A new approach to vague soft toplogical structures concerning soft points

Abstract

This paper concerns the study of the notion of vague soft β-open set and vague soft separation axioms in vague soft topological spaces. By using such notions and that of the vague soft pints, we study the separation axioms β_i (with i = 0, 1, 2, 3, 4) in vague soft topological spaces. We give some peculiar examples about them and we prove some relationships between them. The relationship of β_i (with i = , 1, 2, 3, 4) spaces with the closer of vague soft β-open set by means of soft points, vague soft countable spaces and their relationship with β_i (with i = , 1, 2) spaces by means of soft points are addressed. In continuation, vague soft topological, vague soft inverse topological spaces properties, Bolzano Weirstrass Property(BVP) and its topological characteristics, compact spaces and sequentially compact spaces and their relationship with separation axioms by means soft points are addressed in vague soft topological spaces.

Keywords

vague soft point (VSP)

1 Introduction

During the study towards possible applications in classical and non-classical logic, fuzzy soft sets, vague soft set and neutrosophic soft set is absolutely important. Nowadays, researchers daily deal with the complexities of modelling uncertain data in economics, engineering, environmental science, sociology, medical science, and many other fields. Classical methods are not always successful due to the reason that uncertainties appearing in these domains may be of various types. Zadeh [20] originated a new access of fuzzy set theory, which proved to be the most suitable agenda for dealing with uncertainties. While probability theory, rough sets [21], and other mathematical tools are considered as a useful approaches to designate uncertainty. Each of these theories has its own inherent difficulties as pointed out by Molodtsov [22]. Molodtsov [22, 23] suggested a completely new sophisticated approach of soft sets theory for modelling vagueness and uncertainty which is free from the complications affecting existing methods. In soft set theory the problem of setting the membership function, among other related problems, simply does not arise. Soft sets are considered as neighborhood systems, and are a special case of context-dependent 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., [24] functionalized soft sets in multicriteria decision making problems by applying the technique of knowledge reduction to the information table induced by soft set. In [25], they defined and studied several basic notions of soft set theory. In 2005, Pei and Miao [26] and Chen [27] improved the work of Maji et al. Smarandache [28] 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. Smarandache [28] 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., [1] defined the concept of soft topology on a soft set, and presented its related properties. The authors also discussed the foundations of the theory of soft topological spaces. Shabir and Naz [2] introduced soft topological spaces over an initial universe with a fixed set of parameters and investigated some basic notion. Bayramov and Gunduz [3] investigated some basic notions of soft topological spaces by using soft point approach. Khattak et al., [4] introduced the concept of soft (α, β)-open sets and their characterization in soft single point topology. Zadeh [5] introduced the concept of fuzzy set. Atanassov [6] introduced the concept of ‘intuitionistic fuzzy set’ (IFS) which is an extension of the concept ‘fuzzy set’. The authors discussed various properties including operations and relations over sets. Bayramov and Gunduz [7] 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 [8] addressed similarity measures between vague sets and between elements. Hong and Choi [9] 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 [10] discovered that an improved precision function for a vague sets 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 in some cases are not favorable. Alhazaymeh and Hassan [12] introduced the concept of interval-valued vague soft sets which are an extension of the soft set Alhazaymeh and Hassan [13] generalized intuitionistic fuzzy soft set and its operations. Al-Quran and Hassan [14]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 useful. Selvachandran et al., [15] studied fuzzy set and its extensions, the uncertainties which are present in the data are handled with the help of membership degree which is the subset of real numbers.

Wei et al., [16] introduced five elements, or five pairs, 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., [17] introduced the concept of ordering on vague soft set, and some related properties are discussed. Xu et al., [18] introduced the notion of vague soft set which is an extension to the soft set. The basic properties of vague soft sets are presented and discussed. Wang and Li [19] investigated some basic properties of vague soft topological spaces with usual approach. This reference [19] became source of motivation for my research work. Inthumathi, M. Pavithra [29] introduced some generalization of vague soft open sets in vague soft topological spaces and obtain a decomposition of vague α-soft open sets by using them. Inthumathi, M. Pavithra [30] introduced some weaker forms of vague soft continuous functions and study their characterizations. They also provided a decomposition of $V α \tilde{S}$ -continuous functions. Further, the authors introduced the notion of vague soft irresolute functions and showed the concepts of vague soft continuity and vague soft irresoluteness are independent of each other.

In our study, the intersection, union and difference operations are re-defined on the vague soft sets in contrast to the studies [18] and the properties related to these operations are presented. Then, considering these newly defined processes, unlike [19], vague soft topology is reconstructed.

In the present article, we will present the notion of vague soft topological structures relative to the soft β-open set, which is a generalization of the vague soft p-open set. The rest of this work is organized as follows:

In the next section, we introduced some fundamental concepts, basic definitions, 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. In Section 3, we also introduced some new definitions, which are necessary for our future sections of this article. Vague soft β-open set is defined in vague soft topological spaces. The study of first three separation axioms and other separation axioms with respect to soft points under the definition of vague soft β-open set in vague soft topological spaces is carried out. The relationship between these separation axioms is studied. In Section 4, vague soft countable spaces and their relationship with β_i (with i = , 1, 2) spaces by means of soft points are addressed.

In this section 5, vague soft topological, vague soft inverse topological properties, product of β_i (with i = 0, 1, 2,) spaces, Bolzano Weirstrass Property and its topological characteristics, compact spaces and sequentially compact spaces and their relationship with separation axioms by means soft points are addressed in vague soft topological spaces.

In the final section, some concluding comments are summarized, and future work is included.

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 [11]. Let $X$ be universal set and $P (X)$ denotes the power set of $X$ , E a set a set of parameters, and A ⊆ E. Pair $(f, A)$ is called a soft set over $X$ ,where $f$ a mapping is given by $f : A \to P (X)$ .

Definition 2.2. Let $X$ be universal set and $V (X)$ denotes the set of all vague sets on $X$ , E a set a set of parameters, and A ⊆ E. A pair $(f, A)$ is called a vague soft set over $X$ , where $f$ is a mapping given by $f : A \to V (X)$ . In other words, the vague soft set is a parameterized family of some elements of the set $V (X)$ and therefore it can be written as a set of order pairs $(\tilde{f}, A) = {(𝓃, 〈 𝓍, T_{\tilde{f} {(𝓃)}^{(𝓍)}}, F_{\tilde{f} {(𝓃)}^{(𝓍)}} : 𝓍 \in X 〉) : 𝓃 \in A}$

Definition 2.3. Let $(\tilde{f}, A)$ be vague soft set (VSS) over universal set $X$ . The complement of $(\tilde{f}, A)$ is signified ${(\tilde{f}, A)}^{c}$ and is defined as follows:

${(\tilde{f}, A)}^{c} = {((𝓃, [[𝓍, F_{\tilde{f} {(𝓃)}^{(𝓍)}}, T_{\tilde{f} {(𝓃)}^{(𝓍)}} : 𝓍 \in X]]) : 𝓃 \in A}$ . It’s clear that $({(\tilde{f}, A)}^{c})^{c} = (\tilde{f}, A)$ .

Definition 2.4. Let $(\tilde{f}, A)$ and $(\tilde{ρ}, A)$ two VSS over universal set $X . (\tilde{f}, A)$ is supposed to be vague soft sub-sets (VSSS) of $(\tilde{ρ}, A)$ if $T_{\tilde{f} {(𝓃)}^{(𝓍)}} ≼ T_{\tilde{ρ} {(𝓃)}^{(𝓍)}}$ , $F_{\tilde{f} {(𝓍)}^{(𝓍)}} ≽ F_{\tilde{ρ} {(𝓃)}^{(𝓍)}}$ , for all $𝓃 \in A & \forall 𝓍 \in X$ . It is signified as $(\tilde{f}, A) ⊑ (\tilde{ρ}, A) . (\tilde{f}, A)$ is said to be VSS equal to $(\tilde{ρ}, A)$ if $(\tilde{f}, A)$ is VSSS of $(\tilde{ρ}, A)$ and $(\tilde{ρ}, A)$ is VSSS of if $(\tilde{f}, A)$ . It is symbolized as $(\tilde{f}, A) = (\tilde{ρ}, A)$ .‘

Definition 2.5. Let $({\tilde{f}}_{1}, A)$ and $({\tilde{f}}_{2}, A)$ be two (VSSS) over universal set $X$ such that $({\tilde{f}}_{1}, A) \neq ({\tilde{f}}_{2}, A)$ . Then their union is signified as $({\tilde{f}}_{1}, A) \tilde{⊓} ({\tilde{f}}_{2}, A) = ({\tilde{f}}_{3}, A)$ and is defined as $({\tilde{f}}_{3}, A) = {(𝓃, 〈 𝓍, T_{{\tilde{f}}_{3} {(𝓃)}^{(𝓍)}}, F_{{\tilde{f}}_{3} {(𝓃)}^{(𝓍)}} 〉 : 𝓍 \in X) : 𝓃 \in A}$ , where, $T_{{\tilde{f}}_{3} {(𝓃)}^{(𝓍)}} = \max [T_{{\tilde{f}}_{1} {(𝓃)}^{(𝓍)}}, T_{{\tilde{f}}_{2} {(𝓃)}^{(𝓍)}}]$ and $F_{{\tilde{f}}_{3} {(𝓃)}^{(𝓍)}} = \min [F_{{\tilde{f}}_{1} {(𝓃)}^{(𝓍)}}, F_{{\tilde{f}}_{2} {(𝓃)}^{(𝓍)}}]$ .

Definition 2.6. Let $({\tilde{f}}_{1}, A)$ and $({\tilde{f}}_{2}, A)$ be two VSSS over universal set $X$ such that $({\tilde{f}}_{1}, A) \neq ({\tilde{f}}_{2}, A)$ . Then their intersection is signified as $({\tilde{f}}_{1}, A) \tilde{⊓} ({\tilde{f}}_{2}, A) = ({\tilde{f}}_{3}, A)$ and is defined as $({\tilde{f}}_{3}, A) = {(𝓃, 〈 𝓍, T_{{\tilde{f}}_{3} {(𝓃)}^{(𝓍)}}, F_{{\tilde{f}}_{3} {(𝓃)}^{(𝓍)}} 〉 : 𝓍 \in X) : 𝓃 \in A}$ ,

Where, $T_{{\tilde{f}}_{3} {(𝓃)}^{(𝓍)}} = \min [T_{{\tilde{f}}_{1} {(𝓍)}^{(𝓍)}}, T_{{\tilde{f}}_{2} {(𝓍)}^{(𝓍)}}]$ and $F_{{\tilde{f}}_{3} {(𝓍)}^{(𝓍)}} = \max [F_{{\tilde{f}}_{1} {(𝓍)}^{(𝓍)}}, F_{{\tilde{f}}_{2} {(𝓍)}^{(𝓍)}}]$ .

Definition 2.7. Vague soft set $(\tilde{f}, A)$ over universal $set X$ is said to be a null vague soft set if $T_{f {(𝓃)}^{(𝓍)}} = 0, F_{f {(𝓃)}^{(𝓍)}} = 1$ , for all $\tilde{f}$ ∈ A and for all $𝓍 \in X$ and is signified as $0_{(〈 X 〉, 𝓃)}$ .

Definition 2.8. Vague soft set $(\tilde{f}, A)$ over universal set $〈 X 〉$ is an absolute VSS if $T_{f {(𝓃)}^{(𝓍)}} = 1,, F_{f {(𝓃)}^{(𝓍)}} = 0;$ for all $\tilde{f}$ ∈ A and for all $\in X$ and is signified as $1_{(〈 X 〉, 𝓃)}$ .

Definition 2.9. Let VSS $(〈 \tilde{X} 〉, θ)$ be the family of all VS soft sets and $τ \subset VSS (〈 \tilde{X} 〉, θ)$ then τ is said to be vague soft topology (VST) on $〈 \tilde{X} 〉$ if (1). $0_{(〈 X 〉, 𝓃)}, 1_{(〈 X 〉, 𝓃)} \in τ, (2)$ . The union of any number of VS soft sets in τ belongs to τ, (3). The intersection of a finite number of VS soft sets in τ belongs to τ. Then $(〈 \tilde{X} 〉, τ, θ$ ) is said to be vague soft topological space (VSTS) over $〈 \tilde{X} 〉$ .

Definition 2.10. Let VS be the family of all vague set over $〈 \tilde{X} 〉$ and $𝓃 \in 〈 \tilde{X} 〉$ then $VS 𝓍_{((i, j)}$ is a V point, for $0 < i, j ⩽ 1$ and is defined as follows: $𝓍_{(i, j) (𝓎)} = [\begin{matrix} (i, j) if 𝓎 = x \\ (0, 1) if 𝓎 \neq x \end{matrix}]$ .

It is obvious that every NS is actually the union of its N points.

Definition 2.11. Let $VSS (〈 X 〉) ˜$ be the family of all vague soft sets over universal set $〈 \tilde{X .} 〉$ Then $VSS (𝓍_{((i, j)})^{e}$ is called a VS point if for every $𝓍 \in 〈 \tilde{X} 〉, 0 ≺ i, j ≼ 1, e \in θ$ , and is defined as follows: $𝓍_{{(i, j)}^{e^{squo (𝓎)}}}^{e} = (\begin{matrix} (i, j) if e^{s} quo = e and 𝓎 = 𝓍, \\ (0, 1) if e^{s} quo \neq e or 𝓎 \neq 𝓍 \end{matrix})$ .

