A Novel Framework for Generalizations of Soft Open Sets and Its Applications via Soft Topologies

Abstract

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach inspired by the components of a soft set. This approach opens the door to reformulating the existing soft topological concepts and examining their behaviors. First, we deliberate the main structural properties of this class and detect its relationships with the previous generalizations with the assistance of suitable counterexamples. In addition, we probe some features that are obtained under some specific stipulations and elucidate the properties of the forgoing generalizations that are missing in this class. Next, we initiate the interior and closure operators with respect to the classes of weakly soft semi-open and weakly soft semi-closed subsets and look at some of their fundamental characteristics. Ultimately, we pursue the concept of weakly soft semi-continuity and furnish some of its descriptions. By a counterexample, we elaborate that some characterizations of soft continuous functions are invalid for weakly soft semi-continuous functions.

1. Introduction

In our surroundings, there are various difficulties that involve some uncertainties, which are often seen in many different areas such as medical science, biology, ecology, engineering, social science, economics, and so on. For several years, many philosophers, mathematicians, and computer scientists have been attempting to find adequate instruments to cope with uncertainties. One of these instruments is the soft set (SS), which was first proposed by Molodtsov [1]. Compared to forgoing strategies, the soft set is adequate to process uncertainties and is free from their inherent limitations; for instance, it does not require the preconditions such as a membership function in the fuzzy sets or an equivalence relation in the rough sets.

The first attempt to establish the main characteristics of the soft set theory was conducted by Maji et al. [2]. They put forward the basic notions between soft sets such as soft intersection and soft union operators, a complement of a soft set, absolute and null soft sets. Some shortcomings and drawbacks caused by this study were adjusted and by Ali et al. [3]. Quite recently, it has been proposed that some soft operators and operations are consistent with their analogs in the crisp set theory by Al-shami and El-Shafei [4]. To address the complicated problems, researchers created novel environments by integrating soft sets with some uncertain tools such as fuzzy and rough sets [5,6].

As performed with topology initiated by fuzzy sets, topologists made use of soft sets to introduce the concept of a soft topology (ST). Two approaches were given by Çaǧman et al. [7] and Shabir and Naz [8] to define STSs. The difference between them is the way of selecting the set of parameters for each element of the soft topology as a variable or constant. This study will go along with the line of Shabir and Naz, who stipulate the necessity of a constant set of parameters. Afterward, many researchers, scholars and intellectuals popularized the classical principles and concepts of topology in the frame of STs. Min [9] restated the relationship between the spaces of soft $T_2$ and soft $T_3$. He also showed that the soft closed and open subsets of a soft regular space must be stable; i.e., all components are the same. To preserve almost all properties and characterizations produced by the interaction between separation axioms and other topological concepts as well as the systematic relations of soft $T_i(i=1,2,3,4)$ in the frame of STSs, researchers introduced a new type of soft $T_i$-spaces by El-Shafei et al. [10]. Singh and Noorie [11] made a comparative study among some types of soft separation axioms. The concepts of soft compact and Lindelöf spaces were researched by some authors [12,13]. Interesting applications of these abstract concepts “soft separation axioms” and “soft compactness” were the goals of the ground-breaking manuscripts [14]. Kharal and Ahmad [15] pursued the idea of functions between soft topological spaces and explored main themes. Next, the notions of soft continuity and homeomorphism were examined in [16,17]. It worth noting that Al-shami [18] pioneered soft functions by adopting a binary relation over the family of soft points, which ease the way of computations and justify some definitions of soft themes such as injective and surjective soft functions. Ameen [19] investigated some structural soft sets that are preserved under some non-continuous soft mappings.

In 2013, Chen [20] introduced the idea of soft semi-open sets following a similar technique to that given in the classical topologies. He initiated some topological concepts with respect to this idea and pointed out that they widen their counterparts obtained by the family of soft open sets. In a similar way, the concepts of $\alpha$-open, soft omega open, and soft semi $\omega$-open sets were described by [21,22,23], respectively. Many soft topological principles have been investigated from the view of these generalizations as presented in [24,25,26]. To endorse the importance of generalizations of soft open sets, Al-shami [27] showed how the nutrition systems of individuals are estimated using soft somewhat open sets. Elsayed [28] defined a new type of soft set called a soft equivalent set and probed its main features.

It is well known that an ST produces some classical topologies (named “parametric topologies”), so it is natural to look at the relationships between them in terms of interchangeable topological concepts and properties between them. To depict how topological concepts and properties navigate from ST to classical topology and vice versa, Al-shami and Kočinac [29] fostered the basic conceptions that illustrate this theme, in particular, in the cases of an extended ST. This behavior was discussed for some soft topological notions such as operational characterizations [30], caliber and chain conditions [31] and soft separation axioms [32]. The present work’s contribution lies in this area of ST by inventing a new technique to produce novel generalizations of soft open sets using the classical topologies inspired by the original ST. This technique paves the road to reformulate the topological concepts and establish different classes of soft topologies. In the last, we draw the reader’s attention to the fact that there are various contributions to ST structures such as those defined on specific kinds of STs such as vietoris topology [33], Menger spaces [34], maximal topologies and expandable spaces [35]; also, some extensions of ST such as supra soft topology [36], infra soft topology [36] and weak soft structures [37].

