# Topological Structures via Bipolar Hypersoft Sets

Authors: Sagvan Y. Musa; Baravan A. Asaad
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, we introduce bipolar hypersoft topological spaces over the collection of bipolar hypersoft sets. It is proven that a bipolar hypersoft topological space gives a parametrized family of hypersoft topological spaces, but the converse does not hold in general, and this is shown with the help of an example. Furthermore, we give a condition on a given parametrized family of hypersoft topologies, which assure that there is a bipolar hypersoft topology whose induced family of hypersoft topologies is the given family. The notions of bipolar hypersoft neighborhood, bipolar hypersoft subspace, and bipolar hypersoft limit points are introduced. Finally, we define bipolar hypersoft interior, bipolar hypersoft closure, bipolar hypersoft exterior, and bipolar hypersoft boundary, and the relations between them, differing from the relations on hypersoft topology, are investigated.


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## Body

## 1. Introduction

Many challenges in engineering, artificial intelligence, economics, environmental research, social science, and other fields include ambiguous or unclear data. As a result, traditional methods depending on the specific instance may be ineffective in solving or modeling them. Many theories have been developed to address these issues in light of this. Soft set theory was introduced by Molodtsov [1] as a revolutionary idea for dealing with uncertainty. In comparison to earlier ideas, he demonstrated how effective soft sets are at solving complex issues. Many scholars have since researched soft set theory’s properties, operations, and applications (see, for example, [2–8]). The relevance of topology and its multiple applications, particularly in physics, economics, and computer science, have piqued researchers’ interest in the topological structure of soft sets. There were two definitions of soft topological spaces presented. Shabir and Naz [9] were the first to establish the concept of soft topological space on a universe set. Çaǧman et al. [10] used a soft set to demonstrate the notion of soft topological space. Following that, several academics concentrated on soft topological spaces [11–26].Shabir and Naz [27] first used bipolar soft set to combine bipolarity [28] with soft set theory [1]. According to Dubois and Prade [28], our decision-making is built upon two sides, positive and negative, and we choose based on which is more powerful. Some definitions, operations, and applications on bipolar soft sets were studied in [29–32]. The notion of bipolar soft topological spaces over an ordinary set was first introduced by Shabir and Bakhtawar [33], along with studies into bipolar soft connectedness and bipolar soft compactness. The notions of interior and closure operators, basis, and subspace in bipolar soft topological spaces were studied by Ozturk [34]. By redefining bipolar soft topological spaces on a bipolar soft set, Fadel [35] has expanded the definition of bipolar soft topological spaces introduced in [33]. They have covered the key concepts and properties, as well as some illustrative examples. For additional work on the topological structures on bipolar soft sets, one may study [36, 37].In 2018, Smarandache [38] transformed the argument mapping F into a multiargument mapping to turn generalized soft sets into hypersoft sets. This notion is more adaptable than soft set and more suited to challenges involving decision-making. Under the hypersoft set environment, some vital basics (e.g., elementary properties, set theoretic operations, basic laws, relations, functions, and matrices) as well as applications are conceptualized in [39–46]. Musa and Asaad [47] initiated the study of hypersoft topological spaces. They defined hypersoft topology as a collection of hypersoft sets over the universe X. So, they defined elementary notions of hypersoft topological spaces such as hypersoft open sets, hypersoft closed sets, hypersoft neighborhood, hypersoft limit points, and hypersoft subspace. Further, hypersoft closure, hypersoft interior, and hypersoft boundary were studied, and some of their basic properties were investigated.Recently, Musa and Asaad [48] introduced the notion of bipolar hypersoft sets, which was created by merging the bipolarity [28] and hypersoft set [38]. It was distinguished by two hypersoft sets, one of which provided positive data and the other negative data. In their paper, they defined some basic operations such as bipolar hypersoft subset, bipolar hypersoft complement, and bipolar hypersoft difference. Furthermore, aggregate operations such as intersection, union, AND-operation, and OR-operation of two bipolar hypersoft sets with their characteristics were explored and illustrated with examples.The following is how the rest of the paper is laid out: Section2 contains some basic definitions on bipolar hypersoft sets. In Section 3, we introduce bipolar hypersoft topological spaces, which are defined over an initial universe with a fixed set of parameters. Then, we discuss some basic properties of bipolar hypersoft topological spaces and define bipolar hypersoft neighborhood, bipolar hypersoft subspace, and bipolar hypersoft limit points. In Section 4, bipolar hypersoft interior, bipolar hypersoft closure, bipolar hypersoft exterior, and bipolar hypersoft boundary are defined. Also, the relationship between these concepts is investigated. In Section 5, we summarize the main results and make suggestions for further studies.


## 2. Preliminaries

Here, we recall some basic terminologies regarding bipolar hypersoft sets and hypersoft topological spaces.Throughout this work, we useX as an initial universe, PX as the power set of X, and E1,E2,…,En the pairs of disjoint sets of parameters. For i=1,2,…,n, let Ai,Bi⊆Ei.To keep things simple, forE1×E2×⋯×En, A1×A2×⋯×An, and B1×B2×⋯×Bn we use the notations E, A, and B, respectively. Let A, B⊆E and each element in A, B, and E is obviously an n-tuple element.Definition 1.
[48] A bipolar hypersoft set over X is defined as Λ,Ω,A where Λ and Ω are mappings given by Λ:A⟶PX and Ω:¬A⟶PX such that Λα∩Ω¬α=ϕ for all α∈A.A bipolar hypersoft setΛ,Ω,A can be represented as follows:(1)Λ,Ω,A=α,Λα,Ω¬α:α∈AandΛα∩Ω¬α=ϕ.Definition 2.
[48] Let Λ1,Ω1,A and Λ2,Ω2,B be two bipolar hypersoft sets over a common universe X, we say that Λ1,Ω1,A is a bipolar hypersoft subset of Λ2,Ω2,B if for all α∈A:(1)
A⊆B and;(2)
Λ1α⊆Λ2α and Ω2¬α⊆Ω1¬α.
We symbolize it byΛ1,Ω1,A⊑˜˜Λ2,Ω2,B.IfΛ2,Ω2,B is a bipolar hypersoft subset of Λ1,Ω1,A, then Λ1,Ω1,A is said to be a bipolar hypersoft superset of Λ2,Ω2,B. It is symbolized by Λ1,Ω1,A⊑˜˜Λ2,Ω2,B.Definition 3.
[48] Λ1,Ω1,A and Λ2,Ω2,B are said to be bipolar hypersoft equal if Λ1,Ω1,A⊑˜˜Λ2,Ω2,B and Λ2,Ω2,A⊑˜˜Λ1,Ω1,B.Definition 4.
[48] Λ,Ω,Ac=Λc,Ωc,A is defined as the bipolar hypersoft complement of Λ,Ω,A, where Λc and Ωc are mappings given by Λcα=Ω¬α and Ωc¬α=Λα for all α∈A.Definition 5.
[48] We called Φ,Ψ,A over X a relative null bipolar hypersoft set if for all α∈A, Φα=ϕ and Ψ¬α=X.An absolute null bipolar hypersoft set overX is denoted by Φ,Ψ,E.Definition 6.
[48] We called Ψ,Φ,A over X a relative whole bipolar hypersoft set if for all α∈A, Ψα=X and Φ¬α=ϕ.An absolute whole bipolar hypersoft set overX is denoted by Ψ,Φ,E.Definition 7.
[48] A bipolar hypersoft set Γ,Θ,C, where C=A∩B, is defined as the difference of Λ1,Ω1,A and Λ2,Ω2,B over a common universe X if for all α∈C:
Γα=Λ1α∩Λ2cα=Λ1α∩Ω2¬α and Θ¬α=Ω1¬α∪Ω2c¬α=Ω1¬α∪Λ2α.
We symbolize it byΛ1,Ω1,A\Λ2,Ω2,B=Γ,Θ,C.Definition 8.
[48] A bipolar hypersoft set Γ,Θ,C, where C=A∩B, is defined as the union of Λ1,Ω1,A and Λ2,Ω2,B over a common universe X if for all α∈C:(2)Γα=Λ1α∪Λ2αandΘ¬α=Ω1¬α∩Ω2¬α.
We symbolize it byΛ1,Ω1,A⊔≈Λ2,Ω2,B=Γ,Θ,C.Definition 9.
[48] A bipolar hypersoft set Γ,Θ,C, where C=A∩B, is defined as the intersection of Λ1,Ω1,A and Λ2,Ω2,B over a common universe X if for all α∈C:(3)Γα=Λ1α∩Λ2αandΘ¬α=Ω1¬α∪Ω2¬α.
We symbolize it byΛ1,Ω1,A⊓≈Λ2,Ω2,B=Γ,Θ,C.Definition 10.
[47] If Tℋ is a collection of hypersoft sets over X, then Tℋ is a hypersoft topology on X if:(1)
Φ,E, Ψ,E belong to Tℋ.(2)
The intersection of any two hypersoft sets inTℋ belongs to Tℋ.(3)
The union of any number of hypersoft sets inTℋ belongs to Tℋ.
Then,X,Tℋ,E is called a hypersoft topological space over X.Definition 11.
[47] Suppose that X,Tℋ,E is a hypersoft space over X, then the members of Tℋ are said to be hypersoft open sets in X.Definition 12.
