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DOI: 10.32604/cmes.2022.016967

ARTICLE

A New Approach to Vague Soft Bi-Topological Spaces

Arif Mehmood1, Saleem Abdullah2 and Choonkil Park*,3

1Department of Mathematics & Statistics, Riphah International University, Islamabad, 44000, Pakistan
2Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Pakistan
3Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Korea
*Corresponding Author: Choonkil Park. Email: baak@hanyang.ac.kr
Received: 15 April 2021; Accepted: 27 September 2021

Abstract: Fuzzy soft topology considers only membership value. It has nothing to do with the non-membership value. So an extension was needed in this direction. Vague soft topology addresses both membership and non-membership values simultaneously. Sometimes vague soft topology (single structure) is unable to address some complex structures. So an extension to vague soft bi-topology (double structure) was needed in this direction. To make this situation more meaningful, a new concept of vague soft bi-topological space is introduced and its structural characteristics are attempted with a new definition. In this article, new concept of vague soft bi-topological space (VSBTS) is initiated and its structural behaviors are attempted. This approach is based on generalized vague soft open sets, known as vague soft β open sets. An ample of examples are given to understand the structures. For the non-validity of some results, counter examples are provided to pay the price. Pair-wise vague soft β open and pair-wise vague soft β close sets are also addressed with examples in (VSBTS). Vague soft separation axioms are initiated in (VSBTS) concerning soft points of the space. Other separation axioms are also addressed relative to soft points of the space. Vague soft bi-topological properties are studied with the application of vague soft β open sets with respect to soft points of the spaces. The characterization of vague soft β close as well as vague soft β open sets, characteristics of Bolzano Weirstrass property, vague soft compactness and its marriage with sequences, interconnection between sequential compactness and countable compactness in (VSBTS) with respect to soft β open sets are addressed.

Keywords: Vague soft set; vague soft point; vague soft bi-topological space; vague soft β-open set and vague soft β-separation axioms

1  Introduction

During the study of the potential applications in traditional and non-classical logic, it is essential to have fuzzy soft sets, vague soft sets and neutrosophical soft sets. Researchers now deal with the complications of modeling uncertain information in the economic, engineering, environmental, sociology, medical, and numerous other fields. Classical methods do not always succeed because there may be different types of uncertainties in those fields. Zadeh [1] has created a new approach to the fuzzy set theory that was the most appropriate schedule to address uncertainty. Pawlak [2] pioneered the concept of a rough set. The approximate operations on sets were investigated. Each theory has its own inherent challenges, as Molodtsov has pointed them out [3]. Molodtsov [3] proposed an entirely new and advanced approach to modeling vagueness and uncertainty free of the complications of existing procedures. In soft set theory, among other related issues, the problem of setting the member function simply does not arise. Soft sets are regarded as neighborhood systems and they are a special case of contextual fuzzy sets. Soft set theory has potential applications in many different fields, counting the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory, and measurement theory.

Maji et al. [4] functionalized soft sets in multicriteria decision making problems by applying the technique of knowledge reduction to the information table induced by soft set. Maji et al. [5] discussed different fundamentals of soft set theory. Pei et al. [6] discussed the relationship between soft sets and information systems. Soft set is a kind of particular information system. The more general results show that soft sets and information systems of partitions have the same formally designed structures and that those soft sets and fuzzy information systems are equivalent after soft sets are extended to several classes of general cases.

Chen et al. [7] pointed out some drawbacks in [4]. They improved the work of Maji et al. [4]. Smarandache [8] generalized the soft set to the hyper-soft set by transforming the function F into a multi-attribute function. Further the author introduced the hybrids of crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic hyper-soft set.

Cagman et al. [9] defined and presented the concept of soft topology on a soft set. The authors also discussed the basis of soft topological spaces theory. Shabir et al. [10] introduced soft topological areas and examined some basic notions. Bayramov et al. [11] investigated some basic notions of soft topological spaces by using soft point approach. Khattak et al. [12] ushered the notion of soft (α,β)-open sets and their characterization in soft single point topology. Atanassov [13] initiated the idea of “intuitionistic fuzzy set” (IFS) which is an extension of the concept ‘fuzzy set’. The authors addressed various characteristics comprising of operations and relations over sets. Bayramov et al. [14] introduced some important properties of intuitionistic fuzzy soft topological spaces and defined the intuitionistic fuzzy soft closure and interior of an intuitionistic fuzzy soft set. Chen [15] addressed similarity measures between vague sets and between elements.

Hong et al. [16] discovered new functions to discuss the degree of accuracy in the grades of membership of each substitute relative to a set of conditions embodied by the values of vague. Ye [17] discovered that an improved precision function for a vague set is recommended by considering the effect on the fitting to which every alternative catches the choice maker’s necessities of a vague level, unknown degree. The author added that the precise function is more judicious than the current precise function which is in some cases is not favorable. Alhazaymeh et al. [18] introduced the conception of interval-valued vague soft sets which are an extension of the soft set. Alhazaymeh et al. [19] generalized vague soft set and its operations. Al-Quran et al. [20] extended notion of classical soft sets to neutrosophic vague soft sets by applying the theory of soft sets to neutrosophic vague sets to be more effective and beneficial.

Selvachandran et al. [21] studied vague entropy measure for complex vague soft sets. Wei et al. [22] introduced five elements associated with vague soft sets that enable GML to represent fuzziness and implement vague soft set GML modeling, which solves the problem of lack of fuzzy information expression in GML. Tahat et al. [23] introduced the concept of vague soft set ordering and addressed certain relevant properties. Xu et al. [24] introduced a vague soft set, an extension of the soft set. They presented and deliberated the fundamental features of vague soft sets. With a standard approach, Wang et al. [25] examined a few basic characteristics of vague soft topological areas.

This reference [25] has become a motivating source and it leads me to my research work excellently. Inthumathi et al. [26] introduced some generalization of vague soft open sets in vague soft topological spaces and obtained a decomposition of vague α-soft open sets by using them. In our study, the intersection, union and difference operations are re-defined on the vague soft sets in contrast to the studies [24] and the properties related to these operations are presented. Then, considering these newly defined processes, contrasting [25] vague soft topology is remodeled and further the study is extended to vague soft bitopological spaces with respect to soft points of the spaces under generalized vague soft open sets, known as vague soft β-open sets. An ample of examples are given to understand the structures.

In this article, the concept of vague soft topology is initiated and its structural characteristics are attempted with new definitions. The rest of the article is pictured as follows. Originality begins from this Section 2 because in this section some new operations are defined in vague soft set theory. Main results are addressed in Section 3. In this Section 3, some new definitions of vague soft open sets are given. With the support of this new definition, soft separations axioms and soft other separation axioms are defined in VSBTS with respect to softs. These separation axioms are verified through suitable examples. The engagement of these separation axioms with other results are also addressed.

In Section 4, some results are discussed in two vague soft bi-topological space with respect to vague soft β open sets. Vague soft product spaces are discussed with respect to soft points. The characterization of vague soft β closed as well as soft β open sets, characteristics of Bolzano Weirstrass property, vague soft compactness and its marriage with sequences, interconnection between sequentially compactness and countably compactness in vague soft bi-topology with respect to soft β open sets are addressed. In the final Section 5, some concluding remarks and future work are given.

2  Preliminaries

In this section, some basic definitions, which are soft sets, soft sub-space, soft equal space, soft difference, soft null set, soft absolute set, soft point, soft union, soft intersection, vague soft topology and vague soft neighborhood are addressed.

Definition 2.1. Let 𝓍 be an initial universe set and θ be a set of parameters. Then pair 𝒽,θ is called as vague soft set (VSS) over 𝓍, where 𝒽 is a mapping from θ to V(𝓍).

The set of all NSS over 𝓍 is denoted by VSS(𝓍). A vague set 𝒽,θ can be written as

𝒽,θ={(𝒾,<𝓍,M𝒽(𝒾)(𝓍),N𝒽(𝒾)(𝓍)>:𝓍𝓍,𝒾θ}

Definition 2.2. Let 𝓍 be an initial universe set and θ be a set of parameters. Then the vague soft set 𝓍𝒾(α,γ) defined as

𝓍𝒾(α,γ)(𝒾)(y)={(α,γ) if 𝒾=𝒾 and 𝓍=y(0,1) if 𝒾𝒾 and 𝓍y

for all 𝓍𝓍,0<α,γ1,𝒾θ, is called a vague soft point.

Definition 2.3. Let f~1,θ,f~2,θVSS(𝓍). Then for all 𝓍𝓍

1. Vague Soft Subset: f~1,θf~2,θ if Mf~1(𝒾)(𝓍)Mf~2(𝒾)(𝓍) and Nf~1(𝒾)(𝓍)Nf~2(𝒾)(𝓍) for all 𝒾θ.

2. Vague Soft Equality: f~1,θ=f~2,θ if f~1,θf~2,θ and f~2,θf~1,θ.

3. Vague Soft Intersection:

f~1,θf~2,θ=f~3,θ and is defined as

f~3,θ={(𝒾,{<𝓍,min{Mf~1(𝒾)(𝓍),Mf~2(𝒾)(𝓍)},max{Nf~1(𝒾)(𝓍),Nf~2(𝒾)(𝓍)}>}:𝒾θ}.

