# Homeomorphism and Quotient Mappings in Infrasoft Topological Spaces

Authors: Tareq M. Al-shami
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, we contribute to infrasoft topology which is one of the recent generalizations of soft topology. Firstly, we redefine the concept of soft mappings to be convenient for studying the topological concepts and notions in different soft structures. Then, we introduce the concepts of open, closed, and homeomorphism mappings in the content of infrasoft topology. We establish main properties and investigate the transmission of these concepts between infrasoft topology and its parametric infratopologies. Finally, we define a quotient infrasoft topology and infrasoft quotient mappings and study their main properties with the aid of illustrative examples.


---

## Body

## 1. Introduction

We face vagueness, ambiguity, and representation of imperfect knowledge in different areas such as economics, engineering, medical science, sociality, and environmental sciences. Mathematicians, engineers, and scientists, particularly those who focus on artificial intelligence, are seeking for approaches to solve the problems that contain vagueness. But they experienced a trouble: how they can formulate uncertain concepts that may not involve mathematically definite results. This means that there is a need for alternative mathematical concepts. Therefore, they have begun to look for different fields of research which leads to initiate several set theories as an alternative to George Cantor’s set theory such as fuzzy set (and its generalizations such as intuitionistic fuzzy set and pythagorean fuzzy set), rough set, multiset, and recently soft set.Soft set was proposed by Molodtsov [1] as one of the nonstatistical mathematical approaches targeting to deal with ambiguous, undefined, and imprecise meaning. It is characterized by flexibility and fruitful applications compared with other uncertainty theories. Since Molodtsov put forward the concept of soft sets, many scholars have applied it in several research areas such as decision-making problems [2], systems of linear equations [3], computer science [4], engineering [5], and medical sciences [6].The year 2011 was the beginning point of the interaction between soft set theory and topology. Simultaneously, Shabir and Naz [7] and Çaǧman et al. [8] initiated the concept of soft topology. However, they used different techniques to formulate soft topology. On the one hand, Shabir and Naz formulated soft topology on the collection of soft sets over a universal crisp set and a fixed set of parameters. On the other hand, Çaǧman et al. formulated soft topology on the collection of soft sets over an absolute soft set and different sets of parameters. We conduct this study in the frame of Shabir and Naz’s definition which is more analogous to the classical topology.Kharal and Ahmad [9] defined soft mappings using two ordinary (crisp) mappings, one of them between the sets of parameters and the other between the universal sets. However, we reformulate this definition using the concept of soft points to be convenient for studying in different soft structures. We classify soft mappings by soft spaces into different families such as continuous, open, closed, and homeomorphism mappings. In addition, soft mappings enable us to classify topological concepts in terms of preservation under specific classes of soft mappings. In [10], the authors presented another point of view to study soft mappings with a medical application. Zorlutuna and Çakir [11] investigated continuity between soft topological spaces. The authors of [12] presented new relations between ordinary points and soft sets to define new types of soft separation axioms. Quite recently, Al-shami and Kočinac [13] investigated the conditions under which some concepts are kept between soft topology and its parametric topologies. Kočinac et al. [14] discussed selection principles in the context of soft sets.Some generalizations of soft topology were introduced and studied. For example, El-Sheikh and Abd El-Latif [15] established the concept of suprasoft topological spaces by neglecting a finite intersection condition of a soft topology. Thomas and John [16] formulated the concept of soft generalized topological spaces, and Zakari et al. [17] originated the concepts of soft weak structures. Lately, Al-shami et al. [18] have initiated and investigated the concept of infrasoft topology. Then, Al-shami [19] studied infrasoft compactness and its application to fixed point theorem. Also, Al-shami and Abo-Tabl [20] defined the concepts of infrasoft connected and infrasoft locally connected spaces. As a continuation of this work, we conduct this study. We aim to redefine soft mappings and explore some types of them through the infrasoft topology content.We organize this article as follows. After the introduction section, Section2 mentions some concepts and notions that clarify the investigations of this paper. In Section 3, we show shortcoming of soft mapping defined in [9] and reformulate it simulating to the definition of (crisp) mappings. In Section 4, we define new types of infrasoft mappings and investigate main properties. Among other things, we prove that these infrasoft mappings are preserved under product of infrasoft topological spaces and investigate the transmission of these concepts between infrasoft topology and its parametric infratopologies. We devote Section 5 to study quotient topology and mappings in the content of infrasoft topological spaces. Finally, we outline the fundamental obtained results and suggest some upcoming works in Section 6.


