# An Absolutely New Attempt to Vague Soft Bitopological Spaces Basis at Newly Defined Operations regarding Vague Soft Points

Authors: Arif Mehmood; Farkhanda Afzal; Saleem Abdullah; Muhammad Imran Khan; Saeed Gul
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this study, new operations of union, intersection, and complement are defined with the help of vague soft sets in a new way that is in both true and false statements, union is defined with maximum, and intersection is defined with minimum. On the basis of these operations, vague soft topology is defined. Pairwise vague soft open sets and pairwise vague soft closed sets are defined in vague soft bitopological structures (VSBTS). Moreover, generalized vague soft open sets are introduced in VSBTS concerning soft points of the space. On the basis of generalized vague soft open sets, separation axioms are also introduced. In continuation, these separations axioms are engaged with other important results in VSBTS.


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## Body

## 1. Introduction

Fuzzy set theory [1] is the most importantly effective way to deal with vagueness and incomplete data, and it is being developed and used in various fields of science. Fuzzy set theory is the most attractive approach to address the uncertain situation. It is an entirely different approach in comparison to probability theory. It may be used in such situations where probability theory is getting failed. However, it has a shortcoming, i.e., it only addresses the membership value and is at a loss to address the nonmembership value. Atanassov [2] bridged the gap in fuzzy set theory with a new way and introduced the concept of intuitionistic fuzzy set theory. This approach supposes membership value and nonmembership value.Molodtsov [3] shaked the pillars of set theory and installed the concept of soft set theory (SS) to address the uncertainty in a meaningful way. Soft set theory (SST) has a big hand as an application in many fields like function smoothness, Riemann integration, measurement theory, and game theory [4]. The concept of vague set theory which is generalization of fuzzy set theory was suggested by Gau and Buehrer [5]. A vague set is defined as truth-membership function and false membership function.Wei et al. [6] combined the soft set and vague set structures and leaked out the concept of vague to soft set theory. Huang et al. [7] deeply studied [6] and pointed out some incorrect results. They verified the incorrect result with examples and gave some more new definitions. Wang and Li [8] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. The authors discussed the basic concepts related to vague soft topological and studied the results in vague soft topology. Wang and Li [8] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. They authors discussed the basic concepts related to vague soft topological and studied the results in vague soft topology. For better understanding, the authors provided number of examples.Mukherjee and Das [9] studied the concept of neutrosophic bipolar vague soft sets and some of its operations. It is the combination of neutrosophic bipolar vague sets and soft sets. Furthermore, the authors developed a decision-making method based on neutrosophic bipolar vague soft set. A numerical example has been shown. Some new operations on neutrosophic bipolar vague soft set have also been designed.The organization of the study is as follows. We first attempted the fundamentals definitions of vague soft set including complement, subsets, equal sets, null set, absolute set, union, and intersection in a new approach. On the basis of these concepts, vague soft topological structure is defined that are useful for subsequent discussions. Al-Quran and Hassan [10] extended the notion of classical soft set to neutrosophic vague soft sets by applying the theory of soft sets to neutrosophic vague soft sets to make more effective results. They defined new operations. Finally, they applied the theory to decision-making problems. An ample of examples is provided to make the results more effective in this direction.In our study, we worked with the new operations which are entirely different from references [5, 6]. Then, unlike references [7, 8], vague soft topological structure is reconstructed. In Section 2, some basic operators are introduced in a new way. In Section 3, vague soft bitopology (VSBT) is addressed with ample of examples. Some results, union, and intersection are also studied (VSBT). In Section 4, some main results are addressed. In Section 5, main more results are addressed. In Section 6, some concluding remarks and future work are addressed.


## 2. Preliminaries

In this section, some fundamental concepts are addressed.Definition 1.
Letπ be key set (KS) and θ set of parameters. Let ℒπ signifies power set of all vague sets on π. Then, a vague soft set f˜,θ over π is a set defined by a set-valued function f˜ representing a mapping f˜:θ⟶ℒπ, where f˜ is called the approximate function of the vague soft set f˜,θ. It can be written as a set of ordered pairs: f˜,θ=p,κ,ℸf˜nκ,Ff˜nκ,:κ∈π:p∈θ.Definition 2.
Letf˜,θ be a vague soft set over key set π. The complement of f˜,θ is signified as f˜,θc and is defined as follows: f˜,θc=p,x,1−ℸf˜nx,1−Ff˜nx:x∈π:p∈θ. It is clear that(1)f˜,θcc=f˜,θDefinition 3.
Letf˜,θ and ρ˜,θ be two vague soft subsets over key set π. f˜,θ is supposed to be vague soft subset of ρ˜,θ if ℸf˜nx≼ℸρ˜nx, Ϝf˜nx≼Ϝρ˜nx, ∀p∈θ,and∀x∈π. It is signified as f˜,θ⊑ρ˜,θ.⋅f˜,θ is said to be equal to ρ˜,θ if f˜,θ is a vague soft subset of ρ˜,θ and ρ˜,θ is a vague soft subset of f˜,θ. It is symbolized as f˜,θ=ρ˜,θ.Definition 4.
Letf˜1,θ,f˜2,θ be two vague soft subsets over key set π, so that f˜1,θ≠f˜2,θ. Then, their union is signified as f˜1,θ⊔˜f˜2,θ=f˜3,θ and is defined as f˜3,θ=p,x,ℸf˜3nκ,Ϝf˜3nκ,:κ∈π:p∈θ, where(2)ℸf˜3nκ=maxℸf˜1nκ,ℸf˜2nκ,if˜3nκ=maxϜf˜1nκ,Ϝf˜2nκ.Definition 5.
Iff˜1,θ, f˜2,θ be two vague soft subsets over key set π, such that f˜1,θ≠f˜2,θ. Then, their intersection is signified as f˜1,θ⊓˜f˜2,θ=f˜3,θ and is defined as f˜3,θ=p,x,ℸf˜3nκ,Ϝf˜3nκ,:κ∈π:p∈θ, where(3)ℸf˜3nκ=minℸf˜1nκ,ℸf˜2nκ,if˜3nκ=minϜf˜1nκ,Ϝf˜2nκ.Definition 6.
Iff˜,θ be a vague soft set over key set, π is said to be a null vague soft set (NVSS). If ℸfnκ=0,Ϝfnκ=1;∀n∈θ,∀κ∈π.
It is signified as0π,θ.Definition 7.
Iff˜,θ be a vague soft over key set, π is an absolute vague soft set if Tfnκ=1,Ffnκ=0;∀n∈θ, and ∀κ∈π.Definition 8.
If VSSπ˜,θ be the family of all VS soft sets and τ⊂VSSπ˜,θ, then τ is said to be a VS soft topology on π˜ if 0π,p,1π,p belong to τ, the union of any number of VS soft sets in τ belong to τ, and the intersection of a finite number of vague soft sets in τ belong to τ. Then, π˜,τ,θ is said to be a vague soft topology VST over π˜.Definition 9.
LetVSSπ˜ be the family of all vague soft sets over key set π. Then, κΔ1,Δ2e is called a vague soft point, for every ∈π˜,0≺Δ1,Δ2≼1,e∈θ, and is defined as follows:(4)κΔ1,Δ2e`ye=Δ1,Δ2,ife`=e∧y=κ,0,1,ife`≠e∧y≠κ.Definition 10.
Letf˜,θ be VSS over KS π. κΔ1,Δ2e∈NSSf˜,θ if(5)Δ1≼ℸf˜nκ,Δ2≼Ϝf˜nκ.Definition 11.
π˜,τ,θ be a vague soft topological space over π and f˜,θ be a vague soft over π. Vague soft set f˜,θ in π,τ,θ is called a vague soft nbhd of the vague soft point κΔ1,Δ2⋋∈f˜,θ, if there exists a NS open set g˜,θ, such that κΔ1,Δ2⋋∈g˜,θ.


