# An Innovative Approach ofN-Soft Quotient Topological Space

Authors: G. Kabin Antony; M. Lellis Thivagar; B. Tamilarasan; Mequanint Birhan
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

In the many approaches to soft set theory, the theory of N-soft topological space is a key component of studying topology. The notions of soft mapping inN-soft topological space as well as a novel notion of N-soft continuous mapping, N-soft quotient mapping, and N-soft quotient topology are discussed in this article. We also look at some of its traits and outcomes. It will give the foundation for real-world applications in this subject.


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## Body

## 1. Introduction

The theory of soft sets was initiated in 1999, by Molodtsov [1], as a tool of mathematics to deal with uncertainties of problems with incomplete information in many fields like medical sciences, engineering, economics, physics, computer science, and so on. Later, many researchers established the basic structure of the soft set theory. In 2011, Shabir and Naz [2] and Cagman et al. [3] independently initiated the concepts of soft topological spaces, and they defined their basic notions and properties. We study the Cagman method to develop the theory of soft topological space.Recently, in 2016, Thivagar et al. [4] introduced and established the theory of N-topological space from the geometric existence of topological space. Also, they derived various forms of quotient topology from quotient mappings [5]; in particular, they defined α-quotient mappings, strongly α-quotient mappings, and α∗-quotient mappings between topological spaces. Many researchers established the basic study of quotient topology [6–8].Later, we expand the idea of soft topological space toN-soft topological space and investigate their fundamental features using appropriate examples. We investigate the concepts of N-soft continuous mapping and N-soft quotient mapping in N-soft topological space in this research. N-soft quotient mapping is also used to create the notions of N-soft quotient topology. In addition, we look at the characterization of the N-soft continuous and N-soft quotient mappings. Finally, we apply these ideas to N-soft topology’s probable attributes.


## 2. Preliminaries

Now we would like to present some basic notions of soft set theory, soft image, soft inverse image,N-soft topology, and N-soft topological subspace.Definition 1.
(see [1]). A soft set FA on the universe U is defined by the set of ordered pairs FA=x,fx:x∈E,fAx∈PU, where fA:E⟶PU such that fAx=ϕ if x∈A. Here, fA is called an approximation function of the soft set FA. The value of fAx may be arbitrary. Some of them may be empty, and some may have nonempty intersection. The set of all soft sets over U will be denoted by SU.Definition 2.
(see [3]). Let FA∈SU. The soft power set of FA is defined by P˜FA=FAi:FAi⊆˜FA,i∈I⊆ℕ, and its cardinality is defined by P˜FA=2∑x∈EfAx where fAx is the cardinality of fAx.Definition 3.
LetFA∈SU, τ˜1,τ˜2,…,τ˜N be N-arbitrary soft topologies defined on FA and the collection Nτ˜ be defined by(1)Nτ˜=S˜⊆˜FA:S˜=∪i=1nFAi˜∪​˜∪i=1nFBi˜,FAi,FBi∈τ˜i,satisfying the following axioms:(i)
Fϕ,FA∈Nτ˜.(ii)
∪​˜i=1∞Si˜∈Nτ˜ for all Si∈Nτ˜.(iii)
∩​˜i=1nSi˜∈Nτ˜ for all Si∈Nτ˜.
Then, the pairFA,Nτ˜ is called a N-soft topological space on FA, denoted by Nτ˜f. The elements of the collection Nτ˜ are known as N-soft open sets on FA and the complement of N-soft open on FA is known as N-soft closed sets.Definition 4.
LetFA,Nτ˜ be a N-soft topological space and GB⊆˜FA. Then, the collection Nτ˜'=H∩​˜GB:H∈Nτ˜ is called a N-soft subspace topological space on GB and GB,Nτ˜' or Nτ˜g is called a N-soft topological subspace of FA,Nτ˜.Definition 5.
(see [9]). Let X,E and Y,E′ be soft classes. Let u:X⟶Y and p:E⟶E′ be mappings. Then, a mapping f:X,E⟶Y,E′ is defined as follows: for a soft set F,A in X,E, fF,A,B,B=pA⊆E′, is a soft set Y,E′ given by(2)fF,Aβ=u∪α∈p−1β∩​AFαifp−1β∩​A≠ϕϕ,otherwise,for β∈B⊆E′. fF,A,B is called a soft image of a soft set F,A.Definition 6.
(see [9]). Let X,E and Y,E′ be mappings from a soft class X,E to another soft class Y,E′ and G,C be a soft set in soft class Y,E′, where C⊆E′. Let u:X⟶Y and p:E⟶E′ be mappings. Then, f−1G,C,D,D=p−1C, is a soft set in the soft class X,E, defined as(3)f−1G,Cα=u−1GPα,ifpα∈Cϕ,otherwise,for α∈D⊆E. f−1G,C,D is called a soft inverse image of G,C. Hereafter we shall write f−1G,C,E as f−1G,C.