## 2. Preliminaries

In this section, we mention the main concepts that make this paper self-contained.Definition 1.
(see [1]). We called a pair H,A a soft set over the universal set X≠∅ with a set of parameters A provided that H is a mapping from A to the power set 2X of X. We usually write a soft set H,A as follows H,A=a,Ha:a∈A and Ha∈2X. The symbol SXA denotes the set of all soft sets over X with any subset of A.
We recall some special types of soft sets.(i)
IfHa=∅ for each a∈A, then H,A is called a null soft set [21].(ii)
IfHa=X for each a∈A, then H,A is called an absolute soft set [21].(iii)
IfHa=x and Ha′=∅ for each a′∈A∖a, then H,A is called a soft point. It is denoted by Pax. The set of all soft points over X with A is denoted by PXA. We say that Pax belongs to a soft set F,A, denoted by Pax∈F,A if x∈Fa [22]. More details concerning soft points were given in [23].Definition 2.
(see [24]). A soft set Hc,A is called a relative complement of a soft set H,A provided that a mapping Hc:A⟶2X is defined by Hca=X∖Ha for each a∈A.
Since an infrasoft topological space is defined under a constant set of parameters, we will recall the previous definitions and results under a constant set of parameters.Definition 3.
(see [21, 24, 25]). Let F,A and H,A be two soft sets over X.(i)
F,A is a soft subset of a soft set H,A, symbolized by F,A⊑H,A, if Fa⊆Ha for all a∈A.F,A and H,A are called soft equal if F,A⊑H,A and H,A⊑F,A.(ii)
The soft intersection ofF,A and H,A, symbolized by F,A⊓H,A, is a soft set M,A such that a mapping M:A⟶2X is given by Ma=Fa∩Ha for each a∈A.(iii)
The soft union ofF,A and H,A, symbolized by F,A⊔H,A, is a soft set M,A such that a mapping M:A⟶2X is given by Ma=Fa∪Ha for each a∈A.
More details concerning soft intersection and union were given in [26].Definition 4.
(see [27]). The Cartesian product of H,A and F,B, which are defined over X and Y, respectively, is a soft set, denoted by H×F,A×B, given by H×Fa,b=Ha×Fb for each a,b∈A×B.Definition 5.
(see [9]). Let g:X⟶Y and φ:A⟶B be two crisp mappings. Then, a soft mapping gφ:SXA⟶SYB is defined as follows: the image of a soft set H,M in SXA is a soft set gφH,M=gH,E in SYB such that E=φM⊆B, and a mapping gH is given by(1)gHe=g∪λ∈φ−1e∩MHλ,for eache∈E.Definition 6.
(see [9]). Let gφ:SXA⟶SYB be a soft mapping. Then, the preimage of a soft set U,N in SYB is a soft set gφ−1U,N=g−1U,D in SXA such that D=φ−1N⊆A, and a mapping g−1U is given by(2)g−1Ud=g−1Uφd,for eachd∈D.Definition 7.
(see [9]). We call gφ:SXA⟶SYB an injective (resp. a surjective, a bijective) soft mapping if g and φ are injective (resp. surjective, bijective) mappings.Proposition 1.
(see [9]). Consider gφ:SXA⟶SYB is a soft mapping. Let H1,A and H2,A be two soft sets in SXA and U1,B and U2,B be two soft sets in SYB. Then, we have the following results:(i)
IfH1,A⊑H2,A, then gφH1,A⊑gφH2,A(ii)
IfU1,B⊑U2,B, then gφ−1U1,B⊑gφ−1U2,B(iii)
H1,A⊑gφ−1gφH1,A and the equality relation holds if gφ is injective(vi)
gφgφ−1U1,B⊑U1,B and the equality relation holds if gφ is surjective(v)
gφH1,A⊔H2,A=gφH1,A⊔gφH2,A(vi)
gφH1,A⊓H2,A⊑gφH1,A⊓gφH2,A(vii)
gφ−1U1,B⊔U2,B=gφ−1U1,B⊔gφ−1U2,B(viii)
gφ−1U1,B⊓U2,B=gφ−1U1,B⊓gφ−1U2,BDefinition 8.
(see [18]). The collection τ of soft sets over X under a fixed set of parameters A is called an infrasoft topology on X if it is closed under finite soft intersection as well as it contains Φ.
The tripleX,τ,A is called an infrasoft topological space. The term given to each member of τ is called an infrasoft open set, and the relative complement each member of τ is called an infrasoft closed set.Definition 9.
(see [18]). Let H,A be a subset of X,τ,A.(i)
The intersection of all infrasoft closed subsets ofX,τ,A which contains a soft set H,A is called the infrasoft closure points of H,A. It is denoted by ClH,A.(ii)
The union of all infrasoft open subsets ofX,τ,A which are contained in a soft set H,A is called the infrasoft interior points of H,A. It is denoted by IntH,A.Definition 10.
(see [18]). Let X,τ,A be an infrasoft topological space and U be a nonempty subset of X. A class τU=U˜⊓G,A:G,A∈τ is called an infrasoft relative topology on U, and U,ϑU,A is called an infrasoft subspace of X,τ,A.Proposition 2.
(see [18]). Suppose that Ω=Ωaa∈A is a family of (crisp) classical infratopologies on X. Then,(3)τΩ=a,Ha:a∈A∈SXAsuch thatHa∈Ωafor eacha∈A,which defines an infrasoft topology on X (we called this type of infrasoft topology as an infrasoft topology generated by (crisp) classical infratopologies).Theorem 1.
(see [18]). A soft mapping gφ:X,τ,A⟶Y,θ,B is infrasoft continuous if and only if the inverse image of each infrasoft open (resp. infra soft closed) set is an infrasoft open (resp. infra soft closed) set.Definition 11.
(see [28]). Let g:X⟶Y be a mapping. A subfamily θ of the power set PY of Y is said to be a quotient topology over Y (with respect to g) if θ is the largest topology that makes g continuous.


