# Characterization of Bipolar Vague SoftS-Open Sets

Authors: Farkhanda Afzal; Arif Mehmood; Samer Al Ghour; Mudasar Zafar; Hamzah Sakidin; Saeed Gul
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

This paper concerns the study of the concept of bipolar vague softs-open set, bipolar vague soft s-interior, bipolar vague soft s-closer, and bipolar vague soft s-exterior in bipolar vague soft topological spaces. By using such concepts, some results are addressed in bipolar vague soft topological spaces. The engagements among these results are also addressed by using bipolar vague soft s-open sets.


---

## Body

## 1. Introduction

Zadeh [1] familiarized the notion of fuzzy set theory. Pawlak [2] initiated rough set theory. Molodtsov [3] inaugurated the soft set theory. Maji et al. [4] made soft set theory more powerful by using it in different practical problems. Maji et al. [5] filled up the gaps that exist in [2]. Since the concept of soft set theory was too young, so work was continued on the journey. Pei and Miao [6] and Chen [7] improved the work of Maji et al. Broumi et al. [8] became the founder of the idea interval valued neutrosophic soft relation which is improvement over relations including soft, fuzzy soft, and intuitionistic fuzzy soft relations. Çağman et al. [9] inaugurated the most valuable structure of mathematics known as soft topology. Shabir and Naz [10] also worked on the same structure. Bayramov and Gunduz [11] have driven soft topology with new points known as soft points. Atanassov [12] founded the notion of intuitionistic fuzzy set theory. Some untouched results were left. Bayramov and Gunduz [13] touched the fundamentals and inaugurated the idea of intuitionistic fuzzy topology. Hayat et al. [14] discussed complex structure known as type 2 soft sets. Hayat et al. [15] traditionalized correspondence between a vertex and its neighbors. Hayat et al. [16] ushered in TOPSIS and the Shannon entropy on the idea of soft set. Shabir and Naz [17] made the concept of bipolar soft sets and its fundamentals. Karaaslan and Karatas [18] developed a new access to bipolar soft set. Ozturk [19] organized bipolar soft topology. Al-shami et al. [20] addressed several journaled type of soft semicompact spaces. Al-shami et al. [20] addressed soft compact and Lindelof spaces via soft preopen set.In our study, vague soft set, bipolar vague soft set, bipolar vague soft complements, bipolar vague null set, soft set, absolute bipolar vague soft absolute set, bipolar vague soft subset, bipolar vague soft equal sets, bipolar vague soft union, and bipolar vague soft intersection, some fundamental results are based on these operations. Bipolar vague soft topologyBVST is defined, and related structures are discussed with respect to s-open. This s-open is chosen among eight new definitions which are introduced in bipolar vague soft topology. Examples are given for supporting some results. The concept of interior and closure is inaugurated. On the basis of these concepts, related results are addressed. The results that engage the interior with closure are addressed.


## 2. New Concepts

This section is devoted to the basic notions which are necessary for the upcoming section of this particular piece of work.Definition 1.
LetM be master set, and L be a set of parameters. Let ΡM signify power of vague set VS of M; then a vague soft set f,L over M is a set given by f:L⟶ΡM. In other words, f,L=ν,x,γf~νx,δf~νx:x∈~M:ν∈~L where δfνx∈~0,1 and γfνx∈~0,1 with 0≤~γfνx+δfνx≤~2.Definition 2.
LetM be master set, and L be a set of parameters. A bipolar vague soft set BVSS(1)f~,L=ν,x,γfν⨁x,δfν⨁x,γfν⊖x,δfν⊖x,:x∈~M:ν∈~L,where γf⨁,δf⨁⟶0,1,γf⊖,δf⊖⟶−1,0.Definition 3.
Letf~,L be a BVS set over M, and then, complement of a BVS set f~,L is signified by f~,LC and given as
(2)f~,LC=v,x,δfν⨁x,γfν⨁x,δfν⊖x,γfν⊖x:x∈~M:ν∈~L.Definition 4.
An emptyBVS set f~null,L over M is defined by
(3)f~null,L=e,x,0,0,−1,0:x∈~M:ν∈~L.An absoluteBVS set Maboslute,L over M is defined by
(4)M~aboslute,L=ν,x,1,1,0,−1:x∈~M:ν∈~L.Definition 5.
Letf1~,L and f2~,L be two BVS sets over M. f1~,L is said to be BVS subset of f2~,L if
(5)γ⨁f1~νx≼~γ⨁f2~νx,δ⨁f1~νx≽~δ⨁f2~νx,γ⊖f1~νx≼~γ⊖f2~νx,δ+f1~νx≽~δ+f2~νx∀ν,x∈~L×M.It is denoted byf1~,L⋐~f2~,L.f1~,L is said to be BVS equal to f2~,L if f1~,L is BVS subset of f2~,L and f2~,Lis BVSsubset of f1~,L and is signified byf1~,L=f2~,L.Example 1.