Example 2.12. Suppose that universal set $〈 \tilde{X} 〉$ is assumed to be $〈 \tilde{X} 〉 = {𝓍_{1}, 𝓍_{2}}$ and the set of parameters by θ ={ e₁, e₂ }. Let us consider VS set $(\tilde{f}, θ)$ as follows; $\begin{matrix} (\tilde{f}, θ) \\ = {\begin{matrix} 〈 e_{1} = 〈 x_{1}, 〈 0.3, 0.6 〉〉〉, 〈 x_{2}, 〈 e_{2}, 〈 0.4, 0.8 〉〉〉 \\ 〈 e_{2} = 〈 x_{1}, 〈 0.4, 0.8 〉〉〉, 〈 x_{2}, 〈 e_{2}, 〈 0.3, 0.2 〉〉〉 \end{matrix}} . \end{matrix}$

It is obvious that $(\tilde{f}, θ)$ is the union of its neutrosophic soft points, $𝓍_{1_{(0.3, 0.6)}}^{e_{1}}$ , $𝓍_{1_{(0.4, 0.8)}}^{e_{2}}$ , $𝓍_{2_{(0.4, 0.8)}}^{e_{1}}$ & $𝓍_{2_{(0.3, 0.2)}}^{e_{2}}$ . $\begin{matrix} \begin{matrix} 𝓍_{1_{(0.3,, 0.6)}}^{e_{1}} = [\begin{matrix} e_{1} = {〈 𝓍_{1}, 0.3, 0.6 〉, 〈 𝓍_{2}, 0, 1 〉} \\ e_{2} = {〈 𝓍_{1}, 0, 1 〉, 〈 𝓍_{2}, 0, 1 〉} \end{matrix}], \\ 𝓍_{1_{(0.4, 0.8)}}^{e_{2}} = [\begin{matrix} e_{1} = {〈 𝓍_{1}, 0, 1 〉, 〈 𝓍_{2}, 0, 1 〉} \\ e_{2} = {〈 𝓍_{1}, 0.4, 0.8 〉, 〈 𝓍_{2}, 0, 1 〉} \end{matrix}], \\ 𝓍_{2_{(0.3, 0.6)}}^{e_{1}} = [\begin{matrix} e_{1} = {〈 𝓍_{1}, 0, 1 〉, 〈 𝓍_{2}, 0.4,, 0.8 〉} \\ e_{2} = {〈 𝓍_{1}, 0, 1 〉, 〈 𝓍_{2}, 0, 1 〉} \end{matrix}], \\ 𝓍_{2_{(0.3, 0.2)}}^{e_{2}} = [\begin{matrix} e_{1} = {〈 𝓍_{1}, 0, 1 〉, 〈 𝓍_{2}, 0, 1 〉} \\ e_{2} = {〈 𝓍_{1}, 0, 1 〉, 〈 0.3, 0.2 〉} \end{matrix}] . \end{matrix} \end{matrix}$

Definition 2.13. Let $(\tilde{f}, θ)$ be a VSS over universal set $〈 \tilde{X} 〉$ and θ be the set of parameters or decision variables. We say that $𝓍_{(i, j)}^{e} \in (\tilde{f}, θ)$ read as belonging to the $VSS (\tilde{f}, θ)$ with $i ≼ T_{\tilde{f} {(𝓃)}^{(𝓍)}}$ , $j ≽ F_{\tilde{f} {(𝓍)}^{(𝓃)}}$ for $\tilde{f}$ ∈ θ.

Definition 2.14. $(〈 \tilde{X} 〉, τ, θ$ ) be a VSTS over $〈 \tilde{X} 〉$ and $(\tilde{f}, θ)$ be a VSS over $〈 \tilde{X} 〉$ . $VSS (\tilde{f}, θ)$ in ( $〈 \tilde{X} 〉$ , τ, θ) is called a VS nbhd of the VS point $𝓍_{(i, j)}^{e} \in (\tilde{f}, θ)$ , if there exists a VS open set ( $\tilde{g}$ , θ) such that $𝓍_{(i, j)}^{e} \in (\tilde{g}, θ) ⊏ (\tilde{f}, θ)$ .

3 Vague soft separation axioms by mean of vague soft β-Open sets

In this section we study the first three separation axioms β_i (with i = 0, 1, 2) for the vague soft topological spaces by using soft points and the notion of vague soft β-open set, we give some peculiar examples about them and we prove some relationships between them. Subsequently, further separation axioms for vague soft spaces such as β- regular,β₃, β- normal and β₄ are introduced and investigated also in relation to the previous three separation axioms.

Definition 3.1. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS over $〈 \tilde{X} 〉 & (\tilde{f}, θ)$ be a VS set over $〈 \tilde{X} 〉$ . Then $(\tilde{f}, θ)$ is supposed to be a VS β-open if $(\tilde{f}, θ) \subseteq VScl (VSint (VScl (\tilde{f}, θ)))$ and VS β-closed if $(\tilde{f}, θ) \supseteq VSint (VScl (VScl (\tilde{f}, θ))) .$

Definition 3.2. Let ( $〈 \tilde{X} 〉$ , τ, θ) be a VSTS over $〈 \tilde{X} 〉$ , $𝓍_{(i, j)}^{e} \neq 𝓎_{(i^{/}, j^{/})}^{e^{/}}$ are VS points. If there exist VSβ-open sets $(\tilde{f}, θ) & (\tilde{g}, θ)$ such that $𝓍_{(i, j)}^{e} \in (\tilde{f}, θ), 𝓍_{(i, j)}^{e} ⊓ (\tilde{g}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ or $𝓎_{(i^{/}, j^{/})}^{e^{/}} \in (\tilde{g}, θ)$ , $𝓎_{(i^{/}, j^{/})}^{e^{/}} ⊓ (\tilde{f}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ , Then $(〈 \tilde{X} 〉, τ, θ)$ is called a VSβ₀-space.

Definition 3.3. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS over $〈 \tilde{X} 〉$ , $𝓍_{(i, j)}^{e} \neq 𝓎_{(i^{/}, j^{/})}^{e^{/}}$ are VS points. If there e xi sts VSβ-open sets $(\tilde{f}, θ) & (\tilde{g}, θ)$ such that $𝓍_{(i, j)}^{e} \in (\tilde{f}, θ), 𝓍_{(i, j)}^{e} ⊓ (\tilde{g}, θ) = 0_{(〈 \tilde{X} 〉, θ)} and 𝓎_{(i^{/}, j^{/})}^{e^{/}} \in (\tilde{g}, θ), 𝓎_{(i^{/}, j^{/})}^{e^{/}} ⊓ (\tilde{f}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ , then $(〈 \tilde{X} 〉, τ, θ)$ is called a VSβ₁ space.

Definition 3.4. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS over $〈 \tilde{X} 〉$ , $𝓍_{(i, j)}^{e} \neq 𝓎_{(i^{/}, j^{/})}^{e^{/}}$ are VS points. If there exists VSβ open sets $(\tilde{f}, θ)$ and ( $\tilde{g}$ , θ) such that $𝓍_{(a, c)}^{e} \in (\tilde{f}, θ)$ and $𝓎_{(i^{/}, j^{/})}^{e^{/}} \in (\tilde{g}, θ)$ & $(\tilde{f}, θ) ⊓ (\tilde{g}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ , Then $(〈 \tilde{X} 〉, τ, θ)$ is called a VSβ₂ space.

Definition 3.5. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS. $(\tilde{f}, θ)$ be a VSβ closed set and $(𝓍_{(i, j)}^{e}, θ) ⊓ (\tilde{f}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ . If there exists VSβ-open sets ( $\tilde{g}$ ₁, θ) and ( $\tilde{g}$ ₂, θ) such that $(𝓍_{(i, j)}^{e}, θ) \in ({\tilde{g}}_{1}, θ), (\tilde{f}, θ) \subset ({\tilde{g}}_{2}, θ)$ and $(𝓍_{(i, j)}^{e}, θ) ⊓ ({\tilde{g}}_{1}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ , then ( $〈 \tilde{X} 〉$ , τ, θ) is called a VS β-regular space. $(〈 \tilde{X} 〉, τ, θ)$ is said to be VSβ₃ space, if is both a VS regular and VS β₁ space.

Definition 3.6. Let ( $〈 \tilde{X} 〉$ , τ, θ) be a VSTS. This space is a VSβ normal space, if for every pair of disjoint VSβ close s e ts $({\tilde{f}}_{1}, θ)$ and $({\tilde{f}}_{2}, θ)$ , there exists disjoint VSβ o pe n s e ts ( $\tilde{g}$ ₁, θ) and ( $\tilde{g}$ ₂, θ) such that $({\tilde{f}}_{1}, θ) \subset ({\tilde{g}}_{1}, θ)$ and $({\tilde{f}}_{2}, θ) \subset ({\tilde{g}}_{2}, θ)$ . ( $〈 \tilde{X} 〉$ , τ, θ) is said to be a VS β₄space if it is both a VSβ normal and VS β₁ space.

Example 3.7. Suppose that universal set $〈 \tilde{X} 〉$ is assumed to be $〈 \tilde{X} 〉 = {𝓍_{1}, 𝓍_{2}}$ and the set of parameters by θ ={ e₁, e₂ }. Let us consider VS set $(\tilde{f}, θ)$ and 〈x₁, 〈e₁, 〈0.1, 0.7〉〉〉, 〈x₁, 〈e₂, 〈0.2, 0.6〉〉〉, 〈x₂, 〈e₁, 〈0.3, 0.5〉〉〉, 〈x₂, 〈e₂, 〈0.4, 0.4〉〉〉 be VS points. Then the family $= {0_{(〈 \tilde{X} 〉, θ)}, 1_{(〈 \tilde{X} 〉, θ)}, (\tilde{f_{1}}, θ), (\tilde{f_{2}}, θ), (\tilde{f_{3}}, θ), (\tilde{f_{4}}, θ), (\tilde{f_{5}}, θ), (\tilde{f_{6}}, θ), (\tilde{f_{7}}, θ), (\tilde{f_{8}}, θ)}$ where $(\tilde{f_{1}}, θ) = 〈 x_{1}, 〈 e_{1}, 〈 0.1, 0.7 〉〉〉$ , $(\tilde{f_{2}}, θ) = 〈 x_{1}, 〈 e_{2}, 〈 0.2, 0.6 〉〉〉$ , $(\tilde{f_{3}}, θ) = 〈 x_{2}, 〈 e_{1}, 〈 0.3, 0.5 〉〉〉, (\tilde{f_{4}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{2}}, θ), (\tilde{f_{5}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{3}}, θ), (\tilde{f_{6}}, θ) = (\tilde{f_{2}}, θ) \cup (\tilde{f_{3}}, θ), (\tilde{f_{7}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{2}}, θ) \cup (\tilde{f_{3}}, θ), (\tilde{f_{8}}, θ) = {\begin{matrix} 〈 x_{1}, 〈 e_{2}, 〈 0.2, 0.6 〉〉〉, 〈 x_{2}, 〈 e_{1}, 〈 0.3, 0.5 〉〉〉, \\ 〈〈 x_{2}, e_{2}, 〈 0.4, 0.4 〉〉〉 \end{matrix}}$ is a VSTS. Thus $(\tilde{X}, τ, θ)$ be a VSTS. Also $(〈 \tilde{X} 〉, τ, θ)$ is VSβ₀ structure but it is not VSβ₁ because for VS points 〈x₁, 〈e₁, 〈0.1, 0.7〉〉〉, 〈x₂, 〈e₂, 〈0.4, 0.4〉〉〉, $(〈 \tilde{X} 〉, τ, θ)$ not VSβ₁.

Example 3.8. Suppose universal set $〈 \tilde{X} 〉$ is assumed to be $〈 \tilde{X} 〉 = {𝓍_{1}, 𝓍_{2}}$ and the set of conditions by θ ={ e₁, e₂ }. Let us consider VSS $(\tilde{f}, θ)$ and 〈x₁, 〈e₁, 〈0.1, 0.7〉〉〉, 〈x₁, 〈e₂, 〈0.2, 0.6〉〉〉, 〈x₂, 〈e₁, 〈0.3, 0.5〉〉〉, 〈x₂, 〈e₂, 〈0.4, 0.4〉〉〉 be VS points. Then the family $= {0_{(\tilde{X}, θ)}, 1_{(\tilde{X}, θ)}, (\tilde{f_{1}}, θ), (\tilde{f_{2}}, θ)$ , $(\tilde{f_{3}}, θ)$ , $(\tilde{f_{4}}, θ), (\tilde{f_{5}}, θ), (\tilde{f_{6}}, θ)$ , $(\tilde{f_{7}}, θ), (\tilde{f_{8}}, θ), \dots \dots \dots \dots (\tilde{f_{15}}, θ)}$ , where $(\tilde{f_{1}}, θ) = 〈 x_{1}, 〈 e_{1}, 0.1, 0.7 〉〉$ , $(\tilde{f_{2}}, θ) = 〈 x_{1}, 〈 e_{2}, 〈 0.2, 0.6 〉〉〉, (\tilde{f_{3}}, θ) = 〈 x_{2}, 〈 e_{1}, 〈 0.3, 0.5 〉〉〉, (\tilde{f_{4}}, θ) = 〈 x_{2}, 〈 e_{2}, 〈 0.4, 0.4 〉〉〉, (\tilde{f_{5}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{2}}, θ) = (\tilde{f_{6}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{3}}, θ), (\tilde{f_{7}}, θ) = (\tilde{f_{2}}, θ) \cup (\tilde{f_{4}}, θ), (\tilde{f_{8}}, θ) = (\tilde{f_{2}}, θ) \cup (\tilde{f_{3}}, θ), (\tilde{f_{9}}, θ) = (\tilde{f_{2}}, θ) \cup (\tilde{f_{4}}, θ), (\tilde{f_{10}}, θ) = (\tilde{f_{3}}, θ) \cup (\tilde{f_{4}}, θ) = (\tilde{f_{11}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{2}}, θ) \cup (\tilde{f_{3}}, θ), (\tilde{f_{12}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{2}}, θ) \cup (\tilde{f_{4}}, θ), (\tilde{f_{13}}, θ) = (\tilde{f_{2}}, θ) \cup (\tilde{f_{3}}, θ) \cup (\tilde{f_{4}}, θ) \cup (\tilde{f_{14}}, θ) = (\tilde{f_{1}}, θ) \cup (\tilde{f_{3}}, θ) \cup (\tilde{f_{4}}, θ), (\tilde{f_{15}}, θ) = {\begin{matrix} x_{1}, e_{1}, 0.1, 0.7, x_{1}, e_{2}, 0.2, 0.6, \\ x_{2}, e_{1}, 0.3, 0.5, \\ x_{2}, e_{2}, 0.4, 0.4 \end{matrix}}$ , is a VSTS . Thus $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS. Also $(〈 \tilde{X} 〉, τ, θ)$ is VSβ₂ structure.