## 3. Vague Soft Bitopological Structure

In this part, the concept of vague soft bitopological spaceVSBTS is defined. Furthermore, new types of open and closed sets have been introduced in the same space. Example of vague soft bitopological space is given. Intersection of vague soft bitopological space is addressed. Union of two vague soft bitopological spaces need not be vague soft topological space. This situation is established with example. Pairwise vague soft open and soft closed sets are addressed in vague soft bitopological space. Some theorems related to this concept are also addressed.Definition 12.
Ifπ,τ1,θ and π,τ1,θ are two VSTS, then π,τ1,τ2,θ is called VSBTS. Let π,τ1,τ2,θ be vague soft bitopological space. A vague soft set f,θ is said to be open in π,τ1,τ2,θ if there exists a vague soft open set f1,θ in τ1 and vague soft open set f2,θ in τ2, such that f,θ=f1,θ∪f2,θ. The vague soft set f,θ is said to be close in π,τ1,τ2,θ if its complement is open in π,τ1,τ2,θ.Example 1.
Letπ=κ1,κ2,κ3, θ=e1,e2, τ1=0π,θ,1π,θ,ω1,θ,ω2,θ, and τ1=0π,θ,1π,θ,⋌1,θ,⋌2,θ, where ω1,θ,ω2,θ,⋌1,θ, and ⋌2,θ being vague soft sets are as follows:(6)fω1,θe1=κ1,0210,0310,κ2,0410,0410,κ3,0210,0410,fω1,θe2=κ1,0310,0210,κ2,0110,0510,κ3,0410,0310,fω2,θe1=κ1,0410,0310,κ2,0410,0510,κ3,0310,0510,fω2,θe2=κ1,0310,0410,κ2,0210,0610,κ3,0410,0610,f⋌1,θe1=κ1,0510,0410,κ2,0610,0610,κ3,0410,0610,f⋌1,θe2=κ1,0410,0610,κ2,0310,0710,κ3,0510,0710,f⋌2,θe1=κ1,0110,0210,κ2,0310,0310,κ3,0110,0210,f⋌2,θe2=κ1,0110,0210,κ2,0310,0310,κ3,0110,0210.
Then,ω1,θ∪ω2,θ=ω2,θ, ω1,θ∪⋌1,θ=⋌1,θ, ω1,θ∪⋌2,θ=ω2,θ, ⋌1,θ∪⋌2,θ=⋌1,θ, ω2,θ∪⋌2,θ=ω2,θ and ω1,θ∩ω2,θ=ω2,θ, ω1,θ∩⋌=⋌1,θ, ω1,θ∩⋌2,θ=ω2,θ, ⋌1,θ∩⋌2,θ=⋌1,θ, ω2,θ∩⋌2,θ=⋌2,θ.
Therefore,τ1, τ2 are VSTS on π, so π,τ1,τ2,θ is VSBTS.Theorem 1.
Letπ,τ1,τ2,θ be a VSBTS. Then, τ1∩τ2 is VSBTS on π.Proof.
(1) and (3) are obvious. For (2), we proceed as follows: let ωi,θ;i∈I∈τ1∩τ2. Then, ωi,θ∈τ1 and ωi,θ∈τ2. Since τ2, τ2 are vague soft topologies on π, then ∪iωi,θ∈τ1 and ∪iωi,θ∈τ2. Therefore, ∪iωi,θ∈τ1∩τ2.Remark 1.
Letπ,τ1,τ2,θ be VSBTS; then, τ1∪τ2 need not be VSBTS on π.Example 2.
Letπ=κ1,κ2,κ3,=e1,e2, τ1=0π,θ,1π,θ,ω1,θ,ω2,θ,ω3,θ, and τ2=0π,θ,1π,θ,⋌1,θ,⋌2,θ, where ω1,θ,ω2,θ,⋌1,θ, and ⋌2,θ being vague soft sets are as follows:(7)fω1,θe1=κ1,0210,0310,κ2,0410,0410,κ3,0210,0410,fω1,θe2=κ1,0310,0210,κ2,0110,0510,κ3,0410,0310,fω2,θe1=κ1,0410,0310,κ2,0410,0510,κ3,0310,0510,fω2,θe2=κ1,0310,0410,κ2,0210,0610,κ3,0410,0610,fω3,θe1=κ1,0510,0410,κ2,0610,0610,κ3,0410,0610,fω3,θe2=κ1,0410,0610,κ2,0310,0710,κ3,0510,0710,f⋌1,θe1=κ1,0510,0410,κ2,0610,0610,κ3,0410,0610,f⋌1,θe2=κ1,0410,0610,κ2,0310,0710,κ3,0510,0710,f⋌2,θe1=κ1,0110,0210,κ2,0310,0310,κ3,0110,0210,f⋌2,θe2=κ1,0110,0210,κ2,0310,0310,κ3,0110,0210.
Here,τ1∪τ2=0Y,θ,1π,θ,ω1,θ,ω2,θ,ω3,θ,⋌1,θ,⋌2,θ is not a vague soft topology on π. Since, ω3,θ∪⋌2,θ∉τ1∪τ2.Definition 13.
Letπ,τ1,τ2,θ be VSBTS. Then, a vague soft set(8)⋉,θ=e,x,T⋉ex,F⋉ex:x∈π,e∈θ,is called as a pairwise vague soft open set PVSOS if there exist a vague soft open set ⋉1,θ in τ1 and a vague soft open set ⋉2,θ in τ2, such that for all x∈π,(9)⋉,θ=⋉1,θ∪⋉2,θ=e,x,maxT⋉ex,TGex,maxF⋉ex,FGex:e∈θ.Definition 14.
Letπ,τ1,τ2,θ be VSBTS. Then, a vague soft set(10)⋉,θ=e,x,T⋉ex,,F⋉ex:x∈π,e∈θ,is called as PVSOS if there exist a vague soft open set ⋉1,θ in τ1 and a vague soft open set ⋉2,θ in τ2, such that for all x∈π,(11)⋉,θ=⋉1,θ∪⋉2,θ=e,x,maxTHex,TGex,maxF⋉ex,FGex:e∈θ.
The set of all pairwise vague open sets inπ,τ1,τ2,θ is denoted by PVSO π,τ1,τ2,θ.Definition 15.
Letπ,τ1,τ2,θ be VSBTS. Then, a vague soft set(12)⋉,θ=e,x,T⋉ex,F⋉ex:x∈π,e∈θ,is called as a pairwise vague soft closed set (PVSC) if ⋉,θc is PVSO. It is clear that ⋉,θ is a PVSC set if there exist VSOC⋉1,θ in τ1 and VSOC ⋉2,θ in τ2, such that for all x∈π,(13)⋉,θ=⋉1,θ∩⋉2,θ=e,x,minT⋉x,TGex,minF⋉ex,FGex:e∈θ.
The set of allPVSC in π,τ1,τ2,θ is denoted by PVSCπ,τ1,τ2,θ.Theorem 2.
Letπ,τ1,τ2,θ be VSBTS. In this case,(1)
0π,θ,1π,θ∈PVSOX,τ1,τ2,θ(2)
If⋉i,θ∣i∈I⊆PVSOπ,τ1,τ2,θ, then ∪i∈I⋉i,θ∈PVSOπ,τ1,τ2,θ(3)
IfGi,θ∣i∈I⊆PVSCπ,τ1,τ2,θ, then ∩i∈IGi,θ∈PVSCπ,τ1,τ2,θProof.
(1)
Since0X,θ∪0π,θ=0π,θ, 1π,θ∪1π,θ=1π,θ, then 0π,θ and 1π,θ are PVSC(2)
Since⋉i,θ∈PVSOX,τ1,τ2,θ, there exist ⋉i1,θ∈τ1, ⋉i1,θ∈τ2, such that ⋉i,θ=⋉i1,θ∪⋉i1,θ for all i∈I. Then,(14)∪i∈I⋉i,θ=∪i∈I⋉i1,θ∪⋉i1,θ=∪i∈I⋉i1,θ∪∪i∈I⋉i2,θ.Asτ1, τ2 are VST on π, ∪i∈I⋉i1,θ∈τ1 and ∪i∈I⋉i2,θ∈τ1Therefore,∪i∈I⋉i,θ∈PNSOπ,τ1,τ2,θ(3)
SinceGi,θ∈PVSCX,τ1,τ2,θ, there exist Gi1,θc∈τ1 and Gi2,θc∈τ2, such that Gi,θ=Gi1,θ∩Gi2,θ for all i∈I. Then,(15)∩i∈IGi,θ=∩i∈IGi1,θ∩Gi2,θ=∩i∈IGi1,θ∩∩i∈IGi2,θ.
Then,∩i∈IGi,θ∈PVSCπ,τ1,τ2,θ as ∩i∈IGi1,θc∈τ1, and ∩i∈IGi2,θc∈τ1.Definition 16.
Letπ,τ1,τ2,θ be VSBTS,⋌,θ∈VSSπ. The PVS closure of ⋌,θ, denoted by clPVSS⋌,θ, is the intersection of all PVSC containing ⋌,θ, i.e.,(16)clPVSS⋌,θ=∩ω,θ∈PVSCX∣⋌,θ⊂ω,θ.
It is clear thatclPVSS⋌,θ is the smallest PVSCS containing ⋌,θ.Theorem 3.
Letπ,τ1,τ2,θbe VSBTS and ⋌,θ,ω,θ∈VSSπ. Then,(1)
clPVSS0π,θ=0π,θ and clPVSS1π,θ=1π,θ(2)
⋌,θ⊆clPVSS⋌,θ(3)
⋌,θ is a PVSCS if clPVSS⋌,θ=⋌,θ(4)
clPVSS⋌,θ⊆clPVSSω,θ if ⋌,θ⊆ω,θ(5)
clPVSS⋌,θ∪clPVSSω,θ⊂clPVSS⋌,θ∪ω,θ(6)
clPVSSclPVSS⋌,θ=clPVSS⋌,θ, i.e., clPVSS⋌,θ is PVSCSProof.
It is obvious.Theorem 4.
Letπ,τ1,τ2,θ be VSBTS, ⋌,θ∈VSSπ. Then, xα,βe∈clPVSS⋌,θ if and only if for all Uxα,βe∈τ12xα,βe, where Uxα,βe is any PVSOS containing xα,βe, and τ12xα,βe is the family of all PVSOS containing xα,βe, Uxα,βe∩⋌,θ≠0π,θ.Proof.
Letxα,βe∈clPVSS⋌,θ, and suppose that there exists Uxα,β,γe∈τ12xα,βe, such that Uxα,βe∩⋌,θ=0π,θ. Then, ⋌,θ⊂Uxα,βec. Thus, clPVSS⋌,θ⊂clPVSSUxα,βec=Uxα,βec, which implies clPVSS⋌,θ∩Uxα,βe=0π,θ, a contradiction.
Conversely,xα,βe∉clPNSS⋌,θ; then, xα,βe∈clPVSS⋌,θc∈τ12xα,βe. Therefore, by hypothesis, clPVSS⋌,θc∩⋌,θ≠0π,θ, a contradiction.