LetM=x1,x2 and L=ν1,ν2, if
(6)f1~,L=ν1,x1,06×10−1,05×10−1,−08×10−1,−04×10−1,x2,05×10−1,04×10−1,−06×10−1,−03×10−1,ν2,05×10−1,07×10−1,−06×10−1,−05×10−1,x2,03×10−1,05×10−1,−04×10−1,−02×10−1,f2~,L=ν1,x1,07×10−1,08×10−1,−05×10−1,−06×10−1,x2,06×10−1,06×10−1,−05×10−1,−07×10−1,ν2,x1,06×10−1,09×10−1,−04×10−1,−07×10−1,x2,04×10−1,07×10−1,−03×10−1,−06×10−1.
Then,f1~,L⋐~f2~,L.Definition 6.
Let(7)f1~,L=ν,x,γ⨁f1~νx,δ⨁f1~νx,γ⊖Bi~ex,δ⊖Bi~ex:x∈~M:ν∈~L,
fori=1,2 be two BVS sets over M. Then, their union is signified by f1~,L⋓~f2~,L and it is given as
(8)∐i=12fi~,L=ν,x,maxγ⨁f1~νx,minδ⨁f1~νx,maxγ⊖fi~νx,minδ⊖fi~νx:x∈~M:ν∈~L.Definition 7.
Let(9)fi~,L=ν,x,γ⨁f1~νx,δ⨁f1~νx,γ⊖fi~νx,δ⊖fi~νx:x∈~M:ν∈~L,
fori=1,2be two BVSsets over M.
Then, their intersection is signified byf1~,L⋒~f2~,Land it is given as
(10)∏i=12fi~,L=ν,x,minγ⨁f1~νx,maxδ⨁f1~νx,minγ⊖fi~νx,maxδ⊖fi~νx:x∈~M:e∈~L.Definition 8.
Let(11)fi~,L=ν,x,γ⨁f1~νx,δ⨁f1~νx,γ⊖fi~νx,δ⊖fi~νx:x∈~M:ν∈~L,
fori∈~Ibe a family of BVSsets overM. Then,
(12)∐i∈Ifi~,L=ν,x,supγ⨁f1~νx,infδ⨁f1~νx,supγ⊖fi~νx,infδ⊖fi~νx:x∈~M:ν∈~L,∏i∈Ifi~,L=ν,x,infγ⨁f1~νx,supδ⨁f1~νx,infγ⊖fi~νx,supδ⊖fi~νx:x∈~M:ν∈~L.Proposition 9.
Letf~null,L and M~absolute,L be the empty BVS set and absolute BVSset over M, respectfully. Then,
(1)
f~null,L⋐M~absolute,L(2)
f~null,L⋓~M~absolute,L=M~absolute,L(3)
f~null,L⋒~M~absolute,L=f~null,LProof.
Straightforward.Definition 10.
Letf1~,L and f2~,L be two BVSsets over M. Then, f1~,L difference f2~,L operation on them is given by f1~,L∖f2~,L=f3~,L and is signified by f3~,L=f1~,L⋒~f2~,Lc as follows:
(13)f3~,L=ν,x,γ⨁f3~νx,δ⨁f3~νx,γ⊖f3~νx,δ⊖f3~νx:x∈~M:ν∈~L,where
(14)δ⨁f3~νx=minγ⨁f1~νx,δ⨁f2~νx,γ⊖f3~νx=minγ⊖f1~νx,δ⊖f2~νx,δ⨁f3~νx=maxδ⨁f1~νx,γ⨁f2~νx,δ⊖f3~νx=maxδ⊖f1~νx,γ⊖f2~νx.Definition 11.
Letf1~,L and f2~,L be two BVS sets over M. Then, “AND” operation on them is given by f1~,L∧~f2~,L=f3~,L×L and is signified by
(15)B3~,L×L=ν1,ν2,x,γ⨁f3~ν1,ν2x,δ⨁f3~ν1,ν2xγ⊖f3~ν1,ν2x,δ⊖f3~ν1,ν2x:x∈~M:ν1,ν2∈~L×L,where
(16)γ⨁f3~ν1,ν2x=minγ⨁f1~ν1x,γ⨁f2~ν2x,γ⊖f3~ν1,ν2x=minγ⊖f1~ν1x,δ⊖f2~ν2x,γ⨁f3~ν1,ν2x=maxδ⨁f1~νx,δ⨁f2~νx,F⊖f3~ν1,ν2x=maxδ⊖f1~ν1x,δ⊖f2~ν2x.Definition 12.
Letf1~,L and f2~,L be two BVS sets over M. Then, “OR” operation on them is signified by f1~,L∨~f2~,L=f3~,L×L and is given by
(17)f3~,L×L=ν1,ν2,x,γ⨁f3~ν1,ν2x,δ⨁f3~ν1,ν2x,γ⊖f3~ν1,ν2x,δ⊖f3~ν1,ν2x:x∈~M:ν1,ν2∈~L×L,where
(18)γ⨁f3~ν1,ν2x=maxγ⨁f1~ν1x,γ⨁f2~ν2x,γ⊖f3~ν1,ν2x=maxγ⊖f1~ν1x,γ⊖f2~ν2x,δ⨁f3~ν1,ν2x=minδ⨁f1~ν1x,δ⨁f2~ν2x,γ⊖f3~ν1,ν2x=minδ⊖f1~ν1x,δ⊖f2~ν2x.Example 2.