4. Vague Soft Union:

f~1,θf~2,θ={𝒾,{<𝓍,max{Mf~1(𝒾)(𝓍),Mf~2(𝒾)(𝓍)},min{Nf~1(𝒾)(𝓍),Nf~2(𝒾)(𝓍)}>}:𝒾θ}

More generally, the Vague Soft intersection and the Vague Soft union of a collection of {fi,θ}VSS(𝓍) are defined by

iIf~i,θ={(𝒾,{<𝓍,inf{Mfi(𝒾)(𝓍)},sup{Nfi(𝒾)(𝓍)}>}:𝒾θ}iIf~i,θ={(𝒾,{<𝓍,sup{Mfi(𝒾)(𝓍)},inf{Nfi(𝒾)(𝓍)}>}:𝒾θ}

5. The VSS defined as Mf~1(𝒾)(𝓍)=1 and Nf~1(𝒾)(𝓍)=0, for all 𝒾θ and 𝓍𝓍 is called the universal VSS denoted by 1(𝓍,θ). Also the vague set defined as Mf~1(𝒾)(𝓍)=0 and Nf~1(𝒾)(𝓍)=1 for all 𝒾θ and 𝓍𝓍 is called the empty VSS denoted by 0(𝓍,θ).

6. Vague Soft Complement: f~1,θc=1(𝓍,θ)f~1,θ={(𝒾<e,Nf~1(𝒾)(𝓍),Mf~1(𝒾)(𝓍)>:𝒾θ}

Clearly, the complements of 1(𝓍,θ) and 0(𝓍,θ) are defined

(1(𝓍,θ))c=1(𝓍,θ)1(𝓍,θ)={(𝒾,{<𝓍,1>}:𝒾θ}=0(𝓍,θ)(0(𝓍,θ))c=1(𝓍,θ)0(𝓍,θ)={(𝒾,{<𝓍,0>}:𝒾θ}=1(𝓍,θ)

Definition 2.4. Let τVSS(𝓍) then τ is named a vague soft topology on 𝓍 if the following conditions hold:

(1) 0(𝓍,θ),1(𝓍,θ)τ.

(2) τ is closed under union of VSSs.

(3) τ is closed under finite intersection of VSSs.

Then the order triple 𝓍,τ,θ is called vague soft topology on 𝓍.

3  Main Results

In this section, the notion of vague soft bi-topological space is leaked out. Examples are also reflected to understand the structures. Vague soft β separation axioms are inaugurated in vague soft bi-topological spaces concerning soft points of the space. Other β separation axioms are also addressed relative to soft points of the space in vague soft bi-topological spaces. An ample of examples are provided to secure the results.

Definition 3.1. If 𝓍,τ1,θ and 𝓍,τ2,θ are two VSTS, then 𝓍,τ1,τ2,θ is called VSBTS. If 𝓍,τ1,τ2,θ be VSBTS. A vague soft sub-set (f,θ) is said to be vague soft β open in 𝓍,τ1,τ2,θ if (f,θ)VScl(VSint(VScl(f,θ))). A vague soft sub-set (f,θ) is said to be vague soft β close in 𝓍,τ1,τ2,θ if (f,θ)VSint(VScl(VSint(f,θ))). A vague soft sub-set (f,θ) is said to be vague soft β open in 𝓍,τ1,τ2,θ pair-wisly if there exists a vague soft β open set (f1,θ) in τ1 and vague soft β open set (f2,θ) in τ2 such that (f,θ)=(f1,θ)(f2,θ).

Example 3.2. Let 𝓍={𝓍1,𝓍2,𝓍3}, θ={𝒾1,𝒾2}, τ1={0(𝓍,θ),1(𝓍,θ),(f1,θ),(f2,θ)},

τ2={0(𝓍,θ),1(𝓍,θ),(g1,θ),(g2,θ)}, where (f1,θ),(f2,θ),(g1,θ) and (g2,θ) being VSSs are as following:

[(f1,θ)(𝒾1)={<𝓍1,810,210>,<𝓍2,410,410>,<𝓍3,310,210>}(f1,θ)(𝒾2)={<𝓍1,610,310>,<𝓍2,510,110>,<𝓍3,510,410>}(f2,θ)(𝒾1)={<𝓍1,610,410>,<𝓍2,310,410>,<𝓍3,210,310>}(f2,θ)(𝒾2)={<𝓍1,510,310>,<𝓍2,410,210>,<𝓍3,310,410>}(g1,θ)(𝒾1)={<𝓍1,410,510>,<𝓍2,210,610>,<𝓍3,110,410>}(g1,θ)(𝒾2)={<𝓍1,310,410>,<𝓍2,310,310>,<𝓍3,110,510>}(g2,θ)(𝒾1)={<𝓍1,710,110>,<𝓍2,310,310>,<𝓍3,210,110>}(g2,θ)(𝒾2)={<𝓍1,710,110>,<𝓍2,310,310>,<𝓍3,210,110>}]

Then (f1,θ)(f2,θ)=((f2,θ),(f1,θ)(g1,θ)=(g1,θ),(f1,θ)(g2,θ)=(f2,θ),(g1,θ)(g2,θ)=(g1,θ),(f2,θ)(g2,θ)=(f2,θ) and (f1,θ)(f2,θ)=(f2,θ),(f1,θ)(g1,θ)=(g1,θ),(f1,θ)(g2,θ)=(f2,θ),(g1,θ)(g2,θ)=(g1,θ),(f2,θ)(g2,θ)=(g2,θ).

Therefore τ1 and τ2 are vague soft topologies on 𝓍 and so 𝓍,τ1,τ2,θ is a vague soft bitopological space.

Remark 3.3. Let 𝓍,τ1,τ2,θ be a vague soft bitopological space then τ1τ2 need not necessarily be a vague soft topological space on 𝓍.

Example 3.4. Let 𝓍={𝓍1,𝓍2,𝓍3}, θ={𝒾1,𝒾2}, τ1={0(𝓍,θ),1(𝓍,θ),(f1,θ),(f2,θ)}, τ2={0(𝓍,θ),1(𝓍,θ),(g1,θ),(g2,θ)}, where (f1,θ),(f2,θ),(g1,θ) and (g2,θ) being VSSs are as following:

[(f1,θ)(𝒾1)={<𝓍1,810,210>,<𝓍2,410,410>,<𝓍3,310,210>}(f1,θ)(𝒾2)={<𝓍1,610,310>,<𝓍2,510,110>,<𝓍3,510,410>}(f2,θ)(𝒾1)={<𝓍1,610,410>,<𝓍2,310,410>,<𝓍3,210,310>}(f2,θ)(𝒾2)={<𝓍1,510,310>,<𝓍2,410,210>,<𝓍3,310,410>}(g1,θ)(𝒾1)={<𝓍1,410,510>,<𝓍2,210,610>,<𝓍3,110,410>}(g1,θ)(𝒾2)={<𝓍1,310,410>,<𝓍2,310,310>,<𝓍3,110,510>}(g2,θ)(𝒾1)={<𝓍1,710,110>,<𝓍2,310,310>,<𝓍3,210,110>}(g2,θ)(𝒾2)={<𝓍1,710,110>,<𝓍2,310,310>,<𝓍3,210,110>}.]

Here τ1τ2={0(𝓍,θ),1(𝓍,θ),(f1,θ),(f2,θ),(f3,θ),(g1,θ),(g2,θ)} is not a vague soft topology on 𝓍.

Definition 3.5. Let 𝓍,τ1,τ2,θ be a vague soft bitopological space. Then a VSS

f,θ={(𝒾,{<𝓍,Mf(𝒾)(𝓍),Nf(𝒾)(𝓍)>}:𝓍𝓍,𝒾θ}

is called as a pairwise vague soft β open set if there exist a vague soft open f1,θ in τ1 and a vague soft β open f2,θ in τ2 such that for all 𝓍𝓍.

f,θ=f1,θf2,θ={(𝒾,{<𝓍,max{Mf1𝒾(𝓍),Mf2(𝒾)(𝓍)},min{Nf1(𝒾)(𝓍},Nf2(𝒾)(𝓍)}>}):𝒾θ}

Definition 3.6. Let 𝓍,τ1,τ2,θ be a vague soft bitopological space. Then a VSS

f,θ={(𝒾,{<𝓍,Mf(𝒾)(𝓍),Nf(𝒾)(𝓍)>}):𝓍𝓍,𝒾θ}

is called as a pairwise vague soft β open set if there exist a vague soft open set f1,θ in τ1 and a vague soft β open set f2,θ in τ2 such that for all 𝓍𝓍.

f,θ=f1,θf2,θ={(𝒾,{<𝓍,max{Mf1(𝒾)(𝓍),Mf2(𝒾)(𝓍)},min{Nf1(𝒾)(𝓍),Nf2(𝒾)(𝓍)}>}):𝒾θ}

The set of all pairwise vague open sets in 𝓍,τ1,τ2,θ is denoted by PVSO 𝓍,τ1,τ2,θ.