Theorem 3.9. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS. Then $(〈 \tilde{X} 〉, τ, θ)$ be a VSβ₁ structure if and only if each VS point is a VSβ-close.

Proof. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS over $\tilde{〈 X 〉 .} (𝓍_{(i, j)}^{e}, θ)$ be an arbitrary VS point. We establish $(𝓍_{(i, j)}^{e}, θ)$ is a V soft β-open set. Let $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \neq (𝓍_{(i, j)}^{e}, θ)$ . This means that $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ and $(𝓍_{(i, j)}^{e}, θ)$ are two are distinct VS points. Thus 𝓃 ≠ 𝓎 or e^/ ≠ e. Since $(〈 \tilde{X} 〉, τ, θ)$ be a VSβ₁ structure, there exists a VSβ-open set ( $\tilde{g}$ , θ) so that $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \in (\tilde{g}, θ)$ and $(𝓍_{(i, j)}^{e}, θ) ⊓ (\tilde{g}, θ) = 0_{(〈 \tilde{X} 〉, θ) .}$ Since, $(𝓍_{(i, j)}^{e}, θ) ⊓ (\tilde{g}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ . So $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \in (\tilde{g}, θ) \subset (𝓍_{(i, j)}^{e}, θ)$ . Thus $(𝓍_{(i, j)}^{e}, θ)$ is a NSβ-open set, i.e., $(𝓍_{(i, j)}^{e}, θ)$ is a VSβ-closed set. Suppose that each VS point $(𝓍_{(i, j)}^{e}, θ)$ is a VSβ-closed. Then ${(𝓍_{(i, j)}^{e}, θ)}^{c}$ is a VSβ-open set. Let $(𝓍_{(i, j)}^{e}, θ) ⊓ (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) = 0_{(〈 \tilde{X} 〉, θ) .}$ Thus $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \in {(𝓍_{(i, j)}^{e}, θ)}^{c}$ & $(𝓍_{(i, j)}^{e}, θ) ⊓ {(𝓍_{(i, j)}^{e}, θ)}^{c} = 0_{(〈 \tilde{X} 〉, θ) .}$ So $(〈 \tilde{X} 〉, τ, θ)$ be a VS-β₁ space.

Theorem 3.10. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS over universal $set \tilde{〈 X 〉 .}$ Then $(〈 \tilde{X} 〉, τ, θ)$ is VS-β₂ space if and only if for distinct VS points $(𝓍_{(i, j)}^{e}, θ)$ and $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ , there exists a VSβ-open set $(\tilde{f}, θ)$ containing there exists but not $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ such that $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin \bar{(\tilde{f}, θ) .}$

Proof. Let $(𝓍_{(i, j)}^{e}, θ) \neq (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ be two VS points in VSβ₂ space. Then there exists disjoint VS * β open sets $(\tilde{f}, θ)$ and ( $\tilde{g}$ , θ) such that $(𝓍_{(i, j)}^{e}, θ) \in (\tilde{f}, θ)$ and $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \in (\tilde{g}, θ)$ . Since $(𝓍_{(i, j)}^{e}, θ) ⊓ (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ and $(\tilde{f}, θ) ⊓ (\tilde{g}, θ) = 0_{(〈 \tilde{X} 〉, θ)} . (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin (\tilde{f}, θ)$ implies that $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin \bar{(\tilde{f}, θ) .}$ Next suppose that, $(𝓍_{(i, j)}^{e}, θ) ≻ (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ , there exists a VSβ open set $(\tilde{f}, θ)$ containing $(𝓍_{(i, j)}^{e}, θ)$ but not $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ such that $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin {\bar{(\tilde{f}, θ)}}^{c}$ that is $(\tilde{f}, θ)$ and ${\bar{(\tilde{f}, θ)}}^{c}$ are disjoint VSβ open sets supposing $(𝓍_{(i, j)}^{e}, θ)$ and $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ respectively.

Theorem 3.11. Let ( $〈 \tilde{X} 〉$ , τ, θ) be a VSTS. ( $〈 \tilde{X} 〉$ , τ, θ) is VS β₃ space if and only if for every $(𝓍_{(i, j)}^{e}, θ) \in (\tilde{f}, θ)$ that is ( $\tilde{g}$ , θ)∈ ( $〈 \tilde{X} 〉$ , τ, θ) such that $(𝓍_{(i, j)}^{e}, θ) \in (\tilde{g}, θ) \subset \bar{(\tilde{g}, θ)} \subset (\tilde{f}, θ)$ .

Proof. Let ( $〈 \tilde{X} 〉$ , τ, θ) is VSβ₃ space and $(𝓍_{(i, j)}^{e}, θ) \in (\tilde{f}, θ) \in (〈 \tilde{X} 〉, τ, θ)$ . Since ( $〈 \tilde{X} 〉$ , τ, θ) is VSβ₃ space for the VS point $(𝓍_{(i, j)}^{e}, θ) &$ β closed set ${(\tilde{f}, θ)}^{c}$ , there exists ( $\tilde{g}$ ₁, θ) & ( $\tilde{g}$ ₂, θ) such that $(𝓍_{(i, j)}^{e}, θ) \in ({\tilde{g}}_{2}, θ), {(\tilde{f}, θ)}^{c} \subset ({\tilde{g}}_{2}, θ) & ({\tilde{g}}_{1}, θ) \cap ({\tilde{g}}_{2}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ . Then we have $(𝓍_{(i, j)}^{e}, θ) \in ({\tilde{g}}_{1}, θ) \subset {({\tilde{g}}_{2}, θ)}^{c} \subset (\tilde{f}, θ)$ . Since ( $\tilde{g}$ ₂, θ) ^cVSβ close set. $\bar{({\tilde{g}}_{1}, θ)} \subset {({\tilde{g}}_{2}, θ)}^{c}$ . Conversely, let $(𝓍_{(i, j)}^{e}, θ) \cap (\tilde{H}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ and $(\tilde{H}, θ)$ be a VSβ closed set. $(𝓍_{(i, j)}^{e}, θ) \in {(\tilde{H}, θ)}^{c}$ . So, we have $(𝓍_{(i, j)}^{e}, θ) \in (\tilde{g}, θ) \subset \bar{(\tilde{g}, θ)} \subset {(\tilde{H}, θ)}^{c}$ . Thus $(𝓍_{(i, j)}^{e}, θ) \in (\tilde{g}, θ), (\tilde{H}, S) \subset {\bar{(\tilde{g}, θ)}}^{c}$ and $(\tilde{g}, θ) \cap {\bar{(\tilde{g}, θ)}}^{c} = 0_{(〈 \tilde{X} 〉, θ)}$ . So ( $〈 \tilde{X} 〉$ , τ, θ) is VS β₃ space.

Theorem 3.12. Let ( $〈 \tilde{X} 〉$ , τ, θ) be a VSTS over universal set $\tilde{〈 X 〉 .}$ This space is a VSβ₄ space if and only if for each VSβ closed set $(\tilde{f}, θ)$ and VSβ open set ( $\tilde{g}$ , θ) with $(\tilde{f}, θ) \subset (\tilde{g}, θ)$ , there exists a VSβ open set $(\tilde{π}, θ)$ such that $(\tilde{f}, θ) \subset (\tilde{π}, θ) \subset \bar{(\tilde{π}, θ)} \subset (\tilde{g}, θ)$ .

Proof. Let ( $〈 \tilde{X} 〉$ , τ, θ) be a NS β₄ over universal set $〈 \tilde{\tilde{X}} 〉$ and let $(\tilde{f}, θ) \subset (\tilde{g}, θ)$ . Then ( $\tilde{g}$ , θ) ^c is a VSβ closed set and $(\tilde{f}, θ) \cap (\tilde{g}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ . Since $〈 \tilde{X} 〉$ , τ, θ) be a VS β₄ space, there exists VS β-open sets $({\tilde{π}}_{1}, θ)$ and $({\tilde{π}}_{2}, θ)$ such that $(\tilde{f}, θ) \subset ({\tilde{π}}_{1}, θ), {(\tilde{g}, θ)}^{c} \subset ({\tilde{π}}_{2}, θ) & ({\tilde{π}}_{1}, θ) \cap ({\tilde{π}}_{2}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ . Thus $(\tilde{f}, θ) \subset ({\tilde{π}}_{1}, θ) \subset {({\tilde{π}}_{2}, θ)}^{c} \subset (\tilde{g}, θ), {({\tilde{π}}_{2}, θ)}^{c}$ is a VS β closed set and $\bar{({\tilde{π}}_{1}, θ)} \subset {({\tilde{π}}_{2}, θ)}^{c}$ . So $(\tilde{f}, θ) \subset ({\tilde{π}}_{1}, θ) \subset \bar{({\tilde{π}}_{1}, θ)} \subset (\tilde{g}, θ)$ .

Conversely, $(\tilde{f_{1}}, θ)$ and $(\tilde{f_{2}}, θ)$ be two disjoint VSβ closed sets. Then $(\tilde{f_{1}}, θ) \subset {(\tilde{f_{2}}, θ)}^{c}$ implies there exists VSβ open s et (π, θ) such that $(\tilde{f_{1}}, θ) \subset (\tilde{π}, θ) \subset \bar{({\tilde{π}}_{1}, θ)} \subset {(\tilde{f_{2}}, θ)}^{c}$ . Thus $(\tilde{π}, θ)$ and ${(\tilde{π}, θ)}^{c}$ are VSβ open sets and $(\tilde{f_{1}}, θ) \subset {(\tilde{f_{2}}, θ)}^{c}$ , $(\tilde{f_{2}}, θ) \subset {\bar{(\tilde{π}, θ)}}^{c}$ and $(\tilde{π}, θ)$ and ${\bar{(\tilde{π}, θ)}}^{c} = 0_{(〈 \tilde{X} 〉, θ)}$ . Then ( $〈 \tilde{X} 〉$ , τ, θ) is a VS β₄ space.

4 Some structral results engagement

In this section, vague soft countable spaces and their relationship with β_i (with i = , 1, 2) spaces by means of soft points are addressed.

Theorem 4.1. Let $(𝓍_{(a, c)}^{e}, θ)$ be a point in a VS first countable space $(〈 \tilde{X} 〉, τ, θ)$ and let ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots .}$ , generates a VS countable β open base about the point $H$ , then there exists an infinite soft sub-sequence ${〈 V, θ 〉_{z} : z = 1, 2, 3, \dots .}$ of the VS sequence ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots .}$ , such that (i) for any VSβ-open set $〈 U, θ 〉$ , containing $(𝓍_{(i, j)}^{e}, θ)$ there exists a suffix m such that $〈 V, θ 〉_{i} \subset 〈 U, θ 〉$ for all $z ≽ m;$ and (ii) if $(〈 \tilde{X} 〉, τ, θ)$ be, in particular, a VSβ₁-space, then $\tilde{\cap} {〈 V, θ 〉_{z} : z = 1, 2, 3, \dots} = {(𝓍_{(i, j)}^{e}, θ)}$ .