LetM=x1,x2 and L=ν1,ν2, if
(19)f1~,L=ν1,x1,03×10−1,05×10−1,−05×10−1,−07×10−1,x2,03×10−1,05×10−1,−05×10−1,−08×10−1,ν2,x1,04×10−1,04×10−1,−04×10−1,−03×10−1,x2,05×10−1,08×10−1,−09×10−1,−07×10−1,f2~,L=ν1,x1,4×10−1,06×10−1,−03×10−1,−09×10−1,x2,04×10−1,06×10−1,−02×10−1,−03×10−1,ν2,x1,03×10−1,03×10−1,−06×10−1,−08×10−1,x2,04×10−1,05×10−1,−01×10−1,−03×10−1.
Then,(20)f1~,L⋓~f2~,L=ν1,x1,04×10−1,06×10−1,−03×10−1,−09×10−1,x2,4×10−1,06×10−1,−02×10−1,−03×10−1,ν2,x1,04×10−1,04×10−1,−04×10−1,−08×10−1,x205×10−1,08×10−1,−01×10−1,−07×10−1,f1~,L⋒~f2~,L=ν1,x1,03×10−1,05×10−1,−05×10−1,−07×10−1,x2,03×10−1,05×10−1,−05×10−1,−03×10−1,ν2,x1,03×10−1,03×10−1,−06×10−1,−03×10−1,x2,05×10−1,05×10−1,−09×10−1,−03×10−1,f1~,L∖f2~,L=ν1,x1,03×10−1,04×10−1,−07×10−1,−05×10−1,x2,02×10−1,04×10−1,−08×10−1,−03×10−1,ν2,x1,04×10−1,04×10−1,−04×10−1,−03×10−1,x2,03×10−1,05×10−1,−09×10−1,−06×10−1,f1~,L∧~f2~,L=ν1,ν2,x1,03×10−1,05×10−1,−05×10−1,−07×10−1,x2,03×10−1,05×10−1,−05×10−1,−03×10−1,ν1,ν2,x1,03×10−1,03×10−1,−06×10−1,−07×10−1,x203×10−1,05×10−1,−05×10−1,−03×10−1,ν1,ν2,x104×10−1,04×10−1,−04×10−1,−03×10−1,x2,04×10−1,06×10−1,−02×10−1,−03×10−1,ν1,ν2,x1,03×10−1,03×10−1,−06×10−1,−03×10−1,x2,04×10−1,05×10−1,−09×10−1,−03×10−1,f1~,L∨~f2~,L=ν1,ν2,x1,04×10−1,06×10−1,−03×10−1,−09×10−1,x2,04×10−1,06×10−1,−02×10−1,−08×10−1,ν1,ν2,x1,03×10−1,05×10−1,−05×10−1,−08×10−1,x2,04×10−1,05×10−1,−01×10−1,−08×10−1,ν1,ν2,x1,04×10−1,06×10−1,−03×10−1,−09×10−1,x2,05×10−1,08×10−1,−02×10−1,−07×10−1,ν1,ν2,x1,04×10−1,04×10−1,−04×10−1,−08×10−1,x2,05×10−1,08×10−1,−01×10−1,−07×10−1.Proposition 13.
Letf1~,L, f2~,L, and f3~,L be BVS sets over M. Then,
(1)
f1~,L⋓~f2~,L⋓~f3~,L=f1~,L⋓~f2~,L⋓~f3~,L and f1~,L⋒~f2~,L⋒~f3~,L=f1~,L⋒~f2~,L⋒~f3~,L(2)
f1~,L⋓~f2~,L⋒~f3~,L=f1~,L⋓~f2~,L⋓~f1~,L⋓~f3~,L,f1~,L⋒~f2~,L⋓~f3~,L=f1~,L⋒~f2~,L⋓~f1~,L⋒~f3~,L(3)
f1~,L⋓~f~null,L=f1~,L, f1~,L⋒~f~null,L=f~null,L(4)
f1~,L⋓~M~absolute,L=M~absolute,L and f1~,L⋒~M~absolute,L=f1~,L(5)
f~null,L∖M~absolute,L=f~null,L and M~absolute,L∖f~null,L=M~absolute,LProof.
Straightforward.Proposition 14.
Letf1~,L and f2~,L be two BVS sets over M. Then,
(1)
f1~,L⋓~f2~,LC=f1~,LC⋒~f2~,LC(2)
f1~,L⋒~f2~,LC=f1~,LC⋓~f2~,LCProof.