Definition 3.7. Let 𝓍,τ1,τ2,θ be a vague soft bitopological space. Then a NSS

f,θ={(𝒾,{<𝓍,Mf(𝒾)(x),Nf(𝒾)(𝓍)>}):𝓍𝓍,𝒾θ}

is called as a pairwise vague soft β closed set if f,θc is a pairwise vague soft β open set. It is clear that f,θ is a pairwise neutrosophic soft β closed set if there exist a vague soft β closed set f1,θ in τ1 and a vague soft β closed set f2,θ in τ2 such that for all 𝓍𝓍.

f,θ=f1,θf2,θ={(𝒾,{<𝓍,min{Mf1(𝒾)(𝓍),Mf2(𝒾)(𝓍)},max{Nf1(𝒾)(𝓍),Nf2(𝒾)(𝓍)}>}):𝒾θ}

The set of all pairwise vague β closed sets in 𝓍,τ1,τ2,θ is denoted by PVS β C𝓍,τ1,τ2,θ.

Example 3.8. Let 𝓍={𝓍1,𝓍2,𝓍3},θ={𝒾1,𝒾2},τ1={0(𝓍,θ),1(𝓍,θ),(f1,θ)}, τ2={0(𝓍,θ),1(𝓍,θ),(f2,θ)} where (f1,θ) and (f2,θ) are defined as

[(f1,θ)(𝒾1)={<𝓍1,810,210>,<𝓍2,410,410>,<𝓍3,310,210>}(f1,θ)(𝒾2)={<𝓍1,610,310>,<𝓍2,510,110>,<𝓍3,510,410>}(f2,θ)(𝒾1)={<𝓍1,610,410>,<𝓍2,510,110>,<𝓍3,510,410>}(f2,θ)(𝒾2)={<𝓍1,610,410>,<𝓍2,310,410>,<𝓍3,210,310>}.(f1,θ)(f2,θ)={(𝒾1,{𝓍1,810,210,𝓍2,510,110,𝓍3,510,210}),(𝒾2,{𝓍1,610,310,𝓍2,510,110,𝓍3,510,310})}]

is a pairwise vague soft β open set. Also

[(f1,θ)c(𝒾1)={<𝓍1,210,810>,<𝓍2,610,610>,<𝓍3,210,310>}(f1,θ)c(𝒾2)={<𝓍1,310,610>,<𝓍2,110,510>,<𝓍3,410,510>}(f2,θ)c(𝒾2)={<𝓍1,310,610>,<𝓍2,110,510>,<𝓍3,410,310>}(f2,θ)c(𝒾2)={<𝓍1,410,610>,<𝓍2,410,310>,<𝓍3,310,210>}.(f1,θ)c(f2,θ)c={(𝒾1,{𝓍1,210,810,𝓍2,610,610,𝓍3,710,810}),(𝒾2,{𝓍1,410,710,𝓍2,510,910,𝓍3,110,610})}]

is a pairwise vague soft β close set.

Theorem 3.9. Let 𝓍,τ1,τ2,θ be a vague soft bitopological space. In this case

1.    0(𝓍,θ),1(𝓍,θ)PVSO𝓍,τ1,τ2,θ.

2.    If {(fi,θ)|iΔ}PVSO𝓍,τ1,τ2,θ then iI(fi,θ)PVSO𝓍,τ1,τ2,θ.

3.    If {(gi,θ)|iΔ}PVSC𝓍,τ1,τ2,θ then iI(gi,θ)PVSC𝓍,τ1,τ2,θ.

Proof.

1. Since 0(𝓍,θ)0(𝓍,θ)=0(𝓍,θ) and 1(𝓍,θ)1(𝓍,θ)=1(𝓍,θ) then 0(𝓍,θ) and 1(𝓍,θ) are pairwise vague soft β closed sets.

2. Since (fi,θ)PVSO𝓍,τ1,τ2,θ, there exist (fi1,θ)τ1 and (fi1,θ)τ2 such that (fi1,θ)=(fi1,θ)(fi1,θ) for all iI. Then

iI(fi,θ)=iI((fi1,θ)(fi1,θ))=(iI(fi1,θ))(iI(fi2,θ)).

As τ1 and τ2 are vague soft topologies on X,iI(fi1,θ))τ1, iI(fi2,θ)τ1.

Therefore iI(fi1,θ)PVSO(𝓍,τ1,τ2,θ.

3. Since (gi,θ)PVSC𝓍,τ1,τ2,θ, there exist (gi1,θ)cτ1 and (gi2,θ)cτ2 such that (gi,θ)=(gi1,θ)(gi1,θ) for all iΔ. Then

iI(gi,θ)=iI((gi1,θ)(gi2,θ))=(iI(gi1,θ))(iI(gi2,θ)).

Then iI(gi,θ)PVSC𝓍,τ1,τ2,θ as (iI(gi1,θ))cτ1 and (iI(gi2,θ))cτ1.

Definition 3.10. Let 𝓍,τ1,τ2,θ be a vague soft bitopological space and (f,θ)VSS(𝓍). The pairwise vague soft closure of (f,θ), denoted by clPVSS(f,θ), is the intersection of all pairwise vague soft β closed sets containing (f,θ), i.e.,

clPVSS(f,θ)={(g,θ)PVSC(X)|(f,θ)(g,θ)}

It is clear that  clPVSS(f,θ) is the smallest pairwise vague soft β closed set containing (f,θ).

Example 3.11. Let 𝓍,τ1,τ2,θ be the same as in Example 3.4 and

[(g,θ)={(𝒾1,{<𝓍1,310,710>,<𝓍2,310,410>,<𝓍3,210,410>})(𝒾2,{<𝓍1,210,610>,<𝓍2,110,710>,<𝓍3,310,410})}.]

be a vague soft set over 𝓍. Now, we need to determine pairwise vague soft β closed sets in 𝓍,τ1,τ2,θ to find clPVSS(g,θ) then,

[(f2,θ)(𝒾1)={<𝓍1,610,410>,<𝓍2,310,410>,<𝓍3,210,310>}(f2,θ)(𝒾2)={<𝓍1,510,310>,<𝓍2,410,210>,<𝓍3,310,410>},(f2,θ)c={(𝒾1,{<𝓍1,410,610>,<𝓍2,410,310>,<𝓍3,310,210>}),(𝒾2,{<𝓍1,310,510>,<𝓍2,210,610>,<𝓍3,410,310>})}.]

The pairwise vague soft β closed sets which contains (g,θ) are (f2,θ)c, 1(𝓍,θ). Therefore

clPVSS(g,θ)=(f2,θ)c1(𝓍,θ)=(f2,θ)c.

Definition 3.12. Let 𝓍~,τ1,τ2,θ be vague soft bitopological space, 𝓍𝒾(i,j)𝓎𝒾/(i/,j/) are VS points. If there exist VSβ open sets (f~,θ)τ1τ2 and (~,θ)τ1τ2 such that 𝓍𝒾(i,j)(f~,θ),𝓍𝒾(i,j)(~,θ)=0(𝓍~,θ) or 𝓎𝒾/(i/,j/)(~,θ), 𝓎𝒾/(i/,j/)(f~,θ)=0(𝓍~,θ), Then 𝓍~,τ1,τ2,θ is called a VSβ0.

Definition 3.13. Let 𝓍~,τ1,τ2,θ be a VSTS over 𝓍~, 𝓍𝒾(i,j)𝓎𝒾/(i/,j/) are VS points. If there exists VSβ open sets (f~,θ)τ1τ2 & (~,θ)τ1τ2s.t.𝓍𝒾(i,j)(f~,θ),𝓍𝒾(i,j)(~,θ)=0(𝓍~,θ) and 𝓎𝒾/(i/,j/)(~,θ),𝓎𝒾/(i/,j/)(f~,θ)=0(𝓍~,θ), Then 𝓍~,τ1,τ2,θ is called a VSβ1.

Definition 3.14. Let 𝓍~,τ1,τ2,θ be a VSTS over 𝓍~, 𝓍𝒾(i,j)𝓎𝒾/(i/,j/) are VS points. If VSβ open sets (f~,θ)τ1τ2 and (~,θ)τ1τ2 such that 𝓍𝒾(a,c)(f~,θ) & 𝓎𝒾/(i/,j/)(~,θ) & (f~,θ)(~,θ)=0(𝓍~,θ), Then 𝓍~,τ1,τ2,θ is called a VSβ2.

Theorem 3.15. Let (𝓍~, τ1,τ2, θ) be a VSBTS. Then (𝓍~, τ1,τ2, θ) be a VSβ1 structure if and only if each VS point is a VSβ-close.

Proof. Let (𝓍~, τ1,τ2, θ) be a VSBTS over 𝓍.~(𝓍𝒾(i,j),θ) be an arbitrary VS point. We establish (𝓍𝒾(i,j),θ) is a soft β-open set. Let (𝓎𝒾/(i/,j/),θ)(𝓍𝒾(i,j),θ). This means that (𝓎𝒾/(i/,j/),θ)&(𝓍𝒾(i,j),θ) are two are distinct VS points. Since (𝓍~, τ1,τ2, θ) be a VSβ1 structure, there exists a VSβ-open set (~,θ) so that (𝓎𝒾/(i/,j/),θ)(~,θ)and(𝓍𝒾(i,j),θ)(~,θ)=0(𝓍~,θ). Since, (𝓍𝒾(i,j),θ)(~,θ)=0(𝓍~,θ). So (𝓎𝒾/(i/,j/),θ)(~,θ)(𝓍𝒾(i,j),θ). Thus (𝓍𝒾(i,j),θ) is a NSβ-open set, i.e., (𝓍𝒾(i,j),θ) is a VSβ-close set. Suppose that each VS point (𝓍𝒾(i,j),θ) is a VSβ-close. Then (𝓍𝒾(i,j),θ)c is a VSβ-open set. Let (𝓍𝒾(i,j),θ)(𝓎𝒾/(i/,j/),θ)=0(𝓍~,θ). Thus (𝓎𝒾/(i/,j/),θ)(𝓍𝒾(i,j),θ)c and (𝓍e(i,j),θ)(𝓍𝒾(i,j),θ)c=0(𝓍~,θ). So (𝓍~, τ1,τ2, θ) be a VS-β1 space.