## 4. Main Results

In this section, some new definitions are introduced in vague soft bitopological spaces. All these definitions are the backbone for the upcoming study. Among these definitions, definition (3) became source of motivation for our results which are discussed in this section and in the next section. All the results discussed in this section are discussed with respect to soft points of the spaces.Definition 17.
Letπ,τ1,τ2,θ be VSBTS over π,f˜,θ be a VS set over π. Then, f˜,θ is(1)
Vague soft semiopen iff˜,θ⊆VSclVSintf˜,θ(2)
Vague soft preopeniff˜,θ⊆VSintVSclf˜,θ(3)
Vaguesoft∗bopen iff˜,θ⊆VSclVSintf˜,θ⊔VSintVSclf˜,θ and vague soft ∗b close if f˜,θ⊇VSclVSintf˜,θ⊓VSintVSclf˜,θ.Definition 18.
Letπ,τ1,τ2,θ be VSBTS over π, κ1⋋Δ1,Δ2≠κ2⋋/Δ1/,Δ2/ are VS points . If there exist VS∗b open sets f˜,θ and g˜,θ, such that κ1⋋Δ1,Δ2∈f˜,θ,κ1⋋Δ1,Δ2⊓g˜,θ=0π˜,θ,orκ2⋋/Δ1/,Δ2/∈g˜,θ,κ2⋋/Δ1/,Δ2/⊓f˜,θ=0π˜,θ. Then, π,τ1,τ2,θ is called a NSB∗b0 space.Definition 19.
Letπ,τ1,τ2,θ be VSBTS over X˜, and κ1⋋Δ1,Δ2≠κ2⋋/Δ1/,Δ2/ are VS points. If there exist VS∗b-open sets f˜,θ,g˜,θ, such that κ1⋋Δ1,Δ2∈f˜,θ,κ1⋋Δ1,Δ2⊓g˜,θ=0π˜,θ and κ2⋋/Δ1/,Δ2/∈g˜,θ,κ2⋋/Δ1/,Δ2/⊓f˜,θ=0π˜,θ. Then, π,τ1,τ2,θ is called a VSB∗b1 space.Definition 20.
Letπ,τ1,τ2,θ be VSBTS over π,κ1⋋Δ1,Δ2≠κ2⋋/Δ1/,Δ2/ are VS points. If ∃VS∗b open set f˜,θ, g˜,θ, such that κ1⋋Δ1,Δ2∈f˜,θ, κ2⋋/Δ1/,Δ2/∈g˜,θ, and f˜,θ⊓g˜,θ=0π˜,θ. Then, π,τ1,τ2,θ is called a VSB∗∗b2 space.Example 3.
Letπ˜=κ1,κ2, the set of conditions by θ=⋋1,⋋2. Let us consider κ10.1,0.4⋋1, κ10.2,0.5⋋2, κ20.3,0.3⋋1, and κ20.4,0.4⋋1 be VS points. Then, the family τ1=0π˜,θ,1π˜,θ,f1˜,θ,f2˜,θ,f3˜,θ,f4˜,θ,f5˜,θ,f6˜,θ,f7˜,θ,f8˜,θ,…,f15˜,θ,wheref1˜,θ=κ10.1,0.4⋋1,f2˜,θ=κ10.2,0.5⋋2, f3˜,θ=κ20.3,0.3⋋1,f4˜,θ=κ20.4,0.4⋋2f5˜,θ=f1˜,θ∪f2˜,θ,f6˜,θ=f1˜,θ∪f3˜,θ,f7˜,θ=f2˜,θ∪f4˜,θ,f8˜,θ=f2˜,θ∪f3˜,θ,f9˜,θ=f2˜,θ∪f4˜,θ,f10˜,θ=f3˜,θ∪f4˜,θ,f11˜,θ=f1˜,θ∪f2˜,θ∪f3˜,θ,f12˜,θ=f1˜,θ∪f2˜,θ∪f4˜,θ,f13˜,θ=f2˜,θ∪f3˜,θ∪f4˜,θ,f14˜,θ=f1˜,θ∪f3˜,θ∪f4˜,θ, f15˜,θ=κ10.1,0.4⋋1,κ10.2,0.5⋋2,κ20.3,0.3⋋1,κ20.4,0.4⋋2 is VSTS. Also, τ2=0π˜,θ,1π˜,θ,f1˜,θVSTS. Thus, π,τ1,τ2,θ be VSBTS. Also, π,τ1,τ2,θ is VSB∗b2 structure.Theorem 5.
Letπ,τ1,τ2,θ be VSBTS. Then, π,τ1,τ2,θ be a VSB∗b1 structure if and only if each VS point is VS∗b-closed.Proof.
Letπ,τ1,τ2,θ be VSBTS over and π⋅κ1⋋Δ1,Δ2 be an arbitrary VS point. We establish κ1⋋Δ1,Δ2 is a V soft ∗b-open set. Let κ2⋋/Δ1/,Δ2/∈κ1⋋Δ1,Δ2. Then, either κ2⋋/Δ1/,Δ2/≻κ1⋋Δ1,Δ2 or κ2⋋/Δ1/,Δ2/,Δ3/≺κ2⋋/Δ1/,Δ2/orκ2⋋/Δ1/,Δ2/≻≻κ1⋋Δ1,Δ2 or κ2⋋/Δ1/,Δ2/≺≺κ2⋋/Δ1/,Δ2/. This means that κ2⋋/Δ1/,Δ2/andκ1⋋Δ1,Δ2 are two distinct VS points. Thus, κ1≻κ2 or κ1≺κ2 or ⋋/≻⋋ or ⋋/≺⋋ or κ1≻≻κ2 or κ1≺≺κ2 or ⋋/≻≻⋋ or ⋋/≺≺⋋. Since π˜,τ,θ be a VS∗b1 structure, there exists a VS∗b-open set g˜,θ, so that κ2⋋/Δ1/,Δ2/∈g˜,θ and κ1⋋Δ1,Δ2⊓g˜,θ=0π˜,θ. Since, κ1⋋Δ1,Δ2⊓g˜,θ=0π˜,θ. So, κ2⋋/Δ1/,Δ2/∈g˜,θ⊂κ1⋋Δ1,Δ2. Thus, κ1⋋Δ1,Δ2 is a VS∗b-open set, i.e., κ1⋋Δ1,Δ2 is a VS∗b-closed set. Suppose that each VS point κ1⋋Δ1,Δ2 is VS∗b-closed. Then, κ1⋋Δ1,Δ2c is a VS∗b-open set. Let κ1⋋Δ1,Δ2⊓κ2⋋/Δ1/,Δ2/=0π˜,θ. Thus, κ2⋋/Δ1/,Δ2/,θ∈κ1⋋Δ1,Δ2c, and κ1⋋Δ1,Δ2⊓κ1⋋Δ1,Δ2c=0π˜,θ. So, π,τ1,τ2,θ be a VSB-∗b1 space.Theorem 6.