(1)
For allν∈~L&x∈~M,(21)∐i=12fi~,L=ν,x,maxγ⨁f1~νx,γ⨁f2~νx,minδ⨁f1~νx,δ⨁f2~νx,maxγ⊖f1~νx,γ⊖f2~νx,minδ⊖f1~νx,δ⊖f2~νx:x∈~M:ν∈~L,∐i=12fi~,Lc=ν,v,minδ⨁f1~νx,δ⨁f2~νx,maxγ⨁f1~νx,γ⨁f2~νx,minδ⊖f1~νx,δ⊖f2~νx,maxγ⊖f1~νx,γ⊖f2~νx:x∈~M:ν∈~L.
Now,(22)f1~,Lc=ν,x,δ⨁f1~νx,γ⨁f1~νx,δ⊖f1~νx,γ⊖f2~νx:x∈~M:ν∈~L,f2~,Lc=ν,:x,δ⨁f1~νx,γ⨁f2~νx,δ⊖f1~νx,γ⊖f2~νx:x∈~M:ν∈~L.
Then,(23)∏i=12fi~,Lc=ν,:x,minδ⨁f1~νx,δ⨁f2~νx,maxγ⨁f1~νx,γ⨁f2~νx,minδ⊖f1~νx,δ⊖f2~νx,maxγ⊖f1~νx,γ⊖f2~νx:x∈~M:ν∈~L=ν,:x,minδ⨁f1~νx,δ⨁f2~νx,maxγ⨁f1~νx,γ⨁f2~νx,minδ⊖f1~νx,δ⊖f2~νx,maxγ⊖f1~νx,γ⊖f2~νx:x∈~M:ν∈~L.
Thus,f1~,L⋓~f2~,Lc=f1~,Lc⋒~f2~,Lc.
(2)
ObviousProposition 15.
Letf1~,L and f2~,L be two BVS sets over M. Then,
(1)
f1~,L∨~f2~,Lc=f1~,LC∧~f2~,Lc(2)
f1~,L∧~f2~,Lc=f1~,LC∨~f2~,LcProof.
(1)
For allν1,ν2∈~L×L&x∈~M,(24)⋁i=12fi~,L=ν1,ν2,x,maxγ⨁f1~ν1x,γ⨁f2~ν2x,minδ⨁f1~ν1x,δ⨁f2~ν2x,maxγ⊖f1~ν1x,γ⊖f2~ν2x,minδ⊖f1~ν1x,δ⊖f2~ν2x,⋁i=12fi~,Lc=ν1,ν2,x,minδ⨁f1~ν1x,δ⨁f2~ν2x,maxγ⨁f1~ν1x,γ⨁f2~ν2x,minδ⊖f1~ν1x,δ⊖f2~ν2x,maxγ⊖f1~ν1x,γ⊖f2~ν2x.
Now,(25)f1~,Lc=ν1,x,δ⨁f1~ν1x,γ⨁f2~ν2x,δ⊖f1~ν1x,γ⊖f2~ν2x:ν∈~L,f2~,Lc=ν2,x,δ⨁f1~ν1x,γ⨁f2~ν2x,δ⊖f1~ν1x,γ⊖f2~ν2x:ν∈~L.
Then,(26)⋀i=12fi~,Lc=ν1,ν2,x,minδ⨁f1~ν1x,δ⨁f2~ν2x,maxγ⨁f1~ν1x,γ⨁f2~ν2x,minδ⊖f1~ν1x,δ⊖f2~ν2x,maxγ⊖f1~ν1x,γ⊖f2~ν2x=ν1,ν2,x,minδ⨁f1~ν1x,δ⨁f2~ν2x,maxγ⨁f1~ν1x,γ⨁f2~ν2x,minδ⊖f1~ν1x,δ⊖f2~ν2x,maxγ⊖f1~ν1x,γ⊖f2~ν2x.
Thus,f1~,L∨~f2~,Lc=f1~,LC∧~f2~,LC.
(2)
Following the steps of (1) of this theorem


## 3. Bipolar Vague Soft Structure

This is the most important section. This section has been devoted to few new sets included bipolar vague soft toplogyBVST and eight new definitions. The concept of interior and closure is inaugurated. On the basis of these concepts, related results are addressed. The results that engage the interior with closure are also given a touch.Definition 16.
Let BVSSM~,L be family of all BVS sets over M and TBVSS⋐BNS set M~absolute,L, then TBVSS is said to be a bipolar vague soft topology BVST onM. If
(1)
f~null,L and M~absolute,L∈~TBVSS`(2)
Union of any number ofBVS sets in TBVSS∈~TBVSS(3)
Intersection of finite number ofBVS soft sets in TBVSS∈~TBVSS
Then,M~,TBVSS,L is said to be a BVSTS overM~.Definition 17.
LetM~,TBVSS,L be a BVSTS over M~,f~,L be a BVS set over M~. Then, f~,L is said to be BVS closed set iff its complement is a BVS open set.Definition 18.
LetM~,TBVSS,L be a BVSTS and f~,L be a BVS set over M~, then f~,L is called BVS.
