivalent conditions of soft continuity on infra soft topology with the help of an illustrative example. We close this section by studying “transmission” of continuity between an infra soft topological space and its parametric infra topological spaces.Definition 24.
Definition 24A soft mapfφ from X,ϑ,E to Y,μ,E is said to be infra soft continuous at a soft point Pex∈X˜ if for each infra soft open set UE containing fφPex, there is an infra soft open set VE containing Pex such that fφVE⊑UE.Definition 25.
A soft mapfφ from X,ϑ,E to Y,μ,E is said to be infra soft continuous if it is infra soft continuous for all Pex∈X˜.Theorem 8.
A soft mapfφ:X,ϑ,E⟶Y,μ,E is infra soft continuous if and only if the inverse image of each infra soft open set is an infra soft open set.Proof.
Necessity: letUE be an infra soft open subset of Y,μ,E. Without loss of generality, consider fφ−1UE≠Φ. Then, for each Pex∈fφ−1UE, we have an infra soft open subset VE of X,ϑ,E containing Pex such that fφVE⊑UE. Thus, Pex∈VE⊆fφ−1UE and ⊔VE=fφ−1UE. Hence, fφ−1UE is infra soft open.
Sufficiency: suppose thatPex∈X˜ and UE is an infra soft open set containing fφPex. Then, fφ−1UE is an infra soft open set containing Pex such that fφfφ−1UE⊑UE. Therefore, fφ is infra soft continuous at Pex which we choose arbitrarily; hence, fφ is infra soft continuous.Corollary 2.
A soft mapfφ:X,ϑ,E⟶Y,μ,E is infra soft continuous if and only if the inverse image of each infra soft closed set is an infra soft closed set.Proof.
Necessity: letGE be an infra soft closed subset of Y˜. Then, GEc is infra soft open. Therefore, fφ−1GEc=X˜∖fφ−1GE is an infra soft open set; hence, fφ−1GE is infra soft closed.
Sufficiency: following similar arguments, we prove the sufficient part.The following properties are equivalent to soft continuity on soft topological and supra soft topological spaces.(i)
Clfφ−1HE⊑fφ−1ClHE for each HE⊑Y˜(ii)
fφClFE⊑ClfφFE for each FE⊑X˜(iii)
fφ−1IntHE⊑Intfφ−1HE for each HE⊑Y˜But they are not equivalent to soft continuity on the infra soft topological spaces.Definition 26.
Letfφ:X,ϑ,E⟶Y,μ,E be a soft map and Z be a nonempty subset of X. A soft map fφ∣Z from Z,ϑZ,E to Y,μ,E, which is given by fφ∣ZPez=fφPez for each Pez∈Z˜, is called the restriction soft map of fφ on Z.Theorem 9.
Iffφ:X,ϑ,E⟶Y,μ,E is infra soft continuous, then the restriction soft map fφ∣Z:Z,ϑZ,E⟶Y,μ,E is infra soft continuous.Proof.
LetFE be an infra soft open subset of Y,μ,E. Then, fφ∣Z−1FE=fφ−1FE⊓Z˜. By hypothesis, fφ−1FE is an infra soft open subset of X,ϑ,E; therefore, fφ−1FE⊓Z˜ is an infra soft open subset of Z,ϑZ,E. Hence, fφ∣Z is an infra soft continuous map, as required.It is easy to prove the following two results; thus, their proofs will be omitted.Proposition 18.
Iffφ:X,ϑ,E⟶Y,μ,E and gφ:Y,μ,E⟶Z,ν,E are infra soft continuous maps, then gφ°fφ is an infra soft continuous map.Proposition 19.
Letϑi:i∈I be a family of infra soft topologies on X with a set of parameters E. If fφ:X,ϑi,E⟶Y,μ,E is infra soft continuous for each i, then fφ:X,⊓i∈Iϑi,E⟶Y,μ,E is infra soft continuous.Proposition 20.
Letfφ be a soft map of an infra soft topological space X,ϑ,E to SEY. Then, μ=GE⊑Y˜:fφ−1GE∈ϑ is an infra soft topology on Y. Moreover, μ is the strongest infra soft topology on Y which makes fφ as infra soft continuous.Proof.
Sincefφ−1Φ,fφ−1Y˜∈ϑ, then ΦandY˜∈μ. Let G1EandG2E∈μ. Then, there exist fφ−1G1Eandfφ−1G2E∈ϑ. Since ϑ is infra soft topology, then fφ−1G1E⊓fφ−1G2E=fφ−1G1E⊓G2E∈ϑ. Therefore, G1E⊓G2E∈μ; thus, μ is infra soft topology on Y. To prove that μ is the strongest infra soft topology on Y, it makes fφ as infra soft continuous. Suppose that ν is an infra soft topology on Y such that fφ:X,ϑ,E⟶Y,ν,E is infra soft continuous. Let HE∈ν; then, fφ−1HE∈ϑ. This implies that HE∈μ; thus, μ⊏ν. Hence, the proof is complete.We complete this section by investigating the concept of infra soft continuity between the soft map and crisp map.Theorem 10.