## 3. Note on Soft Mappings

In this section, we target to achieve two goals; first, we update Definition5 of soft mappings to be convenient for studying the concepts of open, closed, and homeomorphism mappings in different soft structures such as soft topology, suprasoft topology, and infrasoft topology. Second, we simplify the formulation of Definition 5 using a soft point as the starting point.We begin by the following example which helps us to clarify the followed approach to achieve the coveted goals.Example 1.
LetX=u,v,w and Y=x,y,z be two universal sets, and let A=a1,a2,a3 and B=b1,b2,b3 be two sets of parameters. Consider a soft mapping gφ from SXA to SYB, where the mappings g:X⟶Y and φ:A⟶B are defined as follows:(4)gu=gv=x,gw=z,φa1=φa2=b2,φa3=b3.
ForH,A=a1,u,a2,X,a3,w, we have gφH,A=b2,x,z,b3,z. Note that, for any infrasoft topology (or any soft structure) on X and Y with the sets of parameters A and B, a soft mapping gφ will not be soft open (soft closed) because the image of any soft sets under gφ is a soft subset of a soft set b2,x,z,b3,x,z. To remove this shortcoming, we make a slight modification for Definition 5 to be appropriate for defining soft open and closed mappings.Definition 12.
The image ofH,M∈SXA under a soft mapping gφ:SXA⟶SYB, where g:X⟶Y and φ:A⟶B are given by gφH,M=gH,B such that(5)gHb=∪a∈φ−1b∩MgHa,φ−1b≠∅,∅,φ−1b=∅,for each b∈B.
If there is no confusion, we simplify the above formulation as follows:(6)gHb=g∪a∈φ−1b∩MHa,for eachb∈B.
The following result is easy, but it will be useful for the investigation.Proposition 3.
(i)
The image of each soft point is a soft point(ii)
The product of two soft points is a soft pointDefinition 13.
A relationΩ on X˜ is a subset of PXA×PXA.
Note that Definition5 does not give meaning of a soft mapping as a self-contained concept. It only gives the method of calculating the image and preimage of soft sets. So, it is nature to wonder what is the formulation of soft mappings that simulates its counterpart on the (crisp) set theory? It is well known that a soft point represents the soft version of an ordinary point so that we redefine a soft mapping between two classes of soft points as follows.Definition 14.
Letg:X⟶Y and φ:A⟶B be two crisp mappings. A soft mapping gφ of PXA into PYB is a relation such that each soft point in PXA is related to one and only one soft point in PYB such that(7)gφPax=Pφagx,for eachPax∈PXA.
In addition,gφ−1Pby=⊔a∈φ−1bx∈g−1yPax for each Pby∈PYB.
From the above definition, we note two matters; first, reduce calculation burden and its difficulty that arises from Definition5. Second, Definition 14 gives a logical explanation (justification) for some soft concepts, for example, it can be easily seen why we determine that gφ is injective or surjective according to its two crisp mappings g and φ.
Now, we prove the following results.Lemma 1.
Letgφ:PXA⟶PYB be a soft mapping. Then, gφH,A=⊔Pax∈H,APφagx.Proof.
Since it can be written a soft set as a soft union of its soft points, we obtaingφH,A=gφ⊔Pax∈H,APax=⊔Pax∈H,AgφPax=⊔Pax∈H,APφagx, as required.Theorem 2.
The image of soft sets obtained from Definitions12 and 14 is identical, i.e., Definitions 12 and 14 are identical.Proof.
LetH,M be a soft set in SXA. Then, gφH,A=gφ⊔Pax∈H,APax=⊔Pax∈H,AgφPax. It follows from (i) of Proposition 3 that gφPax is a soft point in SYB. According to Definition 12, we obtain gφPax=Pφagx. Thus, gφH,M=⊔Pax∈H,APφagx which represents the image of H,M according to Definition 14. Hence, we obtain the coveted result.Corollary 1.
The preimage of soft sets obtained from Definitions6 and 14 is identical.We complete this part by presenting some amendments of some results given in [18]. First, the following result is the correct form of Proposition 7 in [18].Proposition 4.
LetH,A and F,A be subsets of X,τ,A, where X and A are finite sets. If clH,A⊓clF,A=Φ, then intH,A⊔F,A=intH,A⊔intF,A.Proof.
It is clear thatintH,A⊔intF,A⊑intH,A⊔F,A. Conversely, let Pax∈intH,A⊔F,A. Then, τ contains an infrasoft open set G,A such that Pax∈G,A⊑G,A⊔F,A. Since X and A are finite sets, we consider G,A as a smallest infrasoft open set containing Pax. Now, we have three cases:Case 1:G,A⊑G,A. Then, Pax∈intG,A.Case 2:G,A⊑F,A. Then, Pax∈intF,A.Case 3:G,A⊑G,A and G,A⊑F,A. Then, G,A⊓G,A≠Φ and G,A⊓F,A≠Φ. Since G,A is the smallest infrasoft open set containing Pax, Pax∈clG,A and Pax∈clF,A. But this contradicts assumption clG,A⊓clF,A=Φ. Therefore, the only valid cases are Case 1 and Case 2. Thus, intG,A⊔F,A⊑intG,A⊔intF,A. Hence, the proof is complete.
Second, we replace Definition24 by Theorem 8 (which are given in [18]) to keep the systematic line of defining infrasoft continuity, infrasoft openness, and infrasoft closedness. That is, we reformulate Definition 24 of an infrasoft continuous map as follows.Definition 15.
A soft mappinggφ:X,τ,A⟶Y,θ,B is said to be infrasoft continuous if the preimage of each infrasoft open set is infrasoft open.