Letπ,τ1,τ2,θ be VSTBS over the father set π. Then, π,τ1,τ2,θ is VS∗b2 space if and only if for distinct VS points κ1⋋Δ1,Δ2, κ2⋋/Δ1/,Δ2/, there exists a VS∗b-open set f˜,θ, containing but not κ2⋋/Δ1/,Δ2/, such that κ2⋋/Δ1/,Δ2/∉f˜,θ¯.Proof.
Letκ1⋋Δ1,Δ2≻κ2⋋/Δ1/,Δ2/ be two VS points in VS∗b2 space. Then, there exists disjoint VS∗b open sets f˜,θ,g˜,θ, such that κ1⋋Δ1,Δ2∈f˜,θandκ2⋋/Δ1/,Δ2/∈g˜,θ. Since κ1⋋Δ1,Δ2,θ⊓κ2⋋/Δ1/,Δ2/=0π˜,θ, andf˜,θ⊓g˜,θ=0π˜,θ,⋅κ2⋋/Δ1/,Δ2/∉f˜,θ implies κ2⋋/Δ1/,Δ2/∉f˜,θ¯. Next, suppose that κ1⋋Δ1,Δ2≻κ2⋋/Δ1/,Δ2/, there exists a VS∗b-open set f˜,θ containing κ1⋋Δ1,Δ2 but not κ2⋋/Δ1/,Δ2/, such that κ2⋋/Δ1/,Δ2/∉f˜,θ¯c, that is, f˜,θ and f˜,θ¯c are mutually exclusive VS∗b-open sets supposing κ1⋋Δ1,Δ2 and κ2⋋/Δ1/,Δ2/ in turn.Theorem 7.
Letπ,τ1,τ2,θ be VSBTS. Then, π,τ1,τ2,θ is VS∗b1 space if every VS point κ1⋋Δ1,Δ2∈f˜,θ∈π,τ,θ, if there exists a VS∗b open set g˜,θ, such that κ1⋋Δ1,Δ2∈g˜,θ⊂g˜,θ¯⊂f˜,θ. Then, π,τ1,τ2,θ is a VS∗b2 space.Proof.
Supposeκ1⋋Δ1,Δ2,Δ3⊓κ2⋋/Δ1/,Δ2/=0X˜,θ. Since, π,τ1,τ2,θ is VS∗b1 space. κ1⋋Δ1,Δ2 and κ2⋋/Δ1/,Δ2/ are the VS∗b-closed sets in π,τ1,τ2,θ. Then, κ1⋋Δ1,Δ2∈κ2⋋/Δ1/,Δ2/c∈π,τ1,τ2,θ. Thus, there exists a VS∗b-open set g˜,θ∈π,τ1,τ2,θ, such that κ1⋋Δ1,Δ2∈g˜,θ⊂g˜,θ¯⊂κ2⋋/Δ1/,Δ2/c. So, we have κ2⋋/Δ1/,Δ2/∈g˜,θ and g˜,θ⊓g˜,θc=0π˜,θ, that is, π,τ1,τ2,θ is a VS soft ∗b2 space.Definition 21.
LetV be VSBTS. f˜,θ be a VS∗b-closed set, and κ1⋋Δ1,Δ2,Δ3⊓f˜,θ=0π˜,θ. If there exists VS∗b-open sets g1˜,θ and g2˜,θ, such that κ1⋋Δ1,Δ2∈g1˜,θ,f˜,θ⊂g2˜,θ, and κ1⋋Δ1,Δ2⊓g1˜,θ=0π˜,θ, then π,τ1,τ2,θ is called a VBS∗b-regular space. π,τ1,τ2,θ is said to be VS∗b3 space, if it is both a VS regular and VSB∗b1 space.Theorem 8.
Letπ,τ1,τ2,θ be VSBTS.π,τ1,τ2,θ is soft ∗b3 space if and only if for every κ1⋋Δ1,Δ2∈f˜,θ, that is, g˜,θ∈π,τ1,τ2,θ (such that κ1⋋Δ1,Δ2∈g˜,θ⊂g˜,θ¯⊂f˜,θ.Proof.
Letπ,τ1,τ2,θ is VSB∗b3 space and κ1⋋Δ1,Δ2∈f˜,θ∈π,τ1,τ2,θ. Since π,τ1,τ2,θ is VS∗b3 space for the VS point κ1⋋Δ1,Δ2, and VS∗b is a closed set f˜,θc,∃g1˜,θ and g2˜,θ, such that κ1⋋Δ1,Δ2∈g1˜,θ,f˜,θc⊂g2˜,θ, and g1˜,θ∩g2˜,θ=0X˜,θ. Then, we have κ1⋋Δ1,Δ2,θ∈g1˜,θ⊂g2˜,θc⊂f˜,θ. Since g2˜,θcVS∗b is a closed set, g1˜,θ¯⊂g2˜,θc. Conversely, let κ1⋋Δ1,Δ2,θ∩ℋ˜,θ=0X˜,θandℋ˜,θ be a VS∗b closed set. κ1⋋Δ1,Δ2∈ℋ˜,θc, and from the condition of the theorem, we have κ1⋋Δ1,Δ2∈g˜,θ⊂g˜,θ¯⊂ℋ˜,θc. Thus, κ1⋋Δ1,Δ2∈g˜,θ,ℋ˜,θ⊂g˜,θ¯candg˜,θ∩g˜,θ¯c=0π˜,θ. So, π,τ1,τ2,θ is VSB∗b3 space.