[47] Suppose that X,Tℋ,E is a hypersoft space over X. A hypersoft set Λ,E over X is said to be a hypersoft closed set in X, if Λ,Ec belongs to Tℋ.


## 3. Bipolar Hypersoft Topological Spaces

In this section, we introduce the concept of bipolar hypersoft topological spaces and study its relation with hypersoft topological spaces. Then, bipolar hypersoft subspace, bipolar hypersoft neighborhood, and bipolar hypersoft limit point are characterized.Definition 13.
LetΛ,Ω,E be a bipolar hypersoft set over X and x∈X. Then x∈Λ,Ω,E if x∈Λα for all α∈E. If x∈Λ,Ω,E, then automatically x∉Ω¬α for all ¬α∈¬E.
For anyx∈X, x∉Λ,Ω,E, if x∉Λα for some α∈E.Definition 14.
Λ1,Ω1,E and Λ2,Ω2,E are said to be disjoint bipolar hypersoft sets if for all α∈E, Λ1α∩Λ2α=ϕ. We denote it by Λ1,Ω1,E⊓≈Λ2,Ω2,E=Φ,Ω,E where Φα=ϕ for all α∈E and Ω¬α⊆X for all ¬α∈¬E.Definition 15.
LetΥ⊆X. Then, the bipolar hypersoft set Y,Φ,E over X defined by Yα=Υ for all α∈E and Φ¬α=ϕ for all ¬α∈¬E.Definition 16.
LetΥ⊆X. Then, the subbipolar hypersoft set ΛΥ,ΩΥ,E over Υ of a bipolar hypersoft set Λ,Ω,E over X is defined as
ΛΥα=Υ∩Λα for all α∈E and ΩΥ¬α=Υ∩Ω¬α for all ¬α∈¬E.Definition 17.
IfTℬℋ is a collection of bipolar hypersoft sets over X, then Tℬℋ is a bipolar hypersoft topology on X if:(1)
Φ,Ψ,E, Ψ,Φ,E belong to Tℬℋ.(2)
The intersection of any two bipolar hypersoft sets inTℬℋ belongs to Tℬℋ.(3)
The union of any number of bipolar hypersoft sets inTℬℋ belongs to Tℬℋ.
Then,X,Tℬℋ,E,¬E is called a bipolar hypersoft topological space over X.Definition 18.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X; then the members of Tℬℋ are said to be bipolar hypersoft open sets in X.Definition 19.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X. A bipolar hypersoft set Λ,Ω,E over X is said to be a bipolar hypersoft closed set in X, if Λ,Ω,Ec belongs to Tℬℋ.Proposition 1.
LetX,Tℬℋ,E,¬E be a bipolar hypersoft space over X. Then:(1)
Φ,Ψ,E, Ψ,Φ,E are bipolar hypersoft closed set over X.(2)
The union of any two bipolar hypersoft closed sets is a bipolar hypersoft closed set overX.(3)
The intersection of any number of bipolar hypersoft closed sets is a bipolar hypersoft closed set overX.proof.
Follows from Definition17 and De Morgan’s laws.Definition 20.
Suppose thatX is an initial universe, E is the parameters set, ¬E is the not set of E, and Tℬℋ=Φ,Ψ,E,Ψ,Φ,E. Then, Tℬℋ is called the bipolar hypersoft indiscrete topology on X and X,Tℬℋ,E,¬E is said to be a bipolar hypersoft indiscrete space over X.Definition 21.
Suppose thatX is an initial universe, E is the parameters set, ¬E is the not set of E, and Tℬℋ is the collection of all bipolar hypersoft sets, which can be defined over X. Then, Tℬℋ is called the bipolar hypersoft discrete topology on X and X,Tℬℋ,E,¬E is said to be a bipolar hypersoft discrete space over X.The following proposition shows that a bipolar hypersoft topological space gives a parametrized family of hypersoft topological spaces.Proposition 2.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X. Then, the following collections define hypersoft topology on X.(1)
Tℋ=Λ,E|Λ,Ω,E∈˜˜Tℬℋ.(2)
¬Tℋ=Ω,¬E|Λ,Ω,E∈˜˜Tℬℋ (provided that X is finite).Proof.
(1) Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X. Then:(i)
Φ,Ψ,E, Ψ,Φ,E∈˜˜Tℬℋ imply that Φ,E, Ψ,E∈˜Tℋ.(ii)
LetΛ1,E, Λ2,E∈˜Tℋ. Since Λ1,Ω1,E, Λ2,Ω2,E∈˜˜Tℬℋ then Λ1,Ω1,E⊓≈Λ2,Ω2,E∈˜˜Tℬℋ this implies that Λ1,E⊓˜Λ2,E∈˜Tℋ.(iii)
LetΛi,E|i∈I belong to Tℋ. Since Λi,Ωi,E∈˜˜Tℬℋ for all i∈I so that ⊔≈i∈IΛi,Ωi,E∈˜˜Tℬℋ thus ⊔˜i∈IΛi,E∈˜Tℋ.
Hence,Tℋ defines a hypersoft topology over X.
(2) The proof is similar to part (1).Remark 1.
The converse of Proposition2 is incorrect as the next example shows.Example 1.
LetX=χ1,χ2,χ3,χ4, E1=ω1,ω2, E2=ω3, and E3=ω4. Assume that Tℋ=Φ,E, Ψ,E, Λ1,E, Λ2,E, Λ3,E, Λ4,E} and ¬Tℋ=Ω1,¬E, Ω2,¬E, Ω3,¬E, Ω4,¬E} are two hypersoft topologies on X where the hypersoft sets Λ1,E, Λ2,E, Λ3,E, Λ4,E, Ω1,¬E, Ω2,¬E, Ω3,¬E, and Ω4,¬E are defined as follows:(4)Λ1,E=ω1,ω3,ω4,χ2,ω2,ω3,ω4,χ1,Λ2,E=ω1,ω3,ω4,χ1,χ2,ω2,ω3,ω4,χ3,Λ3,E=ω1,ω3,ω4,χ2,ω2,ω3,ω4,ϕ,Λ4,E=ω1,ω3,ω4,χ1,χ2,ω2,ω3,ω4,χ1,χ3,and(5)Ω1,¬E=¬ω1,ω3,ω4,χ1,χ3,χ4,¬ω2,ω3,ω4,χ2,χ4,Ω2,¬E=¬ω1,ω3,ω4,ϕ,¬ω2,ω3,ω4,χ2,Ω3,¬E=¬ω1,ω3,ω4,χ1,¬ω2,ω3,ω4,χ2,Ω4,¬E=¬ω1,ω3,ω4,χ3,χ4,¬ω2,ω3,ω4,χ2,χ4,Then,
Tℬℋ=Φ,Ψ,E,Ψ,Φ,E,Λ1,Ω1,E, Λ2,Ω2,E, Λ3,Ω3,E, Λ4,Ω4,E} where Λ1,Ω1,E, Λ2,Ω2,E, Λ3,Ω3,E, and Λ4,Ω4,E are bipolar hypersoft sets over X, defined as follows:(6)Λ1,Ω1,E=ω1,ω3,ω4,χ2,χ1,χ3,χ4,ω2,ω3,ω4,χ1,χ2,χ4,Λ2,Ω2,E=ω1,ω3,ω4,χ1,χ2,ϕ,ω2,ω3,ω4,χ3,χ2,Λ3,Ω3,E=ω1,ω3,ω4,χ2,χ1,ω2,ω3,ω4,ϕ,χ2,Λ4,Ω4,E=ω1,ω3,ω4,χ1,χ2,χ3,χ4,ω2,ω3,ω4,χ1,χ3,χ2,χ4.
If we take(7)Λ1,Ω1,E⊓≈Λ2,Ω2,E=Λ,Ω,E,then(8)Λ,Ω,E=ω1,ω3,ω4,χ2,χ1,χ3,χ4,ω2,ω3,ω4,ϕ,χ2,χ4,but Λ,Ω,E∉˜˜Tℬℋ. Hence, Tℬℋ is not a bipolar hypersoft topology on X.The following proposition demonstrates when the converse of Proposition2 will be true.Proposition 3.
Suppose thatX,Tℋ,E is a hypersoft space over X. Then, Tℬℋ consisting of bipolar hypersoft sets Λ,Ω,E such that Λ,E∈˜Tℋ and Ω¬α=X\Λα for all ¬α∈¬E, defines a bipolar hypersoft topology over X.Proof.
(i)
SinceΦ,E∈˜Tℋ, then Ψ¬α=X\Φα=X\ϕ=X and hence Φ,Ψ,E∈˜˜Tℬℋ. Also, since Ψ,E∈˜Tℋ then Φ¬α=X\Ψα=X\X=ϕ and hence Ψ,Φ,E∈˜˜Tℬℋ.(ii)
LetΛ1,Ω1,E, Λ2,Ω2,E∈˜˜Tℬℋ. Then Λ1,E∈˜Tℋ and Ω1¬α=X\Λ1α. Also, Λ2,E∈˜Tℋ and Ω2¬α=X\Λ2α. Now, since Tℋ is a hypersoft topology, then Λ1,E⊓˜Λ2,E∈˜Tℋ. Let Λ,E=Λ1,E⊓˜Λ2,E; then Ω¬α=X\Λα=X\ (Λ1α⊓˜Λ2α) = (X\Λ1α) ⊔˜ (X\Λ2α) =Ω1¬α⊔˜Ω2¬α. Hence, Λ1,Ω1,E⊓≈Λ2,Ω2,E∈˜˜Tℬℋ.(iii)
LetΛi,Ωi,E|i∈I∈˜˜Tℬℋ. Then Λi,E|i∈I∈˜Tℋ and Ωi¬α=X\Λiα. Now, since Tℋ is a hypersoft topology, then ⊔˜i∈IΛi,E∈˜Tℋ. Let Λ,E=⊔˜i∈IΛi,E then Ω¬α=X\Λα=X\⊔˜i∈IΛiα=⊓˜i∈IΩi¬α. Thus ⊔≈i∈IΛi,Ωi,E∈˜˜Tℬℋ. Therefore, the proof is completed.Definition 22.