## 4. Infrasoft Homeomorphism Mappings

In this section, we initiate the concepts of infrasoft open, infrasoft closed, and infrasoft homeomorphism mappings. We show the relationships among them and study some properties. We construct some counterexamples to explain some invalid results.Definition 16.
A soft mappinggφ:X,τ,A⟶Y,θ,B is said to be infrasoft open (resp. infra soft closed) if the image of each infrasoft open (resp. infra soft closed) set is an infrasoft open (resp. infra soft closed) set.Proposition 5.
Ifgφ:X,τ,A⟶Y,θ,B is an infrasoft open mapping, then the following statements hold.(i)
gφintH,A⊑intgφH,A for each H,A∈SXA(ii)
gφ−1clF,B⊑clgφ−1F,B for each F,B∈SYBProof.
To prove (i), letPby∈gφintH,A. Then, there exists a soft point Pax∈intH,A such that gφPax=Pby. Therefore, there exists an infrasoft open set U,A such that Pax∈U,A⊑H,A. Obviously, Pby=gφPax∈gφU,A⊑gφH,A. By hypothesis, gφU,A is an infrasoft open set; hence, gφU,A⊑intgφH,A, as required.
One can prove (ii) using a similar technique.Corollary 2.
Ifgφ:X,τ,A⟶Y,θ,B is an infrasoft open mapping, then the image of an infrasoft neighborhood of Pax∈X˜ is an infrasoft neighborhood of gφPax.Proposition 6.
Ifgφ:X,τ,A⟶Y,θ,B is an infrasoft closed mapping, then clgφH,A⊑gφclH,A for each H,A∈SXA.Proof.
Suppose thatPby∉gφclH,A. Then, gφ−1Pby⊑clH,A. This means for each Pax∈gφ−1Pby, there exists an infrasoft open set U,A containing Pax such that U,A⊓H,A=Φ. Therefore, U,A⊓clH,A=Φ. Now, gφH,A⊑gφclH,A⊑gφUc,A. By assumption, Pby∉gφclH,A so that gφPax=Pby∉gφUc,A. Since gφ is an infrasoft closed mapping, then clgφH,A⊑gφUc,A. Thus, Pby∉clgφH,A, as required.
In fact, some characterizations of soft open and closed mappings, which are the counterparts of infrasoft open and infrasoft closed mappings, are losing via the structure of infrasoft topology. The following example illustrates this observation.Example 2.
Consider the following soft sets overX=w1,w2 under a parameter set A=a1,a2 defined as follows:(8)U1,A=a1,w1,a2,w1,U2,A=a1,∅,a2,w2.
Then,τ=Φ,X˜,Ui,A:i=1,2 is an infrasoft topology on X. Consider θ is the indiscrete soft topology (of course, it will be an infrasoft topology) on X. Let a soft mapping gφ:X,θ,A⟶X,τ,A be defined as follows:(9)gφPa1w1=gφPa1w2=Pa1w1,gφPa2w1=Pa2w1,gφPa2w2=Pa2w2.
One can check that the two conditions given in Proposition5 hold. Also, the image of any infrasoft neighborhood of Paw is an infrasoft neighborhood of gφPaw. Moreover, clgφH,A⊑gφclH,A for each H,A∈SXA. On the contrary, the image of an infrasoft clopen set X˜ is a1,w1,a2,X which is neither infrasoft open nor infrasoft closed in τ; hence, gφ is neither an infrasoft open mapping nor an infrasoft closed mapping.Proposition 7.
Letgφ:X,τ,A⟶Y,θ,B be a bijective soft map. Then, gφ is infrasoft open if and only if it is infrasoft closed.Proof.
The proof follows from the fact that a bijective soft mapgφ implies that gφHc,A=gφH,Ac.Proposition 8.
Letgφ:X,τ,A⟶Y,θ,B be an infrasoft open mapping and U˜ be an infrasoft open set in τ. Then, gφ|U:U,τU,A⟶Y,θ,B is an infrasoft open mapping.Proof.
LetG,A be an infrasoft open set in τU. Then, there is an infrasoft open set V,A in τ such that G,A=V,A⊓U˜. This means that G,A is also an infrasoft open set in τ. By hypothesis, gφ|UG,A=gφG,A is an infrasoft open set in θ; hence, gφ|U is an infrasoft open mapping.Definition 17.
The composition of two soft mappingsfψ:X,τ,A⟶Y,θ,B and gφ:Y,θ,B⟶Z,υ,C is a soft mapping gφ∘fψ:X,τ,A⟶Z,υ,C such that gφ∘fψPax=gφfψPax.Proposition 9.
Letgφ:X,τ,A⟶Y,θ,B and fψ:Y,θ,B⟶Z,μ,C be two infrasoft mappings. Then, the following statements hold.(i)
Ifgφ and fψ are infrasoft open mappings, then fψ∘gφ is an infrasoft open mapping(ii)
Iffψ∘gφ is an infrasoft open mapping and gφ is a surjective infrasoft continuous mapping, then fψ is an infrasoft open mapping(iii)
Iffψ∘gφ is an infrasoft open mapping and fψ is an injective infrasoft continuous mapping, then gφ is an infrasoft open mappingProof.
(i)
Straightforward.(ii)
LetG,B be an infrasoft open set in θ. By hypothesis, gφ−1G,B is an infrasoft open set in τ; therefore, fψ∘gφgφ−1G,B is an infrasoft open set in μ. Since gφ is surjective, then fψ∘gφgφ−1G,B=fψgφgφ−1G,B=fψG,B. Thus, fψ is an infrasoft open mapping.(iii)
LetG,A be an infrasoft open set in τ. By hypothesis, fψ∘gφG,A is an infrasoft open set in μ; therefore, fψ−1fψ∘gφG,A is an infrasoft open set in θ. Since fψ is injective, then fψ−1fψ∘gφG,A=fψ−1fψgφG,A=gφG,A. Thus, gφ is an infrasoft open mapping.
Following similar arguments given in the above proof, one can prove the next result.Proposition 10.