Proof. Given $〈 W, θ 〉_{1} \tilde{\cap} 〈 W, θ 〉_{2} \tilde{\cap} 〈 W, θ 〉_{3} \tilde{\cap} \dots . . 〈 \tilde{\cap} W, θ 〉_{k}$ is VSβ open sets, containing the point $(𝓍_{(a, c)}^{e}, θ)$ . As the VS sets ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots .}$ . Forms VSβ open base about $(𝓍_{(a, c)}^{e}, θ)$ , ∃ one among the VS sets ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots .}$ , which we shall denote by $〈 V, θ 〉_{k}$ , such that $(𝓍_{(a, c)}^{e}, θ) \in 〈 V, θ 〉_{k} \subset$ ( $〈 W, θ 〉_{1} \tilde{\cap} 〈 W, θ 〉_{2} \tilde{\cap} 〈 W, θ 〉_{3} \tilde{\cap} \dots . . \tilde{\cap} 〈 W, θ 〉_{k}), fork = 1, 2, 3, \dots$ . The VS sequence ${〈 V, θ 〉_{i} : i = 1, 2, 3, \dots}$ ,

Thus obtained, has the required properties. In fact, if $〈 U, \partial 〉$ is any VSβopen set, containing $(𝓍_{(i, j)}^{e}, θ)$ , then there exist a VS set $〈 W, θ 〉_{m}$ say, belonging to the family ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots .}$ such that $(𝓍_{(i, j)}^{e}, θ) \in 〈 W, θ 〉_{m} \subset 〈 U, θ 〉$ . Also, since $〈 V, θ 〉_{z} \subset 〈 W, \partial 〉_{m}, forall z ≽ m$ . Next, let $(〈 \tilde{X} 〉, τ, θ)$ be VSβ₁-space and let $\tilde{\cap} {〈 V, θ 〉_{z} : z = 1, 2, 3, \dots} = M, θ$ . As $(𝓍_{(i, j)}^{e}, θ)$ is contained in each ${〈 V, θ 〉_{z} : z = 1, 2, 3, \dots}$ that is $(𝓍_{(i, j)}^{e}, θ) \in 〈 V, θ 〉_{1}, (𝓍_{(i, j)}^{e}, θ) \in 〈 V, θ 〉_{1}, (𝓍_{(i, j)}^{e}, θ) \in 〈 V, θ 〉_{2}, (𝓍_{(i, j)}^{e}, θ) \in 〈 V, θ 〉_{3}, (𝓍_{(i, j)}^{e}, θ) \in 〈 V, θ 〉_{4}, \dots \dots . . (𝓍_{(a, c)}^{e}, θ) \in 〈 V, θ 〉_{n},$ it follows that $(𝓍_{(i, j)}^{e}, θ) \in M, θ$ . Let $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ be any point of $〈 \tilde{X} 〉$ , from $(𝓍_{(i, j)}^{e}, θ)$ that is $(𝓍_{(a, c)}^{e}, θ) \neq (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ . By definition of VSβ₁-space, there exists VSβ open set $〈 U, θ 〉$ such that $(𝓍_{(i, j)}^{e}, θ) \in 〈 U, θ 〉$ and $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin 〈 U, θ 〉$ . There exists a suffix m, $suchthat 〈 V, θ 〉_{1} \subset 〈 U, θ 〉, 〈 V, θ 〉_{2} \subset 〈 U, θ 〉, 〈 V, θ 〉_{3} \subset 〈 U, θ 〉, 〈 V, θ 〉_{4} \subset 〈 U, θ 〉, \dots ., 〈 V, θ 〉_{z} \subset 〈 U, θ 〉 for all z ≽ m .$ Consequently, $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, ∥ θ) \notin 〈 V, θ 〉_{z} for all z ≽ m;$ hence $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin 〈 M, θ 〉$ . Thus 〈M, θ〉 cosists of the point $(𝓍_{(i, j)}^{e}, θ)$ only.

Theorem 4.2. VS Second count ability is VS first count-ability.

Proof. Let $(〈 \tilde{X} 〉, τ, θ)$ be a soft 2nd VS countable space. Then this situation permit that there live a VS countable base $W$ for $(〈 \tilde{X} 〉, τ, θ)$ . In order to justify that $(〈 \tilde{X} 〉, τ, θ)$ is VS, we proceed as, let $(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉$ be arbitrary point. Let us assemble those members of $W$ which absorbs $(𝓍_{(i, j)}^{e}, θ)$ and named as $W^{(𝓍_{(i, j)}^{e}, θ)}$ . If 〈 ℵ , θ〉 is soft n-hood of $(𝓍_{(i, j)}^{e}, θ)$ , then this permit there exists NSβ open set 〈𝔣, ∂〉 arresting $(𝓍_{(i, j)}^{e}, θ)$ , in $W$ and so in $W^{(𝓍_{(i, j)}^{e}, θ)}$ such that $(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{g}, θ 〉 \subset 〈 ℵ, θ 〉$ . This justifies that is a VS local base at $(𝓍_{(i, j)}^{e}, θ)$ . One step more, $W^{(𝓍_{(i, j)}^{e}, θ)}$ being a sub-family of a VScountable family $W$ , it is therefore VS countable. Thus every crisp point of $〈 \tilde{X} 〉$ supposes a countable VS local base. This leads us to say $(〈 \tilde{X} 〉, τ, θ)$ is soft first V countable.

Theorem 4.3. A VS countable space in which every VS convergent sequence has a unique soft limit is a VSβ Hausdorff space.

Proof. Let $(〈 \tilde{X} 〉, τ, θ)$ be VS Hausdorff space and let $〈 {(x^{e}_{\tilde{(i, j)}}, θ)}_{n} 〉$ be a soft convergent sequence in $(〈 \tilde{X} 〉, τ, θ)$ . We prove that the limit of this sequence is unique. We prove this result by contradiction. Suppose $〈 {(x^{e}_{\tilde{(i, j)}}, θ)}_{n} 〉$ converges to two soft points $\tilde{l}$ and $\tilde{m}$ such that $\tilde{l} \neq \tilde{m}$ . Then by trichotomy law either $\tilde{l} 〈 \tilde{m} or \tilde{l} 〉 \tilde{m}$ . Since the possess the VSβ Hausdorff charactersitcs, there must happen two VSβ open sets 〈𝔣, θ〉 and 〈ρ, θ〉 such that $〈 𝔣, θ 〉 \tilde{\cap} 〈 ρ, θ 〉 = 0_{(〈 \tilde{X} 〉, θ)}$ . Now, $〈 {(x^{e}_{\tilde{(i, j)}}, θ)}_{n} 〉$ converges to $\tilde{l}$ so there exists an interger n₁ such that ${(x^{e}_{\tilde{(i, j)}}, θ)}_{n}$ $\in 〈 f, θ 〉 \forall n \geq n_{1}$ . Also, $〈 {(x^{e}_{\tilde{(i, j)}}, θ)}_{n} 〉$ converges to $\tilde{m}$ so there exists an interger n₂ such that ${(x^{e}_{\tilde{(i, j)}}, θ)}_{n}$ $\in 〈 ρ, \partial 〉 \forall n \geq n_{2}$ . 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. Max (n₁, n₂) = n₀. Which leads to the situation ${(x^{e}_{\tilde{(i, j)}}, θ)}_{n} \in 〈 ρ, θ 〉 \forall n \geq n_{0}$ and ${(x^{e}_{\tilde{(i, j)}}, θ)}_{n} \in 〈 f, θ 〉 \forall n \geq n_{0}$ . This implies that ${(𝓍_{(i, j)}^{e}, θ)}_{n} \in 〈 𝔣, θ 〉$ and ${(x_{\tilde{(i, j)}}^{e}, θ)}_{n} \in 〈 ρ, θ 〉 \forall n \geq n_{0}$ . This guarantees that ${(x_{\tilde{(i, j)}}^{e}, θ)}_{n} \in (〈 f, θ 〉 \tilde{\cap} 〈 ρ, θ 〉) \forall n \geq n_{0}$ . Which beautifully contradict the fact that $〈 𝔣, θ 〉 \tilde{\cap} 〈 ρ, θ 〉 = 0 (\tilde{X}, θ) ˜$ . Hence, the limit of the VS sequence must be unique.

Theorem 4.4. The cardinality of all VS open sets in a VS second countable space is at most equal to complement (the power of the conntinum).

Proof. Let $(〈 \tilde{X} 〉, τ, θ)$ be VSST such that is VS second countable space. Let $〈 G, θ 〉$ be any soft set of $(〈 \tilde{X} 〉, τ, θ)$ then $〈 G, θ 〉$ is the soft union of a certain soft sub-collection of the VS countable collection ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots .}$ . Hence the cardinality of the set of all soft open sets in $(〈 \tilde{X} 〉, τ, θ)$ is not greater than the cardinality of the soft set of all soft sub-collections of the VS countable collection ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots .}$ thus the cardinality of $(〈 \tilde{X} 〉, τ, θ) ≺ ∁$ .

Theorem 4.5. Any collection of mutually exclusive VSβ open sets in a VSβ second countable space is at most VSβ countable.

Proof. Let $(〈 \tilde{X} 〉, τ, θ)$ be VSST such that it is VS second countable. Let 〈 ℶ , θ〉 signifies any collection of mutually exclusive VSβ open sets in $(〈 \tilde{X} 〉, τ, θ)$ . Let $〈 U, θ 〉 \in 〈 ℶ, θ 〉$ , then there exists at least one soft set $〈 W, θ 〉_{m}$ , belonging to the collection ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots}$ such that $〈 W, θ 〉_{m} \subset 〈 U, θ 〉$ . Let n be the smallest suffix for which $〈 W, θ 〉_{n} \subset 〈 U, θ 〉$ . Since the soft sets $〈 U, θ 〉$ in 〈 ℶ , θ〉 are mutually disjoint, it follows that, for different soft sets $〈 U, θ 〉 \in 〈 ℶ, θ 〉$ , there exists soft sets $〈 W, θ 〉_{n}$ with different suffices n. Hence the soft sets in 〈 ℶ , θ〉 are in a one to one correspondence with a soft sub-collection of the VS countable collection ${〈 W, θ 〉_{z} : z = 1, 2, 3, \dots}$ ; consequently, the cardinality of 〈 ℶ , θ〉 is less than or equal to C.

Theorem 4.6. Let $(〈 \tilde{X} 〉, τ, θ)$ be VSTS such that it is VS second countably VSβ Hausdorff space. Then set of all VSβ open sets has the cardinality C.

Proof. Let $(〈 \tilde{X} 〉, τ, θ)$ be second countable VSβ Hausdorff space then there exists in $VSST (〈 \tilde{X} 〉, τ, θ)$ an infinite soft sequence of VSβ open sets $〈 G, θ 〉_{1}, 〈 G, θ 〉_{2}, 〈 G, θ 〉_{3}, 〈 G, θ 〉_{4}, 〈 G, θ 〉_{5}, \dots ., 〈 G, θ 〉_{n}, \dots \dots$ such that $〈 G, θ 〉_{1} \neq 0_{(〈 \tilde{X} 〉, θ)}, 〈 G, θ 〉_{2} \neq 0_{(〈 \tilde{X} 〉, θ)}, 〈 G, θ 〉_{3} \neq 0_{(〈 \tilde{X} 〉, θ)}, 〈 G, θ 〉_{4} \neq 0_{(〈 \tilde{X} 〉, θ)}, \dots . 〈 G, θ 〉_{n} \neq 0_{(〈 \tilde{X} 〉, θ)} \dots \dots$ $〈 G, θ 〉_{1} \tilde{\cap} 〈 G, θ 〉_{2} \tilde{\cap} 〈 G, θ 〉_{3} \tilde{\cap} 〈 G, θ 〉_{4} \tilde{\cap} \dots . \tilde{\cap} 〈 G, θ 〉_{n} \tilde{\cap} \dots = 0_{(〈 \tilde{X} 〉, θ)}$ . Different soft sub-sequences of the sequences $〈 G, θ 〉_{1}, 〈 G, θ 〉_{2}, 〈 G, θ 〉_{3}, 〈 G, \partial 〉_{4}, 〈 G, θ 〉_{5}, \dots ., 〈 G, θ 〉_{n}$ will determine, as their unions, different VSβ open sets. But since the soft set of all soft sub-sets of a countable soft set has the cardinality C, it follows that soft set of all the VSβ open sets in $(〈 \tilde{X} 〉, τ, θ)$ has the cardinality ≽C. Again the cardinality of the soft set of all VSβ open sets in $(〈 \tilde{X} 〉, τ, θ) ≼ C$ . consequently, the cardinality of all VSβ open sets in $(〈 \tilde{X} 〉, τ, θ)$ is exactly equal to C.

Theorem 4.7. Let $(〈 \tilde{X} 〉, τ, θ)$ be VS STsuch that it is VS second countable β₁space the the soft set of all VS points in this space has the cardinality at most equal to C (i.e, posses the power of the continuum at most).

Proof. Given that $(〈 \tilde{X} 〉, τ, θ)$ is VS second countable β₁ space, for which the VS sets $〈 W, θ 〉_{1}, 〈 W, θ 〉_{2}, 〈 W, θ 〉_{3}, 〈 W, θ 〉_{4}, 〈 W, θ 〉_{5}, \dots$ . Forms a soft countable VSβ open base of $(〈 \tilde{X} 〉, τ, θ)$ . Then, for any given point $(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉$ $〈 W, θ 〉_{ς 1}, 〈 W, θ 〉_{ς 2}, 〈 W, θ 〉_{ς 3}, 〈 W, θ 〉_{ς 4}, 〈 W, θ 〉_{ς 5}, \dots .$ generate countable VSβ open base about the point ς in $VSTS (〈 \tilde{X} 〉, τ, θ)$ , where $ς z, z = 1, 2, 3, \dots . .,$ is a soft sub-sequence of the soft sequence $〈 W, θ 〉_{n} : n = 1, 2, 3, \dots$ , consisting of all those $〈 W, θ 〉_{n}$ which contain the point ς. Since a second VS countable space is necessary first VS countable. So corresponding to the point ς, there exists an infinite soft sequence of VSβ open sets ${〈 𝒷, θ 〉_{z} : z = i, 2, 3, \dots}$ , which is a soft sub-sequence of the soft sequence ${〈 W, θ 〉_{pn} : n = 1, 2, 3, . .}$ , and therefore also a soft sub-sequence of the soft sequence ${〈 W, θ 〉_{n} : n = 1, 2, 3, . .}$ , such that $\tilde{\cap} {〈 𝒷, θ 〉_{z} : z = i, 2, 3, \dots} = {(𝓍_{(i, j)}^{e}, θ)}$ . Thus to each point $(𝓍_{(i, j)}^{e}, θ)$ in $〈 \tilde{X} 〉$ , there corresponds a soft sub-sequence ${〈 𝒷, θ 〉_{z} : z = i, 2, 3, \dots}$ of the soft sequence ${〈 W, θ 〉_{n} : n = 1, 2, 3, . .}$ ; and two ndifferent points there correspond two such different soft sub-spaces ${𝒷, θ_{z} : z = i, 2, 3, \dots}$ . Hence the soft set of all points in $〈 \tilde{X} 〉$ , i.e, the crisp set $〈 \tilde{X} 〉$ , has the same cardinality as that of a certain soft sub-collection of the soft collection of all soft sub-space of the sequence ${〈 W, θ 〉_{n} : n = 1, 2, 3, . .}$ . Thus the cardinality of $〈 \tilde{X} 〉$ is less than or equal to complement. In other words the crisp set $〈 \tilde{X} 〉$ has the power of the continuum at most.