LetX,Tℬℋ1,E,¬E and X,Tℬℋ2,E,¬E be two bipolar hypersoft topological spaces over X. If Tℬℋ1⊑˜˜Tℬℋ2, then Tℬℋ2 is said to be finer than Tℬℋ1. If Tℬℋ1⊑˜˜Tℬℋ2 or Tℬℋ2⊑˜˜Tℬℋ1, then Tℬℋ1 and Tℬℋ2 are said to be comparable bipolar hypersoft topologies over X.Proposition 4.
LetX,Tℬℋ1,E,¬E and X,Tℬℋ2,E,¬E be two bipolar hypersoft topological spaces on X, and then X,Tℬℋ1⊓≈Tℬℋ2,E,¬E is a bipolar hypersoft topological space over X.Proof.
(i)
Φ,Ψ,E, Ψ,Φ,E belong to Tℬℋ1⊓≈Tℬℋ2.(ii)
LetΛ1,Ω1,E, Λ2,Ω2,E∈˜˜Tℬℋ1⊓≈Tℬℋ2. Then Λ1,Ω1,E, Λ2,Ω2,E∈˜˜Tℬℋ1 and Λ1,Ω1,E, Λ2,Ω2,E∈˜˜Tℬℋ2. Since Λ1,Ω1,E⊓≈Λ2,Ω2,E∈˜˜Tℬℋ1 and Λ1,Ω1,E⊓≈Λ2,Ω2,E∈˜˜Tℬℋ2, so Λ1,Ω1,E⊓≈Λ2,Ω2,E∈˜˜Tℬℋ1⊓≈Tℬℋ2.(iii)
LetΛi,Ωi,E|i∈I be a family of bipolar hypersoft sets in Tℬℋ1⊓≈Tℬℋ2. Then Λi,Ωi,E∈˜˜Tℬℋ1 and Λi,Ωi,E∈˜˜Tℬℋ2, for all i∈I, so ⊔≈i∈IΛi,Ωi,E∈˜˜Tℬℋ1 and ⊔≈i∈IΛi,Ωi,E∈˜˜Tℬℋ2. Therefore, ⊔≈i∈IΛi,Ωi,E∈˜˜Tℬℋ1⊓≈Tℬℋ2.
ThusTℬℋ1⊓≈Tℬℋ2 defines a bipolar hypersoft topology on X and X,Tℬℋ1⊓≈Tℬℋ2,E,¬E is a bipolar hypersoft topological space over X.Remark 2.
The union of any two bipolar hypersoft topologies onX may not be a bipolar hypersoft topology on X.Example 2.
LetX=χ1,χ2,χ3,χ4, E1=ω1,ω2, E2=ω3, and E3=ω4. Let Tℬℋ1=Φ,Ψ,E, Ψ,Φ,E, Λ1,Ω1,E, Λ2,Ω2,E, Λ3,Ω3,E} and Tℬℋ2=Φ,Ψ,E, Ψ,Φ,E, Γ1,Θ1,E, Γ2,Θ2,E, Γ3,Θ3,E} be two bipolar hypersoft topologies defined on X where Λ1,Ω1,E, Λ2,Ω2,E, Λ3,Ω3,E, Γ1,Θ1,E, Γ2,Θ2,E, and Γ3,Θ3,E are bipolar hypersoft sets over X, defined as follows:(9)Λ1,Ω1,E=ω1,ω3,ω4,χ3,χ4,χ2,ω2,ω3,ω4,χ2,χ3,χ1,Λ2,Ω2,E=ω1,ω3,ω4,χ1,χ2,χ3,ϕ,ω2,ω3,ω4,χ1,χ4,χ2,χ3,Λ3,Ω3,E=ω1,ω3,ω4,χ3,χ2,ω2,ω3,ω4,ϕ,χ1,χ2,χ3,and(10)Γ1,Θ1,E=ω1,ω3,ω4,χ3,χ4,χ1,ω2,ω3,ω4,χ1,χ3,χ4,χ2,Γ2,Θ2,E=ω1,ω3,ω4,χ1,χ2,χ3,ω2,ω3,ω4,χ2,χ4,χ1,Γ3,Θ3,E=ω1,ω3,ω4,ϕ,χ1,χ3,ω2,ω3,ω4,χ4,χ1,χ2,Then,
Tℬℋ1⊔≈Tℬℋ2=Φ,Ψ,E, Ψ,Φ,E, Λ1,Ω1,E, Λ2,Ω2,E, Λ3,G3,E,Γ1,Θ1,E, Γ2,Θ2,E, Γ3,Θ3,E}.
If we take(11)Λ1,Ω1,E⊔≈Γ1,Θ1,E=Λ,Ω,E,then(12)Λ,Ω,E=ω1,ω3,ω4,χ3,χ4,ϕ,ω2,ω3,ω4,X,ϕ,but Λ,Ω,E∉˜˜Tℬℋ1⊔≈Tℬℋ2. Hence, Tℬℋ1⊔≈Tℬℋ2 is not a bipolar hypersoft topology on X.Definition 23.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X, and Λ,Ω,E is a bipolar hypersoft set over X and x∈X. Then, Λ,Ω,E is said to be a bipolar hypersoft neighborhood of x if there exists a bipolar hypersoft open set Γ,Θ,E such that x∈Γ,Θ,E⊑˜˜Λ,Ω,E.Proposition 5.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X; then:(1)
IfΛ,Ω,E is a bipolar hypersoft neighborhood of x∈X, then x∈Λ,Ω,E.(2)
Eachx∈X has a bipolar hypersoft neighborhood.(3)
IfΛ,Ω,E and Γ,Θ,E are bipolar hypersoft neighborhoods of some x∈X, then Λ,Ω,E⊓≈Γ,Θ,E is also a bipolar hypersoft neighborhood of x.(4)
IfΛ,Ω,E is a bipolar hypersoft neighborhood of x∈X and Λ,Ω,E⊑˜˜Γ,Θ,E, then Γ,Θ,E is also a bipolar hypersoft neighborhood of x∈X.Proof.
(1)
It follows from Definition23.(2)
For anyx∈X, x∈Ψ,Φ,E and since Ψ,Φ,E∈˜˜Tℬℋ, so x∈Ψ,Φ,E⊑˜˜Ψ,Φ,E. Thus, Ψ,Φ,E is a bipolar hypersoft neighborhood of x.(3)
LetΛ,Ω,E and Γ,Θ,E be the bipolar hypersoft neighborhoods of x∈X, then there exist Λ1,Ω1,E and Λ2,Ω2,E∈˜˜Tℬℋ such that x∈Λ1,Ω1,E⊑˜˜Λ,Ω,E and x∈Λ2,Ω2,E⊑˜˜Γ,Θ,E. Now x∈Λ1,Ω1,E and x∈Λ2,Ω2,E imply that x∈Λ1,Ω1,E⊓≈Λ2,Ω2,E and Λ1,Ω1,E⊓≈Λ2,Ω2,E∈˜˜Tℬℋ. So, we have x∈Λ1,Ω1,E⊓≈Λ2,Ω2,E⊑˜˜Λ,Ω,E⊓≈Γ,Θ,E. Thus, Λ,Ω,E⊓≈Γ,Θ,E is a bipolar hypersoft neighborhood of x.(4)
LetΛ,Ω,E be a bipolar hypersoft neighborhood of x∈X and Λ,Ω,E⊑˜˜Γ,Θ,E. By definition, there exists a bipolar hypersoft open set Λ1,Ω1,E such that x∈Λ1,Ω1,E⊑˜˜Λ,Ω,E⊑˜˜Γ,Θ,E. Thus, x∈Λ1,Ω1,E⊑˜˜Γ,Θ,E. Hence, Γ,Θ,E is a bipolar hypersoft neighborhood of x.Proposition 6.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ,Ω,E is any bipolar hypersoft open set over X; then Λ,Ω,E is a bipolar hypersoft neighborhood of each point of ∩α∈EΛα, that is, of each of its points.Proof.
LetΛ,Ω,E∈˜˜Tℬℋ. For x∈∩α∈EΛα, we obtain x∈Λα for each α∈E. Therefore, x∈Λ,Ω,E⊑˜˜Λ,Ω,E and hence Λ,Ω,E is a bipolar hypersoft neighborhood of x.Remark 3.
The converse of Proposition6 is incorrect as the next example shows.Example 3.