If a soft mapfφ:X,ϑ,E⟶Y,μ,E is infra soft continuous, then a map f:X,ϑe⟶Y,μφe is infra continuous.Proof.
LetU be an infra open subset of Y,μφe. Then, there exists an infra soft open subset GE of Y,μ,E such that Gφe=U. Since fφ is an infra soft continuous map, then fφ−1GE is an infra soft open subset of X,ϑ,E. It follows from Definition 10 that fφ−1GE=f−1GE, where f−1Ge=f−1Gφe; this implies that f−1U=f−1Gφe in the infra open subset of X,ϑe. Hence, we obtain the desired result.We explain that the converse of the above theorem fails in the example below.Example 14.
LetE=e1,e2 and ϑ=Φ,X˜,FE and μ=Φ,X˜,HE be two infra soft topologies on X=a,b, where(15)FE=e1,a,e2,∅,HE=e1,∅,e2,a.
Considerφ:E⟶E and f:X⟶X are identity maps. Then, f:X,ϑe⟶X,μφe=e is infra continuous for each e∈E. But fφ:X,ϑ,E⟶X,μ,E is not an infra soft continuous map because fφ−1FE=FE∉ϑ.We show under which condition the converse of Theorem10 holds.Theorem 11.
Letϑ be an infra soft topology generated from the crisp infra topologies. Then, a soft map fφ:X,ϑ,E⟶Y,μ,E is infra soft continuous if and only if a map f:X,ϑe⟶Y,μφe is infra continuous.Proof.
The necessary part is proved in Theorem10.
To prove the sufficient part, letUE be an infra soft open subset of Y,μ,E. Then, fφ−1UE=f−1UE is a soft subset of X,ϑ,E such that f−1Ue=f−1Uφe for each e∈E. Since Uφe is an infra open subset of Y,μφe and a map f is infra continuous, then f−1Uφe is the infra open subset of X,ϑe. By hypothesis, ϑ is generated from the crisp infra topologies, so fφ−1UE is an infra soft open subset of X,ϑ,E, as required.


## 6. Conclusion

This study has introduced the concept of an infra soft topology as a new structure is weaker than a soft topology. The most important goal of investigating this concept is to keep some soft topological properties under fewer conditions than topology.We have contributed to improve the knowledge about this area in three aspects. First, we have established the basic concepts of infra soft topological spaces and scrutinized properties. We have noted that most properties of interior and closure operators are valid on infra soft topological spaces, while most of them are losing on other generalizations of soft topology such as supra soft topology. Second, we have proposed some techniques of producing infra soft topologies such as soft maps, soft neighbourhood systems, infra soft basis, infra soft subspace, and crisp infra topologies. In fact, the techniques of soft neighbourhood systems and soft operators initiate soft topology which is due to the identical between their properties on soft topology and infra soft topology. Third, we have introduced and investigated the concept of continuity between infra soft topological spaces. We have described this concept using infra soft open and infra soft closed sets. Moreover, we have showed that some characterizations of continuity on soft topology are losing on the frame of infra soft topology, especially those that are based on the interior and closure operators.In future works, we plan to formulate the soft topological concepts such as separation axioms, compactness, and connectedness on the frame of infra soft topology. In particular, we shed light on discovering which ones of their properties are still valid on the infra soft topologies.



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