## 5. More Main Results

In this section, more main results are addressed. These results are discussed between two structures. The behavior of one structure can safely be transferred from one space into another space through soft functions by putting some conditions on the soft function. Similarly, through soft function, the engagement between structures and soft closed sets can be studied. The engagement is between second countable and soft limit points. The study of B. V. property, soft vague compactness, sequentially compactness, and countably compactness results are addressed with respect to soft points of the spaces.Theorem 9.
Letπ,τ1,τ2,θ be VSBTS, such that it is VSB∗b Hausdorff space and Y˜,F1,F2,θ be another VSBTS. Let f,θ: π,τ1,τ2,θ⟶andY˜,F1,F2,θbe another VSBTS which is a soft function, such that it is soft monotone and continuous. Then, Y˜,F1,F2,θ is also the characteristics of VSB∗b Hausdorffness.Proof.
Supposeκ1⋋Δ1,Δ21,κ1⋋Δ1,Δ22∈π˜, such that either κ1⋋Δ1,Δ21≻κ1⋋Δ1,Δ22orκ1⋋Δ1,Δ21≺κ1⋋Δ1,Δ22. Since, f,θ is soft monotone. Let us suppose the monotonically increasing case. So, κ1⋋Δ1,Δ21≻κ1⋋Δ1,Δ22orκ1⋋Δ1,Δ21≺κ1⋋Δ1,Δ22 imply that fκ1⋋Δ1,Δ21≻fκ1⋋Δ1,Δ22 or fκ1⋋Δ1,Δ21≺fκ1⋋Δ1,Δ22, respectively. Suppose κ2⋋/Δ1/,Δ2/1,κ2⋋/Δ1/,Δ2/2∈Y˜, such that κ2⋋/Δ1/,Δ2/1≻κ2⋋/Δ1/,Δ2/2orκ2⋋/Δ1/,Δ2/1≺κ2⋋/Δ1/,Δ2/2; so, κ2⋋/Δ1/,Δ2/1≻κ2⋋/Δ1/,Δ2/2orκ2⋋/Δ1/,Δ2/1≺κ2⋋/Δ1/,Δ2/2, respectively, such that κ2⋋/Δ1/,Δ2/1=fκ1⋋Δ1,Δ21,κ2⋋/Δ1/,Δ2/2=fκ1⋋Δ1,Δ22. Since, π,τ1,τ2,θ is VSB∗b Hausdorff space, so there exist mutually disjoint VS∗b open sets k1,θ and k2,θ∈π,τ1,τ2,θ⇒fk1,θ and fk2,θ∈Y˜,F,θ. We claim that ff1,θ∩˜fk2,θ=0π˜,θ˜. Otherwise, fk1,θ∩˜fk2,θ≠0π˜,θ˜. Suppose there exists κ3⋋//Δ1//,Δ2//,Δ3//1∈fk1,θ∩˜fk2,θ, there exists κ3⋋//Δ1//,Δ2//1∈fk1,θandκ3⋋//Δ1//,Δ2//1∈fk2,θ. κ3⋋//Δ1//,Δ2//1∈fk1,θ,f is soft one-one, there exists κ3⋋//Δ1//,Δ2//2∈k1,θ, such that κ3⋋//Δ1//,Δ2//1=fκ3⋋//Δ1//,Δ2//2,κ3⋋//Δ1//,Δ2//1∈fk2,θ imply that there exists κ3⋋//Δ1//,Δ2//3∈k2,θ, such that fκ3⋋//Δ1//,Δ2//3f⇒κ3⋋//Δ1//,Δ2//2=fκ3⋋//Δ1//,Δ2//3. Since, f is soft one-one, ta//,b//e//,θ2=κ3⋋//Δ1//,Δ2//3 implies that κ3⋋//Δ1//,Δ2//2∈fk1,θ,κ3⋋//Δ1//,Δ2//2∈fk2,θ implies that κ3⋋//Δ1//,Δ2//2∈fk1,θ∩˜fk2,θ. This is contradiction because k1,θ∩˜k2,θ=0π˜,θ. Therefore, fk1,θ∩˜fk2,θ=0π˜,θ. Finally, κ1⋋Δ1,Δ21≻κ1⋋Δ1,Δ22orκ1⋋Δ1,Δ21≺κ1⋋Δ1,Δ22⇒κ1⋋Δ1,Δ21≠κ1⋋Δ1,Δ22 implies fκ1⋋Δ1,Δ21≻fκ1⋋Δ1,Δ22orfκ1⋋Δ1,Δ21≺fκ1⋋Δ1,Δ22f⇒κ1⋋Δ1,Δ21≠fκ1⋋Δ1,Δ22. Given a pair of points κ2⋋/Δ1/,Δ2/1,κ2⋋/Δ1/,Δ2/2∈Y˜, there exists κ2⋋/Δ1/,Δ2/1≠κ2⋋/Δ1/,Δ2/2. We are able to find out mutually exclusive VS∗b-open sets fk1,θ,fk2,θ∈Y˜,F1,F2,θ s.t., κ2⋋/Δ1/,Δ2/1∈fk1,θ,κ2⋋/Δ1/,Δ2/2∈fk2,θ. This proves that Y˜,F1,F2,θ is VSB∗b Hausdorff space.Theorem 10.
Letπ˜,τ1,τ2,θ be VSBTS and Y˜,F1,F2,θ be another VSBTS which satisfies one more condition of VSB∗b Hausdorffness. Let f,θ: π˜,τ1,τ2,θ⟶Y˜,F1,F2,θ be a soft function, s.t., it is soft monotone and continuous. Then, π,τ1,τ2,θ is also characteristics of VSB∗b Hausdorffness.Proof.
Supposeκ1⋋Δ1,Δ21,κ1⋋Δ1,Δ22∈π˜, such that either κ1⋋Δ1,Δ21≻κ1⋋Δ1,Δ22orκ1⋋Δ1,Δ21≺κ1⋋Δ1,Δ22. Let κ1⋋Δ1,Δ21≻κ1⋋Δ1,Δ22orκ1⋋Δ1,Δ21≺κ1⋋Δ1,Δ22 implies that fκ1⋋Δ1,Δ21≻fκ1⋋Δ1,Δ22 or fκ1⋋Δ1,Δ21≺fκ1⋋Δ1,Δ22, respectively. Suppose κ2⋋/Δ1/,Δ2/1,κ2⋋/Δ1/,Δ2/2∈Y˜, such thatκ2⋋/Δ1/,Δ2/1≻κ2⋋/Δ1/,Δ2/2orκ2⋋/Δ1/,Δ2/1≺κ2⋋/Δ1/,Δ2/2. So, κ2⋋/Δ1/,Δ2/1≻κ2⋋/Δ1/,Δ2/2orκ2⋋/Δ1/,Δ2/1≺κ2⋋/Δ1/,Δ2/2, respectively, such that κ2⋋/Δ1/,Δ2/2=fκ1⋋Δ1,Δ21,κ2⋋/Δ1/,Δ2/2=fκ1⋋Δ1,Δ22, such that κ1⋋Δ1,Δ21=f−1κ2⋋/Δ1/,Δ2/1 and κ1⋋Δ1,Δ22=f−1κ2⋋/Δ1/,Δ2/2. Since κ2⋋/Δ1/,Δ2/1,κ2⋋/Δ1/,Δ2/2∈Y˜, but Y˜,F1,F2,θ is VSB∗b Hausdorff space. So according to definition, κ2⋋/Δ1/,Δ2/1≻κ2⋋/Δ1/,Δ2/2 or κ2⋋/Δ1/,Δ2/1≺κ2⋋/Δ1/,Δ2/2. There definitely exists VS∗b-open sets k1,θ and k2,θ∈Y˜,F1,F2,θ, such that κ2⋋/Δ1/,Δ2/1∈k1,θ and κ2⋋/Δ1/,Δ2/2∈k2,θ, and these two VS∗b-open sets are disjoint. Since, f−1k1,θ and f−1k2,θ are VS∗b open in π˜,τ1,τ2,θ. Now, f−1k1,θ∩˜f−1k1,θ=f−1k1,θ∩˜k2,θ=f−1∅˜=0π˜˜,∂˜ and κ2⋋/Δ1/,Δ2/1∈k1,θ⇒f−1κ2⋋/Δ1/,Δ2/1∈f−1k1,θ⇒κ1⋋Δ1,Δ21∈k1,θ, κ2⋋/Δ1/,Δ2/2∈k2,θ⇒f−1κ2⋋/Δ1/,Δ2/2∈f−1k2,θ imply that κ1⋋Δ1,Δ22∈k2,θ. k1,θ, κ1⋋Δ1,Δ22∈f−1k2,θ. Accordingly, VSBTS is ∗b Hausdorff space.Theorem 11.
Letπ˜,τ1,τ2,θ be VSBTS and Y˜,F1,F2,θ be another VSBTS. Let f,θ: π˜,τ1,τ2,θ⟶Y˜,F1,F2,θ be a soft mapping. Let Y˜,F1,F2,θ be VSB∗b Hausdorff space; then, it is guaranteed that κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=fκ2⋋/Δ1/,Δ2/ is a VS∗b-closed subset of π˜,τ1,τ2,θ×Y˜,F1,F2,θ.Proof.