ConsiderTℬℋ1 given in Example 2 and let Λ,Ω,E be any bipolar hypersoft set defined as follows:(13)Λ,Ω,E=ω1,ω3,ω4,χ1,χ3,χ4,χ2,ω2,ω3,ω4,χ2,χ3,χ1,Then, Λ,Ω,E is a bipolar hypersoft neighborhood of each point of ∩α∈EΛα, that is, of each of its points, but it is not a bipolar hypersoft open set.Definition 24.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ,Ω,E is a bipolar hypersoft set over X. A point x∈X is called a bipolar hypersoft limit point of Λ,Ω,E if Λ,Ω,E⊓≈Γ,Θ,E\x≠Φ,Ψ,E for every bipolar hypersoft open set Γ,Θ,E containing x. The set of all bipolar hypersoft limit points of Λ,Ω,E is denoted by Λ,Ω,Ed.Proposition 7.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ1,Ω1,E, Λ2,Ω2,E are bipolar hypersoft sets over X. Then,(1)
Λ1,Ω1,E⊑≈Λ2,Ω2,E implies Λ1,Ω1,Ed⊑≈Λ2,Ω2,Ed.(2)
Λ1,Ω1,E⊓≈Λ2,Ω2,Ed⊑≈Λ1,Ω1,Ed⊓≈Λ2,Ω2,Ed.(3)
Λ1,Ω1,E⊔≈Λ2,Ω2,Ed=Λ1,Ω1,Ed⊔≈Λ2,Ω2,Ed.Proof.
(1)
Letx∈Λ1,Ω1,Ed so that x is a bipolar hypersoft limit point of Λ1,Ω1,E. Then, by definition Λ1,Ω1,E⊓≈Γ,Θ,E\x≠Φ,Ψ,E for every bipolar hypersoft open set Γ,Θ,E containing x. But Λ1,Ω1,E⊑˜˜Λ2,Ω2,E, and it follows that Λ2,Ω2,E⊓≈Γ,Θ,E\x≠Φ,Ψ,E. Thus, x∈Λ2,Ω2,Ed. Therefore, Λ1,Ω1,Ed⊑˜˜Λ2,Ω2,Ed.(2)
SinceΛ1,Ω1,E⊓≈Λ2,Ω2,E⊑˜˜Λ1,Ω1,E and Λ1,Ω1,E⊓≈Λ2,Ω2,E⊑˜˜Λ2,Ω2,E, it follows from (1.) that (Λ1,Ω1,E⊓≈Λ2,Ω2,Ed⊑˜˜Λ1,Ω1,Ed and (Λ1,Ω1,E⊓≈Λ2,Ω2,Ed⊑˜˜Λ2,Ω2,Ed. So, Λ1,Ω1,E⊓≈Λ2,Ω2,Ed⊑˜˜Λ1,Ω1,Ed⊓≈Λ2,Ω2,Ed.(3)
SinceΛ1,Ω1,E⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E and Λ2,Ω2,E⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E, by (1.), we have Λ1,Ω1,Ed⊑˜˜ (Λ1,Ω1,E⊔≈Λ2,Ω2,Ed and Λ2,Ω2,Ed⊑˜˜ (Λ1,Ω1,E⊔≈Λ2,Ω2,Ed. So, Λ1,Ω1,Ed⊔≈Λ2,Ω2,Ed⊑˜˜ (Λ1,Ω1,E⊔≈ (Λ2,Ω2,E))d. Now, let x∈ (Λ1,Ω1,E⊔≈Λ2,Ω2,Ed. Then, (Λ1,Ω1,E⊔≈Λ2,Ω2,E) ⊓≈Γ,Θ,E\x≠Φ,Ψ,E for every bipolar hypersoft open set Γ,Θ,E containing x. Therefore, Λ1,Ω1,E⊓≈Γ,Θ,E\x≠Φ,Ψ,E or Λ2,Ω2,E⊓≈Γ,Θ,E\x≠Φ,Ψ,E. Thus, x∈Λ1,Ω1,Ed or x∈Λ2,Ω2,Ed and then x∈Λ1,Ω1,Ed⊔≈Λ2,Ω2,Ed. Therefore, (Λ1,Ω1,E⊔≈Λ2,Ω2,Ed) ⊑˜˜Λ1,Ω1,Ed⊔≈Λ2,Ω2,Ed. Now, we have Λ1,Ω1,E⊔≈Λ2,Ω2,Ed=Λ1,Ω1,Ed⊔≈Λ2,Ω2,Ed.Remark 4.
The equality in Proposition7 (2.) does not hold in general as shown in the next example.Example 4.
Let us consider the bipolar hypersoft topological spaceX,Tℬℋ1,E,¬E in Example 2 and let Λ,Ω,E and Γ,Θ,E be bipolar hypersoft sets defined as follows:(14)Λ,Ω,E=ω1,ω3,ω4,χ1,χ2,χ4,ω2,ω3,ω4,χ1,χ3,χ2,Γ,Θ,E=ω1,ω3,ω4,χ2,χ1,χ3,ω2,ω3,ω4,χ2,χ1,χ3,χ4,Then, Λ,Ω,Ed=Γ,Θ,Ed=χ1,χ3. But Λ,Ω,E⊓≈Γ,Θ,E=Φ,Ψ,E and (Λ,Ω,E⊓≈Γ,Θ,Ed=Φ,Ψ,Ed=ϕ. Hence, Λ1,Ω1,E⊓≈Λ2,Ω2,Ed≠Λ1,Ω1,Ed⊓≈Λ2,Ω2,Ed.Definition 25.
LetX,Tℬℋ,E,¬E be a bipolar hypersoft space over X and let Υ be a nonempty subset of X. Then,(15)TℬℋΥ=ΛΥ,ΩΥ,E|Λ,Ω,E∈˜˜Tℬℋ,is said to be the relative bipolar hypersoft topology on Υ and Υ,TℬℋΥ,E,¬E is called a bipolar hypersoft subspace of X,Tℬℋ,E,¬E.It is easy to verify thatTℬℋΥ is a bipolar hypersoft topology on Υ.Example 5.
Any bipolar hypersoft subspace of a bipolar hypersoft indiscrete topological space is a bipolar hypersoft indiscrete topological space.Example 6.
Any bipolar hypersoft subspace of a bipolar hypersoft discrete topological space is a bipolar hypersoft discrete topological space.Proposition 8.
Assume thatΥ,TℬℋΥ,E,¬E and Z,TℬℋZ,E,¬E are bipolar hypersoft subspaces of X,Tℬℋ,E,¬E and Υ⊆Z. Then, Υ,TℬℋΥ,E,¬E is a bipolar hypersoft subspace of Z,TℬℋZ,E,¬E.Proof.
SinceΥ⊆Z, then Υ=Υ∩Z. From Definition 16, each bipolar hypersoft open set ΛΥ,ΩΥ,E is defined as ΛΥα=Υ∩Λα and ΩΥ¬α=Υ∩Ω¬α, for each α∈E where Λ,Ω,E is a bipolar hypersoft open set of X,Tℬℋ,E,¬E. Now, for each α∈E,(16)ΛΥα=Υ∩Z∩Λα,ΩΥ¬α=Υ∩Z∩Ω¬α,⇒ΛΥα=Υ∩Z∩Λα,ΩΥ¬α=Υ∩Z∩Ω¬α,⇒ΛΥα=Υ∩ΛZα,ΩΥ¬α=Υ∩ΩZ¬α,where ΛZ,ΩZ,E is a bipolar hypersoft open set of Z. So, Υ,TℬℋΥ,E,¬E is a bipolar hypersoft subspace of Z,TℬℋZ,E,¬E.


## 4. Bipolar Hypersoft Closure, Bipolar Hypersoft Interior, Bipolar Hypersoft Exterior, and Bipolar Hypersoft Boundary

In this section, we introduce the notions of bipolar hypersoft closure, bipolar hypersoft interior, bipolar hypersoft exterior, and bipolar hypersoft boundary supported with some properties and relations between them.Definition 26.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space, and Λ,Ω,E is a bipolar hypersoft set over X. The bipolar hypersoft closure of Λ,Ω,E is denoted by Λ,Ω,E¯ and is defined as the intersection of all bipolar hypersoft closed supersets of Λ,Ω,E.
In other words,Λ,Ω,E¯=⊓≈ \{Γ,Θ,E|Γ,Θ,Ec∈˜˜Tℬℋ, Γ,Θ,E⊑˜˜Λ,Ω,E\}.Proposition 9.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space, and Λ,Ω,E is a bipolar hypersoft set over X. Then:(1)
Λ,Ω,E¯ is the smallest bipolar hypersoft closed set containing Λ,Ω,E.(2)
Λ,Ω,E is a bipolar hypersoft closed set if and only if Λ,Ω,E=Λ,Ω,E¯.Proof.