Theorem 4.8. Let $(〈 \tilde{X} 〉, τ, θ)$ be VSTS such that it is VS second countable. Every soft uncountable soft subsets contains a point of condensation.

Proof. Since $(〈 \tilde{X} 〉, τ, θ)$ is VS second countable space and ${〈 W, θ 〉_{z} : z = 1, 2, 3, . .}$ be a soft countable VSβ open base of $(〈 \tilde{X} 〉, τ, θ)$ . Let 〈𝔣, θ〉 be a VS sub-set of $〈 \tilde{X} 〉$ such that 〈𝔣, θ〉 does not contain any point of condensation. For each point $(𝓍_{(i, j)}^{e}, θ) \in 〈 𝔣, θ 〉$ , $(𝓍_{(i, j)}^{e}, θ)$ is not point of condensation of 〈𝔣, θ〉. Hence there exists VS open β set $〈 G, θ 〉$ , containing $(𝓍_{(i, j)}^{e}, θ)$ such that $〈 G, θ 〉 \tilde{\cap} 〈 𝔣, θ 〉$ is soft countable at most so there exists a suffix $n ((𝓍_{(i, j)}^{e}, θ))$ , such that $(𝓍_{(i, j)}^{e}, θ) \in 〈 W, θ 〉_{n (p)} \subset 〈 G, θ 〉;$ and then $〈 𝔣, θ 〉 \tilde{\cap} 〈 W, θ 〉_{n (p)}$ is also NS countable at most. But we can express 〈𝔣, θ〉 in the form $〈 𝔣, θ 〉 = \tilde{\cup} {(𝓍_{(i, j)}^{e}, θ) : (𝓍_{(i, j)}^{e}, θ) \in 〈 𝔣, θ 〉} \tilde{\cup} {〈 𝔣, θ 〉 \tilde{\cap} 〈 W, θ 〉_{n ((𝓍_{(i, j)}^{e}, θ))} : (𝓍_{(i, j)}^{e}, θ) \in 〈 𝔣, θ 〉}$ , and there can be at most a VS countable number of different suffices. So, 〈𝔣, θ〉 is at most a VS union of VS uncountable soft sub-set; that is 〈𝔣, θ〉 is at most a VS countable soft sub-set of $〈 \tilde{X} 〉$ . Consequently, if 〈𝔣, θ〉 is soft uncountable soft subset, then it must possess a point of condensation.

5 More structural results

In this section, vague soft topological, vague soft inverse topological properties, product of β_i (with i = 0, 1, 2,) spaces, Bolzano Weirstrass Property and its topological characteristics, compact spaces and sequentially compact spaces and their relationship with separation axioms by means soft points are addressed in vague soft topological spaces.

Theorem 5.1. Let $〈 \tilde{X} 〉, τ, θ)$ be NSTS such that it is VSβ Hausdorff space and $〈 \tilde{Y} 〉, F, θ)$ be any VSTS. Let $〈 𝔣, θ 〉 : 〈 \tilde{X} 〉, τ, θ) \to 〈 \tilde{Y} 〉, F, θ)$ be a soft fuction such that it is soft monotone and continuous. Then $〈 \tilde{Y} 〉, F, θ)$ is also of characteristics of VSβ Hausdorffness.

Proof. Suppose ${(𝓍_{(i, j)}^{e}, θ)}_{1}, {(𝓍_{(i, j)}^{e}, θ)}_{2} \in 〈 \tilde{X} 〉$ such that either ${(𝓍_{(i, j)}^{e}, θ)}_{1} \neq {(𝓍_{(i, j)}^{e}, θ)}_{2}$ . Since 〈𝔣, θ〉 is soft monotone. Let us suppose the monotonically increasing case. Suppose ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1}, {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} \in 〈 \tilde{Y} 〉$ such that ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} \neq {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2}$ respectively such that $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) = 𝔣_{{(𝓍_{(i, j)}^{e}, θ)}_{1}}, {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} = 𝔣_{{(𝓍_{(i, j)}^{e}, θ)}_{2}}$ . Since, $〈 \tilde{X} 〉, τ, θ)$ is VSβ Hausdorff space so there exists mutually disjoint VSβ open sets 〈𝔣₁, θ〉 and $𝓀_{2}, θ \in 〈 \tilde{X} 〉, τ, θ) \Rightarrow 𝔣 (𝓀_{1}, θ)$ and $𝔣 (𝓀_{2}, θ) \in 〈 \tilde{Y} 〉, F, θ)$ . We claim that $𝔣 (𝓀_{1}, \partial) \tilde{\cap} 𝔣 (𝓀_{2}, θ) = 0 (〈 \tilde{X} 〉, θ ˜$ . Otherwise $𝔣 (𝓀_{1}, θ) \tilde{\cap} 𝔣 (𝓀_{2}, θ) \neq 0 (〈 \tilde{X} 〉, θ) ˜$ . Suppose there exists ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{1} \in 𝔣 (𝓀_{1}, θ) \tilde{\cap} 𝔣 (𝓀_{2}, θ)$ implies that ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{1} \in 𝔣 (〈 𝓀_{1}, θ 〉) and {(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{1} \in 𝔣 (〈 𝓉_{2}, θ 〉),$ ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{1} \in 𝔣 (𝓀_{1}, θ), 𝔣$ is soft one to one there exists ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{2} \in 〈 𝓀_{1}, θ 〉$ such that ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{1} = 𝔣 ({(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{2}), {(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{1} \in 𝔣 (𝓀_{2}, θ) implies that$ there exists ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{3} \in 𝓀_{2}, θ$ such that $𝔣 ({(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{3})$ implies that $𝔣 ({(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{2}) = 𝔣 ({(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{3})$ . Since, 𝔣 is soft one to one implies that ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{2} = {(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{3}$ implies that ${(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{2} \in 𝔣 (𝓀_{1}, θ), {(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{2} \in 𝔣 (𝓀_{2}, θ) \Rightarrow {(𝓉_{(i^{/ /}, j^{/ /})}^{e^{/ /}}, θ)}_{2} \in 𝔣 (𝓀_{1}, θ) \tilde{\cap} 𝔣 (𝓀_{2}, θ)$ . This is contradiction because $〈 𝓀_{1}, θ 〉 \tilde{\cap} 〈 𝓀_{2}, θ 〉 = 0_{(〈 \tilde{X} 〉, θ)}$ . Therefore, $𝔣 (𝓀_{1}, θ) \tilde{\cap} 𝔣 (𝓀_{2}, θ) = 0_{(〈 \tilde{X} 〉, θ)}$ . Finally, ${(𝓍_{(i, j)}^{e}, θ)}_{1} ≻ {(𝓍_{(i, j)}^{e}, θ)}_{2} or {(𝓍_{(i, j)}^{e}, θ)}_{1} ≺ {(𝓍_{(i, j)}^{e}, θ)}_{2} \Rightarrow {(𝓍_{(i, j)}^{e}, θ)}_{1} \neq {(𝓍_{(i, j)}^{e}, θ)}_{2}$ implies that $𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{1}) ≻ 𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{2}) or 𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{1}) ≺ 𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{2}) \Rightarrow 𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{1}) \neq 𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{2}) .$ Given a pair of points ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1}, {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} \in 〈 \tilde{Y} 〉$ such that ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} \neq {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2}$ We are able to find out mutually exclusive VSβ open sets $𝔣 (〈 𝓀_{1}, θ 〉), 𝔣 (〈 𝓀_{2}, θ 〉) \in (〈 \tilde{Y} 〉, F, θ)$ such that ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} \in 𝔣 (𝓀_{1}, θ), {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} \in 𝔣 (𝓀_{2}, θ)$ this proves that $\tilde{(Y}, F, θ)$ is VSβ Husdorff space.

Theorem 5.2. Let $(〈 \tilde{X} 〉, τ, θ)$ be VSST and $(〈 \tilde{Y} 〉, F, θ)$ be an-other VSTS which satisfies one more condition of VSβ Hausdorffness. Let 〈𝔣, θ〉: $(〈 \tilde{X} 〉, τ, θ) \to (\tilde{Y}, F, θ)$ be a soft fuction such that it is soft monotone and continuous. Then $(〈 \tilde{X} 〉, τ, θ)$ is also of characteristics of VSβ Hausdorfness.

Proof. Suppose ${(𝓍_{(i, j)}^{e}, θ)}_{1}, {(𝓍_{(i, j)}^{e}, θ)}_{2} \in 〈 \tilde{X} 〉$ such that either ${(𝓍_{(i, j)}^{e}, θ)}_{1} \neq {(𝓍_{(i, j)}^{e}, θ)}_{2}$ . Let us suppose the VS monotonically increasing case. So, ${(𝓍_{(i, j)}^{e}, θ)}_{1} \neq {(𝓍_{(i, j)}^{e}, θ)}_{2}$ implies that $𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{1}) \neq 𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{2})$ respectively. Suppose ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1}, {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} \in 〈 \tilde{Y} 〉$ such that ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} \neq {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2}$ . So, ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} \neq {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2}$ respectively such that ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} = 𝔣 ((𝓍_{(i, j)}^{e}, {θ)}_{1}), {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} = 𝔣 ({(𝓍_{(i, j)}^{e}, θ)}_{2})$ such that ${(𝓍_{(i, j)}^{e}, θ)}_{1} = 𝔣^{- 1} (𝓎_{1})$ and ${(𝓍_{(i, j)}^{e}, θ)}_{2} = 𝔣^{- 1} ({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2})$ . Since ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1}, {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} \in 〈 \tilde{Y} 〉$ but $(〈 \tilde{Y} 〉, F, θ)$ is VSβ Hausdorff space. So according to definition ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} \neq (𝓎_{(i^{/}, j^{/})}^{e^{/}}, {θ)}_{2}$ . There definitely exists VSβ open sets 〈𝓀₁, θ〉 and $〈 𝓀_{2}, θ 〉 \in (〈 \tilde{Y} 〉, F, θ)$ such that ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1} \in 〈 𝓀_{1}, θ 〉$ and ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} \in 〈 𝓀_{2}, θ 〉$ and these two VSβ open sets are guaranteedly mutually exclusive because the possibility of one rules out the possibility of other. Since 𝔣^-1 (𝓀₁, θ) and 𝔣^-1 (𝓀₂, θ) are VSβ open in $(〈 \tilde{X} 〉, τ, θ)$ . Now, $𝔣^{- 1} (〈 𝓀_{1}, θ 〉) \tilde{\cap} 𝔣^{- 1} (〈 𝓀_{1}, θ 〉) = 𝔣^{- 1} (〈 𝓀_{1}, θ 〉 \tilde{\cap} 〈 𝓀_{2}, θ 〉) = 𝔣^{- 1} (\tilde{\emptyset}) = 0 \tilde{(〈 \tilde{X} 〉, θ)}$ and ${(𝓎_{(a^{/}, c^{/})}^{e^{/}}, θ)}_{1} \in 〈 𝓀_{1}, θ 〉 \Rightarrow 𝔣^{- 1} ({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{1}) \in 𝔣^{- 1} (𝓀_{1}, θ) \Rightarrow {(𝓍_{(i, j)}^{e}, θ)}_{1} \in (𝓀_{1}, \partial)$ , ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2} \in 〈 𝓀_{2}, θ 〉 \Rightarrow 𝔣^{- 1} ({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{2}) \in 𝔣^{- 1} (𝓀_{2}, θ)$ implies that ${(𝓍_{(i, j)}^{e}, θ)}_{2} \in (〈 𝓀_{2}, θ 〉)$ . We see that it has been shown for every pair of distinct points of $〈 \tilde{X} 〉$ , there suppose disjoint NSβ open sets namely, 𝔣^-1 (〈𝓀₁, θ〉) and 𝔣^-1 (𝓀₂, θ) belong to $(〈 \tilde{X} 〉, τ, θ)$ such that ${(𝓍_{(i, j)}^{e}, θ)}_{1} \in 𝔣^{- 1} (〈 𝓀_{1}, θ 〉)$ and ${(𝓍_{(i, j)}^{e}, θ)}_{2} \in 𝔣^{- 1} (〈 𝓀_{2}, θ 〉)$ . Accordingly, VSTS is VSβ Hausdorff space.

Theorem 5.3. Let $(〈 \tilde{X} 〉, τ, θ)$ be VSTS and $(〈 \tilde{Y} 〉, F, θ)$ be an-other VSTS. Let 〈𝔣, θ〉: $(〈 \tilde{X} 〉, τ, θ) \to (〈 \tilde{Y} 〉, F, θ)$ be a soft mapping such that it is continuous mapping. Let $(〈 \tilde{Y} 〉, F, θ)$ is VSβ Hausdorff space then it is guaranteed that ${((i, j), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 ((𝓍_{(i, j)}^{e}, θ)) = 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))}$ is a VSβ closed sub-set of $(〈 \tilde{X} 〉, τ, θ) \times (〈 \tilde{Y} 〉, F, θ)$ .