(1)
Semiopen iff~,L⋐~BVSclBVSintf~,L and BVS semiclose if f~,θ⋑~BVSintBVSclf~,θ(2)
Preopen iff~,L⋐~BVSintBVSclf~,L and BVS preclose if f~,L⋑~BVSclBVSintf~,L(3)
α-open if f~,L⋐~BVSintBVSclBVSintf~,L and BVSα-close if f~,L⋑~BVSclBVSintBVSclf~,L(4)
β-open if f~,L⋐~BVSclBVSintBVSclf~,L and BVSβ-close if f~,L⋑~BVSintBVSclBVSintf~,θ(5)
b-open if f~,L⋐~BVSclBVSintf~,L∪BVSintNBVSclf~,L and BVSb-close if f~,L⋑~BVSintBVSclf~,θ⋒~BVSclBVSintf~,L(6)
∗b-open if f~,L⋐~BVSclBVSintf~,L⋒~BVSintBVSclf~,L and BVS∗b-close if f~,L⋑~BVSintBVSclf~,L⋓~BVSclBVSintf~,L(7)
b∗∗-open if f~,L⋐~BVSintBVSclBVSintf~,L⋓~BVSclBVSintBVSclf~,L and BVSb∗∗-close if f~,L⋑~BVSclBVSintBVSclf~,L⋒~BVSintBVSclBVSintf~,L(8)
∗∗b-open if f~,L⋐~BVSintBVSclBVSintf~,θ⋒~NSclBVSintBVSclf~,L and BVS soft ∗∗b-close if f~,L⋑~BVSclBVSintBVSclf~,L⋓~BVSintBVSclNSintf~,LProposition 19.
LetM~,TBVSS,L be a BVSTS overM~. Then,
(1)
f~null,L, M~absolute,L are BVSs-closed sets over M~(2)
⋒~ of any number of BVSs-closed sets CS is a BVSs-closed sets CS overM~(3)
⋓~ of finite number of BVSs-closed sets 〈〈CS〉〉 is a BVSs-closed sets CS over M~Proof.
Obvious.Definition 20.
LetBNSSM~absolute,L be the family of all BVS sets over M~.
(1)
IfTBVSS=f~null,L,M~absolute,L, then TBVSS is said to be BVS indiscrete topology, M~,TBVSS,E is said to be a BVS indiscrete topological space over M~(2)
IfTBVSS=BVSSM~absolute,L, then TBVSS is said to be BVS discrete topology, M~,TBVSS,L is said to be a BVS discrete topological space over M~Proposition 21.
LetM~,TBVSS1,L and X,TBVSS2,L be two BVSTS over M~. Then, M~,TBVSS1⋒~TBVSS2,L is BVSTS over M~.Proof.
(1)
Sincef~null,L,M~absolute,L∈~TBVSS1 and f~null,L,M~absolute,L∈~TBVSS2, then f~null,L,M~absolute,L∈~TBVSS1⋒~TBVSS2(2)
Letf~,L:i∈I be a family of BVS sets in TBVSS1⋒~TBVSS2. Then, fi~,L∈~TBVSS1, fi~,L∈~TBVSS2∀i∈I, so ∐i∈Ifi~,L∈~TBVSS1&∐i∈Ifi~,L∈τ2BN. Thus, ∐i∈Ifi~,L∈~TBVSS1⋒~TBVSS2(3)
Letf~,L:i=1,n¯ be a family of finite number of BVS sets in TBVSS1⋒~TBVSS2. Then, fi~,L∈~TBVSS1,fi~,L∈~TBVSS2 for i=1,n¯, so ∏i=1nfi~,L∈TBVSS1 and ∏i=1nfi~,L∈~TBVSS2. Thus, ∏i=1nfi~,L∈~TBVSS1⋒~TBVSS2Remark 22.
The union of twoBVSTS over Mmay not be a BVSTS on M.Example 3.
LetM=x1,x2, L=ν1,ν2 be a set of parameters, TBVSS1=f~null,L,M~absolute,L,f1~,L,f2~,L,f3~,L, TBVSS2=f~null,L,M~absolute,L,f2~,L,f4~,L be two BVSTS over M. Here, BVS sets f1~,L,f2~,L,f3~,L, and f4~,L over Mare as succeeding:
(27)f1~,L=ν1,x1,09×10−1,04×10−1,−03×10−1,−07×10−1,x2,05×10−1,06×10−1,−02×10−1,−08×10−1,ν2,x1,07×10−1,03×10−1,−05×10−1,−04×10−1,x2,06×10−1,06×10−1,−07×10−1,−05×10−1,f2~,L=ν1,x1,07×10−1,04×10−1,−04×10−1,−06×10−1,x2,04×10−1,05×10−1,−03×10−1,−07×10−1,ν2,x1,06×10−1,02×10−1,−06×10−1,−03×10−1,x2,05×10−1,04×10−1,−08×10−1,−04×10−1,f3~,L=ν1,x1,05×10−1,03×10−1,−05×10−1,−05×10−1,x2,03×10−1,04×10−1,−04×10−1,−06×10−1,ν2,x1,04×10−1,01×10−1,−07×10−1,−02×10−1,x2,04×10−1,03×10−1,−09×10−1,−03×10−1,f4~,L=ν1,x1,08×10−1,05×10−1,−02×10−1,−08×10−1,x2,05×10−1,06×10−1,−01×10−1,−09×10−1,ν2,x1,07×10−1,03×10−1,−04×10−1,−05×10−1,x2,06×10−1,05×10−1,−06×10−1,−06×10−1.