(1)
It follows from Definition26.(2)
Suppose thatΛ,Ω,E is a bipolar hypersoft closed set. So, Λ,Ω,E itself is the smallest bipolar hypersoft closed set over X containing Λ,Ω,E and hence Λ,Ω,E=Λ,Ω,E¯. Conversely, let Λ,Ω,E=Λ,Ω,E¯. By (1.), Λ,Ω,E¯ is a bipolar hypersoft closed, and Λ,Ω,E is also a bipolar hypersoft closed set over X.Proposition 10.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ1,Ω1,E, Λ2,Ω2,E are bipolar hypersoft sets over X. Then:(1)
Φ,Ψ,E¯=Φ,Ψ,E and Ψ,Φ,E¯=Ψ,Φ,E.(2)
Λ1,Ω1,E⊑˜˜Λ1,Ω1,E¯.(3)
Λ1,Ω1,E⊑˜˜Λ2,Ω2,E implies Λ1,Ω1,E¯⊑˜˜Λ2,Ω2,E¯.(4)
Λ1,Ω1,E⊔≈Λ2,Ω2,E¯=Λ1,Ω1,E¯⊔≈Λ2,Ω2,E¯.(5)
Λ1,Ω1,E⊓≈Λ2,Ω2,E¯⊑˜˜Λ1,Ω1,E¯⊓≈Λ2,Ω2,E¯.(6)
Λ1,Ω1,E¯¯=Λ1,Ω1,E¯.Proof.
(1)
SinceΦ,Ψ,E and Ψ,Φ,E are bipolar hypersoft closed sets, then by Proposition 9 (2.), we have Φ,Ψ,E¯=Φ,Ψ,E and Ψ,Φ,E¯=Ψ,Φ,E.(2)
By Proposition9 (1.), Λ1,Ω1,E¯ is the smallest bipolar hypersoft closed set containing Λ1,Ω1,E and so Λ1,Ω1,E⊑˜˜Λ1,Ω1,E¯.(3)
By (2.),Λ2,Ω2,E⊑˜˜Λ2,Ω2,E¯. Since Λ1,Ω1,E⊑˜˜Λ2,Ω2,E, we have Λ1,Ω1,E⊑˜˜Λ2,Ω2,E¯. But Λ2,Ω2,E¯ is a bipolar hypersoft closed set. Thus, Λ2,Ω2,E¯ is a bipolar hypersoft closed set containing Λ1,Ω1,E. Since Λ1,Ω1,E¯ is the smallest bipolar hypersoft closed set over X containing Λ1,Ω1,E, so we have Λ1,Ω1,E¯⊑˜˜Λ2,Ω2,E¯.(4)
SinceΛ1,Ω1,E⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E and Λ2,Ω2,E⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E, by (3.), Λ1,Ω1,E¯⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E¯ and Λ2,Ω2,E¯⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E¯. Hence, Λ1,Ω1,E¯⊔≈Λ2,Ω2,E¯⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E¯. Now, since Λ1,Ω1,E¯ and Λ2,Ω2,E¯ are bipolar hypersoft closed sets, Λ1,Ω1,E¯⊔≈Λ2,Ω2,E¯ is also bipolar hypersoft closed. Also, Λ1,Ω1,E⊑˜˜Λ1,Ω1,E and Λ2,Ω2,E⊑˜˜Λ2,Ω2,E¯ imply that Λ1,Ω1,E⊔≈Λ2,Ω2,E⊑˜˜Λ1,Ω1,E¯⊔≈Λ2,Ω2,E¯. Thus, Λ1,Ω1,E¯⊔≈Λ2,Ω2,E¯ is a bipolar hypersoft closed containing Λ1,Ω1,E⊔≈Λ2,Ω2,E. SinceΛ1,Ω1,E⊔≈Λ2,Ω2,E¯ is the smallest bipolar hypersoft closed set containing Λ1,Ω1,E⊔≈Λ2,Ω2,E, we have Λ1,Ω1,E⊔≈Λ2,Ω2,E¯) ⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E¯. So, Λ1,Ω1,E⊔≈Λ2,Ω2,E¯=Λ1,Ω1,E⊔≈Λ2,Ω2,E¯.(5)
SinceΛ1,Ω1,E⊓≈Λ2,Ω2,E⊑˜˜Λ1,Ω1,E and Λ1,Ω1,E⊓≈Λ2,Ω2,E⊑˜˜Λ2,Ω2,E, then Λ1,Ω1,E⊓≈Λ2,Ω2,E¯) ⊑˜˜Λ1,Ω1,E andΛ1,Ω1,E⊓≈Λ2,Ω2,E¯⊑˜˜Λ2,Ω2,E¯. Therefore, Λ1,Ω1,E⊓≈Λ2,Ω2,E¯⊑˜˜Λ1,Ω1,E¯⊓≈Λ2,Ω2,E¯.(6)
SinceΛ1,Ω1,E¯ is a bipolar hypersoft closed set, therefore, by Proposition 9 (2.), we have Λ1,Ω1,E¯¯=Λ1,Ω1,E¯.Remark 5.
The equality does not hold in Proposition10 (5.) as the following example shows.Example 7.
Let us consider the bipolar hypersoft topological spaceX,Tℬℋ1,E,¬E in Example 2 and let Λ,Ω,E and Γ,Θ,E be bipolar hypersoft sets defined as follows:(17)Λ,Ω,E=ω1,ω3,ω4,ϕ,X,ω2,ω3,ω4,ϕ,χ1,χ2,χ3,Γ,Θ,E=ω1,ω3,ω4,χ2,χ1,χ3,ω2,ω3,ω4,χ2,χ4,Then, Λ,Ω,E¯=Λ1,Ω1,Ec and Γ,Θ,E¯=Λ3,Ω3,Ec and Λ,Ω,E¯⊓≈Γ,Θ,E¯=Λ1,Ω1,Ec. Now, Λ,Ω,E⊓≈Γ,Θ,E=Φ,Ψ,E and Λ,Ω,E⊓≈Γ,Θ,E¯) =Φ,Ψ,E¯=Φ,Ψ,E. Hence, Λ,Ω,E⊓≈Γ,Θ,E¯≠Λ,Ω,E¯⊓≈Γ,Θ,E¯.Definition 27.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X, and Λ,Ω,E is a bipolar hypersoft set over X and x∈X. Then, x is said to be a bipolar hypersoft interior point of Λ,Ω,E if there exists a bipolar hypersoft open set Γ,Θ,E such that x∈Γ,Θ,E⊑˜˜Λ,Ω,E.Definition 28.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X. The bipolar hypersoft interior of bipolar hypersoft set Λ,Ω,E is denoted by Λ,Ω,Eo and is defined as the union of all bipolar hypersoft open set contained in Λ,Ω,E.
In other words,Λ,Ω,Eo=⊔≈Γ,Θ,E|Γ,Θ,E∈˜˜Tℬℋ, Γ,Θ,E⊑≈Λ,Ω,E}.Proposition 11.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space and Λ,Ω,E is a bipolar hypersoft set over X. Then:(1)
Λ,Ω,Eo is the largest bipolar hypersoft open set contained in Λ,Ω,E.(2)
Λ,Ω,E is a bipolar hypersoft open set if and only if Λ,Ω,E=Λ,Ω,Eo.Proof.
(1)
It follows from Definition28.(2)
Suppose thatΛ,Ω,E is a bipolar hypersoft open set. Then, Λ,Ω,E is surely identical with the largest bipolar hypersoft open subset of Λ,Ω,E. But by (1.), Λ,Ω,Eo is the largest bipolar hypersoft open subset of Λ,Ω,E. Hence, Λ,Ω,E=Λ,Ω,Eo. Conversely, let Λ,Ω,E=Λ,Ω,Eo. By (1.), Λ,Ω,Eo is a bipolar hypersoft open set, and therefore, Λ,Ω,E is also bipolar hypersoft open set.Proposition 12.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ1,Ω1,E, Λ2,Ω2,E are bipolar hypersoft sets over X. Then:(1)
Φ,Ψ,Eo=Φ,Ψ,E and Ψ,Φ,Eo=Ψ,Φ,E.(2)
Λ1,Ω1,Eo⊑˜˜Λ1,Ω1,E.(3)
Λ1,Ω1,E⊑˜˜Λ2,Ω2,E implies Λ1,Ω1,Eo⊑˜˜Λ2,Ω2,Eo.(4)
Λ1,Ω1,Eo⊓≈Λ2,Ω2,Eo=Λ1,Ω1,E⊓≈Λ2,Ω2,Eo.(5)
Λ1,Ω1,Eo⊔≈Λ2,Ω2,Eo⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,Eo.(6)
Λ1,Ω1,Eoo=Λ1,Ω1,Eo.Proof.