Proof. Given that $(〈 \tilde{X} 〉, τ, θ) squo$ be VSTS and $(〈 \tilde{Y} 〉, F, θ)$ be an-other VSST. Let $〈 𝔣, \partial 〉 : (〈 \tilde{X} 〉, τ, θ) \to (〈 \tilde{Y} 〉, F, θ)$ be a soft mapping such that it is continuous mapping. $(〈 \tilde{Y} 〉, F, θ)$ is VSβ Hausdorff space Then we will prove that ${((𝓍_{(i, j)}^{e}, θ squo), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 ((𝓍_{(i, j)}^{e}, θ)) = 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))}$ is a VSβ closed sub-set of $(〈 \tilde{X} 〉, τ, θ) \times (〈 \tilde{Y} 〉, F, θ)$ . Equivalently, we will prove that ${((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 ((𝓍_{(i, j)}^{e}, θ)) {= (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}}^{c}$ is VSβ open sub-set of $(〈 \tilde{X} 〉, τ, θ) \times (〈 \tilde{Y} 〉, F, θ)$ . Let $((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) \in {((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) with (𝓍_{(i, j)}^{e}, θ) ≻ {(𝓎 e_{(i^{/}, j^{/})}^{/}, θ) : 𝔣 (𝓍_{(i, j)}^{e}, θ) ≻ 𝔣 (𝓎 e_{(i^{/}, j^{/})}^{/}, θ)}}^{c} or ((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) \in {((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) with (𝓍_{(i, j)}^{e}, θ) ≺ (𝓎 e_{(i^{/}, j^{/})}^{/}, θ) : 𝔣 {(𝓍_{(i, j)}^{e}, θ) ≺ 𝔣 (𝓎 e_{(i^{/}, j^{/})}^{/}, θ)}}^{c}$ Then, $𝔣 ((𝓍_{(i, j)}^{e}, θ)) ≻ 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))$ or $≺ 𝔣 ((𝓍_{(i, j)}^{e}, θ)) ≺ 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))$ accordingly. Since, $(〈 \tilde{Y} 〉, F, θ)$ is VSβ-Hausdorff space. Certainly, $𝔣 ((𝓍_{(i, j)}^{e}, θ)), 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))$ are points of $(〈 \tilde{Y} 〉, F, θ)$ , there exists VSβ open sets $〈 G, θ 〉, 〈 𝓀, θ 〉 \in (〈 \tilde{Y} 〉, F, θ)$ such that $𝔣 ((𝓍_{(i, j)}^{e}, θ)) \in 〈 G, θ 〉$ & $𝔣 ((𝓍_{(i, j)}^{e}, θ)) \in 〈 𝓀, θ 〉$ if $〈 G, θ 〉 \tilde{\cap} 〈 𝓀, θ 〉 = 0 \tilde{{(〈 \tilde{X} 〉, θ)}_{Y}}$ . Since, 〈𝓀, θ〉 is soft continuous, $𝔣^{- 1} (〈 G, θ 〉$ and 𝔣^-1 (〈𝓀, θ〉 are VSβ open sets in $(〈 \tilde{X} 〉, τ, θ)$ supposing $(𝓍_{(i, j)}^{e}, θ)$ and $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ respectively and so $𝔣^{- 1} (〈 G, θ 〉 \times 𝔣^{- 1} (〈 H, θ 〉$ is basic VSβ open set in $(〈 \tilde{X} 〉, τ, θ) \times (〈 \tilde{Y} 〉, F, θ)$ containing $((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))$ . Since $〈 G, θ 〉 \tilde{\cap} 〈 𝓀, θ 〉 = 0 \tilde{{(〈 \tilde{X} 〉, θ)}_{Y}}$ , it is clear by the definition of ${((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 ((𝓍_{(i, j)}^{e}, θ)) = 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))}$ that and $\tilde{\cap} {((𝓍_{((i, j))}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 (𝓍) = 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))} = 0_{(〈 \tilde{X} 〉, θ)}$ , that is $𝔣^{- 1} (〈 G, θ 〉 \times 𝔣^{- 1} (〈 𝓀, θ 〉 \subset {((𝓍_{(i, j)}^{e}, θ), {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 ((𝓍_{(i, j)}^{e}, θ)) = 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))}}^{c}$ . Hence, ${((𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 ((𝓍_{(a, c)}^{e}, θ)) {= 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))}}^{c}$ implies that ${((𝓍_{(a, c)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)) : 𝔣 ((𝓍_{(i, j)}^{e}, θ)) = 𝔣 ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ))}$ is VSβ closed.

Theorem 5.4. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VS second countable space then it is guaranteed that every family of non-empty dis-joint VSβ open subsets of a VS second countable space $(〈 \tilde{X} 〉, τ, θ)$ is VS countable.

Proof. Given that $(〈 \tilde{X} 〉, τ, θ)$ be a VS second countable space.

Then, there exists a VS countable base $W = 〈 B^{1}, B^{2}, B^{3}, B^{4}, \dots . . B^{n} : n \in N 〉 for (〈 \tilde{X} 〉, τ, θ)$ . Let $〈 C, θ 〉$ be a family of non-vacuous mutually exclusive VSβ open sub-sets of $(〈 \tilde{X} 〉, τ, θ)$ . Then, for each 〈𝔣, θ〉 of in $〈 C, θ 〉$ there exists a soft $B^{n} \in W$ in such a way that $B^{n} \subset 〈 𝒻, θ 〉$ . Let us attach with 〈𝔣, θ〉, the smallest positive interger n such that $B^{n} \subset 〈 𝒻, θ 〉$ . Since the candidates of $〈 C, θ 〉$ are mutully exclusive because of this behavior distinct candidates will be associated with distinct positive integers. Now, if we put the elements of $〈 C, θ 〉$ in order so that the order is increasing relative to the positive integers associated with them, we obtain a sequence of of candidates of $〈 C, θ 〉$ . This verifies that $〈 C, θ 〉$ is VS countable.

Theorem 5.5. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VS second countable space and let 〈𝔣, θ〉 be VS uncountable sub set of $(〈 \tilde{X} 〉, τ, θ)$ . Then, at least one point of 〈𝔣, θ〉 is a soft limit point of 〈𝔣, θ〉.

Proof. Let $W = 〈 B_{1}, B_{2}, B_{3}, B_{4}, \dots . . B_{n} : n \in N 〉 for (〈 \tilde{X} 〉, τ, θ)$ .

Let, if possible, no point of 〈𝔣, θ〉 be a soft limit point of 〈𝔣, θ〉. Then, for each $(𝓍_{(i, j)}^{e}, θ) \in 𝔣, θ$ there exists VSβ open set $〈 ρ, θ 〉_{(𝓍_{(i, j)}^{e}, θ)}$ such that $(𝓍_{(i, j)}^{e}, θ) \in 〈 ρ, θ 〉_{(𝓍_{(i, j)}^{e}, θ)}$ , $〈 ρ, θ 〉_{(𝓍_{(i, j)}^{e}, θ)} \tilde{\cap} 〈 𝔣, θ 〉 = {(𝓍_{(i, j)}^{e}, θ)}$ . Since $W$ is soft base there exists $B_{n_{(𝓍^{e} (i, j), θ)}} \in W$ such that $(𝓍_{(i, j)}^{e}, θ) \in B_{n_{(𝓍_{(i, j)}^{e}, θ)}} \subset 〈 ρ, θ 〉_{(𝓍_{(i, j)}^{e}, θ)}$ . Therefore, $B_{n_{(𝓍^{e} (i, j), θ)}} \tilde{\cap} 〈 𝒻, θ 〉 \subset 〈 ρ, θ 〉_{(𝓍^{e} (i, j), θ)} \tilde{\cap} 〈 𝒻, θ 〉 = {(𝓍^{e} (i, j), θ)}$ . More-over, if ${(𝓍_{(i, j)}^{e}, θ)}_{1}$ and ${(𝓍_{(i, j)}^{e}, θ)}_{2}$ be any two VS points such that ${(𝓍_{(i, j)}^{e}, θ)}_{1} \neq {(𝓍_{(i, j)}^{e}, θ)}_{2}$ which means either ${(𝓍_{(i, j)}^{e}, θ)}_{1} ≻ {(𝓍_{(i, j)}^{e}, θ)}_{2}$ or ${(𝓍_{(i, j)}^{e}, θ)}_{1} ≺ {(i, j)}_{2}$ then $\exists B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{1}}}$ and $B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{2}}}$ in $W$ such that $B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{1}}} \tilde{\cap} 〈 𝒻, θ 〉 = {{(𝓍_{(i, j)}^{e}, θ)}_{1}}$ and $B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{2}}} \tilde{\cap} 〈 𝒻, θ 〉 = {{(𝓍_{(i, j)}^{e}, θ)}_{2}}$ . Now, ${(𝓍_{(i, j)}^{e}, θ)}_{1} \neq (𝓍_{(i, j)}^{e}, {θ)}_{2}$ which guarantees that ${{(𝓍_{(i, j)}^{e}, θ)}_{1}} \neq {{(𝓍_{(i, j)}^{e}, θ)}_{2}}$ which implies $B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{1}}} \tilde{\cap} 〈 𝒻, θ 〉 \neq B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{2}}} \tilde{\cap} 〈 𝒻, θ 〉$ which implies $B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{1}}} \neq B_{n_{{(𝓍_{(i, j)}^{e}, θ)}_{2}}}$ . Thus, there exists a one to one soft correspondence of 〈𝔣, θ〉 on to ${B_{n_{(𝓍_{(i, j)}^{e}, θ)}} : (𝓍_{(i, j)}^{e}, θ) \in 〈 𝔣, θ 〉}$ . Now, 〈𝔣, θ〉 being VS uncountable, it follows that ${B_{n_{(𝓍_{(i, j)}^{e}, θ)}} : (𝓍_{(i, j)}^{e}, θ) \in 〈 𝔣, θ 〉} .$ is VS uncountable. But, this is contradiction, since ${B_{n_{(𝓍_{(i, j)}^{e}, θ)}} : (𝓍_{(i, j)}^{e}, θ) \in 〈 𝒻, θ 〉} .$ benig 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 5.6. Let $(〈 \tilde{X} 〉, τ, θ) VSTS$ such that is is VS countably compact then this space has the characteristics of Bolzano Weirstrass Property (BWP).

Proof. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VS countably compact space and suppose, if possible, that it negate the (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, θ〉⊂〈𝔣, θ〉. Then this guarantees 〈ρ, θ〉 has no soft limit poit. It follows that 〈ρ, θ〉 is VS soft β closed set. Also for each ${({\tilde{f}}^{e} (i, j) ˜, θ)}_{n} \in ρ, θ, {({\tilde{f}}^{e} (i, j) ˜, θ)}_{n}$ is not a soft limit point of 〈ρ, θ〉. Hence there exists VSβ open set $〈 G_{n}, θ 〉$ , such that ${(𝓍^{e} (i, j) ˜, θ)}_{n} \in G_{n}, θ$ and $〈 G_{n}, θ 〉 \tilde{\cap} 〈 ρ, θ 〉 = {{(𝓍^{e} (i, j) ˜, θ)}_{n}}$ . The the collection ${〈 G_{n}, θ 〉 : n \in N} \tilde{\cap} 〈 ρ, θ 〉^{c}$ is countable VSβ open cover of $(〈 \tilde{X} 〉, τ, θ)$ this soft cover has no finite sub-cover. For this we exhaust a single $〈 G_{n}, θ 〉$ , it would not be a soft cover of $(〈 \tilde{X} 〉, τ, θ)$ since then ${(𝓍^{e} (i, j) ˜, θ)}_{n}$ would be covered. Result in $(〈 \tilde{X} 〉, τ, θ)$ is not VS countably compact. But this contradicts the given. Hence, we are compelled to accept $(〈 \tilde{X} 〉, τ, θ)$ must have Bolzano Weirstrass Property.

Theorem 5.7. Let $(〈 \tilde{X} 〉, τ, θ)$ and $(〈 \tilde{Y} 〉, F, θ)$ be two VSTS and suppose $f, θ$ be a VS continuous function such that $〈 f, θ 〉 : (〈 \tilde{X} 〉, τ, θ) \to (〈 \tilde{Y} 〉, F, θ)$ is VS continuous function and let $〈 L, θ 〉 \subset (〈 \tilde{X} 〉, τ, θ)$ has the B.V.P. then safely $f (θ)$ has the B . V . P.

Proof. Suppose $〈 L, θ 〉$ be an infinite VS sub-set of $〈 f, θ 〉$ , so that $〈 L, θ 〉$ contains an enumerable VS set $〈 {(𝓍_{(i, j)}^{e}, θ)}_{n} : n \in N 〉$ then there exists enumerable VSβ set $〈 {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{n} : n \in N 〉 \subset 〈 L, θ 〉$ such that $f ({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{n}) = {(𝓍_{(i, j)}^{e}, θ)}_{n} . 〈 L, θ 〉$ has B.V.P implies that every infinite soft subset of $〈 L, θ 〉$ has soft accumulation point belonging to $〈 L, θ 〉$ implies that $〈 {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{n} : n \in N 〉$ has soft vague limit poit, say, ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0} \in 〈 L, θ 〉 .$ implies that the limit of soft sequence $〈 {(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{n} : n \in N 〉$ is ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0} \in 〈 L, θ 〉$ implies that ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{n}$ tends to ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0} \in 〈 L, θ 〉 . f$ is soft continuous implies that it is soft continuous. Furthermore ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{n}$ tends ${(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0} \in 〈 L, θ 〉$ implies that $f (({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{n}))$ tends $f (({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0})) \in f (〈 L, θ 〉) impliesthat {(𝓍_{(i, j)}^{e}, θ)}_{n}$ tends $f ((𝓎_{(i^{/}, j^{/})}^{e^{/}}, {θ)}_{0}) \in f (〈 L, θ 〉)$ implies that limit of a soft sequence $〈 {(𝓍_{(i, j)}^{e}, θ)}_{n} : n \in N 〉$ is $f ({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0}) \in f (〈 L, θ 〉)$ implies that limit of a soft sequence $〈 (𝓍_{(i, j)}^{e}, {θ)}_{n} : n \in N 〉 is f ({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0}) \in 〈 f, θ 〉 (〈 L, θ 〉)$ . Finally we have shown that there exists infinite soft subset $〈 {(𝓍_{(i, j)}^{e}, θ)}_{n} : n \in N 〉$ of $f (〈 L, θ 〉)$ containing a limit point $f ({(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)}_{0}) \in f (〈 L, θ 〉)$ . This guarantees that $f (〈 L, θ 〉)$ has B . V . P .