Sincef1~,L⋓~f2~,L=TBVSS1⋓~TBVSS2, then TBVSS1⋓~TBVSS2 is not a BVVS over M.Proposition 23.
LetM,TBVSS,L be a BVSTS over M,TBVSS=fi~,L:fi~,L∈~BVSSM,L where
(28)fi~,L=ν,x,T⨁fi~νx,F⨁fi~νx,T⊖fi~νx,F⊖fi~νx:x∈~M:ν∈~L,
fori∈~I.
Then,(29)TBVSS=f⨁i~,L=v,x,T⨁fi~νx,F⨁fi~νx:x∈~M:ν∈L:f⨁i~,L∈~VSSM~,Ldefine BVST on M.Proof.
Obvious.Definition 24.
LetM,TBVSS,L be a BVSTS over M,f~,L∈BNSSM~,L be a BVS set. Then, BVS interior of f~,L, denoted f~,L∘, is defined as BVS union of all BVSs-open subsets of f~,L.Clearly,f~,L∘ is the biggest BVSs-open set contained by f~,L.Example 4.
Let us considerBVSTS, i.e., TBVSS1 given in Example 3. Let f~,L∈~BVSSM~,L be defined as succeeding:
(30)f~,L=ν1,x1,08×10−1,04×10−1,−02×10−1,−06×10−1,x2,04×10−1,07×10−1,−01×10−1,−09×10−1,ν2,x1,09×10−1,02×10−1,−06×10−1,−05×10−1,x2,07×10−1,05×10−1,−06×10−1,−06×10−1.
Then,f~null,L,f2~,L,f3~,L⋐f~,L; therefore, f~,L∘=f~null,L⋓~f2~,L⋓~f3~,L=f2~,L.Theorem 25.
LetM,TBVSS,L be a BVSTS over M, f~,L∈BVSSM~,L. f~,L is a BVSs-open set iff f~,L=f~,L∘.Proof.
Letf~,Lbe a BVSs-open set. Then, the biggest BVSs-open set that is contained by f~,L is equal to f~,L. Hence, f~,L=f~,L∘.
Contrariwise, it is known thatf~,L∘ is a BVSs-open set; if f~,L=f~,L∘, then f~,L is a BVS set.Theorem 26.
LetM,TBVSS,L be a BVSTS over M, f1~,L,f2~,L∈~BVSSM~,L. Then,
(1)
f1~,L∘∘=f1~,L∘(2)
f~null,L∘=f~null,L&M~absolute,L∘=M~absolute,L(3)
f1~,L⋐f2~,L⇒f1~,L°⋐f2~,L°(4)
f1~,L⋒~f2~,L∘=f1~,L∘⋒~f2~,L∘(5)
f1~,L∘⋓~f2~,L∘⋐f1~,L⋓~f2~,L∘Proof.
(1)
Letf1~,L∘=f2~,L. Then, f2~,L∈τBN iff f2~,L=f2~,L∘. So, f1~,L∘°=f1~,L°(2)
Obvious(3)
Letf1~,L∘⋐f1~,L⋐f2~,L,f2~,L∘⋐f2~,L. Since f2~,L∘ is the biggest BVSs-open set covered in f2~,L, so f1~,L∘⋐f2~,L∘(4)
Sincef1~,L⋒~f2~,L⋐f1~,L&f1~,L⋒~f2~,L⋐f2~,L, then f1~,L⋒~f2~,L∘⋐f1~,L∘,f1~,L⋒~f2~,L∘⋐f2~,L∘ and so f1~,L⋒~f2~,L∘⋐f1~,L∘⋒~f2~,L∘. On the other hand, since f1~,L∘⋐f1~,L and f2~,L∘⋐f2~,L, then f1~,L∘⋒~f2~,L∘⋐f1~,L⋒~f2~,L; besides, f1~,L⋒~f2~,L∘⋐f1~,L⋒~f2~,L and it is the biggest BVSs-open set. Therefore, f1~,L∘⋒~f2~,L∘⋐f1~,L⋒~f2~,L∘. Thus, f1~,L⋒~f2~,L∘=f1~,L∘⋒~f2~,L∘(5)
Sincef1~,L⋐f1~,L⋓~f2~,L&f2~,L⋐f1~,L⋓~f2~,L, then f1~,L∘⋐f1~,L⋓~f2~,L∘ and f2~,L∘⋐f1~,L⋓~f2~,L∘. Therefore, f1~,L∘⋒~f2~,L∘⋐f1~,L⋓~f2~,L∘Definition 27.