(1)
SinceΦ,Ψ,E and Ψ,Φ,E are bipolar hypersoft open sets, then by Proposition 11 (2.), we have Φ,Ψ,Eo=Φ,Ψ,E and Ψ,Φ,Eo=Ψ,Φ,E.(2)
Letx∈Λ1,Ω1,Eo; then x is a bipolar hypersoft interior point of Λ1,Ω1,E. This implies that Λ1,Ω1,E is a bipolar hypersoft neighborhood of x. Then, x∈Λ1,Ω1,E. Hence, Λ1,Ω1,Eo⊑˜˜Λ1,Ω1,E.(3)
Letx∈Λ1,Ω1,Eo. Then, x is a bipolar hypersoft interior point of Λ1,Ω1,E and so Λ1,Ω1,E is a bipolar hypersoft neighborhood of x. Since Λ1,Ω1,E⊑˜˜Λ2,Ω2,E, Λ2,Ω2,E is also a bipolar hypersoft neighborhood of x. This implies that x∈Λ2,Ω2,Eo. Thus, Λ1,Ω1,Eo⊑˜˜Λ2,Ω2,Eo.(4)
SinceΛ1,Ω1,E⊓≈Λ2,Ω2,E⊑˜˜Λ1,Ω1,E and Λ1,Ω1,E⊓≈Λ2,Ω2,E⊑˜˜Λ2,Ω2,E, we have, by (3.), (Λ1,Ω1,E⊓≈Λ2,Ω2,Eo⊑˜˜Λ1,Ω1,Eo and (Λ1,Ω1,E⊓≈Λ2,Ω2,Eo⊑˜˜Λ2,Ω2,Eo, implying (Λ1,Ω1,E⊓≈Λ2,Ω2,Eo⊑˜˜Λ1,Ω1,Eo⊓≈Λ2,Ω2,Eo. Now, let x∈Λ1,Ω1,Eo⊓≈Λ2,Ω2,Eo. Then, x∈Λ1,Ω1,Eo and x∈Λ2,Ω2,Eo. Hence, x is a bipolar hypersoft interior point of each of the bipolar hypersoft sets Λ1,Ω1,E and Λ2,Ω2,E. It follows that Λ1,Ω1,E and Λ2,Ω2,E are bipolar hypersoft neighborhoods of x so that their intersection Λ1,Ω1,E⊓≈Λ2,Ω2,E is also a bipolar hypersoft neighborhood of x. Hence, x∈ (Λ1,Ω1,E⊓≈Λ2,Ω2,Eo. Thus, Λ1,Ω1,Eo⊓≈Λ2,Ω2,Eo⊑˜˜ (Λ1,Ω1,E⊓≈Λ2,Ω2,E)o. Therefore, Λ1,Ω1,Eo⊓≈Λ2,Ω2,Eo=Λ1,Ω1,E⊓≈Λ2,Ω2,Eo.(5)
By (3.),Λ1,Ω1,E⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E implies Λ1,Ω1,Eo⊑˜˜ (Λ1,Ω1,E⊔≈Λ2,Ω2,E)o and Λ2,Ω2,E⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,E implies Λ2,Ω2,Eo⊑˜˜ (Λ1,Ω1,E⊔≈Λ2,Ω2,E)o. Hence, Λ1,Ω1,Eo⊔≈Λ2,Ω2,Eo⊑˜˜Λ1,Ω1,E⊔≈Λ2,Ω2,Eo.(6)
By Proposition11 (1.), Λ1,Ω1,Eo is the bipolar hypersoft open set. Hence, by (2.) of the same proposition Λ1,Ω1,Eoo=Λ1,Ω1,Eo.Remark 6.
The equality does not hold in Proposition12 (5.) as shown in the following example.Example 8.
Let us consider the bipolar hypersoft topological spaceX,Tℬℋ1,E,¬E in Example 2 and let Λ,Ω,E and Γ,Θ,E be bipolar hypersoft sets defined as follows:(18)Λ,Ω,E=ω1,ω3,ω4,χ1,χ3,χ4,χ2,ω2,ω3,ω4,χ2,χ3,χ1,Γ,Θ,E=ω1,ω3,ω4,X,ϕ,ω2,ω3,ω4,χ1,χ4,χ2,Then, Λ,Ω,Eo=Λ1,Ω1,E and Γ,Θ,Eo=Λ2,Ω2,E and Λ,Ω,Eo⊔≈Γ,Θ,Eo=Λ3,Ω3,E. Now, Λ,Ω,E⊔≈Γ,Θ,E=Ψ,Φ,E and (Λ,Ω,E⊔≈Γ,Θ,E)o=Ψ,Φ,Eo=Ψ,Φ,E. Hence, Λ,Ω,E))⊔≈Γ,Θ,Eo≠Λ,Ω,Eo⊔≈Γ,Θ,Eo.Proposition 13.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ,Ω,E is a bipolar hypersoft set over X. Then, Λ,Ω,Eo⊑˜˜Λ,Ω,E⊑˜˜Λ,Ω,E¯.Proof.
It follows from Proposition10 (2.) and Proposition12 (2.).Proposition 14.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ1,Ω1,E, Λ2,Ω2,E are bipolar hypersoft sets over X. Then:(1)
Λ1,Ω1,E¯c=Λ1,Ω1,Eco.(2)
Λ1,Ω1,Eoc=Λ1,Ω1,Ec¯.(3)
Λ1,Ω1,E¯=Λ1,Ω1,Ecoc.(4)
Λ1,Ω1,Eo=Λ1,Ω1,Ec¯c.(5)
Λ1,Ω1,E\Λ2,Ω2,Eo⊑˜˜Λ1,Ω1,Eo\Λ2,Ω2,Eo.Proof.
From the definitions of bipolar hypersoft closure and bipolar hypersoft interior, we have:(1)
Λ1,Ω1,E¯=⊓≈ {Γ,Θ,E∣Γ,Θ,Ec∈˜˜Tℬℋ, Γ,Θ,E⊑˜˜Λ1,Ω1,E}. Λ1,Ω1,E¯c= [⊓≈ {Γ,Θ,E∣Γ,Θ,Ec∈˜˜Tℬℋ, Γ,Θ,E⊑˜˜Λ1,Ω1,Ec. Λ1,Ω1,E¯c=⊔≈ {Γ,Θ,Ec∣Γ,Θ,Ec∈˜˜Tℬℋ, Γ,Θ,Ec⊑≈Λ1,Ω1,Ec=Λ1,Ω1,Eco.(2)
Λ1,Ω1,Eo=⊔≈ {Γ,Θ,E∣Γ,Θ,E∈˜˜Tℬℋ, Γ,Θ,E⊑˜˜Λ1,Ω1,E}. Λ1,Ω1,Eoc= [⊔≈ {Γ,Θ,E∣Γ,Θ,E∈˜˜Tℬℋ, Γ,Θ,E⊑˜˜Λ1,Ω1,Ec. Λ1,Ω1,Eoc=⊓≈ {Γ,Θ,Ec∣Γ,Θ,E∈˜˜Tℬℋ, Λ1,Ω1,Ec⊑≈Γ,Θ,Ec=Λ1,Ω1,Ec¯.(3)
Obtained from (1.) by taking the bipolar hypersoft complement.(4)
Obtained from (2.) by taking the bipolar hypersoft complement.(5)
(Λ1,Ω1,E\Λ2,Ω2,E)o= (Λ1,Ω1,E⊓≈Λ2,Ω2,Ec)o=Λ1,Ω1,Eo⊓≈Λ2,Ω2,Eco=Λ1,Ω1,Eo⊓≈Λ2,Ω2,E¯c=Λ1,Ω1,Eo\Λ2,Ω2,E¯⊑˜˜Λ1,Ω1,Eo\Λ2,Ω2,Eo.Definition 29.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ,Ω,E is a bipolar hypersoft set over X. A point x∈X is said to be a bipolar hypersoft exterior point of Λ,Ω,E if and only if it is a bipolar hypersoft interior point of Λ,Ω,Ec, that is, if and only if there exists a bipolar hypersoft open set Γ,Θ,E such that x∈Γ,Θ,E⊑˜˜Λ,Ω,Ec. The set of all bipolar hypersoft exterior points of Λ,Ω,E is called the bipolar hypersoft exterior of Λ,Ω,E and is denoted by Λ,Ω,Ee.Thus,Λ,Ω,Ee=Λ,Ω,Eco. It follows that Λ,Ω,Ece=Λ,Ω,Ecco=Λ,Ω,Eo.We also haveΛ,Ω,E⊓≈Λ,Ω,Ee=Φ,Ω,E; that is, no point of Λ,Ω,E can be a bipolar hypersoft exterior point of Λ,Ω,E.Remark 7.
SinceΛ,Ω,Ee is the bipolar hypersoft interior of Λ,Ω,Ec, it follows that Λ,Ω,Ee is the bipolar hypersoft open and is the largest bipolar hypersoft open set contained in Λ,Ω,Ec.Proposition 15.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space and Λ,Ω,E is a bipolar hypersoft set over X. Then,(19)Λ,Ω,Ee=⊔˜˜Γ,Θ,E|Γ,Θ,E∈˜˜Tℬℋ,Γ,Θ,E⊑≈Λ,Ω,Ec.Proof.
From the definitions of bipolar hypersoft interior and bipolar hypersoft exterior, we have(20)Λ,Ω,Eco=⊔˜˜Γ,Θ,E|Γ,Θ,E∈˜˜Tℬℋ,Γ,Θ,E⊑˜˜Λ,Ω,Ec=Λ,Ω,Ee.Proposition 16.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ1,Ω1,E, Λ2,Ω2,E are bipolar hypersoft sets over X. Then:(1)
Ψ,Φ,Ee=Φ,Ψ,E and Φ,Ψ,Ee=Ψ,Φ,E.(2)
Λ1,Ω1,Ee⊑˜˜Λ1,Ω1,Ec.(3)
Λ1,Ω1,Ee=Λ1,Ω1,Eece.(4)
Λ1,Ω1,E⊑˜˜Λ2,Ω2,E implies Λ2,Ω2,Ee⊑˜˜Λ1,Ω1,Ee.(5)
Λ1,Ω1,Eo⊑˜˜Λ1,Ω1,Eee.(6)
Λ1,Ω1,E⊔≈Λ2,Ω2,Ee=Λ1,Ω1,Ee⊓≈Λ2,Ω2,Ee.(7)
Λ1,Ω1,E⊓≈Λ2,Ω2,Ee⊑˜˜Λ1,Ω1,Ee⊔≈Λ2,Ω2,Ee.Proof.