Theorem 3.16. Let (𝓍~, τ1,τ2, θ) be a VSBTS over universal set𝓍.~ Then (𝓍~, τ1,τ2, θ) is VS-β2 space iff for distinct VS points (𝓍𝒾(i,j),θ)&(𝓎𝒾/(i/,j/),θ), there exists a VSβ-open set (f~,θ) containing there exists but not (𝓎𝒾/(i/,j/),θ) such that (𝓎i/(i/,j/),θ)(f~,θ).¯

Proof. Let (𝓍𝒾(i,j),θ)(𝓎𝒾/(i/,j/),θ) be two VS points in VSβ2 space. Then there exists disjoint VSβ open sets (f~,θ) and (~,θ) such that (𝓍𝒾(i,j),θ)(f~,θ) and (𝓎𝒾/(i/,j/),θ)(~,θ). Since(𝓍𝒾(i,j),θ)(𝓎𝒾/(i/,j/),θ)=0(𝓍~,θ) and (f~,θ)(~,θ)=0(𝓍~,θ).(𝓎𝒾/(i/,j/),θ)(f~,θ)(𝓎𝒾/(i/,j/),θ)(f~,θ).¯ Next suppose that, (𝓍𝒾(i,j),θ)(𝓎𝒾/(i/,j/),θ), there exists a VSβ open set (f~,θ) containing (𝓍𝒾(i,j),θ) but not (𝓎𝒾/(i/,j/),θ) such that (𝓎𝒾/(i/,j/),θ)(f~,θ)¯c that is (f~,θ) and (f~,θ)¯c are mutually exclusive VSβ open sets supposing (𝓍𝒾(i,j),θ) and (𝓎𝒾(i/,j/),θ), respectively.

Theorem 3.17. Let (𝓍~, τ1,τ2, θ) be a VSBTS. Then (𝓍~, τ1,τ2, θ) is VSβ1 space if every VS point (𝓍𝒾(i,j),θ)(f~,θ)(𝓍~,τ,θ). If there exists a VSβ open set (~,θ) such that (𝓍e(i,j),θ)(~,θ)(~,θ)¯(f~,θ), Then (𝓍~, τ, θ) a VSβ2 space.

Proof. Suppose (𝓍𝒾(i,j),θ)(𝓎𝒾/(i/,j/),θ)=0(𝓍~,θ). Since (𝓍~, τ1,τ2, θ) is NSβ1 space. (𝓍𝒾(i,j),θ)&(𝓎𝒾/(i/,j/),θ)areVSβ close sets in (𝓍~, τ, θ). Then (𝓍𝒾(i,j),θ)((𝓎𝒾/(i/,j/),θ))c(𝓍~,τ,θ). Thus there exists a VSβ open set (~,θ)(𝓍~,τ,θ) such that (𝓍𝒾(i,j),θ)(~,θ)(~,θ)¯((𝓎e/(i/,j/),θ))c. So (𝓎𝒾/(i/,j/),θ)(~,θ) and (~,θ)((~,θ))c=0(𝓍~,θ) that is (𝓍~, τ1,τ2, θ) is a VS soft β2space.

Definition 3.18. Let (𝓍~, τ1,τ2, θ) be a VSBTS. (f~,θ) be a VSβ closed set and (𝓍𝒾(i,j),θ)(f~,θ)=0(𝓍~,θ). If there exists VSβ-open sets (1~,θ)&(2~,θ) such that (𝓍𝒾(i,j),θ)(1~,θ),(f~,θ)(2~,θ) and (𝓍e(i,j),θ)(1~,θ)=0(𝓍~,θ), then (𝓍~, τ1,τ2, θ) is called a VSβ-regular space. (𝓍~, τ1,τ2, θ) is said to be VSβ3space, if is both a VS regular and VSβ1space.

Theorem 3.19. Let (𝓍~, τ1,τ2, θ) be a VSBTS. (𝓍~, τ1,τ2, θ) is VS β3space iff for every (𝓍𝒾(i,j),θ)(f~,θ) that is (~,θ)((𝓍~,τ1,τ2,θ) such that (𝓍𝒾(i,j),θ)(~,θ)(~,θ)¯(f~,θ).

Proof. Let (𝓍~, τ1,τ2, θ) is VSβ3 space and (𝓍𝒾(i,j),θ)(f~,θ)(𝓍~,τ,θ). Since (𝓍~, τ1,τ2, θ) is VSβ3 space for the VS point (𝓍𝒾(i,j),θ) and β closed set (f~,θ)c, there exists (1~,θ) and (2~,θ) such that (𝓍𝒾(i,j),θ)(1~,θ),(f~,θ)c(2~,θ)&(1~,θ)(2~,θ)=0(𝓍~,θ). Then we have (𝓍𝒾(i,j),θ)(1~,θ)(2~,θ)c(f~,θ), since (2~,θ)cVSβ close set (1~,θ)¯(2~,θ)c. Conversely, let (𝓍𝒾(i,j),θ)(h~,θ)=0(𝓍~,θ) and (h~,θ) be a VSβ close set. (𝓍𝒾(i,j),θ)(h~,θ)c and from the condition of the theorem, we have (𝓍𝒾(i,j),θ)(~,θ)(~,θ)¯(h~,θ)c. Thus (𝓍𝒾(i,j),θ)(~,θ),(h~,θ) and (~,θ)(~,θ)¯c=0(𝓍~,θ). So (𝓍~, τ, θ) is VSβ3 space.

Definition 3.20. Let (𝓍~, τ1,τ2, θ) be a VSBTS. This space is a VSβ normal space, if for every pair of disjoint VSβ close sets (f~1,θ) and (f~2,θ), there exists disjoint VSβ open sets (~1,θ) and (~2,θ) such that (f~1,θ)(~1,θ) and (f~2,θ)(~2,θ).

(𝓍~, τ, θ) is said to be a VSβ4 space if it is both a VSβ normal and VSβ1 space.

Theorem 3.21. Let (𝓍~, τ1,τ2, θ) be a VSBTS over universal set 𝓍.~ This space is a VSβ4 space if and only if for each VSβ closed set (f~,θ) and VSβ open set (~,θ) with (f~,θ)(~,θ), there exists a VSβ open set (π~,θ) such that (f~,θ)(π~,θ)(π~,θ)¯(~,θ).

Proof. Let (𝓍~, τ1,τ2, θ) be a NSβ4 over universal set 𝓍~ and let (f~,θ)(~,θ). Then (~,θ)c is a VSβ close set and (f~,θ)(~,θ)=0(𝓍~,θ). Since (𝓍~,τ1,τ2,θ) be a VSβ4 space, there exists VSβ-open sets (π~1,θ) and (π~2,θ) such that (f~,θ)(π~1,θ),(~,θ)c(π~2,θ) and (π~1,θ)(π~2,θ)=0(𝓍~,θ). Thus (f~,θ)(π~1,θ)(π~2,θ)c(~,θ),(π~2,θ)c is a VSβ close set and (π~1,θ)¯(π~2,θ)c. So (f~,θ)(π~1,θ)(π~1,θ)¯(~,θ).

Conversely, (f1~,θ) and (f2~,θ) be two disjoint VSβ close sets. Then (f1~,θ)(f2~,θ)c implies there exists VSβ open set (π,θ) such that (f1~,θ)(π~,θ)(π~1,θ)¯(f2~,θ)c. Thus (π~,θ) and (π~,θ)c are VSβ open sets and (f1~,θ)(f2~,θ)c, (f2~,θ)(π~,θ)¯c and (π~,θ)and(π~,θ)¯c=0(𝓍~,θ). (𝓍~, τ, θ) be a VSβ4 space.

Theorem 3.22. A VBS countable space in which every VS convergent sequence has a unique soft limit is a VSBβ Hausdorff space.

Proof. Let (𝓍~, τ1,τ2, θ) be VBS Hausdorff space and let (𝓍𝒾(i,j),θ)n~ be a soft convergent sequence in (𝓍~, τ1,τ2, θ). We prove that the limit of this sequence is unique. We prove this result by contradiction. Suppose (𝓍𝒾(i,j),θ)n~ converges to two soft points l~ and m~ such that l~m~. Then by trichotomy law either l~<m~ or l~>m~. Since the possess the VSBβ Hausdorff characterstics, there must happen two VSβ open sets 𝒻,θ and ρ,θ such that 𝒻,θ~ρ,θ=0(𝓍~,θ). Now, (𝓍𝒾(i,j),θ)n~ converges to l~ so there exists an integer n1 such that (𝓍𝒾(i,j),θ)n~𝒻,θnn1. Also, (𝓍𝒾(i,j),θ)n~ converges to m~ so there exists an integer n2 such that (𝓍𝒾(i,j),θ)n~ρ,θnn2. We are interested to discuss the maximum possibility, for that we must suppose maximum of both the integers which will enable us to discuss the soft sequence for single soft number now max(n1, n2) = n0. Which leads to the situation (𝓍𝒾(i,j),θ)n~𝒻,θnn0 and (𝓍𝒾(i,j),θ)n~ρ,θnn0. This implies that (𝓍𝒾(i,j),θ)n~𝒻,θ and (𝓍𝒾(i,j),θ)n~ρ,θnn0. Implies that (𝓍𝒾(i,j),θ)n~(𝒻,~ρ,θ)nn0. Which beautifully contradict the fact that 𝒻,θ~ρ,θ=0(𝓍~,θ)~. Hence, the limit of the VS sequence must be unique.