Theorem 5.8. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSTS and $〈 f, H, θ 〉$ be VS sub-space of $(〈 \tilde{X} 〉, τ, θ)$ . The necessary and sufficient condition for $〈 f, θ 〉$ to be VSβ compact relative to $〈 f, T, θ 〉$ is that $〈 f, θ 〉$ is VSβ compact relative to $(〈 \tilde{X} 〉, τ, θ)$ .

Proof. First we prove that $〈 f, θ 〉$ relative to $(〈 \tilde{X} 〉, τ, θ)$ . Let ${〈 k, θ 〉_{i} : i \in I}$ that is ${〈 k, θ 〉_{1}, 〈 k, θ 〉_{2}, 〈 k, θ 〉_{3}, 〈 k, θ 〉_{4}, \dots}$ be $(〈 \tilde{X} 〉, τ, θ) - VS β$ open cover of $〈 f, θ 〉$ , then $〈 f, θ 〉 \subset \tilde{\cup_{z}} 〈 k, θ 〉_{z} . 〈 k, θ 〉_{z} \in (〈 \tilde{X} 〉, τ, θ)$ implies that there exists $〈 g, θ 〉_{z} \in (\tilde{X}, τ, θ)$ such that $. 〈 k, θ 〉_{z} = 〈 g, θ 〉_{z} \tilde{\cap} 〈 f, θ 〉 \subset 〈 g, θ 〉_{z}$ implies that there exists $〈 g, θ 〉_{z} \in (〈 \tilde{X} 〉, τ, θ)$ such that $〈 k, θ 〉_{z} \subset 〈 g, θ 〉_{z}$ implies that $\tilde{\cup_{z}} v_{z} \subset z$ but $〈 f, θ 〉 \subset 〈 k, θ 〉_{z}$ . So that $〈 f, θ 〉 \subset \tilde{\cup_{i}} 〈 k, θ 〉_{z}$ . This guarantees that ${〈 g, θ 〉_{z} : z \in I}$ is a $(〈 \tilde{X} 〉, τ, θ) - VS β$ open cover of $〈 f, θ 〉$ which is known to be VSβ compact relative $(〈 \tilde{X} 〉, τ, θ)$ and hence the soft cover ${z : z \in I (𝓎_{(i, j)}^{e}, θ)}$ must be freezable to a finite soft cub cover, say, ${〈 g, θ 〉_{z r} : r = 1, 2, 3, 4, \dots ., n}$ , Then $〈 f, θ 〉 \subset \cup_{r = 1}^{n} \tilde{〈 G, θ 〉_{z r}}$ implies that $\begin{matrix} 〈 f, θ 〉 \tilde{\cap} 〈 f, θ 〉 \subset 〈 f, θ 〉 \tilde{\cap} [\cup_{r = 1}^{n} 〈 \tilde{G, θ} 〉_{z r}] \end{matrix}$

$= \cup \tilde{{r = 1}_{n}} (〈 f, θ 〉 \tilde{\cap} 〈 g, θ 〉_{z r} = \cup_{r = 1}^{n} \tilde{〈 k, θ 〉_{z r}}$ or $〈 f, θ 〉 \subset \cup_{r = 1}^{n} \tilde{〈 k, θ 〉_{ir}}$ implies that ${〈 k, θ 〉_{z r} : 1 ⩽ r ⩽ n}$ is a $(〈 \tilde{X} 〉, τ, θ) - VS β$ open cover of $〈 f, θ 〉$ . Steping from an arbitrary $(〈 \tilde{X} 〉, τ, θ) - β$ open cover of $〈 f, θ 〉$ , we are able to show that the VSβ cover is freezable to a finite soft sub cover ${〈 k, θ 〉_{z r} : 1 ⩽ r ⩽ n}$ of $〈 f, θ 〉$ , meaning there by $〈 f, θ 〉$ is $(〈 \tilde{X} 〉, τ, θ) - VS β$ compact. The condition is sufficient: Suppose $〈 f, H, θ 〉$ be soft sub-space of $(〈 \tilde{X} 〉, τ, θ)$ and also $f, θ$ is $(〈 \tilde{X} 〉, τ, θ) - VS β$ compact. We have to prove that $〈 f, θ 〉,$ is $(〈 \tilde{X} 〉, τ, θ) - VS β$ compact. Let ${〈 k, θ 〉_{1}, 〈 k, θ 〉_{2}, 〈 k, θ 〉_{3}, 〈 k, θ 〉_{4}, \dots}$ be soft $(〈 \tilde{X} 〉, τ, θ) - VS β$ open cover of $〈 f, θ 〉$ , so that $〈 f, θ 〉 \subset \tilde{\cup_{i}} 〈 g, θ 〉_{i}$ from which $〈 f, θ 〉 \tilde{\cap} 〈 f, θ 〉 \subset 〈 f, θ 〉 \tilde{\cap} (\tilde{\cup_{i}} 〈 g, θ 〉_{z}) or, 〈 f, θ 〉 \subset \tilde{\cup_{i}} (〈 f, θ 〉 \tilde{\cap} 〈 g, θ 〉_{z})$ . On taking $〈 k, θ 〉_{z} = 〈 g, θ 〉_{i} \tilde{\cap} 〈 f, θ 〉$ , we get $〈 f, θ 〉 \subset \tilde{\cup} 〈 k, θ 〉_{i}, 〈 g, θ 〉_{i} \in (〈 \tilde{X} 〉, τ, θ)$ implies that $〈 k, θ 〉_{z} = 〈 g, θ 〉_{i} \tilde{\cap} 〈 f, θ 〉 \in 〈 f, H, θ 〉 \dots . (1)$ . Now from (1) it is clear that ${〈 k, θ 〉_{1}, 〈 k, θ 〉_{2}, 〈 k, θ 〉_{3}, 〈 k, θ 〉_{4}, \dots}$ is $〈 f, H, θ 〉 - VS β$ open soft cover of $〈 f, θ 〉$ which is known to be $〈 f, H, θ 〉 - VS β$ compact hence this sof cover must be reducible to a finite soft sub-cover.say, ${〈 k, θ 〉_{z r} : 1 ⩽ r ⩽ n}$ . This implies that $〈 f, θ 〉 \subset \tilde{\cup_{r = 1}^{n} 〈 k, θ 〉_{ir}} = \cup_{r = 1}^{n} ((g, θ_{z r}) \tilde{\cap} 〈〈 f, θ 〉〉 \in (〈 \tilde{X} 〉, τ, θ),$ or $〈 f, θ 〉 \subset (\cup_{r = 1}^{n} ((〈 g, θ 〉_{ir}) \tilde{\cap} 〈 f, θ 〉) \subset \tilde{\cup_{r = 1}^{n} 〈 g, θ 〉_{ir,}}$ or $〈 f, θ 〉 \cup_{r = 1}^{n} 〈 g, θ 〉_{z r}$ . This proves that ${〈 g, θ 〉_{z r} : 1 ⩽ r ⩽ n}$ is a finite soft sub-cover ot the soft cover $〈, θ 〉_{z}$ . Starting from an arbitrary $(〈 \tilde{X} 〉, τ, θ) - VS β$ 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 $(〈 \tilde{X} 〉, τ, θ) - V β$ compact.

Theorem 5.9. Let $(〈 \tilde{X} 〉, τ, θ) VSTS$ and let ${(𝓍^{e} \tilde{(i, j),} θ)}_{n}$ be a VS sequence in $(〈 \tilde{X} 〉, τ, θ)$ such that it converges to a point ${(𝓍_{(i, j)}^{e}, θ)}_{0}$ then the soft set 〈g, θ〉 consisting of the points ${(𝓍_{(i, j)}^{e}, θ)}_{n_{0}}$ and ${(𝓍_{(i, j)}^{e}, θ)}_{n} (n = 1, 2, 3, \dots \dots)$ is soft VSβ compact.

Proof. Given $(〈 \tilde{X} 〉, τ, θ) VSTS$ and let ${(𝓍^{e} \tilde{(i, j),} θ)}_{n}$ be a VS sequence in $(〈 \tilde{X} 〉, τ, θ)$ such that it converges to a point ${(𝓍_{(i, j)}^{e}, θ)}_{n_{0}}$ that is ${(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n} \to (𝓍_{(i, j)}^{e}, {θ)}_{n_{0}} \in 〈 \tilde{X} 〉$ . let $〈 g, θ 〉 = 〈 {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{1}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{2}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{3}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{4}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{5}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{7}, \dots 〉$ . Let ${〈 S, θ 〉_{α} : α \in Δ}$ be VSβ open covering of 〈g, θ〉 so that $〈 g, θ 〉 \subset \tilde{\cup} {〈 S, θ 〉_{α} : α \in Δ}, {(𝓍_{(i, j)}^{e}, θ)}_{n_{0}} \in 〈 g, θ 〉$ implies that there exists α₀ ∈ Δ such that $. {(𝓍_{(i, j)}^{e}, θ)}_{n_{0}} \in 〈 S, θ 〉_{α_{0}}$ . According to the definition of soft convergence, ${(𝓍_{(i, j)}^{e}, θ)}_{n_{0}} \in 〈 S, θ 〉_{α_{0}} \in (〈 \tilde{X} 〉, τ, θ) suchthat$ there exists n₀ ∈ Vsuchthatn ⩾ n₀ and ${(𝓍_{(i, j)}^{e}, θ)}_{n} \in 〈 S, θ 〉_{α_{0}}$ . Evidently, $〈 S, θ 〉_{α_{0}}$ contains the points $\begin{matrix} \begin{matrix} {(𝓍_{(i, j)}^{e}, θ)}_{n_{0}}, {(𝓍_{(i, j)}^{e}, θ)}_{n_{0 + 1}}, {(𝓍_{(i, j)}^{e}, θ)}_{n_{0 + 2}}, \\ {(𝓍_{(i, j)}^{e}, θ)}_{n_{0 + 3}}, (𝓍_{(i, j)}^{e}, θ), \dots . \end{matrix} \end{matrix}$

${(𝓍_{(i, j)}^{e}, θ)}_{n_{0 + n}}, \dots . . .$ Look carefully at the points and train them in a way as, ${(𝓍_{(i, j)}^{e}, θ)}_{1}, {(𝓍_{(i, j)}^{e}, θ)}_{2}, {(𝓍_{(i, j)}^{e}, θ)}_{3}, {(𝓍_{(i, j)}^{e}, θ)}_{4}, \dots .$ ${(𝓍_{(i, j)}^{e}, θ)}_{i} \in 〈 g, θ 〉 .$

$(𝓍_{(i, j)}^{e}, θ), \dots . . .$ generating a finite soft set. Let 1 ⩽ n_0-1. Then For this $i, {(𝓍_{(i, j)}^{e}, θ)}_{i} \in 〈 g, θ 〉$ . Hence, there exists α_i ∈ Δ such that ${(𝓍_{(i, j)}^{e}, θ)}_{i} \in 〈 S, θ 〉_{α_{i}}$ . Evidently $〈 g, θ 〉 \subset \tilde{\cup_{r = 0}^{n_{0 - 1}} 〈 S, θ 〉_{α_{i}}}$ . This shows that ${〈 S, θ 〉_{α_{i}} : 0 ⩽ n_{0 - 1}}$ is VSβ open cover of 〈g, θ〉. Thus an arbitrary VS β open cover ${〈 S, θ 〉_{α} : α \in Δ}$ of 〈g, θ〉 is reducible to a finite VS cub-cover ${〈 S, θ 〉_{α i} : i = 0, 1, 2, 3, \dots n_{0 - 1}}$ , it follows that 〈g, θ〉 is soft VSβ compact.

Theorem 5.10. If $(〈 \tilde{X} 〉, τ, θ) VSTS$ such that it has the characteristics of VS β sequentially compactness. Then $(〈 \tilde{X} 〉, τ, θ)$ is VSβ countably compact.

Proof. Let $(〈 \tilde{X} 〉, τ, θ) VSTS$ and let 〈ρ, θ〉 be finite soft sub-set of $(〈 \tilde{X} 〉, τ, θ)$ . Let $〈 \begin{matrix} {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{1}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{2}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{3}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{4}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{5}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{6}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{7}, \dots \end{matrix} 〉$ 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 ${(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}$ must be duplicated an in-finite number of times in the VS sequence. Hence, $〈 \begin{matrix} {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}, \dots \end{matrix} 〉$ is soft sub-sequence of ${(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n}$ such that it is soft constant sequence and repeatedly constructed by single soft number ${(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}$ and we know that a soft constant sequence converges on its self. So it converges to ${(𝓍^{e} \tilde{_{(i, j)},} θ)}_{0}$ which belongs to ρ, θ . Hence, 〈ρ, θ〉 is soft sequentially VSβ compact.