(1)
Ψ,Φ,Ee=Ψ,Φ,Eco=Φ,Ψ,Eo=Φ,Ψ,E.(21)Φ,Ψ,Ee=Φ,Ψ,Eco=Ψ,Φ,Eo=Ψ,Φ,E.(2)
By definition,Λ1,Ω1,Ee=Λ1,Ω1,Eco and by Proposition 12 (2.), we have Λ1,Ω1,Eco⊑˜˜Λ1,Ω1,Ec. Hence, Λ1,Ω1,Ee⊑˜˜Λ1,Ω1,Ec.(3)
Λ1,Ω1,Eece=Λ1,Ω1,Ecoce=Λ1,Ω1,Ecocco=Λ1,Ω1,Ecoo=Λ1,Ω1,Eco=Λ1,Ω1,Ee.(4)
Λ1,Ω1,E⊑˜˜Λ2,Ω2,E then Λ2,Ω2,Ec⊑˜˜Λ1,Ω1,Ec, implying Λ2,Ω2,Eco⊑˜˜Λ1,Ω1,Eco. So, Λ2,Ω2,Ee⊑˜˜Λ1,Ω1,Ee.(5)
By (2.), we haveΛ1,Ω1,Ee⊑˜˜Λ1,Ω1,Ec. Then, (4.) gives Λ1,Ω1,Ece⊑˜˜Λ1,Ω1,Eee. But, Λ1,Ω1,Eo=Λ1,Ω1,Ece. Hence, Λ1,Ω1,Eo⊑˜˜Λ1,Ω1,Eee.(6)
(Λ1,Ω1,E⊔≈Λ2,Ω2,E)e= (Λ1,Ω1,E⊔≈Λ2,Ω2,Ec)o= (Λ1,Ω1,Ec⊓≈Λ2,Ω2,Ec)o=Λ1,Ω1,Eco⊓≈Λ2,Ω2,Eco=Λ1,Ω1,Ee⊓≈Λ2,Ω2,Ee.(7)
(Λ1,Ω1,E⊓≈Λ2,Ω2,E)e= (Λ1,Ω1,E⊓≈Λ2,Ω2,Ec)o= (Λ1,Ω1,Ec⊔≈Λ2,Ω2,Ec)o⊑˜˜Λ1,Ω1,Eco⊓≈Λ2,Ω2,Eco=Λ1,Ω1,Ee⊓≈Λ2,Ω2,Ee.Definition 30.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X; then, bipolar hypersoft boundary of bipolar hypersoft set Λ,Ω,E over X is denoted by Λ,Ω,Eb and is defined as Λ,Ω,Eb=Λ,Ω,E¯⊓≈Λ,Ω,Ec¯.Remark 8.
From Definition30 it follows that the bipolar hypersoft sets Λ,Ω,E and Λ,Ω,Ec have the same bipolar hypersoft boundary.Proposition 17.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space and Λ,Ω,E is a bipolar hypersoft set over X. Then:(1)
Λ,Ω,Eb⊑˜˜Λ,Ω,E¯.(2)
Λ,Ω,Eb=Λ,Ω,E¯\Λ,Ω,Eo.(3)
Λ,Ω,Ebc=Λ,Ω,Eo⊔≈Λ,Ω,Ee.(4)
Λ,Ω,Eo⊑˜˜Λ,Ω,E\Λ,Ω,Eb.(5)
Λ,Ω,Eob⊑˜˜Λ,Ω,Eb.(6)
Λ,Ω,E¯b⊑˜˜Λ,Ω,Eb.Proof.
(1)
By definition,Λ,Ω,Eb=Λ,Ω,E¯⊓≈Λ,Ω,Ec¯. Hence, Λ,Ω,Eb⊑˜˜Λ,Ω,E¯.(2)
Λ,Ω,Eb=Λ,Ω,E¯⊓≈Λ,Ω,Ec¯=Λ,Ω,E¯⊓≈Λ,Ω,Eoc=Λ,Ω,E¯\Λ,Ω,Eo.(3)
Λ,Ω,Ebc= [Λ,Ω,E¯⊓≈Λ,Ω,Ec¯c=Λ,Ω,E¯c⊔≈Λ,Ω,Ec¯c=Λ,Ω,Eco⊔≈Λ,Ω,Eo=Λ,Ω,Ee⊔≈Λ,Ω,Eo.(4)
Λ,Ω,E\Λ,Ω,Eb=Λ,Ω,E⊓≈Λ,Ω,Ebc=Λ,Ω,E⊓≈ (Λ,Ω,Eo⊔≈Λ,Ω,Ee) = (Λ,Ω,E⊓≈Λ,Ω,Eo) ⊔≈ (Λ,Ω,E⊓≈Λ,Ω,Ee)=Λ,Ω,Eo⊔˜˜Φ,Ω,E⊓˜˜Λ,Ω,Eo.(5)
Λ,Ω,Eob=Λ,Ω,Eo¯⊓≈Λ,Ω,Eoc¯=Λ,Ω,Eo¯⊓≈Λ,Ω,Ec¯¯⊑≈Λ,Ω,E¯⊔≈Λ,Ω,Ec¯=Λ,Ω,Eb.(6)
Λ,Ω,E¯b=Λ,Ω,E¯¯⊓≈Λ,Ω,E¯)c¯⊑≈Λ,Ω,E¯⊔≈Λ,Ω,Ec¯=Λ,Ω,Eb.Remark 9.
The next example shows that the equality of Proposition17 (4.) is incorrect, whereas it is equal in hypersoft topology.Example 9.
LetX=χ1,χ2,χ3, E1=ω1,ω2, E2=ω3, and E3=ω4. Let Tℬℋ=Φ,Ψ,E, Ψ,Φ,E, Λ1,Ω1,E, Λ2,Ω2,E, Λ3,Ω3,E be a bipolar hypersoft topology defined on X where Λ1,Ω1,E, Λ2,Ω2,E, and Λ3,Ω3,E are bipolar hypersoft sets over X, defined as follows:(22)Λ1,Ω1,E=ω1,ω3,ω4,χ3,χ1,ω2,ω3,ω4,χ3,χ1,χ2,Λ2,Ω2,E=ω1,ω3,ω4,ϕ,χ2,χ3,ω2,ω3,ω4,χ1,χ3,Λ3,Ω3,E=ω1,ω3,ω4,χ3,ϕ,ω2,ω3,ω4χ1,χ3,ϕ.
LetΛ,Ω,E be any bipolar hypersoft set defined as(23)Λ,Ω,E=ω1,ω3,ω4,χ1,χ3,ω2,ω3,ω4,χ1,χ3,Then,(24)Λ,Ω,Eo=ω1,ω3,ω4,ϕ,χ2,χ3,ω2,ω3,ω4,χ1,χ3,Λ,Ω,Eb=ω1,ω3,ω4,ϕ,χ3,ω2,ω3,ω4,ϕ,χ1,χ3,Hence,(25)Λ,Ω,Eo≠Λ,Ω,E\Λ,Ω,Eb.Proposition 18.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space over X and Λ1,Ω1,E, Λ2,Ω2,E are bipolar hypersoft sets over X. Then:(1)
Λ1,Ω1,E⊔≈Λ2,Ω2,Eb⊑˜˜Λ1,Ω1,Eb⊔≈Λ2,Ω2,Eb.(2)
Λ1,Ω1,E))⊓≈Λ2,Ω2,Eb⊑˜˜Λ1,Ω1,Eb⊔≈Λ2,Ω2,Eb.Proof.
(1)
(Λ1,Ω1,E⊔≈Λ2,Ω2,E)b=Λ1,Ω1,E⊔≈Λ2,Ω2,E¯⊓≈Λ1,Ω1,E⊔≈Λ2,Ω2,Ec¯= [Λ1,Ω1,E¯⊔≈Λ2,Ω2,E¯] ⊓≈Λ1,Ω1,Ec⊓≈Λ2,Ω2,Ec¯⊑˜˜ [Λ1,Ω1,E¯⊔≈Λ2,Ω2,E¯] ⊓≈Λ1,Ω1,Ec¯⊓≈Λ2,Ω2,Ec¯= [Λ1,Ω1,E¯⊓≈ (Λ1,Ω1,Ec¯⊓≈Λ2,Ω2,Ec¯)] ⊔≈ [Λ2,Ω2,E¯⊓≈ (Λ1,Ω1,Ec¯⊓≈Λ2,Ω2,Ec¯)] = [(Λ1,Ω1,E¯⊓≈Λ1,Ω1,Ec¯) ⊓≈Λ2,Ω2,Ec¯] ⊔≈ [(Λ2,Ω2,E¯⊓≈Λ2,Ω2,Ec¯) ⊓≈Λ1,Ω1,Ec¯] ⊑˜˜ [Λ1,Ω1,Eb¯⊓≈Λ2,Ω2,Ec¯] ⊔≈Λ2,Ω2,Eb⊓≈Λ1,Ω1,Ec¯⊑˜˜Λ1,Ω1,Eb⊔≈Λ2,Ω2,Eb.(2)
Λ1,Ω1,E⊓≈Λ2,Ω2,Eb=Λ1,Ω1,E⊓≈Λ2,Ω2,E¯⊓≈Λ1,Ω1,E⊓≈Λ2,Ω2,Ec¯⊑˜˜ [Λ1,Ω1,E¯⊓≈Λ2,Ω2,E¯] ⊓≈Λ1,Ω1,E⊓≈Λ2,Ω2,Ec¯= [Λ1,Ω1,E¯⊓≈Λ2,Ω2,E¯] ⊓≈Λ1,Ω1,Ec¯⊔≈Λ2,Ω2,Ec¯= [Λ1,Ω1,E¯⊓≈Λ2,Ω2,E¯⊓≈Λ1,Ω1,Ec¯] ⊔≈ [Λ1,Ω1,E¯⊓≈Λ2,Ω2,E¯⊓≈Λ2,Ω2,Ec¯] =Λ1,Ω1,E¯⊓≈Λ1,Ω1,Ec¯)⊓≈Λ2,Ω2,E¯⊔≈ [Λ1,Ω1,E¯⊓≈Λ2,Ω2,E¯⊓≈Λ2,Ω2,Ec)¯] = [Λ1,Ω1,Eb⊓≈Λ2,Ω2,E¯⊔≈Λ1,Ω1,E¯⊓≈Λ2,Ω2,Eb⊑˜˜Λ1,Ω1,Eb⊔≈Λ2,Ω2,Eb.Proposition 19.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space and Λ,Ω,E is a bipolar hypersoft set over X. Then:(1)
Λ,Ω,Eo⊔≈Λ,Ω,Eb⊑˜˜Λ,Ω,E¯.(2)
Λ,Ω,Eo⊔≈Λ,Ω,Ee⊔≈Λ,Ω,Eb⊑˜˜Ψ,Φ,E.Proof.