LetM,TBVSS,L be a BVSTS over M, f~,L∈BVSSM~,L be a BVS set. Then, BVSs-closure of f~,L,denoted f~,L¯, is defined as BVS soft intersection of all BVSs-closed supersets of f~,L.Clearly,f~,L¯ is the smallest BVSs-closed set covering by f~,L.Definition 28.
LetM,TBVSS,L be a BVSTS over M, f~,L∈BVSSM~,L be a BVS set then the boundary of f~,L is dented by Fr. f~,L is defined as a BVS point. t1mp1,p2 is called boundary of f~,L if every BVSs-open set containing t1mp1,p2 contains at least one point of f~,Land least one BVS point of f~,Lc.Definition 29.
LetM,TBVSS,L be a BVSTS over M, f~,L∈BVSSM~,L be a BVS set then BVS exterior of f~,L is dented by Ext. f~,L is defined as a BVS point. tmp1,p2 is called exterior of f~,Lif tmp1,p2 if BVS point tmp1,p2 is BVS interior of f~,Lc, that is, there exists BVSs-open set g~,Lsuch that tmp1,p2∈g~,L⊆f~,Lc.Example 5.
LetBVS topology TBVSS given in Example 3. Suppose f~,L∈~BVSSM~,L is defined as succeeding:
(31)f~,L=ν1,x1,01×10−1,04×10−1,−09×10−1,−01×10−1,x2,04×10−1,02×10−1,−08×10−1,−01×10−1,ν2,x1,02×10−1,03×10−1,−07×10−1,−02×10−1,x2,01×10−1,02×10−1,−07×10−1,−04×10−1.
Obviously,f~null,L,M~absolute,L,f1~,LC,f2~,LC and f3~,LC are all BVSs-closed sets over M,TBVSS,L. They are given as follows:
(32)f~null,LC=M~absolute,L,M~absolute,LC=f~null,L,f1~,LC=ν1,x1,03×10−1,06×10−1,−07×10−1,−02×10−1,x2,05×10−1,04×10−1,−08×10−1,−01×10−1,ν2,x1,04×10−1,07×10−1,−05×10−1,−04×10−1,x2,02×10−1,04×10−1,−03×10−1,−06×10−1,f2~,LC=ν1,x1,05×10−1,06×10−1,−07×10−1,−03×10−1,x2,05×10−1,05×10−1,−07×10−1,−02×10−1,ν2,x1,04×10−1,08×10−1,−04×10−1,−05×10−1,x2,04×10−1,07×10−1,−02×10−1,−07×10−1,f3~,LC=ν1,x1,06×10−1,07×10−1,−05×10−1,−03×10−1,x2,07×10−1,06×10−1,−06×10−1,−03×10−1,ν2,x1,05×10−1,09×10−1,−03×10−1,−06×10−1,x2,04×10−1,07×10−1,−02×10−1,−07×10−1.
Then,M~absolute,LC,f1~,LC,f2~,LC,f3~,LC⋑~f~,L; therefore, f~,L¯=f1~,LC⋒~f2~,LC⋒~f3~,LC=f1~,LC.Theorem 30.
LetM,TBVSS,L be a BVSTS over M~, f~,L∈BNSSM~,L. f~,L is a BVSs-open set iff f~,L=f~,L¯.Proof.
Obvious.Theorem 31.
LetM,TBVSS,L be a BVSTS over M,f1~,L,f2~,L∈BVSSM~,L. Then,
(1)
f1~,L¯¯=f1~,L¯(2)
f~null,L¯=f~null,L and M~absolute,L¯=M~absolute,L(3)
f1~,L⋐f2~,L⇒f1~,L¯⋐f2~,L¯(4)
f1~,L⋒~f2~,L¯⋐f1~,L¯⋒~f2~,L¯(5)
f1~,L⋓~f2~,L¯=f1~,L¯⋓~f2~,L¯Proof.
(1)
Letf1~,L¯=f2~,L, then f2~,L is a BVSs-closed set. Hence, f2~,L=f1~,L¯. So, f1~,L¯¯=f1~,L¯(2)
Obvious(3)
It is known thatf1~,L⋐f1~,L¯,f2~,L⋐f2~,L¯. Since f1~,L⋐f2~,L⋐f2~,L¯. Since f1~,L¯ is the smallest BVSs-closed set covering f1~,L, then so f1~,L¯⋐f2~,L¯(4)
Sincef1~,L⋐f1~,L⋓~f2~,L,f2~,L⋐f1~,L⋓~f2~,L, then f1~,L¯⋐f1~,L⋓~f2~,L¯,f2~,L¯⋐f1~,L⋓~f2~,L¯ and so f1~,L¯⋓~f2~,L¯⋐f1~,L⋓~f2~,L¯. Contrariwise, since f1~,L⋐f1~,L¯,f2~,L⋐f2~,L¯, then f1~,L⋓~f2~,L⋐f1~,L¯⋓~f2~,L¯. Besides, f1~,L⋓~f2~,L¯ is the smallest BVSs-closed set covering f1~,L⋓~f2~,L. Therefore, f1~,L⋓~f2~,L¯⋐f1~,L¯⋓~f2~,L¯. Thus, f1~,L⋓~f2~,L¯⋐f1~,L¯⋓~f2~,L¯(5)
Sincef1~,L⋒~f2~,L⋐~f1~,L¯⋒~f2~,L¯ and f1~,L⋒~f2~,L¯ is then smallest BVSs-closed set covering f1~,L⋒~f2~,L, then f1~,L⋒~f2~,L¯⋐f1~,L¯⋒~f2~,L¯Theorem 32.