Letgφ:X,τ,A⟶Y,θ,B and fψ:Y,θ,B⟶Z,μ,C be two infrasoft mappings. Then, the following statements hold.(i)
Ifgφ and fψ are infrasoft closed mappings, then fψ∘gφ is an infrasoft closed mapping(ii)
Iffψ∘gφ is an infrasoft closed mapping and gφ is a surjective infrasoft continuous mapping, then fψ is an infrasoft closed mapping(iii)
Iffψ∘gφ is an infrasoft closed mapping and fψ is an injective infrasoft continuous mapping, then gφ is an infrasoft closed mappingProposition 11.
LetXi,τi,Ai:i∈I be a family of infrasoft topological spaces. Then, τ=∏i∈IUi,Ai:Ui,Ai∈τi is an infrasoft topology on X=∏i∈IXi under a set of parameters A=∏i∈IAi.Proof.
It is clear thatX˜ and Φ are members in τ. Now, let ∏i∈IUi,Ai and ∏i∈IVi,Ai are two members in τ. Since Ui,Ai⊓Vi,Ai∈τi for each i, then it follows from the fact ∏i∈IUi,Ai⊓∏i∈IVi,Ai=∏i∈IUi,Ai⊓Vi,Ai that τ is closed under finite soft intersection. Hence, the proof is complete.
We callτ given in the proposition above a product of infra soft topologies, and X,τ,A a product of infrasoft spaces.Definition 18.
Letgiφ:Xi,τi,Ai⟶Yi,θi,Bi:i∈I be a family of soft mappings. The product of these soft mappings is given by gφ:X,τ,A⟶Y,θ,B such that gφPaii∈Ixii∈I=gφPaixii∈I.
IfI is countable, then we write it as follows:(10)gφPai0,ai1,…xi0,xi1,…=gφPai0xi0,gφPai1xi1,….Theorem 3.
The product of infrasoft open mappings is an infrasoft open mapping.Proof.
Letgφ:X,τ,A⟶Y,θ,B be the product of infrasoft open mappings of the family giφ:Xi,τi,Ai⟶Yi,θi,Bi:i∈I. Let U,A be an infrasoft open set in τ. Since U,A=∏i∈IUi,Ai where Ui,Ai∈τi for each i, then gφU,A=gφ∏i∈IUi,Ai=∏i∈IgiφUi,Ai. By hypothesis, giφUi,Ai is an infrasoft open set in θi for each i. According to the definition of the product of infrasoft spaces, gφU,A is an infrasoft open set in θ. Hence, we obtain the coveted result.Corollary 3.
The product of infrasoft closed mappingsgiφ:i∈I is an infrasoft closed mapping provided that giφ is bijective for each i.Theorem 4.
The product of infrasoft continuous mappings is an infrasoft continuous mapping.Proof.
Similar to the proof of Theorem3, one can prove this result.Theorem 5.
Ifgφ:X,τ,A⟶Y,θ,B is an infrasoft open (resp. infrasoft closed) mapping such that φ is injective, then g:X,τa⟶Y,θφa is an infraopen (resp. infraclosed) mapping.Proof.
LetU be an infraopen set in τa. Then, there exists an infrasoft open set H,A in τ such that Ha=U. Since gφ is an infrasoft open mapping, then gφH,A=gH,B is an infrasoft open set in θ. Therefore, gHb=φa=g∪a∈φ−1bHa is infrasoft open in θφa. Since φ is injective, then gHb=φa=g∪a=φ−1bHa=gU. Thus, gU is an infraopen set in θφa. Hence, we obtain the coveted result.
Following the same arguments, the case between parenthesis can be proved.
Example below explains that the converse of the above theorem fails.Example 3.
LetA=a1,a2, and consider the two infrasoft topologies τ=Φ,X˜,U,A and θ=Φ,X˜,V,A on X=w1,w2, where(11)U,A=a1,w1,a2,∅,V,A=a1,∅,a2,w1.
Takinggφ:X,τ,A⟶X,θ,A as the soft identity mapping, it is clear that g:X,τa⟶X,τφa=a is an infraopen mapping and an infraclosed mapping for each a∈A. But gφ is neither an infrasoft open mapping nor an infrasoft closed mapping because gφU,A=U,A is not an infrasoft open set in θ and gφUc,A=Uc,A is not an infrasoft closed set in θ.
We show under which condition the converse of Theorem5 is true.Theorem 6.
Letθ be an infrasoft topology induced from the (crisp) classical infratopologies and gφ:X,τ,A⟶Y,θ,B be a soft mapping such that φ is injective. Then, gφ:X,τ,A⟶Y,θ,B is infrasoft open (resp. infrasoft closed) if and only if g:X,τa⟶Y,θφa is infraopen (resp. infraclosed).Proof.
Necessity: it follows from Theorem5.
Sufficiency: letH,A be an infrasoft open set in τ. Since g:X,τa⟶Y,θφa is an infraopen mapping, then gHa is an infraopen set in θφa for each a∈A. Now, gφH,A=gH,B such that either gHb=φa=g∪a=φ−1bHa=gHa is an infraopen set in θφa or gHb=g∪φ−1b=∅Ha=∅ is an infraopen set in θb. Since θ is generated from the crisp infratopologies, then gφH,A is an infrasoft open set in θ. Hence, the proof is complete.
Following the same arguments, the case between parenthesis can be proved.Definition 19.
A bijective soft mappinggφ:X,τ,A⟶Y,θ,B is said to be an infrasoft homeomorphism if it is infrasoft continuous and infrasoft open.
We cancel the proofs of the next two propositions because they are easy.Proposition 12.
Letgφ:X,τ,A⟶Y,θ,B and fψ:Y,θ,B⟶Z,μ,C be two infrasoft homeomorphism mappings. Then, fψ∘gφ is an infrasoft homeomorphism mapping.Proposition 13.
Ifgφ:X,τ,A⟶Y,θ,B is a bijective soft mapping, then the following statements are equivalent.(i)
gφ is an infrasoft homeomorphism(ii)