(1)
Λ,Ω,Eo⊔≈Λ,Ω,Eb=Λ,Ω,Eo⊔≈ [Λ,Ω,E¯⊓≈Λ,Ω,Ec¯] = [Λ,Ω,Eo⊔≈Λ,Ω,E¯] ⊓≈ [Λ,Ω,Eo⊔≈Λ,Ω,Ec¯] =Λ,Ω,E¯⊓≈ [Λ,Ω,Eo⊔≈Λ,Ω,Eoc⊑˜˜Λ,Ω,E¯⊓≈Ψ,Φ,E=Λ,Ω,E¯.(2)
By Proposition17 (3.), Λ,Ω,Eo⊔≈Λ,Ω,Eo=Λ,Ω,Ebc, then Λ,Ω,Eo⊔≈Λ,Ω,Ee⊔≈Λ,Ω,Eb=Λ,Ω,Ebc⊔≈Λ,Ω,Eb⊑˜˜Ψ,Φ,E.Remark 10.
The quality of Proposition19 (1.) and (2.) does not hold in general, whereas it is equal in hypersoft topology.Example 10.
ConsiderTℬℋ and Λ,Ω,E given in Example 9. We found that(26)Λ,Ω,Eo=ω1,ω3,ω4,ϕ,χ2,χ3,ω2,ω3,ω4,χ1,χ3,Λ,Ω,Eb=ω1,ω3,ω4,ϕ,χ3,ω2,ω3,ω4,ϕ,χ1,χ3.
Now,(27)Λ,Ω,E¯=Λ1,Ω1,Ec=ω1,ω3,ω4,χ1,χ3,ω2,ω3,ω4,χ1,χ2,χ3,Λ,Ω,Ee=ω1,ω3,ω4,χ3,χ1,ω2,ω3,ω4,χ3,χ1,χ2,Then,(28)Λ,Ω,Eo⊔≈Λ,Ω,Eb=ω1,ω3,ω4,ϕ,χ3,ω2,ω3,ω4,χ1,χ3≠Λ,Ω,E¯,Also,(29)Λ,Ω,Eo⊔˜˜Λ,Ω,Ee⊔˜˜Λ,Ω,Eb=ω1,ω3,ω4,χ3,ϕω2,ω3,ω4,χ1,χ3,ϕ≠Ψ,Φ,E.Proposition 20.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space and Λ,Ω,E is a bipolar hypersoft set over X. If Λ,Ω,E is a bipolar hypersoft open set, then Λ,Ω,E and Λ,Ω,Eb are disjoint bipolar hypersoft sets, that is, Λ,Ω,E⊓≈Λ,Ω,Eb=Φ,Ω,E.proof.
LetΛ,Ω,E be a bipolar hypersoft open set. By Proposition 17 (3.), Λ,Ω,Eo⊑˜˜Λ,Ω,Ebc. But Λ,Ω,Eo=Λ,Ω,E since Λ,Ω,E is a bipolar hypersoft open set. Hence, Λ,Ω,E⊑˜˜Λ,Ω,Ebc. This implies that Λ,Ω,E and Λ,Ω,Eb are disjoint bipolar hypersoft sets, that is, Λ,Ω,E⊓≈Λ,Ω,Eb=Φ,Ω,E.Remark 11.
The next example illustrates that the opposite of Proposition20 does not hold in general, whereas it is true in hypersoft topology.Example 11.
ConsiderTℬℋ, Λ,Ω,E, and Λ,Ω,Eb given in Example 9. It is easy to see that Λ,Ω,E⊓≈Λ,Ω,Eb=Φ,Ω,E. Hence, Λ,Ω,E and Λ,Ω,Eb are disjoint bipolar hypersoft sets, but Λ,Ω,E is not a bipolar hypersoft open set.Proposition 21.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space and Λ,Ω,E is a bipolar hypersoft set over X. If Λ,Ω,E is a bipolar hypersoft closed set, then Λ,Ω,Eb⊑˜˜Λ,Ω,E.proof.
Suppose thatΛ,Ω,E is a bipolar hypersoft closed set. By Proposition 5 (1.), Λ,Ω,Eb⊑˜˜Λ,Ω,E¯. Since Λ,Ω,E is a bipolar hypersoft closed set, then Λ,Ω,E¯=Λ,Ω,E. This implies that Λ,Ω,Eb⊑˜˜Λ,Ω,E.Remark 12.
The converse of Proposition21 is false in general, while it is true in hypersoft topology.Example 12.
ConsiderTℬℋ, Λ,Ω,E, and Λ,Ω,Eb given in Example 9. It is easy to see that Λ,Ω,Eb⊑˜˜Λ,Ω,E but Λ,Ω,E is not a bipolar hypersoft closed set.Proposition 22.
Suppose thatΛ,Ω,E is a bipolar hypersoft set of a bipolar hypersoft space over X. If Λ,Ω,E is a bipolar hypersoft open set and a bipolar hypersoft closed set, then Λ,Ω,Eb=Φ,Ω,Eproof.
Suppose thatΛ,Ω,E is a bipolar hypersoft open set and a bipolar hypersoft closed set. Then Λ,Ω,Eb=Λ,Ω,E¯⊓≈Λ,Ω,Ec¯=Λ,Ω,E¯⊓≈Λ,Ω,Eoc=Λ,Ω,E⊓≈Λ,Ω,Ec=Φ,Ω,E.Remark 13.
The next example shows that the opposite of Proposition22 is incorrect in general, while it is hold in hypersoft topology.Example 13.
ConsiderTℬℋ, Λ,Ω,E, and Λ,Ω,Eb given in Example 9. We saw that Λ,Ω,Eb=Φ,Ω,E, but Λ,Ω,E is neither a bipolar hypersoft open set nor a bipolar hypersoft closed set.Proposition 23.
Suppose thatX,Tℬℋ,E,¬E is a bipolar hypersoft space and Λ,Ω,E is a bipolar hypersoft set over X. Then:(1)
Λ,Ω,Eo⊓≈Λ,Ω,Eb=Φ,Ω,E.(2)
Λ,Ω,Ee⊓≈Λ,Ω,Eb=Φ,Ω,E.proof.
(1)
Λ,Ω,Eo⊓≈Λ,Ω,Eb=Λ,Ω,Eo⊓≈Λ,Ω,E¯⊓≈Λ,Ω,Ec¯=Λ,Ω,Eo⊓≈Λ,Ω,E¯] ⊓≈Λ,Ω,Ec¯=Λ,Ω,Eo⊓≈Λ,Ω,Ec¯=Λ,Ω,Eo⊓≈Λ,Ω,Eoc=Φ,Ω,E.(2)
Λ,Ω,Ee⊓≈Λ,Ω,Eb=Λ,Ω,Eco⊓≈ [Λ,Ω,E¯⊓≈Λ,Ω,Ec¯] =Λ,Ω,E¯c⊓≈ [Λ,Ω,E¯⊓≈Λ,Ω,E¯c] = [Λ,Ω,E¯c⊓≈Λ,Ω,E¯] ⊓≈Λ,Ω,Ec¯=Φ,Ω,E⊓≈Λ,Ω,E¯c=Φ,Ω,E.


## 5. Conclusions

We have introduced bipolar hypersoft topological spaces over the collection of bipolar hypersoft sets. The notions of bipolar hypersoft neighborhood, bipolar hypersoft subspace, and bipolar hypersoft limit points are introduced, and their elementary characteristics are investigated. In addition, we have defined bipolar hypersoft interior, bipolar hypersoft closure, bipolar hypersoft exterior, and bipolar hypersoft boundary, and the relations between them are studied. In the end, it is necessary to establish more topological structures such as compactness, connectedness, and separation axioms for the practical applications.



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