4  More Results in Vague Soft Bitopological Spaces

In section, some results are discussed in two vague soft bi-topological space with respect to vague soft β open sets. Vague soft product spaces are discussed with respect to soft points. The characterization of vague soft β closed as well as vague soft β open sets, characteristics of Bolzano Weirstrass property, vague soft compactness and its marriage with sequences, interconnection between sequentially compactness and countably compactness in vague soft bi-topology with respect to soft β open sets are addressed.

Theorem 4.1. Let (𝓍~, τ1,τ2, θ) be VSBTS such that it is VSβ Hausdorff space and (Y~,F1,F2,θ) be any VSBST. Let 𝒻,θ: (𝓍~,τ1,τ2,θ)(Y~,F1,F2,θ) be a soft function such that it is soft monotone and continuous. Then (Y~,F1,F2,θ) is also of characteristics of VBSβ Hausdorffness.

Proof. Suppose (𝓍𝒾(i,j),θ)1,(𝓍𝒾(i,j),θ)2𝓍~ such that either (𝓍𝒾(i,j),θ)1(𝓍𝒾(i,j),θ)2. Since 𝒻,θ is soft monotone. Let us suppose the monotonically increasing case. Suppose (𝓎𝒾/(i/,j/),θ)1,(𝓎𝒾/(i/,j/),θ)2Y~ such that (𝓎𝒾/(i/,j/),θ)1(𝓎𝒾/(i/,j/),θ)2 respectively such that (𝓎𝒾/(i/,j/),θ)=𝒻(𝓍𝒾(i,j),θ)1,(𝓎𝒾/(i/,j/),θ)2=𝒻(𝓍𝒾(i,j),θ)2. Since, (𝓍~, τ1,τ2, θ) is VSBβ Hausdorff space so there exists mutually disjoint VSβ open sets 𝓀1,θ and 𝓀2,θ(𝓍~,τ1,τ2,θ)𝒻(𝓀1,θ) and 𝒻(𝓀2,θ)(Y~,F1,F2,θ). We claim that 𝒻(𝓀1,θ)~𝒻(𝓀2,θ)=0(𝓍~,θ~. Otherwise 𝒻(𝓀1,θ)~𝒻(𝓀2,θ)0(𝓍~,θ)~. Suppose there exists (𝓉𝒾//(i//,j//),θ)1𝒻(𝓀1,)~𝒻(𝓀2,θ)(𝓉𝒾//(i//,j//),θ)1𝒻(𝓀1,θ) and (𝓉𝒾//(i//,j//),θ)1𝒻(𝓀2,θ),(𝓉𝒾//(i//,j//),θ)1𝒻(𝓀1,θ),𝒻 is soft one-one there exists (𝓉𝒾//(i//,j//),θ)2𝓀1,θ such that (𝓉𝒾//(i//,j//),θ)1=𝒻((𝓉𝒾//(i//,j//),θ)2),(𝓉𝒾//(i//,j//),θ)1 𝒻(𝓀2,θ) implies there exists (𝓉𝒾//(i//,j//),θ)3𝓀2,θ such that 𝒻((𝓉𝒾//(i//,j//),θ)3)𝒻((𝓉𝒾//(i//,j//),θ)2)=𝒻((𝓉𝒾//(i//,j//),θ)3). Since, 𝒻 is soft one-one (𝓉𝒾//(i//,j//),θ)2=(𝓉𝒾//(i//,j//),θ)3 (𝓉𝒾//(i//,j//),θ)2𝒻(𝓀1,θ),(𝓉𝒾//(i//,j//),θ)2𝒻(𝓀2,θ)(𝓉e//(i//,j//),θ)2𝒻(𝓀1,θ)~𝒻(𝓀2,θ). This is contradiction because 𝓀1,θ~𝓀2,θ=0(𝓍~,θ). Therefore 𝒻(𝓀1,θ)~𝒻(𝓀2,θ)=0(𝓍~,θ). Finally, (𝓍𝒾(i,j),θ)1(𝓍𝒾(i,j),θ)2 or (𝓍𝒾(i,j),θ)1(𝓍𝒾(i,j),θ)2(𝓍𝒾(i,j),θ)1(𝓍𝒾(i,j),θ)2𝒻((𝓍𝒾(i,j),θ)1)𝒻((𝓍𝒾(i,j),θ)2) or 𝒻((𝓍𝒾(i,j),θ)1)𝒻((𝓍𝒾(i,j),θ)2)𝒻((𝓍𝒾(i,j),θ)1)𝒻((𝓍𝒾(i,j),θ)2). Given a pair of points (𝓎𝒾/(i/,j/),θ)1,(𝓎𝒾/(i/,j/),θ)2Y~ such that (𝓎i/(i/,j/),θ)1(𝓎𝒾/(i/,j/),θ)2. We are able to find out mutually exclusive VSβ open sets 𝒻(𝓀1,θ),𝒻(𝓀2,θ)(Y~,F1,F2,θ) such that (𝓎𝒾/(i/,j/),θ)1𝒻(𝓀1,θ),(𝓎𝒾/(i/,j/),θ)2𝒻(𝓀2,θ). This proves that (Y~,F1,F2,θ) is VSBβ Husdorff space.

Theorem 4.2. Let (𝓍~, τ1,τ2, θ) be VSBTS and (Y~,F1,F2,θ) be an-other VSBTS which satisfies one more condition of VSBβ Hausdorffness. Let 𝒻,θ: (𝓍~,τ1,τ2,θ)(Y~,F1,F2,θ) be a soft fuction such that it is soft monotone and continuous. Then (𝓍~, τ1,τ2, θ) is also of characteristics of VSβ Hausdorfness.

Proof. Suppose (𝓍𝒾(i,j),θ)1,(𝓍𝒾(i,j),θ)2𝓍~ such that either (𝓍𝒾(i,j),θ)1(𝓍𝒾(i,j),θ)2. Let us suppose the VS monotonically increasing case. So, (𝓍𝒾(i,j),θ)1(𝓍𝒾(i,j),θ)2 implies that 𝒻((𝓍𝒾(i,j),θ)1)𝒻((𝓍𝒾(i,j),θ)2) respectively. Suppose (𝓎𝒾/(i/,j/),θ)1,(𝓎𝒾/(i/,j/),θ)2Y~ such that (𝓎𝒾/(i/,j/),θ)1(𝓎𝒾/(i/,j/),θ)2. So, (𝓎𝒾/(i/,j/),θ)1(𝓎𝒾/(i/,j/),θ)2 respectively such that (𝓎𝒾/(i/,j/),θ)1=𝒻((𝓍𝒾(i,j),θ)1),(𝓎𝒾/(i/,j/),θ)2=𝒻((𝓍𝒾(i,j),θ)2) such that (𝓍𝒾(i,j),θ)1=𝒻1(𝓎1) and (𝓍𝒾(i,j),θ)2=𝒻1((𝓎𝒾/(i/,j/),θ)2). Since (𝓎𝒾/(i/,j/),θ)1,(𝓎𝒾/(i/,j/),θ)2Y~ but (Y~,F1,F2,θ) is VSβ Hausdorff space. So according to definition (𝓎𝒾/(i/,j/),θ)1(𝓎𝒾/(i/,j/),θ)2. There definitely exists VSβ open sets 𝓀1,θ and 𝓀2,θ(Y~,F1,F2,θ) such that (𝓎𝒾/(i/,j/),θ)1𝓀1,θ and (𝓎𝒾/(i/,j/),θ)2𝓀2,θ and these two VSβ open sets are guaranteedly mutually exclusive because the possibility of one rules out the possibility of other. Since 𝒻1(𝓀1,θ) and 𝒻1(𝓀2,θ) are VSβ open in (𝓍~, τ1,τ2, θ). Now, 𝒻1(𝓀1,θ)~𝒻1(𝓀1,θ)=𝒻1(𝓀1,θ~𝓀2,θ)=𝒻1(~)=0(𝓍~,θ)~ and (𝓎𝒾/(a/,c/),θ)1𝓀1,θ𝒻1((𝓎𝒾/(i/,j/),θ)1)𝒻1(𝓀1,θ)(𝓍𝒾(i,j),θ)1(𝓀1,θ), (𝓎𝒾/(i/,j/),θ)2𝓀2,θ𝒻1((𝓎e/(i/,j/),θ)2)𝒻1(𝓀2,θ) implies that (𝓍𝒾(i,j),θ)2(𝓀2,θ). We see that it has been shown for every pair of distinct points of 𝓍~, there suppose disjoint NSβ open sets namely, 𝒻1(𝓀1,θ) and 𝒻1(𝓀2,θ) belong to (𝓍~, τ1,τ2, θ) such that (𝓍𝒾(i,j),θ)1𝒻1(𝓀1,θ) and (𝓍𝒾(i,j),θ)2𝒻1(𝓀2,θ). Accordingly, VSBTS is VSBβ Hausdorff space.