gφ and gφ−1 are infrasoft continuous(iii)
gφ is infrasoft closed and infrasoft continuous
The proofs of the following two theorems follow from Theorems5 and 6, respectively.Theorem 7.
If a soft mappingfφ:X,τ,A⟶Y,θ,B is infrasoft homeomorphism, then a mapping f:X,τa⟶Y,θφa is infrahomeomorphism.Theorem 8.
Letθ be an infrasoft topology induced from classical (crisp) infratopologies. Then, a soft mapping fφ:X,τ,A⟶Y,θ,B is infrasoft homeomorphism if and only if a mapping f:X,τa⟶Y,θφa is infrahomeomorphism.Definition 20.
A soft subsetH,A of an infrasoft topological space X,τ,A is called an isolated soft set if there exists a soft point Pax∈H,A such that Pax∉H,Ai′.Proposition 14.
Ifgφ:X,τ,A⟶Y,θ,B is an infrasoft homeomorphism mapping, then the following statements hold for each H,A∈SXA.(i)
gφintH,A=intgφH,A(ii)
gφclH,A=clgφH,A(iii)
gφH,Ai′=gφH,Ai′Proof.
We prove (i), and one can prove the other two cases similarly.
It follows from (i) of Proposition5 that gφintH,A⊑intgφH,A. Conversely, let Pby∈intgφH,A. Then, there exists an infrasoft open set U,B such that Pby∈U,B⊑gφH,A. By hypothesis, Pax=gφ−1Pby∈gφ−1U,B⊑H,A such that gφ−1U,B is an infrasoft open set in τ. Therefore, Pax∈intH,A. This means that Pby∈gφintH,A, as required.Definition 21.
A property is said to be an infrasoft topological invariant if the property possessed by an infratopological spaceX,τ,A is also possessed by each an infrasoft homeomorphic to X,τ,A.Theorem 9.
The property of an infrasoft dense set (isolated soft set) is an infrasoft topological invariant.Proof.
Letgφ:X,τ,A⟶Y,θ,B be an infrasoft homeomorphism mapping, and let H,A an infrasoft dense subset of X,τ,A, i.e., clH,A=X˜. It follows from (ii) of Proposition 14 that clgφH,A=gφclH,A=gφX˜=clY˜=Y˜. Therefore, gφH,A is an infrasoft dense subset of Y,θ,B. Hence, the proof is complete.Theorem 10.
The product of two infrasoft homeomorphism mappings is an infrasoft homeomorphism mapping.Proof.
It is clear that the product of bijective soft mappings is bijective. Then, Theorem3 and Corollary 4.15 finish the coveted result.Proposition 15.
LetXi,τi,A:i∈I be a family of pairwise disjoint infrasoft topological spaces. Then, the family τ=U,A:U,A⊓Xi˜ is an infrasoft open set in Xi,τi,A for each i∈I produces an infrasoft topology on X=∪i∈IXi with a constant set of parameters A.Proof.
It is clear thatX˜ and Φ are members of τ. To prove that τ is closed under finite soft intersections, let U1,A and U2,A be two members of τ. Then, U1,A⊓Xi˜∈τi and U2,A⊓Xi˜∈τi for each i∈I. Therefore, U1,A⊓U2,A⊓Xi˜∈τi for each i∈I. Thus, U1,A⊓U2,A∈τ. Hence, τ is an infrasoft topology on X.
We call the infrasoft topological space given in the above proposition a sum of infrasoft topological spaces and is denoted by⊕Xi,τ,A.Proposition 16.
A soft subsetH,A of ⊕Xi,τ,A is infrasoft closed if and only if H,A⊓Xi˜ is an infrasoft closed set in τi for each i∈I.Proof.
H,Ais an infra soft closed set in τ⇔Hc,A⊓Xi˜ is an infra soft open set in τi for any i∈I⇔H,A⊓Xi˜ is an infra soft closed set inτi for any i∈I.Definition 22.
Letgiφ:Xi,τi,A⟶Yi,θi,B:i∈I be a family of soft mappings. Then, we define a soft mapping gφ:⊕Xi,τ,A⟶⊕Yi,θ,B as follows: the image of each U,A⊑⊕Xi and the image of each V,B⊑⊕Yi are given by(1)
gφU,A=⊔i∈IgiφU,A⊓Xi˜(2)
gφ−1V,B=⊔i∈Igiφ−1V,B⊓Xi˜Theorem 11.
A soft mappinggφ:⊕Xi,τ,A⟶⊕Yi,θ,B is infrasoft open (resp. infrasoft closed) if and only if all soft mappings giφ:Xi,τi,A⟶Yi,θi,B are infrasoft open (resp. infrasoft closed).Proof.
We prove the theorem in case of the soft mapping is infra soft open.
Necessity: letgφ:⊕Xi,τ,A⟶⊕Yi,θ,B be an infrasoft open mapping. Taking an arbitrary soft mapping gjφ:Xj,τj,A⟶Yj,θj,B, where j∈I, let U,A be an infrasoft open set in τj. Then, U,A is an infrasoft open set in τ. Therefore, gφU,A is an infrasoft open set in θ. Since U,A⊓Xi˜=Φ for each i≠j, then gφU,A=gjφU,A. Thus, gjφU,A is an infrasoft open set in θj, as required.
Sufficiency: letgiφ:Xi,τi,A⟶Yi,θi,B be an infrasoft open mapping for each i∈I, and let U,A be an infrasoft open set in τ. Since U,A⊓Xi˜ is an infrasoft open set in τi for each i∈I, then giφU,A⊓Xi˜ is an infrasoft open set in θi for each i∈I. According to the definition of sum of infrasoft topologies, we obtain ⊔i∈IgiφU,A⊓Xi˜ is an infrasoft open set in θ. Now, gφU,A=⊔i∈IgiφU,A⊓Xi˜ so that gφU,A is an infrasoft open set in ⊕Yi,θ,A.
Following similar arguments, one can prove the case between the parenthesis.Corollary 4.
A soft mappinggφ:⊕Xi,τ,A⟶⊕Yi,θ,B is infrasoft homeomorphism if and only if all soft mappings giφ:Xi,τi,A⟶Yi,θi,B are infrasoft homeomorphism.