Given thatπ˜,τ1,τ2,θ be VSBTS and Y˜,F1,F2,θ be another VSBTS, let f,θ : π˜,τ1,τ2,θ×Y˜,F1,F2,θ be a soft mapping, such that it is continuous mapping. Y˜,F1,F2,θ is VSB∗b Hausdorff space. Then, we will prove that κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=fκ2⋋/Δ1/,Δ2/ is a VS∗b-closed subset of π˜,τ1,τ2,θ×Y˜,F1,F2,θ. Equivalently, we will prove that κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=κ2⋋/Δ1/,Δ2/c is a VS∗b-open subset of π˜,τ1,τ2,θfY˜,F1,F2,θ. Let κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∈κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/corκ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∈κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/c. Then, fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/ or ≺fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/ accordingly. Since, Y˜,F1,F2,θ is ∗b V-Hausdorff space. Certainly, fκ1⋋Δ1,Δ2,fκ2⋋/Δ1/,Δ2/ are points of Y˜,F1,F2,θ, and there exists VS∗b-open sets G,θ,k,θ∈Y˜,F1,F2,θ, such that fκ1⋋Δ1,Δ2∈G,θ, fκ1⋋Δ1,Δ2∈k,θ, provided G,θ∩˜k,θ=0π˜˜,θY˜. Since f,θ is soft continuous, f−1G,θ and f−1k,θ are VS∗b-open sets in π˜,τ1,τ2,θ, supposing κ1⋋Δ1,Δ2 and κ2⋋/Δ1/,Δ2/, respectively, and so, f−1G,θ×f−1ℋ,θ is a basic VS∗b-open set in π˜,τ1,τ2,θ×Y˜,F1,F2,θ containing κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/. Since G,θ∩˜k,θ=0θY˜, it is clear by the definition of κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=fκ2⋋/Δ1/,Δ2/, that is, f−1G,θandf−1k,θ∩˜κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ=fκ2⋋/Δ1/,Δ2/=0π,θ, that is, f−1G,θ×f−1k,θ⋐κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=fκ2⋋/Δ1/,Δ2/c. Hence, κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=fκ2⋋/Δ1/,Δ2/c implies that κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=fκ2⋋/Δ1/,Δ2/ is VS∗b closed.Theorem 12.
Letπ˜,τ1,τ2,θ be VSBTS and Y˜,F1,F2,θ be another VSBTS. Let f,θ : π˜,τ1,τ2,θ×Y˜,F1,F2,θ be VS∗b open mapping, such that it is onto. If the soft set κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2=fκ2⋋/Δ1/,Δ2/ is NS∗b closed in π˜,τ1,τ2,θ×Y˜,F1,F2,θ, then π˜,τ1,τ2,θ will behave as VSB∗b Hausdorff space.Proof.
Supposefκ1⋋Δ1,Δ2,fκ2⋋/Δ1/,Δ2/ be two points of Y˜, such that either fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/ or ≺fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/. Then, κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∉κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≻κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/ orκ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∉κ1⋋Δ1,Δ2withκ1⋋Δ1,Δ2≺κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/, that is, κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∉κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≻κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/ or κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∉κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≺κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/c. Since, κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∈κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≻κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/c or κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∈κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≺κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/c is soft in π˜,τ1,τ2,θ×Y˜,F1,F2,θ, then there exists VS∗b-open sets G,θ and G,θ in π˜,τ1,τ2,θ, such that κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∈G,θ×k,θ⋐κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≻κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/c or κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/∈G,θ×k,θ⋐κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≺κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/c. Then, since f is VS∗b open, fG,θ) and fk,θ are VS∗b-open sets in Y˜,F1,F2,θ containing fκ1⋋Δ1,Δ2 and fκ2⋋/Δ1/,Δ2/, respectively, and fG,θ∩˜fk,θ=0π˜,θ˜; otherwise, fG,θ×fk,θ∩˜κ1⋋Δ1,Δ2,withκ1⋋Δ1,Δ2≻κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≻fκ2⋋/Δ1/,Δ2/ or κ1⋋Δ1,Δ2,Δ3,κ2⋋/Δ1/,Δ2/∉κ1⋋Δ1,Δ2,κ2⋋/Δ1/,Δ2/withκ1⋋Δ1,Δ2≺κ2⋋/Δ1/,Δ2/:fκ1⋋Δ1,Δ2≺fκ2⋋/Δ1/,Δ2/=0π˜,θ˜. It follows that Y˜,F1,F2,θ is VSB∗b Hausdorff space.Theorem 13.
Letπ˜,τ1,τ2,θ be VSB second countable space, and let f,θ be a VS uncountable subset of π˜,τ1,τ2,θ. Then, at least one point of f,θ is a soft limit point of f,θ.Proof.
LetW=ℬ1,ℬ2,ℬ3,ℬ4,…,ℬn:n∈ℕforπ˜,τ1,τ2,θ.
Let, if possible, no point off,θ be a soft limit point of f,θ. Then, for each κ1⋋Δ1,Δ2∈f,θ, there exists a VS∗b-open set ρ,θκ1⋋Δ1,Δ2, such that κ1⋋Δ1,Δ2∈ρ,θκ1⋋Δ1,Δ2 and ρ,θκ1⋋Δ1,Δ2∩˜f,θ=κ1⋋Δ1,Δ2. Since W is soft base ∃ℬnκ1⋋Δ1,Δ2∈W, such that κ1⋋Δ1,Δ2∈ℬnκ1⋋Δ1,Δ2⋐ρ,θκ1⋋Δ1,Δ2. Therefore, ℬnκ1⋋Δ1,Δ2∩˜f,θ⋐ρ,θκ1⋋Δ1,Δ2∩˜f,θ=κ1⋋Δ1,Δ2. Moreover, if κ1⋋Δ1,Δ21 and κ1⋋Δ1,Δ22 be any two VS points, such that κ1⋋Δ1,Δ21≠κ1⋋Δ1,Δ22 which means either κ1⋋Δ1,Δ21≻κ1⋋Δ1,Δ22 or κ1⋋Δ1,Δ21≺κ1⋋Δ1,Δ22, then there exist ℬnκ1⋋Δ1,Δ21 and ℬnκ1⋋Δ1,Δ22 in W, such that ℬnκ1⋋Δ1,Δ21∩˜f,θ=κ1⋋Δ1,Δ21 and ℬnκ1⋋Δ1,Δ22∩˜f,θ=κ1⋋Δ1,Δ22. Now, κ1⋋Δ1,Δ21≠κ1⋋Δ1,Δ22, which guarantees that κ1⋋Δ1,Δ21≠κ1⋋Δ1,Δ22, which implies that ℬnκ1⋋Δ1,Δ21∩˜f,θ≠ℬnκ1⋋Δ1,Δ22∩˜f,θ, which implies ℬnκ1⋋Δ1,Δ21≠ℬnκ1⋋Δ1,Δ22. Thus, there exists a one-to-one soft correspondence of f,θ on to ℬnκ1⋋Δ1,Δ2:κ1⋋Δ1,Δ2∈f,θ. Now, f,θ being VS uncountable, it follows that ℬnκ1⋋Δ1,Δ2:κ1⋋Δ1,Δ2∈f,θ is NS uncountable.Theorem 14.