LetM,TBVSS,L be a BVSTS over M, f~,L∈BVSSM~,L.(1)
f~,L¯c=f~,Lc°(2)
f~,L°c=f~,LC¯Proof.
(1)
f~,L¯=∏i∈Ifi~,L∈~TBVSSC:fi~,L⋑~f~,L⇒f~,L¯c=∏i∈Ifi~,L∈~TBVSSc:fi~,L⋑~f~,L∀i∈~Ic=∐i∈Ifi~,Lc∈~TBVSS:fi~,Lc⋐~f~,Lc=f~,Lc°(2)
f~,L°=∐i∈Ifi~,L∈TBVSS:fi~,L⋐~f~,L⇒f~,L°C=∐i∈Ifi~,L∈~TBVSS:fi~,L⋐~f~,Lc=∏i∈Ifi~,Lc∈~TBVSSc:fi~,Lc⋑~f~,Lc=f~,Lc¯Theorem 33.
LetM,TBVSS,L be a BVSTS over M, f1~,L,f2~,L∈BVSSM~,L, then
(1)
Extf1~,L⋓~f2~,L=Extf1~,L⋓~Extf2~,L(2)
Extf1~,L⋒~f2~,L⋑~Extf1~,L⋓~Extf2~,L(3)
Frf1~,L⋓~g,ℓ⋐~Frf1~,L⋓~Frf2~,L(4)
Frf1~,L⋒~f2~,L⋐~Frf1~,L⋓~Frf2~,LProof.
(1)
SinceExtf1~,L⋓~f2~,L=f1~,L⋓~f2~,Lc∘=f1~,Lc⋒~f2~,Lc∘=f1~,Lc∘⋒~f2~,Lc∘=Extf1~,L⋒~Extg,ℓ(2)
Extf1~,L⋒~f2~,L=f1~,L⋒~f2~,Lc∘=f1~,Lc⋓~f2~,Lc∘⋑~f1~,Lc∘⋓~f2~,Lc∘=Extf1~,L⋓~Extf2~,L that is Extf1~,L⋒~f2~,L⋑~Extf1~,L⋓~Extf2~,L(3)
Frf1~,L⋓~f2~,L=f1~,L⋓~f2~,L¯⋒~f1~,L⋓~f2~,Lc¯=f1~,L⋓~f2~,L¯⋒~f1~,Lc⋒~f2~,Lc¯⋐~f1~,L¯⋓~f2~,L¯⋒~f1~,Lc¯⋒~f2~,Lc¯f,ℓ¯⋓~f2~,L¯⋒~f1~,Lc¯⋒~g,ℓc¯=f1~,L¯⋒~f1~,Lc¯⋓~f2~,L¯⋒~f1~,Lc¯⋓~f2~,Lc¯⋒~f1~,Lc¯⋒~f2~,Lc¯=Frf1~,L⋒~f2~,Lc¯⋓~Frf2~,L⋒~f1~,Lc¯⋐~Frf1~,L⋓~Frf2~,L(4)
Frf1~,L⋒~f2~,L=f1~,L⋒~f2~,L¯⋒~f1~,L⋒~f2~,Lc¯⋐~f1~,L¯⋒~f2~,L¯⋒~f1~,Lc⋓~f2~,Lc=f1~,L¯⋒~f2~,L¯⋒~f1~,Lc¯⋓~f1~,L¯⋒~f2~,L¯⋒~f2~,Lc¯=Frf1~,L⋒~f2~,L¯⋓~f1~,L¯⋒~Frg,ℓ⋐~Frf1~,L⋓~Frf2~,L


## 4. Conclusions

During the study, we have gone into detail about defining and finding out the characteristics of bipolar vague soft sets and fundamental operations in a new way. These operations are discussed with examples. On the basis of these operations, vague soft topology is defined. Some structures are discussed with respect to bipolar vague softs-open sets. These bipolar vague soft s-open sets are chosen among eight new definitions which are introduced in bipolar vague soft topology. Concepts of interior and closure with respect to s-open set are inaugurated. On the basis of these concepts, related results are addressed. The results that engage the interior with closure are also addressed. In the future, we will extend the study to bipolar vague soft bitopology with respect to bipolar vague soft s-open sets and bipolar vague softβ-open sets.



---

*Source: 5964872-2022-05-05.xml*