## 3.N-Soft Continuous and N-Soft Quotient Mappings

In this section, we define the notion ofN-soft continuous and N-soft quotient mappings and their characterization.Definition 7.
LetFX,E and FY,K be soft sets on U and V, respectively. A mapping f:FX,Nτ˜,E⟶FY,Nσ˜,K is said to be N-soft mapping if FX,Nτ˜,EandFY,Nσ˜,K are N-soft topological spaces, and u:U⟶V and v:E⟶K are mappings.
Throughout this work, we represent the spaceFX,Nτ˜,E as F˜X and the space FY,Nσ˜,K as F˜Y.Definition 8.
A soft mappingf:F˜X⟶F˜Y is said to be N-soft continuous if the inverse image of every N-soft open of F˜Y is N-soft open in F˜X.Example 1.
Let the universeU=w1,w2,w3,w4,w5,w6 and the parameters E=x1,x2,x3,x4. Let X=x2,x3,x4⊆E and FX=x2,w1,w3,x3,w5,x4,w2,w6. Then, all the soft subsets of FX are(4)FX1=x2,w1,FX2=x2,w3,FX3=x2,w1,w3,FX4=x3,w5FX5=x2,w1,x3,w5,FX6=x2,w3,x3,w5,FX7=x2,w1,w3,x3,w5,FX8=x4,w2,FX9=x4,w6,FX10=x4,w2,w6,FX11=x2,w1,x4,w2,FX12=x2,w3,x4,w2,FX13=x2,w1,w3,x4,w2,FX14=x3,w5,x4,w2,FX15=x2,w1,x3,w5,x4,w2,FX16=x2,w3,x3,w5,x4,w2,FX17=x2,w1,w3,x3,w5,x4,w2,FX18=x2,w1,x4,w6,FX19=x2,w3,x4,w6,FX20=x2,w1,w3,x4,w6,FX21=x3,w5,x4,w6,FX22=x2,w1,x3,w5,x4,w6,FX23=x2,w3,x3,w5,x4,w6,FX24=x2,w1,w3,x3,w5,x4,w6,FX25=x2,w1,x4,w2,w6,FX26=x2,w3,x4,w2,w6,FX27=x2,w1,w3,x4,w2,w6,FX28=x3,w5,x4,w2,w6,FX29=x2,w1,x3,w5,x4,w2,w6,FX30=x2,w3,x3,w5,x4,w2,w6,FX31=FX,FX32=ϕ.
Letτ˜1=ϕ,FX,FX3,FX7 and τ˜2=ϕ,FX,FX4,FX7 be two soft topologies. Then, 2τ˜=ϕ,FX,FX3,FX4,FX7 is a bi-soft topology on FX.
Let another universeV=h1,h2,h3,h4,h5 and the parameters K=x1′,x2′,x3′,x4′. Let Y=x1′,x2′,x4′⊆K and FY=x2′,h1,x4′,h3,h4.
Then, all the soft subsets ofFY are(5)FY1=x1′,h2,FY2=x2′,h1,FY3=x1′,h2,x2′,h1,FY4=x4′,h3,FY5=x4′,h4,FY6=x4′,h3,h4,FY7=x1′,h2,x4′,h3,FY8=x1′,h2,x4′,h4,FY9=x2′,h1,x4′,h3,FY10=x2′,h1,x4′,h4,FY11=x1′,h2,x4′,h3,h4,FY12=x1′,h2,x4′,h3,h4,FY13=x1′,h2,x2′,h1,x4′,h3,FY14=x1′,h2,x2′,h1,x4′,h4,FY15=FY,FY16=ϕ.
Letσ˜1=ϕ,FY,FY2,FY12 and σ˜2=ϕ,FY,FY6,FY11 be two soft topologies. Then, 2σ˜=ϕ,FY,FY2,FY6,FY11,FY12 is a bi-soft topology on FY.