## 5. Infrasoft Quotient Mappings

In this section, we define the concepts of quotient infrasoft topologies and infrasoft quotient mappings. We establish their main properties and investigate transmission of them to (crisp) mappings defined between parametric infrasoft topological spaces.Definition 23.
Letgφ:X,τ,A⟶PYB be a soft mapping. A family θ⊑SYB is said to be a quotient infrasoft topology over Y (with respect to gφ) if θ is the largest infrasoft topology that makes gφ infrasoft continuous.
Note thatθ=U,B⊑Y˜:gφ−1U,B∈τ.
The following example points out how a quotient infrasoft topology is constructed.Example 4.
Letτ=Φ,X˜,H1,A,H2,A be an infrasoft topology on X=u,v,w with a set of parameters A=a1,a2 such that(12)H1,A=a1,u,a2,X,H2,A=a1,v,a2,∅.
LetY=x,y,z be another universal set with a set of parameters B=b1,b2, and consider a soft mapping gφ from X,τ,A to PYB, where g:X⟶Y and φ:A⟶B are defined as follows:(13)gu=x,gv=gw=y,φa1=b1,φa2=b2.
Then,θ=Φ,Y˜,Ui,B:i=1,2,…,10 is the quotient infrasoft topology on Y (with respect to gφ), where(14)U1,B=b1,x,b2,x,y,U2,B=b1,x,b2,Y,U3,B=b1,x,z,b2,x,y,U4,B=b1,x,z,b2,Y,U5,B=b1,x,y,b2,x,y,U6,B=b1,Y,b2,x,y,U7,B=b1,x,y,b2,Y,U8,B=b1,z,b2,∅,U9,B=b1,∅,b2,z,U10,B=b1,z,b2,z.
Note that, for any infrasoft topology onY is proper finer than θ, gφ is not infrasoft continuous.Theorem 12.
Letgφ:X,τ,A⟶Y,θ,B be an infrasoft continuous mapping. Then, the following statements are equivalent.(i)
θ is a quotient infrasoft topology(ii)
U,B is an infrasoft open set in θ if gφ−1U,B is an infrasoft open set in τ(iii)
V,B is an infrasoft closed set if gφ−1V,B is an infrasoft closed setProof.
(i)
⟹ (ii): since gφ is infrasoft continuous, then gφ−1U,B is an infrasoft open set in τ for each infrasoft open set U,B in θ. Conversely, let gφ−1U,B be an infrasoft open set in τ. Since θ is a quotient infrasoft topology, then U,B∈θ, as required.(ii)
⟹ (i): suppose that there exists an infrasoft topology ν that makes gφ infrasoft continuous such that θ⊑ν. Now, let U,B∈ν. Then, gφ−1U,B∈τ. By hypothesis, U,B∈θ. Thus, θ=ν, as required.(ii)
⟹ (iii): straightforward.Corollary 5.
Ifgφ is an infrasoft continuous mapping from X,τ,A onto Y,θ,B such that gφ is either infrasoft open or infrasoft closed, then θ is a quotient infrasoft topology.Proof.
It is clear thatgφ−1U,B is an infrasoft open set in τ for each infrasoft open set U,B in θ on one hand. On the other hand, gφ is a surjective; then, U,B=gφgφ−1U,B, and since gφ is an infrasoft open mapping, then gφgφ−1U,B is an infrasoft open set so that U,B is infrasoft open. It follows from (ii) of the above theorem that θ is a quotient infrasoft topology, as required.
The case ofgφ is an infrasoft closed mapping is proved in a similar manner.Proposition 17.
Letfψ be an infrasoft continuous mapping from X,τ,A onto a quotient infrasoft topological space Y,θ,B. Then, gφ:Y,θ,B⟶Z,μ,C is infrasoft continuous if gφ∘fψ is infrasoft continuous.Proof.
Necessity: it follows from the fact that the composition of two infrasoft continuous mappings is an infrasoft continuous mapping.
Sufficiency: suppose thatU,C is an infrasoft open set in μ. Since gφ∘fψ is infrasoft continuous, then gφ∘fψ−1U,C=fψ−1gφ−1U,C is an infrasoft open set in τ. Since θ is a quotient infrasoft topology, it follows from Theorem 12 that gφ−1U,C is an infrasoft open set in θ. Hence, we obtain the coveted result.
Recall that a mappingg:X⟶Y is called a quotient inframapping if it is surjective and Y is equipped with the quotient infratopology with respect to g.Definition 24.