Theorem 4.3. Let (𝓍~,τ1,τ2,θ) be VSBTS and (Y~,F1,F2,θ) be an-other VSBTS. Let 𝒻,θ: (𝓍~,τ1,τ2,θ)(Y~,F1,F2,θ) be a soft mapping such that it is continuous mapping. Let (Y~,F1,F2,θ) is VSβ Hausdorff space then it is guaranteed that {((i,j),(𝓎𝒾/(i/,j/),θ)):𝒻((𝓍𝒾(i,j),θ))=𝒻((𝓎𝒾/(i/,j/),θ))} is a VSβ close sub-set of (𝓍~,τ1,τ2,θ)×(Y~,F1,F2,θ).

Proof. Given that (𝓍~,τ1,τ2,θ) be VSBTS and (Y~,F1,F2,θ) be an-other VSBTS. Let 𝒻,θ: (𝓍~,τ1,τ2,θ)(Y~,F1,F2,θ) be a soft mapping such that it is continuous mapping. (Y~,F1,F2,θ) is VSBβ Hausdorff space Then we will prove that {((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)):𝒻((𝓍𝒾 (i,j),θ))=𝒻((𝓎𝒾/(i/,j/),θ))} is a VSβ closed sub-set of (𝓍~,τ1,τ2,θ)×(Y~,F1,F2,θ). Equvalently, we will prove that {((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)):𝒻((𝓍𝒾(i,j),θ))=(𝓎𝒾/(i/,j/),θ)}c is VSβ open sub-set of (𝓍~,τ1,τ2,θ)×(Y~,F1,F2,θ). Let ((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)){((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)) with (𝓍𝒾(i,j),θ) (𝓎𝒾/(i/,j/),θ):𝒻((𝓍𝒾(i,j),θ))𝒻((𝓎𝒾/(i/,j/),θ))}cor((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)){((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)) with (𝓍𝒾(i,j),θ)(𝓎𝒾/(i/,j/),θ):𝒻((𝓍𝒾(i,j),θ))𝒻((𝓎𝒾/(i/,j/),θ))}c. Then, 𝒻((𝓍𝒾(i,j),θ))𝒻((𝓎𝒾/(i/,j/),θ)) or 𝒻((𝓍𝒾(i,j),θ))𝒻((𝓎𝒾/(i/,j/),θ)) accordingly. Since, (Y~,F1,F2,θ) is VSBβ-Hausdorff space. Certainly, 𝒻((𝓍𝒾(i,j),θ)),𝒻((𝓎𝒾/(i/,j/),θ)) are points of (Y~,F1,F2,θ), there exists VSβ open sets G,θ,𝓀,θ(Y~,F1,F2,θ) such that 𝒻((𝓍𝒾(i,j),θ))G,θ & 𝒻((𝓍𝒾(i,j),θ))𝓀,θ provided G,θ~𝓀,=0(𝓍~,θ)Y~~. Since, 𝒻,θ is soft continuous, 𝒻1(G,θ & 𝒻1(𝓀,θ are VSβ open sets in (𝓍~, τ1,τ2, θ) supposing (𝓍𝒾(i,j),θ) and (𝓎𝒾/(i/,j/),θ) respectively and so 𝒻1(G,θ×𝒻1(h,θ is basic VSβ open set in (𝓍~,τ1,τ2,θ)×(Y~,F1,F2,θ) containing ((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)). Since G,θ~𝓀,θ=0(𝓍~,θ)Y~, it is clear by the definition of {((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)):𝒻((𝓍𝒾(i,j),θ))=𝒻((𝓎𝒾/(i/,j/),θ))} that {𝒻1(G,θ&𝒻1(𝓀,θ}~{((𝓍𝒾((i,j)),θ), (𝓎𝒾/(i/,j/),θ)):𝒻(𝓍)=𝒻((𝓎𝒾/(i/,j/),θ))}=0(𝓍~,θ), that is 𝒻1(G,θ×𝒻1(𝓀,θ{((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/), θ)):𝒻((𝓍𝒾(i,j),θ))=𝒻((𝓎𝒾/(i/,j/),θ))}c. Hence {((𝓍𝒾(i,j),θ),(𝓎𝒾/(i/,j/),θ)):𝒻((𝓍𝒾(a,c),θ))=𝒻((𝓎𝒾/(i/,j/),θ))}c implies that {((𝓍𝒾(a,c),θ),(𝓎𝒾/(i/,j/), θ)):𝒻((𝓍𝒾(i,j),θ))=𝒻((𝓎𝒾/(i/,j/),θ))} is VSβ close.

Theorem 4.4. Let (𝓍~, τ1,τ2, θ) a VSB second countable space and let 𝒻,θ be VSB uncountable sub set of (𝓍~,τ1,τ2,θ). Then, at least one point of 𝒻,θ is a soft limit point of 𝒻,θ.

Proof. Let W=B1,B2,B3,B4,..Bn:nN for (𝓍~,τ1,τ2,θ).

Let, if possible, no point of 𝒻,θ be a soft limit point of 𝒻,θ. Then, for each (𝓍𝒾(i,j),θ)𝒻, there exists VSβ open set ρ,θ(𝓍𝒾(i,j),θ) such that (𝓍𝒾(i,j),θ)ρ,θ(𝓍𝒾(i,j),θ), ρ,θ(𝓍𝒾(i,j),θ)~𝒻,θ={(𝓍𝒾(i,j),θ)}. Since W is soft base there exists Bn(𝓍𝒾(i,j),θ)W such that (𝓍𝒾(i,j),θ)Bn(𝓍𝒾(i,j),θ)ρ,(𝓍𝒾(i,j),θ). Therefore, Bn(𝓍𝒾(i,j),θ)~𝒻,θρ,θ(𝓍𝒾(i,j),θ)~𝒻,θ={(𝓍𝒾(i,j),θ)}. More-over, if (𝓍𝒾(i,j),θ)1 and (𝓍e(i,j),θ)2 be any two VS points such that (𝓍v(i,j),θ)1(𝓍𝒾(i,j),θ)2 which means either (𝓍𝒾(i,j),θ)1(𝓍e(i,j),θ)2 or (𝓍𝒾(i,j),θ)1(i,j)2 then there exists Bn(𝓍𝒾(i,j),θ)1 and Bn(𝓍𝒾(i,j),θ)2 in W such that Bn(𝓍𝒾(i,j),θ)1~𝒻,θ={(𝓍𝒾(i,j),θ)1} and Bn(𝓍𝒾(i,j),θ)2~𝒻,θ={(𝓍𝒾(i,j),θ)2}. Now, (𝓍𝒾(i,j),θ)1(𝓍𝒾(i,j),θ)2 which guarantees that {(𝓍𝒾(i,j),θ)1}{(𝓍𝒾(i,j),θ)2} which implies that Bn(𝓍𝒾(i,j),θ)1~𝒻,θBn(𝓍𝒾(i,j),θ)2~𝒻,θ which implies Bn(𝓍𝒾(i,j),θ)1Bn(𝓍𝒾(i,j),θ)2. Thus, there exists a one to one soft correspondence of 𝒻, on to {Bn(𝓍𝒾(i,j),θ):(𝓍e(i,j),θ)𝒻,θ}. Now, 𝒻,θ being VS uncountable, it follows that {Bn(𝓍𝒾(i,j),θ):(𝓍𝒾(i,j),θ)𝒻,θ} is VS uncountable. But, this is purely a contradiction, since {Bn(𝓍𝒾(i,j),θ):(𝓍𝒾(i,j),θ)𝒻,θ} being a  VS sub-family of the NS countable collection W. This contradiction is taking birth that on point of 𝒻,θ is a soft limit point of 𝒻,θ, so at least one point of 𝒻,θ is a soft limit point of 𝒻,θ.

Theorem 4.5. Let (𝓍~,τ1,τ2,θ)VSBTS such that is is VSB countably compact then this space has the characteristics of Bolzano Weirstrass property.

Proof. Let (𝓍~, τ1,τ2, θ) be a VSB countably compact space and suppose, if possible, that it negates the Bolzano Weierstrass Property (BWP). Then there must exists an infinite VSβ set 𝒻,θ supposing no soft limit point. Further suppose ρ,θ be soft countability infinite soft sub-set 𝒻,θ that is ρ,θ𝒻,θ. Then the guarantee ρ,θ has no soft limit point. It follows that ρ,θ is VS soft β closed set. Also for each (𝓍𝒾(i,j),θ)n~ρ,θ,(𝓍𝒾(i,j),θ)n~ is not a soft limit point of ρ,θ. Hence there exists VSβ open set Gn,θ, such that (𝓍𝒾(i,j),θ)n~Gn,θ, Gn,θ~ρ,θ={(𝓍𝒾(i,j),θ)n~}. The the collection {Gn,θ:nN}~ρ,θc is countable VSβ open cover of (𝓍~,τ1,τ2,θ). this soft cover has no finite sub-cover. For this we exhaust a single Gn,θ, it would not be a soft cover of (𝓍~, τ1,τ2, θ) since then (𝓍𝒾(i,j),θ)n~ would be covered. Result in (𝓍~, τ1,τ2, θ) is not VS countably compact. But this contradicts the given. Hence, we are compelled to accept (𝓍~, τ1,τ2, θ) must have Bolzano Weirstrass Property.