Define mappingsu:U⟶V and v:E⟶K. uw1=h1,uw2=h4,uw3=h1, uw4=h3,uw5=h2,uw6=h3 and vx1=x3′,vx2=x2′,vx3=x1′, vx4=x3′. The bi-soft inverse image is f−1FY,Yx2=u−1FYPx2=u−1FYx2′=u−1h1=w1,w3,f−1FY,Yx3=u−1FYx1′=w5, where D=p−1Y=x2,x3, and thus we have that bi-soft inverse image f−1FY,YD=x2,w1,w3x3,w5 is bi-soft open. Hence, the given soft mapping is bi-soft continuous mapping.Definition 9.
LetF˜X and F˜Y be N-soft topological spaces and let f:FX⟶FY be a surjective map. Then, f is said to be a N-soft quotient map if a N-soft subset A of F˜Y is N-soft open if and only if f−1A is N-soft open in F˜X.Example 2.
Consider the soft set ofFX and FY from Example 1. Let τ˜1=ϕ,FX,FX3,FX27, τ˜2=ϕ,FX,FX4,FX28, and τ˜3=ϕ,FX,FX4,FX10,FX28 be three soft topologies. Then, 3τ˜=ϕ,FX,FX3,FX4,FX10,FX27,FX28 is a tri-soft topology on FX and let σ˜1=ϕ,FX,FY1,FY11, σ˜2=ϕ,FX,FY2,FY12, and σ˜3=ϕ,FX,FY6,FY12 be three soft topologies. Then, 3σ˜=ϕ,FX,FY2,FY6,FY11,FY12 is a tri-soft topology on FY.
Define mappingsu:U⟶V and v:E⟶K. uw1=h2,uw2=h3,uw3=h2, uw4=h5,uw5=h1,uw6=h4 and vx1=x3′,vx2=x1′,vx3=x2′, vx4=x4′. The tri-soft inverse image is f−1FY,Yx2=u−1FYx1′=u−1h2=w1,w3, f−1FY,Yx3=u−1FYx2′=u−1h1=w5, f−1FY,Yx4=u−1FYx4′ = u−1h3,h4=w2,w6 where D=p−1Y=x2,x3,x4, and thus we have that tri-soft inverse image f−1FY,YD=x2,w1,w3,x3,w5,x4,w2,w6 is tri-soft open and the tri-soft image is fFX,Xx1′=u∪​Fx2=uw1,w3=h2,fFX,Xx2′ = u∪​Fx3=uw5=h1,fFX,Xx4′=u∪​Fx4=uw2,w6=h3,h4 where B=PX=x1′,x2′,x4′. Thus, the tri-soft image fFX,XD=x1′,h2,x2′,h1, x4′,h3,h4 is also a tri-soft open. So, the given soft mapping is tri-soft quotient mapping.Theorem 1.
Letf:F˜X⟶F˜Y be a relation preserving, N-quotient map. Then, the map f∗:FX/R,Nτ˜p,E⟶FY/R′,Nσ˜q,K given by f∗A=fA for every A∈FX/R is also a quotient map.Proof.
TheN-soft quotient topology Nτ˜p on FX/R is given by Nτ˜p=U⊆˜FX/R:p−1U∈Nτ˜ where p:FX⟶FX/R is the project mapping. Similarly, the N-soft quotient topology Nσ˜q on FY/R′ is given by Nσ˜q=V⊆˜FY/R′:q−1V∈Nσ˜q where q:FY⟶FY/R′ is the projection mapping. Since f is relation preserving and N-soft continuous, f∗ is N-soft continuous. That is, whenever a soft subset U⊆˜FY/R′ is N-soft open, f∗−1U is N-soft open in FX/R. If U⊆˜FY/R′ such that f∗−1U is N-soft open in FX/R; then, p−1f∗−1U is N-soft open in FX. That is, f∗−1U is N-soft open in FX. q∘f−1U is N-soft open