A soft mappinggφ:PXA⟶PYB is said to be a quotient infrasoft mapping if gφ is surjective and Y is equipped with the quotient infrasoft topology with respect to gφ.
In other words,gφ:PXA⟶PYB is said to be a quotient infrasoft mapping if gφ is surjective and a subset U,B of Y˜ is infrasoft open if gφ−1U,B is infrasoft open.Proposition 18.
Ifgφ:X,τ,A⟶Y,θ,B be a quotient infrasoft mapping such that τ generated from crisp infratopologies, then g:X,τa⟶Y,θφa is a quotient inframapping.Proof.
Firstly, it is clear thatg is a surjective mapping. To prove that θφa is a quotient infratopology, let U be an infraopen set in θφa. Then, there is an infrasoft open set H,B in θ such that Hφa=U. Therefore, gφ−1H,B=g−1H,A is an infrasoft open set in τ; thus, g−1Ha=g−1Hφa=g−1U is an infrasoft open set in τa. Conversely, let g−1U be an infrasoft open set in τa. Then, there is an infrasoft open set F,A in τ such that Fa=g−1U. Since τ generated from crisp infratopologies, we can write Fa′=∅ for each a′≠a. Now, F,A=gφ−1H,B, where Hφa=U and Hb=∅ for each b≠φa. By hypothesis, H,B is infrasoft open set in θ. Thus, U is infrasoft open set in θφa. Hence, the proof is complete.
The following example shows that a condition imposed on the infrasoft topology of the domain is indispensable.Example 5.
Letθ=Φ,Y˜ be an infrasoft topology on Y=x,y with a set of parameters B=b1,b2, and let τ=Φ,X˜,H1,A,H2,A be an infrasoft topology on X=u,v,w with a set of parameters A=a1,a2, where(15)H1,A=a1,u,a2,v,H2,A=a1,v,a2,∅.
Consider a soft mappinggφ:X,τ,A⟶Y,θ,B, where g:X⟶Y and φ:A⟶B are defined as follows:(16)gu=x,gv=gw=y,φa1=b1,φa2=b2.
It is clear thatgφ is a quotient infrasoft mapping. Now, we have τa1=∅,X,u,v and τa2=∅,X,v are two parametric infratopologies on X, and θb1=θb2=∅,Y is a parametric infratopology on Y. On the contrary, g:X,τa1⟶Y,θφa1=b1 is not a quotient inframapping because g−1x=u is an infrasoft open set in τa1 in spite of x is not an infrasoft open set in θb1.


## 6. Conclusion

We study some extensions of soft topology, which are defined by reducing the stipulations of soft topology, for various purposes such as obtaining appropriate models to handle some real-life issues, or building some paradigms that demonstrate the relations among some topological notions and ideas, or keeping certain properties under fewer conditions of those given on soft topology. To this end, we have recently defined a new generalization of soft topology, namely, infrasoft topology.The principal focus of the article was revising the definition of soft mappings and studying some types of soft mapping in the frame of infrasoft topological structures. The main contributions of this paper are listed as follows.(1)
Improve the definition of soft mapping given in [9] using soft points(2)
Introduce the concepts of infrasoft open, infrasoft closed, and infrasoft homeomorphism mappings, and study several properties with the help of examples(3)
Present the concepts of quotient infrasoft topology and quotient infrasoft mappings and investigate main featuresTo complete building the infrasoft topological structure, we plan to do the following studies in the frame of infrasoft topological spaces.(1)
Define some types of separation axioms and show the relationships among them(2)
Explore the concepts of compactness and Lindelöffness and establish main characterizations(3)
Initiate the concept of connectedness and research fundamental properties



---

*Source: 3388288-2021-08-10.xml*