Theorem 4.6. Let (𝓍~, τ1,τ2, θ) be a VSTS and (Y~,F1,F2,θ) be VS sub-space of (𝓍~, τ1,τ2, θ). The necessary and sufficient condition for (Y~,F1,F2,θ) to be VSβ compact relative to (Y~,F1,F2,θ) is that f,θ is VSβ compact relative to (𝓍~, τ1,τ2, θ).

Proof. First we prove that f,θ relative to (𝓍~, τ1,τ2, θ) Let {k,θi:iI} that is {k,θ1,k,θ2,k,θ3,k,θ4,} be (𝓍~,τ1,τ2,θ)VSβ open cover of f,θ, then f,θz~k,θz.k,θz(𝓍~,τ1,τ2,θ),θz(𝓍~,τ1,τ2,θ) such that k,θz=,θz~f,θ,θz,θz(𝓍~,τ1,τ2,θ) such that k,θz,θzz~k,θzz but f,θk,θz. So that f,θi~k,θz. This guarantees that {,θz:zI} is a (𝓍~,τ1,τ2,θ)VSβ open cover of k,θ which is known to be VSβ compact relative (𝓍~, τ1,τ2, θ) and hence the soft cover {z:zI} must be freezable to a finite soft cub cover, say, {,θzr:r=1,2,3,4,.,n}, Then f,θr=1nG,θzr~

f,θ~f,θf,θ~[r=1nG,θzr~]

=r=1n(~f,θ~,θzr=r=1nf,θzr~ or f,θr=1nf,θir~{k,zr:1rn} is a (𝓍~,τ1,τ2,θ)VSβ open cover of f,θ. Steping from an arbitrary (𝓍~,τ1,τ2,θ)β open cover of (Y~,F1,F2,θ), we are able to show that the VSβ cover is freezable to a finite soft sub cover {k,zr:1rn} of f,θ, meaning there by f,θ is (𝓍~,τ1,τ2,θ)VSβ compact. The condition is sufficient: Suppose (Y~,F1,F2,θ) be soft sub-space of (𝓍~, τ1,τ2, θ) and also f,θ is (𝓍~,τ1,τ2,θ)VS compact. We have to prove that f,θ is (𝓍~,τ1,τ2,θ)VSβ compact. Let {k,θ1,k,θ2,k,θ3,k,θ4,} be soft (𝓍~,τ1,τ2,θ)VSβ open cover of f,θ, so that f,θi~,θi from which f,θ~f,θf,θ~(i~,θz) or,f,θi~(f,θ~,θz). On taking k,θz=,θi~f,θ, we get f,θ~k,θi,,θi(𝓍~,τ1,τ2,θ)k,θz=,θi~f,θ(Y~,F1,F2,θ).(1). Now from (1) it is clear that {k,θ1,k,θ2,k,θ3,k,θ4,} is (Y~,F1,F2,θ)VSβopen soft cover of f,θ which is known to be (Y~,F1,F2,θ)VSβ compact hence this soft cover must be reducible to a finite soft sub-cover. say, {k,θzr:1rn}. Thisf,θr=1nk,θir~=r=1n((,θzr)~f,θ(𝓍~,τ1,τ2,θ),or

f,θ(r=1n((,θir)~f,θ))r=1n,θir,~ or

f,θr=1n,θzr. This proves that {,θzr:1rn} is a finite soft sub-cover to the soft cover ,θz. Starting from an arbitrary (𝓍~,τ1,τ2,θ)VSβ open soft cover of f,θ, we are able to show that this soft vague open cover is freezable to a finite soft sub-cover, showing there by f,θ is (𝓍~,τ1,τ2,θ)Vβ compact.

Theorem 4.7. Let (𝓍~,τ1,τ2,θ)VSBTS and let (𝓍𝒾(i,j),θ)n~ be a VS sequence in (𝓍~, τ1,τ2, θ) such that it converges to a point (𝓍𝒾(i,j),θ)n0 then the soft set ,θ consisting of the points (𝓍𝒾(i,j),θ)n0 and (𝓍𝒾(i,j),θ)n(n=1,2,3,) is soft VSβ compact.

Proof. Given (𝓍~,τ1,τ2,θ)VSBTS and let (𝓍𝒾(i,j),θ)n~ be a VS sequence in (𝓍~, τ1,τ2, θ) such that it converges to a point (𝓍𝒾(i,j),θ)n0 that is (𝓍𝒾(i,j),θ)n~(𝓍𝒾(i,j),θ)n0𝓍~. Let ,θ=(𝓍𝒾(i,j),θ)1~,(𝓍𝒾(i,j),θ)2~,(𝓍𝒾(i,j),θ)3~,(𝓍𝒾(i,j),θ)4~,(𝓍𝒾(i,j),θ)5~,(𝓍𝒾(i,j),θ)7~,. Let {S,α:αΔ} be VSβ open covering of ,θ so that ,θ~{S,θα:αΔ},(𝓍𝒾(i,j),θ)n0,θ implies that there exists α0Δs.t.(𝓍𝒾(i,j),θ)n0S,θα0. According to the definition of soft convergence, (𝓍𝒾(i,j),θ)n0S,θα0(𝓍~,τ1,τ2,θ) implies there exists n0V such that nn0 and (𝓍𝒾(i,j),θ)nS,θα0. Evidently, S,α0 contains the points (𝓍𝒾(i,j),θ)n0,(𝓍𝒾(i,j),θ)n0+1,(𝓍𝒾(i,j),θ)n0+2,(𝓍𝒾(i,j),θ)n0+3,(𝓍𝒾(i,j),θ),.(𝓍𝒾(i,j),θ)n0+n, Look carefully at the points and train them in a way as, (𝓍𝒾(i,j),θ)1,(𝓍𝒾(i,j),θ)2,(𝓍𝒾(i,j),θ)3,(𝓍𝒾(i,j), θ)4,.(𝓍𝒾(i,j),θ), generating a finite soft set. Let 1n01. Then (𝓍𝒾(i,j),θ)i,θ. For this i,(𝓍𝒾(i,j),θ)i,θ. Hence there exists αiΔ such that (𝓍𝒾(i,j),θ)iS,θαi. Evidently ,r=0n01S,θαi~. This shows that {S,θαi:0n01} is VSβ open cover of ,θ. Thus an arbitrary VS β open cover {S,θα:αΔ} of ,θ is reducible to a finite VS cub-cover {S,θαi:i=0,1,2,3,n01}, it follows that ,θ is soft VSβ compact.

Theorem 4.8. Let (𝓍~,τ1,τ2,θ)VSBTS and (Y~,F1,F2,θ) be another VSBTS. Let 𝒻,θ be a soft continuous mapping of a soft vague sequentially compact VSβ space (𝓍~,τ1,τ2,θ) into (Y~,F1,F2,θ).Then, 𝒻,θ(𝓍~,τ1,τ2,θ) is VSβ sequentially compact.

Proof. Given (𝓍~,τ1,τ2,θ)VSBTS and (Y~,F1,F2,θ) be another VSBTS. Let 𝒻,θ be a soft continuous mapping of a VS sequentially compact space (𝓍~,τ1,τ2,θ) into (Y~,F1,F2,θ) then we have to prove 𝒻,θ(𝓍~,τ1,τ2,θ)VS sequentially. For this we proceed as. Let (𝓎𝒾/(i/,j/),θ)1~,(𝓎𝒾/(i/,j/),θ)2~,,(𝓎𝒾/(i/,j/),θ)5~,(𝓎𝒾/(i/,j/),θ)6~,(𝓎𝒾/(i/,j/),θ)7~,(𝓎𝒾/(i/,j/),θ)n~, be a soft sequence of VS points in 𝒻,θ(𝓍~,τ1,τ2,θ),