in FX. That is, f−1q−1U is N-soft open in FX and q−1⊆˜FY. Therefore, q−1U is N-soft open FY,Nσ˜, since f is a N-soft quotient mapping. That is, q−1U∈Nσ˜, and hence U∈Nσ˜q. That is, U is N-soft open in FY/R′. Thus, a soft subset U is N-soft open in FY/R′ if and only if f∗−1U is N-soft open in FX/R. Therefore, f∗ is a N-soft quotient map.Theorem 2.
Letp:FX⟶FY be a N-soft quotient map; let A be a soft subspace of FX, that is, N-saturated with respect to p; let q:A⟶pA be the map obtained by restricting p.(i)
IfA is either a N-soft open map or a N-soft closed map, then q is a N-soft quotient map.(ii)
Ifp is either a N-soft open map or a N-soft closed map, then q is a N-soft quotient map.Proof.
(i)
We shall prove the following:(a)
q−1V=p−1 if V⊂˜pA.(b)
pU∩​˜A=pU∩​˜pA if U⊂˜FX.To prove (i) of (a), first let us assume thatV⊂˜PA and A is N-saturated, p−1V⊂˜A. It implies that both p−1V and q−1V equal all points of A that are mapped by p into V. To examine the second equation, we observe that for any two soft subsets U and A of FX, we have the inclusion pU∩​˜A⊂˜pU∩​˜pA. To verify the contrary, suppose y=pu=pa, for u∈U and a∈A. Since A is N-saturated, A contains p−1pa, so that A⊂˜U. Then, y=pu, where u∈U∩​˜A.To prove (i) of (b), let us supposeA is N-soft open or p is N-soft open. Given the soft subset V of pA, we assume that q−1V is N-soft open in A and show that V is N-soft open in pA. Suppose first that A is N-soft open. Since q−1V is N-soft open in A and A is N-soft open in FX, the soft set q−1V is N-soft open in FX. Since q−1V=p−1V, the latter set is N-soft open in FX, so that V is N-soft open in FY because p is a N-soft quotient map. In particular, V is N-soft open in pA. Now suppose p is N-soft open. Since q−1V=p−1V and q−1V is N-soft open in A, we have p−1V=U∩​˜A for some soft set U, N-soft open in FX (FX be a soft set on the universe U). Now, pp−1V=V because p is surjective; then, V=pp−1V=pU∩​˜A=pU∩​˜pA. The soft set pU is N-soft open in FY because p is an N-soft open map; hence, V is N-soft open in pA.(ii)
WhenA or p is N-soft closed, this may be demonstrated by substituting N-soft open with N-soft closed in step (i) of the proof (b).