Letπ˜,τ1,τ2,θ and Y˜,F1,F2,θ be two VSBTS and suppose f,θ be a NS continuous function, such that f,θ:π˜,τ1,τ2,θ⟶Y˜,F1,F2,θ is a NS continuous function, and let ℒ,θ⋐π˜,τ1,τ2,θ suppose the B.V.P. Then, safely, fℒ,θ has the B.V.P..Proof.
Supposeℒ,θ be an infinite VS subset of f,θ, so that ℒ,θ contains an enumerable NS set κ1⋋Δ1,Δ2n:n∈N; then, there exists a enumerable NS set κ2⋋/Δ1/,Δ2/n:n∈N⋐ℒ,θ s.t., fκ2⋋/Δ1/,Δ2/n=κ1⋋Δ1,Δ2n. ℒ,θ has B.V.P which implies that every infinite soft subset of ℒ,θ supposes soft accumulation point belonging to ℒ,θ; this implies that κ2⋋/Δ1/,Δ2/n:n∈N has soft vague limit point, say, κ2⋋/Δ1/,Δ2/0∈ℒ,θ implies that the limit of soft sequence κ2⋋/Δ1/,Δ2/n:n∈N is κ2⋋/Δ1/,Δ2/0∈ℒ,θ⇒κ2⋋/Δ1/,Δ2/n⟶κ2⋋/Δ1/,Δ2/0∈ℒ,θ. f is soft continuous implying that it is soft continuous. Furthermore, κ2⋋/Δ1/,Δ2/n⟶κ2⋋/Δ1/,Δ2/0∈ℒ,θ⇒fκ2⋋/Δ1/,Δ2/n⟶fκ2⋋/Δ1/,Δ2/0∈fℒ,θ⇒κ1⋋Δ1,Δ2n⟶fκ2⋋/Δ1/,Δ2/0∈fℒ,θ implies that limit of a soft sequence κ1⋋Δ1,Δ2n:n∈N is fκ2⋋/Δ1/,Δ2/0∈fℒ,θ, implying that limit of a soft sequence κ1⋋Δ1,Δ2n:n∈Nisfκ2⋋/Δ1/,Δ2/0∈f,θℒ,θ. Finally, we have shown that there exists an infinite soft subset κ1⋋Δ1,Δ2n:n∈N of fℒ,θ containing a limit point fκ2⋋/Δ1/,Δ2/0∈fℒ,θ. This guarantees that fℒ,θ has B.V.P.Theorem 15.
Letπ˜,τ1,τ2,θVSBTSand letκ1⋋Δ1,Δ2n˜ be a VS sequence in π˜,τ1,τ2,θ, such that it converges to a point κ1⋋Δ1,Δ20; then, the soft set g,θ consisting of the points κ1⋋Δ1,Δ2n0 and κ1⋋Δ1,Δ2nn=1,2,3,… is soft VSB compact.Proof.
Givenπ˜,τ1,τ2,θVSBTS and let κ1⋋Δ1,Δ2n˜ be a VS sequence in π˜,τ1,τ2,θ, such that it converges to a point κ1⋋Δ1,Δ2n0, that is, κ1⋋Δ1,Δ2n˜⟶κ1⋋Δ1,Δ2n0∈π˜. Let g,θ=κ1⋋Δ1,Δ2˜,κ1⋋Δ1,Δ22˜,κ1⋋Δ1,Δ23˜,κ1⋋Δ1,Δ24˜,κ1⋋Δ1,Δ25˜,κ1⋋Δ1,Δ26˜,κ1⋋Δ1,Δ27˜,…. Let S,θα:α∈Δ be VS∗b open covering of g,θ, so that g,θ⋐∪˜S,θα:α∈Δ,κ1⋋Δ1,Δ2n0∈g,θ implies that ∃α0∈Δ, such that κ1⋋Δ1,Δ2n0∈S,θα0. According to the definition of soft convergence, κ1⋋Δ1,Δ2n0∈S,θα0∈π˜,τ1,τ2,θ implies that there exists n0∈Ns.t.n≥n0 and κ1⋋Δ1,Δ2n∈S,θα0. Evidently, S,θα0 contains the points κ1⋋Δ1,Δ2n0,κ1⋋Δ1,Δ2n0+1,κ1⋋Δ1,Δ2n0+2,(17)κ1⋋Δ1,Δ2n0+3,κ1⋋Δ1,Δ2,….κ1⋋Δ1,Δ2n0+n. Look carefully at the points and train them in a way as κ1⋋Δ1,Δ21,κ1⋋Δ1,Δ22,κ1⋋Δ1,Δ23,(18)κ1⋋Δ1,Δ24,….κ1⋋Δ1,Δ2n, generating a finite soft set. Let 1≤n0−1. Then, κ1⋋Δ1,Δ2i∈g,θ. For this, i,κ1⋋Δ1,Δ2i∈g,θ. Hence, there exists αi∈Δ, such that κ1⋋Δ1,Δ2i∈S,θαi. Evidently, g,θ⋐∪r=0n0−1S,θαi˜. This shows that S,θαi:0≤n0−1 is VS∗b open cover of g,θ. Thus, an arbitrary VS ∗b open cover S,θα:α∈Δ of g,θ is reducible to a finite VS subcover S,θαi:i=0,1,2,3,…n0−1, and it follows that g,θ is soft VSB∗b compact.Theorem 16.
Ifπ˜,τ1,τ2,θVSBTS, it has the characteristics of NS ∗b sequentially compactness. Then, π˜,τ1,τ2,θ is safely VSB∗b countably compact.Proof.
LetVSB∗b, and let ρ,θ be finite soft subset of π˜,τ1,τ2,θ. Let κ1⋋Δ1,Δ21˜,κ1⋋Δ1,Δ22˜,κ1⋋Δ1,Δ23˜,κ1⋋Δ1,Δ24˜,κ1⋋Δ1,Δ25˜,κ1⋋Δ1,Δ26˜,κ1⋋Δ1,Δ27˜,… be a soft sequence of soft points of ρ,θ. Then, ρ,θ being finite, at least one of the elements. In ρ,θ, say κ1⋋Δ1,Δ20˜ must be duplicated in an infinite number of times in the VS sequence. Hence, κ1⋋Δ1,Δ20˜,κ1⋋Δ1,Δ20˜,κ1⋋Δ1,Δ20˜,κ1⋋Δ1,Δ20˜,κ1⋋Δ1,Δ20˜,κ1⋋Δ1,Δ20˜,κ1⋋Δ1,Δ20˜,… is a soft subsequence of κ1⋋Δ1,Δ2n˜, such that it is soft constant sequence and repeatedly constructed by single soft number κ1⋋Δ1,Δ20˜, and we know that a soft constant sequence converges on itself. So, it converges to κ1⋋Δ1,Δ20˜, which belongs to ρ,θ. Hence, ρ,θ is soft sequentially VSB∗b compact.Theorem 17.
Letπ˜,τ1,τ2,θVSBTS and Y˜,F1,F2,θ be another VSBTS. Let f,θ be a soft continuous mapping of a soft vague sequentially compact NS∗b space π˜,τ1,τ2,θ into Y˜,F1,F2,θ. Then, f,θπ˜,τ1,τ2,θ is VSB∗b sequentially compact.Proof.
Givenπ˜,τ1,τ2,θVSBTS and Y˜,F1,F2,θ be another VSBTS. Let f,θ be a soft continuous mapping of a VSB sequentially compact space π˜,τ1,τ2,θ into Y˜,F1,F2,θ; then, we have to prove f,θπ˜,τ1,τ2,θ VS sequentially. For this, we proceed as let κ2⋋/Δ1/,Δ2/1˜,κ2⋋/Δ1/,Δ2/2˜,κ2⋋/Δ1/,Δ2/5˜,κ2⋋/Δ1/,Δ2/6˜,κ2⋋/Δ1/,Δ2/7˜,…κ2⋋/Δ1/,Δ2/n˜,…. be a soft sequence of VS points in f,θπ˜,τ1,τ2,θ. Then, for each n∈N, there exists κ1⋋Δ1,Δ21˜,κ1⋋Δ1,Δ22˜,κ1⋋Δ1,Δ24˜,κ1⋋Δ1,Δ25˜,κ1⋋Δ1,Δ27˜,…κ1⋋Δ1,Δ2n˜,…∈π˜,τ1,τ2,θ, such that f,θκ1⋋Δ1,Δ21˜,κ1⋋Δ1,Δ22˜,κ1⋋Δ1,Δ24˜,κ1⋋Δ1,Δ25˜,κ1⋋Δ1,Δ27˜,…κ1⋋Δ1,Δ2n˜,…∈π˜,τ1,τ2,θ=κ2⋋/Δ1/,Δ2/1˜,κ2⋋/Δ1/,Δ2/2˜,κ2⋋/Δ1/,Δ2/3˜,κ2⋋/Δ1/,Δ2/4˜,κ2⋋/Δ1/,Δ2/6˜,κ2⋋/Δ1/,Δ2/7˜,…κ2⋋/Δ1/,Δ2/n˜,….. Thus, we obtain a soft sequence κ1⋋Δ1,Δ21˜,κ1⋋Δ1,Δ22˜,κ1⋋Δ1,Δ23˜,κ1⋋Δ1,Δ24˜,κ1⋋Δ1,Δ25,κ1⋋Δ1,Δ26˜,κ1⋋Δ1,Δ27˜,…κ1⋋Δ1,Δ2n˜,…. in π˜,τ1,τ2,θ. But π˜,τ1,τ2,θ being soft sequentially VSB∗b compact, there is a VS subsequence κ1⋋Δ1,Δ2ni˜ of