Then for each nN there exists (𝓍𝒾(i,j),θ)1~,(𝓍𝒾(i,j),θ)2~,(𝓍𝒾(i,j),θ)4~,(𝓍𝒾(i,j),θ)5~,(𝓍𝒾(i,j),θ)7~,(𝓍𝒾(i,j),θ)n~,.(𝓍~,τ1,τ2,θ) such that 𝒻,θ((𝓍𝒾(i,j),θ)1~,(𝓍𝒾(i,j),θ)2~,(𝓍𝒾(i,j),θ)3~,(𝓍𝒾(i,j),θ)7~,(𝓍𝒾(i,j),θ)n~,)=(𝓎𝒾/(i/,j/),θ)1~,(𝓎𝒾/(i/,j/),θ)2~,(𝓎𝒾/(i/,j/),θ)3~,(𝓎𝒾/(i/,j/),θ)4~,(𝓎𝒾/(i/,j/),θ)6~,(𝓎𝒾/(i/,j/),θ)7~,(𝓎𝒾/(i/,j/),θ)n~,. Thus we obtain a soft sequence (𝓍𝒾(i,j),θ)1~,(𝓍𝒾(i,j),θ)2~,(𝓍𝒾(i,j),θ)3~,(𝓍𝒾(i,j),θ)4~,(𝓍𝒾(i,j),θ)~,(𝓍𝒾(i,j),θ)6~,(𝓍𝒾(i,j),θ)7~,(𝓍𝒾(i,j),θ)n~, in (𝓍~, τ1,τ2, θ). But (𝓍~,τ1,τ2,θ) being soft sequentially VSBβ compact, there is a VSB sub-sequence (𝓍𝒾(i,j),θ)ni~ of (𝓍𝒾(i,j),θ)n~ such that (𝓍𝒾(i,j),θ)ni~(𝓍𝒾(i,j),θ)~(𝓍~,τ1,τ2,θ). So, by VSβ continuity of 𝒻,θ,(𝓍𝒾(i,j),θ)ni~(𝓍𝒾(i,j),θ)~ implies that 𝒻,θ((𝓍𝒾(i,j),θ)ni~)𝒻,θ((𝓍𝒾(i,j),θ)n~)𝒻,(𝓍~,τ1,τ2,θ). Thus, 𝒻,((𝓍𝒾(i,j),θ)ni~) is a soft sub-sequence of (𝓎𝒾/(i/,j/),θ)1~,(𝓎𝒾/(i/,j/),θ)2~,(𝓎𝒾/(i/,j/),θ),(𝓍𝒾(i,j),θ)4~,(𝓎𝒾/(i/,j/),θ)5~,(𝓎𝒾/(i/,j/),θ)6~,(𝓎𝒾/(i/,j/),θ)7~,(𝓎𝒾/(i/,j/),θ)n~, converges to (𝒻,θ)((𝓍𝒾(i,j),θ)~) in 𝒻,θ(𝓍~,τ1,τ2,θ). Hence, 𝒻,θ(𝓍~,τ1,τ2,θ) is VSβ sequentially compact.

5  Conclusion

Fuzzy soft topology considers only membership value. It has nothing to do with non-membership value. So extension was needed in this direction. The concept of vague soft topology was introduced to address the issue with fuzzy soft topology. Vague soft topology addresses both membership and non-membership values simultaneously. To make this object more meaningful, the conception of vague soft bi-topological structure is ushered in and its structural characteristics are attempted with new definitions. An ample of examples are also given to understand the structures. For the non-validity of some results, counter examples are provided. Pair-wise vague open and pair-wise vague soft closed sets are also addressed with examples in vague soft bi-topological spaces. Vague soft β separation axioms are inaugurated in vague soft bitopological spaces concerning soft points of the space. Other β separation axioms are also addressed relative to soft points of the space. Vague soft bi-topological properties from one space to another and then from other space to the first space with respect to vague soft β open sets are addressed. Vague soft product spaces are discussed with respect to soft points. In future, we will try to address [27,28] in soft setting with respect to soft points of the spaces. Also, we will try to convert the work of this manuscript to hyper soft and plithogenic hyper soft set based on reference [29].

Acknowledgement: We wish to express our extra sincere appreciation to the PI of our combined project, Professor Choonkil Park, who has the substance (figure) of a genius: he became the driving force, convincingly guided and encouraged us to be professional and do the right things even when the road got tough. Without his persistent help, the goal of this project would not have been realized.

Author Contributions: All authors read and approved the final manuscript.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

References

  1.  1.  Zadeh, L. A. (1965). Fuzzy sets. Information and Control Journal, 8(3), 338–353. DOI 10.1016/S0019-9958(65)90241-X.
  2.  2.  Pawlak, Z. (1982). Rough sets. International Journal of Computer Science, 11(5), 341–356. DOI 10.1007/BF01001956.
  3.  3.  Molodtsov, D. (1999). Soft set theory–-First results. Computers & Mathematics with Applications, 4(5), 19–31. DOI 10.1016/S0898-1221(99)00056-5.
  4.  4.  Maji, K. P., Biswas, R., Roy, R. (2002). An application of soft sets in a decision making problem. Computers & Mathematics with Applications, 8(9), 1077–1083. DOI 10.1016/S0898-1221(02)00216-X.
  5.  5.  Maji, K. P., Biswas, R., Roy, R. (2003). Soft set theory. Computers & Mathematics with Applications, 45(4–5), 555–562. DOI 10.1016/S0898-1221(03)00016-6.
  6.  6.  Pie, D., Miao, D. (2005). From soft sets to information systems. IEEE International Conference on Granular Computing, 2, 617–621. DOI 10.1109/GRC.2005.1547365.
  7.  7.  Chen, D., Tsang, C. C. E., Yeung, S. D., Wang, X. (2005). The parametrization reduction of soft sets and its applications. Computers & Mathematics with Applications, 49(6), 757–763. DOI 10.1016/j .camwa. 2004.10.036.
  8.  8.  Smarandache, F. (2018). Extension of soft set to hyper-soft set and then to plithogenic hyper-soft set. Neutrosophic Sets and Systems, 22(1), 168–171.
  9.  9.  Çağman, N., Karatas, S., Enginoglu, S. (2011). Soft topology. Computers and Mathematics with Application, 62(1), 351–358. DOI 10.1016/j.camwa.2011.05.016.
  10. 10. Shabir, M., Naz, M. (2011). On soft topological spaces. Computers and Mathematics with Application, 61(7), 1786–1799. DOI 10.1016/j.camwa.2011.02.006.
  11. 11. Bayramov, S., Gunduz, C. (2018). A new approach to separability and compactness in soft topological spaces. Turkish World Mathematical Society (TWMS) Journal of Pure and Applied Mathematics, 9(1), 82–93.
  12. 12. Khattak, M. A., Khan, G. A., Ishfaq, M., Jama, F. (2017). Characterization of soft α-separation axioms and soft β-separation axioms in soft single point spaces and in soft ordinary spaces. Journal of New Theory, 19(19), 63–81.
  13. 13. Atanassov, K. (1986). Intuitionistic fuzzy set. Fuzzy Sets and Systems, 20(1), 87–96. DOI 10.1016/S0165-0114(86)80034-3.
  14. 14. Bayramov, S., Gunduz, C. (2014). On intuitionistic fuzzy soft topological spaces. Turkish World Mathematical Society (TWMS) Journal of Pure and Applied Mathematics, 5(1), 66–79.
  15. 15. Chen, M. S. (1997). Similarity measures between vague sets and between elements. IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics, 27(1), 153–158. DOI 10.1109/3477.552198.
  16. 16. Hong, H. D., Choi, H. C. (2000). Multicriteria fuzzy decision making problems based on vague set theory. Fuzzy Sets and Systems, 114(1), 103–113. DOI 10.1016/S0165-0114(98)00271-1.
  17. 17. Ye, J. (2010). Using an improved measure function of vague sets for multicriteria fuzzy decision-making. Expert Systems with Applications, 37(6), 4706–4709. DOI 10.1016/j.eswa.2009.11.084.
  18. 18. Alhazaymeh, K., Hassan, N. (2012). Interval-valued vague soft sets and its application. Advances in Fuzzy Systems, 2012(4–5), 1–7. DOI 10.1155/2012/208489.
  19. 19. Alhazaymeh, K., Hassan, N. (2012). Generalised vague soft set and its applications. International Journal of Pure and Applied Mathematics, 77(3), 391–401.
  20. 20. Al-Quran, A., Hassan, N. (2017). Neutrosophic vague soft set and its applications. Malaysian Journal of Mathematical Sciences, 11(2), 141–163.
  21. 21. Selvachandran, G., Garg, H., Quek, G. S. (2018). Vague entropy measure for complex vague soft sets. Entropy, 20(6), 1–19. DOI 10.3390/e20060403.
  22. 22. Wei, B., Xie, Q., Meng, Y., Zou, Y. (2017). Fuzzy GML modeling based on vague soft sets. International, Journal of Geo-Information, 6(1), 1–18. DOI 10.3390/ijgi6010010.
  23. 23. Tahat, N., Ahmad, B. F., Alhazaymeh, K., Hassan, N. (2015). Ordering on vague soft set. Global Journal of Pure and Applied Mathematics, 11(5), 3189–3193.
  24. 24. Xu, W., Ma, J., Wang, S., Hao, G. (2010). Vague soft sets and their properties. Computers and Mathematics with Applications, 59(2), 787–794. DOI 10.1016/j.camwa.2009.10.015.
  25. 25. Wang, C., Li, Y. (2014). Topological structure of vague soft sets. Abstract and Applied Analysis, 2014(3), 1–8. DOI 10.1155/2014/504021.
  26. 26. Inthumathi, V., Pavithra, M. (2018). Decomposition of vague α-soft open sets in vague soft topological spaces. Global Journal of Pure and Applied Mathematics, 14(3), 501–515.
  27. 27. Akram, M., Gulzar, H., Smarandache, F., Broumi, S. (2018). Certain notion of neutrosophic topological K-algebra. Mathematics, 6(11), 234. DOI 10.3390/math6110234.
  28. 28. Akram, M., Dar, K. H. (2010). Intuitionistic fuzzy topological K-algebras. Journal of Fuzzy Mathematics, 17(1), 19–34. DOI 10.3390/math6110234.
  29. 29. Smarandache, F. (2019). Extension of soft set to hyper soft set, and then to plithogenic hyper soft set. Octogon Mathematical Magazine, 27(1), 413–418.
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