Theorem 5.11. Let $(〈 \tilde{X} 〉, τ, θ) VSTS$ and $〈 Y, T, θ 〉$ be another VSTS. Let 〈𝔣, θ〉 be a soft continuous mapping of a soft vague sequentially compact VSβ space $(〈 \tilde{X} 〉, τ, θ)$ into $〈 Y, T, θ 〉$ . Then, $〈 𝔣, θ 〉 ((〈 \tilde{X} 〉, τ, θ))$ is VSβ sequentially compact.

Proof. Given $(〈 \tilde{X} 〉, τ, θ) VSTS$ and $〈 Y, T, θ 〉$ be another VSTS. Let 〈𝔣, θ〉 be a soft continuous mapping of a VS sequentially compact space $(〈 \tilde{X} 〉, τ, θ)$ into $〈 Y, T, θ 〉$ . Then we have to prove $〈 𝔣, θ 〉 ((〈 \tilde{X} 〉, τ, θ)) VS$ sequentially. For this we proceed as. Let $〈 \begin{matrix} {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{1}, {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{2}, \\ , {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{5}, {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{6}, \\ {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{7}, \dots {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{n}, \dots . \end{matrix} 〉$ be a soft sequence of VS points in $〈 𝔣, θ 〉 ((〈 \tilde{X} 〉, τ, θ))$ , Then for each n ∈ N there exists $〈 \begin{matrix} {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{1}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{2}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{4}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{5}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{7}, \dots {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n}, \dots . \in (〈 \tilde{X} 〉, τ, θ) \end{matrix} 〉$ such that $\begin{matrix} 〈 𝔣, \partial 〉 (〈 \begin{matrix} {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{1}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{2}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{3}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{7}, \dots {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n}, \dots . \end{matrix} 〉) = \\ 〈 \begin{matrix} {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{1}, {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{2}, \\ {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{3}, {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{4}, \\ {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{6}, \dots {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{7}, \end{matrix} 〉 \end{matrix}$ . Thus we obtain a soft sequence $〈 \begin{matrix} {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{1}, {(𝓍^{e}_{(i, j)}, ˜ θ)}_{2}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{3}, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{4}, \\ (𝓍^{e} \tilde{_{(i, j)},} θ), {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{6}, \\ {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{7}, \dots {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n}, \dots . \end{matrix} 〉$ in $(〈 \tilde{X} 〉, τ, θ)$ . But $(〈 \tilde{X} 〉, τ, θ)$ being soft sequentially VSβ compact, there is a VS sub-sequence $〈 {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n_{i}} 〉$ of $〈 {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n} 〉$ such that $〈 {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n_{i}} 〉$ tends to $(𝓍^{e} \tilde{_{(i, j)},} θ) \in (〈 \tilde{X} 〉, τ, θ)$ . So, by VSβ contiuity of $〈 𝔣, θ 〉, {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n_{i}}$ tends to $(𝓍^{e} \tilde{_{(i, j)},} θ)$ implies that $〈 𝔣, θ 〉 ({(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n_{i}})$ tends to $〈 𝔣, \partial 〉 (〈 (𝓍^{e} \tilde{_{(i, j)}}, θ) 〉) \in 〈 𝔣, θ 〉 ((〈 \tilde{X} 〉, τ, θ))$ . Thus, $〈 𝔣, θ 〉 (〈 {(𝓍^{e} \tilde{_{(i, j)},} θ)}_{n_{i}} 〉)$ is a soft sub-sequence of $〈 \begin{matrix} {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{1}, {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{2}, \\ (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ), {(𝓎^{e}_{(i^{/}, j^{/})}, ˜ θ)}_{4}, \\ {(𝓎^{e^{/}}_{(i^{/}, j^{/})}, ˜ θ)}_{5}, {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{6}, \\ {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{7}, \dots {(𝓎^{e^{/}} \tilde{_{(i^{/}, j^{/})},} θ)}_{n}, \dots . \end{matrix} 〉$ converges to $(〈 𝔣, θ 〉) (\tilde{𝔣})$ in $〈, θ 〉 ((〈 \tilde{X} 〉, τ, θ))$ . Hence, $〈, θ 〉 ((〈 \tilde{X} 〉, τ, θ))$ is VSβ sequentially compact.

Theorem 5.12. Let $(〈 \tilde{X} 〉, τ, θ)$ be a VSβ₁ space and $(𝓍_{(i, j)}^{e}, θ), (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \in X^{crip}$ such that $(𝓍_{(i, j)}^{e}, θ) \neq (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ . If $W_{(𝓍_{(i, j)}^{e}, θ)}$ is a VS local base at $(𝓍_{(i, j)}^{e}, θ)$ , then there exists at least one member of $W_{(𝓍_{(i, j)}^{e}, θ)}$ which does not supposes $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ .

Proof. Since $(〈 \tilde{X} 〉, τ, θ)$ be a VSβ₁ space and $(𝓍_{(i, j)}^{e}, θ) \neq (𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ)$ , there exists VSβ open sets $〈 G, θ 〉$ and $〈 H, θ 〉$ such that $(𝓍_{(i, j)}^{e}, θ) \in 〈 G, θ 〉$ but $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin 〈 G, θ 〉$ and $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \in 〈 H, θ 〉$ but $(𝓎_{(i, j)}^{e}, θ) \notin 〈 H, θ 〉$ . Since, $W_{(𝓍_{(i, j)}^{e}, θ)}$ is VS local base at $(𝓍_{(i, j)}^{e}, θ)$ there exists $(𝓍_{(i, j)}^{e}, θ) \in B \subset 〈 G, θ 〉$ . Since $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin 〈 G, θ 〉$ and $B \subset 〈 G, θ 〉$ , so $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin B$ . Thus, $B \in W_{(𝓍_{(i, j)}^{e}, θ)}$ such that $(𝓎_{(i^{/}, j^{/})}^{e^{/}}, θ) \notin B$ .

Theorem 5.13. Let $(〈 \tilde{X} 〉, τ, θ) VSTS$ and suppose 〈𝔣, θ〉, 〈g, θ〉 be two VS continuous function on a VS TS $(〈 \tilde{X} 〉, τ, θ)$ in to a $VSTS 〈 Y, F, θ 〉$ which is VSβ Hausdorff. Then, soft set ${(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉 : (𝔣) ((𝓍_{(i, j)}^{e}, θ)) = (g) ((𝓍_{(i, j)}^{e}, θ))}$ is VSβ closed of $(〈 \tilde{X} 〉, τ, θ)$ .

Proof: If ${(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉 : (𝔣) ((𝓍_{(i, j)}^{e}, θ)) = (g) ((𝓍_{(i, j)}^{e}, θ))}$ is a VS set of function. If ${(𝓍_{(i, j)}^{e}, θ) {\in 〈 \tilde{X} 〉 : (𝔣) ((𝓍_{(i, j)}^{e}, θ)) = (𝔣) (𝓍)}}^{c} = \tilde{\emptyset},$ it is clearly VSβ open and therefore, ${(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉 : (𝔣) ((𝔣_{(i, j)}^{e}, θ)) = (g) ((𝓍_{(i, j)}^{e}, θ))}$ is VS β closed, that is nothing is proved in this case. Let us consider the case when ${(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉 : {(𝔣) ((𝓍_{(i, j)}^{e}, θ)) = (g) ((𝓍_{(i, j)}^{e}, θ))}}^{c} \neq (𝓍_{(i, j)}^{e}, θ) .$ let $ρ \in {(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉 : (𝔣) ((𝓍_{(i, j)}^{e}, θ)) = (g) {((𝓍_{(i, j)}^{e}, θ))}}^{c}$ . Then ρ does not belong ${(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉 : (𝔣) ((𝓍_{(i, j)}^{e}, θ)) = (g) (𝓍)}$ . Result in (𝔣) (ρ) ≠ (g) (ρ) . Now, $〈 Y, F, θ 〉$ being VSβ Hausdorff space so there exists VSβ open sets 〈g, θ〉 and $〈 H, θ 〉$ of (𝔣) (ρ) and (g) (ρ) respectively such that 〈g, θ〉 and $〈 H, θ 〉$ such that these VS sets such that the possibility of one rules out the possibility of other. By soft continuity of 〈𝔣, θ〉, 〈g, θ〉, 〈𝔣, θ〉^-1 as well as 〈g, θ〉^-1 is VSβ open nhd of ρ and therefore, so is $〈 𝔣, θ 〉^{- 1} \tilde{\cap} 〈 g, θ 〉^{- 1}$ is contained in ${(𝓍_{(i, j)}^{e}, θ) \in 〈 \tilde{X} 〉 : (𝔣) ((𝓍_{(i, j)}^{e}, θ)) = (g) ((𝓍_{(i, j)}^{e}, θ))}$ , for, ${(^{e}_{(i, j)}, θ)}_{} \in (, θ^{- 1} \tilde{\cap}, θ^{- 1})$ implies that $(x) ({(^{e}_{(i, j)}, θ)}_{}) \in, \partial$ and $(x) (x) ({(^{e}_{(i, j)}, θ)}_{}) \neq (x) ({(^{e}_{(i, j)}, θ)}_{})$ because , θ and $H, θ$ are disjoint. This implies that x does not belong to ${(^{e}_{(i, j)}, θ) \in \tilde{X} : (x) ((^{e}_{(i, j)}, θ)) = (x) ((^{e}_{(i, j)}, θ))}$ . Therefore, $ρ \in {(x)}^{- 1} (, θ) \tilde{\cap} {(x)}^{- 1} (, θ) left faced u double {(^{e}_{(i, j)}, θ) \in \tilde{X} : (x) ((^{e}_{(i, j)}, θ)) = (x) ((^{e}_{(i, j)}, θ))}^{c}$ . This shows that ${(^{e}_{(i, j)}, θ) \in \tilde{X} : (x) ((^{e}_{(i, j)}, θ)) = (x) ((^{e}_{(i, j)}, θ))}^{c}$ is nhd.of each of its points. So, ${(^{e}_{(i, j)}, θ) \in \tilde{X} : (x) ((^{e}_{(i, j)}, θ)) = (x) ((^{e}_{(i, j)}, θ))}^{c}$ open and hence ${(^{e}_{(i, j)}, θ) \in \tilde{X} : (x) ((^{e}_{(i, j)}, θ)) = (x) ((^{e}_{(i, j)}, θ))}^{c}$ is VSβ closed.

Theorem 5.14. Let $(\tilde{X}, τ, θ) VSTS$ such that it is VSβ Hausdorff space and let (𝒻) be soft continuous function of $(\tilde{X}, τ, θ)$ into itself. Then, the VSβ set of fixed points under (𝒻) is a VSβ closed set.

Proof. Let $δ = {(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}$ . If $δ^{c} = \tilde{\emptyset}$ , Then is VSβ open and therefore ${(x) ((^{e}_{(i, j)}, θ)) = (^{e}_{(i, j)}, θ)}$ closed. So, let ${(x) ((^{e}_{(i, j)}, θ)) = (^{e}_{(i, j)}, θ)}^{c} \neq \tilde{\emptyset}$ and let $(_{(i^{/}, j^{/})}^{e^{/}}, θ) \in {(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}^{c}$ . Then, $(_{(i^{/}, j^{/})}^{e^{/}}, θ)$ does not belong to ${(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}$ . and therefore $(x) ((_{(i^{/}, j^{/})}^{e^{/}}, θ)) \neq (_{(i^{/}, j^{/})}^{e^{/}}, θ) ..$ Now, $(_{(i^{/}, j^{/})}^{e^{/}}, θ)$ and $(x) ((_{(i^{/}, j^{/})}^{e^{/}}, θ))$ being two distinct points of the VSβ Hausdorff space $(\tilde{X}, τ, θ)$ , so there exists VSβ open sets , θ and $H, θ$ such that $(_{(i^{/}, j^{/})}^{e^{/}}, θ) \in, θ, (x) ((_{(i^{/}, j^{/})}^{e^{/}}, θ)) \in h, θ$ and $, θ, H, θ$ are disjoint. Also, by the VS continuity of (𝒻), (𝒻) ^-1 (H, θ is VSβ open set containing. We prtend that $, θ \tilde{\cap} {(x)}^{- 1} (h, θ) \subset {(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}^{c}$ . Snice $μ \in, θ \tilde{\cap} {(𝓍)}^{- 1} (H, θ)$ implies that μ ∈ , θ d μ ∈ (𝒻) ^-1 implies that μ ∈ , θ and $(𝓍) (μ) \in H, θ$ implies that μ ≠ (𝒻) (μ). As $, θ \tilde{\cap} H, θ = \tilde{\emptyset}$ implies that μ does not belong to ${(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}$ implies that $μ \in {(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}^{c}$ . Therefore, $(_{(i, j)}^{e}, θ) \in, θ \tilde{\cap} {(x)}^{- 1} (h, θ) \subset {(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}^{c}$ . Thus, ${(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}^{c}$ is the VS nhd of each of its points. So, ${(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}^{c}$ is VSβ open and hence ${(x) ((_{(i, j)}^{e}, θ)) = (_{(i, j)}^{e}, θ)}$ is VSβ closed.

6 Conclusion

Fuzzy soft topology consider 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. Vague Soft topology is a theory which contains both the characters of applied mathematics and pure mathematics and this is why it is supposed to be a very important branch of mathematics. In this our particular study we have first defined the notion of neutrosophic soft β-open set and, by means of soft points we have given some peculiar examples. Furthermore, using such a new notion, we studied the first three separation axioms β_i (with i = 0, 1, 2) for the neutrosophic soft topological spaces and we have also extended the investigation to other separation axioms such as β_i (with i = 3, 4), by proving some relationships between them and some other relevant results concerning topological properties.

In future, we will try to generate the above structures with respect to vague soft near open sets and soft very near open sets. We will try to extend the study of domain to vague soft bi topology. We will try to use the concept of crisp points and soft pints to reproduce these structures.

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