κ1⋋Δ1,Δ2ni˜, such that κ1⋋Δ1,Δ2ni˜⟶κ1⋋Δ1,Δ2˜∈π˜,τ1,τ2,θ. So, by VS∗b continuity of f,θ,κ1⋋Δ1,Δ2ni˜⟶κ1⋋Δ1,Δ2θ˜ implies that f,θκ1⋋Δ1,Δ2ni˜⟶f,θκ1⋋Δ1,Δ2n˜∈f,θπ˜,τ1,τ2,θ. Thus, f,θκ2⋋/Δ1/,Δ2/ni˜ is a soft subsequence of κ2⋋/Δ1/,Δ2/1˜,κ2⋋/Δ1/,Δ2/2˜,κ2⋋/Δ1/,Δ2/3,Δ1/,Δ2/5˜,κ2⋋/Δ1/,Δ2/5˜,κ2⋋/Δ1/,Δ2/6˜,κ2⋋/Δ1/,Δ2/7˜,…κ2⋋/Δ1/,Δ2/n˜,…. that converges to f,θκ1˜ in f,θπ˜,τ1,τ2,θ. Hence, f,θπ˜,τ1,τ2,θ is VS∗b sequentially compact.Theorem 18.
Letπ˜,τ1,τ2,θVSBTS, and suppose f,θ,g,θ be two NS continuous function on VS BTS π˜,τ1,τ2,θ into VSBTSY˜,F1,F2,θ, which is VSB∗b Hausdorff. Then, soft set κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2 is VS∗b closed of π˜,τ1,τ2,θ.Proof.
Letκ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2 is a VS set of function. If κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1c=∅˜, it is clearly VS∗b open, and therefore, κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2 is VS∗b closed, that is, nothing is proved in this case. Let us consider the case when κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2c≠κ1⋋Δ1,Δ2. Let ρ∈κ1⋋Δ1,Δ2∈π˜˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2c. Then, ρ does not belong to κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1. Result in fρ≠gρ. Now, Y˜,F1,F2,θ be VSB∗b Hausdorff space, so there exists VS∗b open sets g,θ, ℌ,θ of fρ and gρ, respectively, such that g,θ and ℌ,θ, and these NS sets are mutually disjoint. By soft continuity of f,θ,g,θ, f,θ−1 as well as g,θ−1 is VS∗∗b open neighborhood of ρ, and therefore, so is f,θ−1∩˜g,θ−1 contained in κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2 for κ1⋋Δ1,Δ2∈f,θ−1∩˜g,θ−1⇒fκ1⋋Δ1,Δ2∈g,θ and gfκ1⋋Δ1,Δ2≠gκ1⋋Δ1,Δ2 because g,θ and ℌ,θ are mutually exclusive. This implies that κ1 does not belong to κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2. Therefore, ρ∈f−1g,θ∩˜g−1g,θ⋐κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2c. This shows that κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2c is neighborhood of each of its points. So, κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2cNS∗b open, and hence, κ1⋋Δ1,Δ2∈π˜:fκ1⋋Δ1,Δ2=gκ1⋋Δ1,Δ2 is VS∗b closed.Theorem 19.
Letπ˜,τ1,τ2,θVSBTS, such that it is VSB∗b Hausdorff space, and let f be soft continuous function of π˜,τ1,τ2,θ into itself. Then, the VS set of fixed points under f is a VSB∗bclosed set.Proof.
Letδ=fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2. If δc=∅˜, then is VS∗b open, and therefore, fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2VS∗b closed. So, let fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2c≠∅˜, and let κ2⋋/Δ1/,Δ2/∈fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2c. Then, κ2⋋/Δ1/,Δ2/ does not belong to fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2ɛ, and therefore, fκ2⋋/Δ1/,Δ2/≠κ2⋋/Δ1/,Δ2/. Now, κ2⋋/Δ1/,Δ2/, fκ2⋋/Δ1/,Δ2/ being two distinct points of the VSB∗b Hausdorff space π˜,τ1,τ2,θ, there exists VS∗b open sets g,θ, ℌ,θ, such that κ2⋋/Δ1/,Δ2/∈g,θ,fκ2⋋/Δ1/,Δ2/∈ℌ,θ and g,θ,ℌ,θ are disjoints. Also, by the VS continuity of f, f−1H,θ is a VS∗b-open set. We pretend that g,θ∩˜f−1ℌ,θ⋐fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2c. Since μ∈g,θ∩˜f−1ℌ,θ⇒μ∈g,θ,μ∈f−1⇒μ∈g,θ,fμ∈ℌ,θ⇒μ≠fμ. g,θ∩˜ℌ,θ=∅˜ implies that μ does not belong to fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2⇒μ∈fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2c. Therefore, κ2⋋/Δ1/,Δ2/∈g,θ∩˜f−1ℌ,θ⋐fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2c. Thus, fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2c is the NS nhd of each of its points. So, fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2c is NS∗b open, and hence, fκ1⋋Δ1,Δ2=κ1⋋Δ1,Δ2 is VSB∗b closed.


## 6. Conclusion

Vague soft topology is an extension of fuzzy soft topology. Fuzzy soft topology is directly an extension of the crisp topology which has membership functional candidates bound by length one. However, it has a shortcoming, i.e., it only addresses a membership value and is unable to address the nonmembership value. Vague soft topology is dominant over fuzzy soft topology because it supposes all the two information that is true and false at the same time. Vague soft topology has narrowed domain as compared to vague soft bitopology. Some problems are very hard to discuss in vague soft topology. So, need of vague soft bitopology is felt. In our work, we regenerated some structures in vague soft bitopological spaces with new definition, that is,∗b open sets concerning soft points. In future, we would try to see the validity of the given structures in vague soft tritopological structures under soft points of the space under the application of some more generalized vague soft open sets.



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