Computer Modeling in Engineering & Sciences

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

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

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

for all 𝓍∈𝓍,0<α,γ≤1,𝒾∈θ, is called a vague soft point.

Definition 2.3. Let ⟨f~1,θ⟩,⟨f~2,θ⟩∈VSS(𝓍). Then for all 𝓍∈𝓍

1. Vague Soft Subset: ⟨f~1,θ⟩⊂⟨f~2,θ⟩ if Mf~1(𝒾)(𝓍)≤Mf~2(𝒾)(𝓍) and Nf~1(𝒾)(𝓍)≥Nf~2(𝒾)(𝓍) for all 𝒾∈θ.

2. Vague Soft Equality: ⟨f~1,θ⟩=⟨f~2,θ⟩ if ⟨f~1,θ⟩⊂⟨f~2,θ⟩ and ⟨f~2,θ⟩⊂⟨f~1,θ⟩.

3. Vague Soft Intersection:

⟨f~1,θ⟩∩⟨f~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)}>}:𝒾∈θ}.

4. Vague Soft Union:

⟨f~1,θ⟩∪⟨f~2,θ⟩={𝒾,{<𝓍,max{Mf~1(𝒾)(𝓍),Mf~2(𝒾)(𝓍)},min{Nf~1(𝒾)(𝓍),Nf~2(𝒾)(𝓍)}>}:𝒾∈θ}

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

5. The VSS defined as Mf~1(𝒾)(𝓍)=1 and Nf~1(𝒾)(𝓍)=0, for all 𝒾∈θ and 𝓍∈𝓍 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 𝒾∈θ and 𝓍∈𝓍 is called the empty VSS denoted by 0(𝓍,θ).

6. Vague Soft Complement: ⟨f~1,θ⟩c=1(𝓍,θ)∖⟨f~1,θ⟩={(𝒾<e,Nf~1(𝒾)(𝓍),Mf~1(𝒾)(𝓍)>:𝒾∈θ}

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

Definition 2.4. Let τ⊂VSS(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:

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 τ1∪τ2 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:

Here τ1∪τ2={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

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 𝓍∈𝓍.

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

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 𝓍∈𝓍.

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

is called as a pairwise vague soft β closed set if ⟨f,θ⟩c 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 𝓍∈𝓍.

The set of all pairwise vague β closed sets in ⟨𝓍,τ1,τ2,θ⟩ is denoted by PVS β C⟨𝓍,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

is a pairwise vague soft β open set. Also

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)∈PVSO⟨𝓍,τ1,τ2,θ⟩.

2.    If {(fi,θ)|i∈Δ}⊆PVSO⟨𝓍,τ1,τ2,θ⟩ then ⋃i∈I(fi,θ)∈PVSO⟨𝓍,τ1,τ2,θ⟩.

3.    If {(gi,θ)|i∈Δ}⊆PVSC⟨𝓍,τ1,τ2,θ⟩ then ⋂i∈I(gi,θ)∈PVSC⟨𝓍,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)∈PVSO⟨𝓍,τ1,τ2,θ⟩, there exist (fi1,θ)∈τ1 and (fi1,θ)∈τ2 such that (fi1,θ)=(fi1,θ)∪(fi1,θ) for all i∈I. Then

As τ1 and τ2 are vague soft topologies on X,⋃i∈I(fi1,θ))∈τ1, ⋃i∈I(fi2,θ)∈τ1.

Therefore ⋃i∈I(fi1,θ)∈PVSO(⟨𝓍,τ1,τ2,θ⟩.

3. Since (gi,θ)∈PVSC⟨𝓍,τ1,τ2,θ⟩, there exist (gi1,θ)c∈τ1 and (gi2,θ)c∈τ2 such that (gi,θ)=(gi1,θ)∩(gi1,θ) for all i∈Δ. Then

Then ⋂i∈I(gi,θ)∈PVSC⟨𝓍,τ1,τ2,θ⟩ as (⋂i∈I(gi1,θ))c∈τ1 and (⋂i∈I(gi2,θ))c∈τ1.

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.,

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

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,