## 4.N-Soft Quotient Topology

Here, we discuss the concept ofN-soft quotient topological space from N-soft quotient mapping and their properties along with N-soft continuous mapping.Definition 10.
LetF˜X be a N-soft quotient topological space. Let FY be a nonempty soft set and f:FX⟶FY be a surjective map. Then, defined Nσ˜=G⊂˜FY:f−1G is open in Nτ˜ is a N-soft topology on FY relative to which f is a N-soft quotient topology on FY induced by f.Definition 11.
LetF˜X be a N-soft topology and R be an equivalence relation on FX. Let P:FX⟶FX/R:Px=X∀x∈FX; then, the family FX/R with N-soft quotient topology is called N-soft quotient space.Remark 1.
N-soft quotient topology for FX/R is the strongest N-soft topology for which P is continuous, and this N-soft topology is given by Nτ˜p=H∈FX/R; then, P−1H∈Nτ˜.Theorem 3.
LetF˜X be a N-soft topological space and let FY be a nonempty soft set. Let f:FX⟶FY be a N-soft quotient map. Then, the N-soft quotient topology FX on FY induced by f is a N-soft topology.Proof.
We shall verify each of the three conditions for a topology:(i)
The soft setf−1ϕ=ϕ, which is Nτ˜-soft open in FX. f−1FY=FX, which is Nτ˜-soft open in FX. So, ϕ and F_Y are Nτ˜-soft open in the N-soft quotient topology.(ii)
Suppose each of the soft sets in the collectionGββ∈B is Nτ˜-soft open in the N-soft quotient topology on FY. Then, f−1∪​˜Gβ=∪​˜f−1Gβ, which is a union of Nτ˜-soft open sets in FX, and therefore it is Nτ˜-soft open in FX. Thus, ∪​˜Gβ is Nτ˜-soft open in the N-soft quotient topology, which implies that the arbitrary union of Nτ˜-soft open sets in the N-soft quotient topology is a Nτ˜-soft open set in the N-soft quotient topology.(iii)
Suppose each of the soft setsGi,i=1,2,…,n, is Nτ˜-soft open in the N-soft quotient topology on FY. Then, f−1∩​˜Gi=∩˜∩​˜f−1Gi, which is a finite intersection of Nτ˜-soft open sets in FX, and therefore it is Nτ˜-soft open in FX. Hence, ∩​˜Gi is Nτ˜-soft open in the N-soft quotient topology, and it follows that the finite intersection of open sets in the N-soft quotient topology is Nτ˜-soft open set. Hence, the N-soft quotient topology is a topology on FY.Theorem 4.
Iff is a N-soft continuous and N-soft open mapping of a N-soft topological space F˜X onto a N-soft topological space F˜Y, then Nσ˜ must be the N-soft quotient topology for FY.Proof.
LetNτ˜f denote the N-soft quotient topology for FY. Since the N-soft quotient topology is the largest for which f is N-soft continuous, Nσ˜⊆˜Nτ˜f. Again let H∈Nτ˜f; then, f−1H∈Nτ˜. Since f is N-soft open, it follows that ff−1H∈Nσ˜, i.e., H∈Nσ˜ (since f is onto). Thus, H∈Nτ˜f⇒H∈Nσ˜, i.e., Nτ˜f⊆˜Nσ˜. Hence, Nσ˜=Nτ˜f, the N-soft quotient topology on FY.Theorem 5.
Letf be a N-soft continuous mapping of a N-soft topological space FX onto a N-soft topological space FY and let FY have a N-soft quotient topology. Then, a mapping g from FY onto a space Nτ˜ is N-soft continuous if and only if the composite mapping g∘f is N-soft continuous.Proof.
Assume the mappingg:FY⟶Nτ˜ to be N-soft continuous. The soft mapping f:FX⟶FY is N-soft continuous, and since the composite of N-soft continuous mapping IS composite, g∘f is N-soft continuous as well. Conversely, let g∘f be N-soft continuous and let H be a N-soft open subset of Nτ˜. Then, g∘f−1H=f−1∘g−1H=f−1g−1H, which is N-soft open in FX because of the N-soft continuity of g∘f. By Definition 8, it follows that g−1H is a N-soft open subset of FY. Hence, g is N-soft continuous.Theorem 6.
Letp:FX⟶FY be a N-soft quotient map. Let FZ be a space and let g:FX⟶FZ such that f∘p=g. The induced map f is N-soft continuous if and only if g is N-soft continuous; f is N-soft quotient map if and only if g is a N-soft quotient map.Proof.
For eachy∈FY, the soft set gp−1y is a one-point soft set in FZ (since g is constant on p−1y). If we let fy denote this point, then we have defined a soft map f:FY⟶FZ such that for each x∈FX, fpx=gx. If f is N-soft continuous, then g=f∘p is N-soft continuous. Conversely, suppose that g is N-soft continuous. Given an N-soft open set V of Z, g−1V is N-soft open in FX. But g−1V=p−1f−1V; because p is a N-soft quotient map, it follows that f−1V is N-soft open in FY. Hence, f is N-soft continuous. If f is a N-soft quotient map, then g is the composite of two N-soft quotient maps and is thus a N-soft quotient map. Conversely, suppose that g is a N-soft quotient map. g is surjective, and so is f. Let V be a soft subset of FZ; we show that V is N-soft open in FZ if f−1V is N-soft open in FY. Now the set p−1f−1V is N-soft open in FX because g is N-soft continuous. Since this soft set equals g−1V the latter is N-soft open in FX. Then, because g is a N-soft quotient map, V is N-soft open in FZ.


## 5. Conclusion

The concepts of quotient mapping and quotient topology are very seldom studied. We attempted to study quotient topology andN-soft topology, resulting in N-soft quotient topology. We are here to explain how this notion is relevant in most pure and practical mathematical domains, as well as explore the mapping of N-soft continuous and N-soft quotient. We also hope that the study of N-soft quotient topology will pave the way for future research into a broader field and a more in-depth investigation.



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