To know the content of this work, we mention its layout in the following. The fundamentals to make this manuscript self-contained are reviewed in Section 2. We adopt a new approach to study generalizations of soft open sets in Section 3. We define the concepts of weakly soft semi-open and weakly soft semi-closed subsets using their analog obtained from a parametric topology instead of the soft interior and closure operators. In Section 4, we introduce the operators of a weakly soft semi-interior, weakly soft semi-closure, weakly soft semi-boundary, and weakly soft semi-limit, and scrutinized their master features. In Section 5, we familiarize the concept of weakly soft semi-continuity and construct an example that elaborates that some equivalent conditions of soft continuity are invalid for this type of continuity. Finally, we dedicate Section 6 to summarizing the main contributions and suggest some future work.

2. Preliminaries

We invoke in this segment the basic definitions and results necessitated to comprehend this work.

Definition 1 ([1]).

Let $\mathbf{O}\ne \varnothing$ be a universal set, $2^{\mathbf{O}}$ be the family of all subsets of $\mathbf{O}$ and $\mathsf{B}\ne \varnothing$ be a set of parameters (or attributes). A set-valued function $\mathtt{F}:\mathsf{B}\to 2^{\mathbf{O}}$ is said to be an SS. For simplicity, we symbol an SS by $(\mathtt{F},\mathsf{B})$ and write as

$(\mathtt{F},\mathsf{B})=\left\{\right(\mathsf{b},\mathtt{F}\left(\mathsf{b}\right)):\mathsf{b}\in \mathsf{B}$ and $\mathtt{F}\left(\mathsf{b}\right)\in 2^{\mathbf{O}}\}$;

That is, an SS $(\mathtt{F},\mathsf{B})$ over $\mathbf{O}\ne \varnothing$ institutes a parameterized families of subsets of $\mathbf{O}$ such that we call each $\mathtt{F}\left(\mathsf{b}\right)$ a $\mathsf{b}$-component of $(\mathtt{F},\mathsf{B})$.

Through this work, the family of all SSs over $\mathbf{O}$ with a set of parameters $\mathsf{B}$ is designated by $2^{{\mathbf{O}}_{\mathsf{B}}}$.

Definition 2 ([2,3,38,39]).

Let $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ be SSs over $\mathbf{O}$ and $\mathbf{\nu }\in \mathbf{O}$. Then,

(i) 

If $\mathtt{G}\left(\mathsf{b}\right)=\mathbf{O}-\mathtt{F}\left(\mathsf{b}\right)$ for all $\mathsf{b}\in \mathsf{B}$, then $(\mathtt{G},\mathsf{B})$ is called a complement of $(\mathtt{F},\mathsf{B})$. We write $(\mathtt{G},\mathsf{B})={(\mathtt{F},\mathsf{B})}^c=({\mathtt{F}}^c,\mathsf{B})$.

(ii) 

If $\mathtt{F}\left(\mathsf{b}\right)=\mathbf{O}$ for every $\mathsf{b}\in \mathsf{B}$, then we call $(\mathtt{F},\mathsf{B})$ an absolute soft set; it is symbolized by $\tilde{\mathbf{O}}$. The complement of an absolute soft set is called a null soft set; it is symbolized by ϕ.

(iii) 

If $\mathtt{F}\left(\mathsf{b}\right)=\left\{\mathbf{\nu }\right\}$ and $\mathtt{F}\left({\mathsf{b}}^{*}\right)=\varnothing$ for all ${\mathsf{b}}^{*}\in \mathsf{B}-\left\{\mathsf{b}\right\}$, then we call $(\mathtt{F},\mathsf{B})$ a soft point and symbolized by ${\nu }_{\mathsf{b}}$. We write ${\nu }_{\mathsf{b}}\in (\mathtt{F},\mathsf{B})$ if $\mathbf{\nu }\in \mathtt{F}\left(\mathsf{b}\right)$.

(iv) 

If $\mathtt{F}\left(\mathsf{b}\right)=\mathbf{O}$ or ∅ for all $\mathsf{b}\in \mathsf{B}$, then we call $(\mathtt{F},\mathsf{B})$ a pseudo constant soft set. Note that there are $2^{\left|\mathsf{B}\right|}$ pseudo constant soft sets.

Definition 3 ([40]).

If $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ are SSs over $\mathbf{O}$ such that $\mathtt{F}\left(\mathsf{b}\right)\subseteq \mathtt{G}\left(\mathsf{b}\right)$ for each $\mathsf{b}\in \mathsf{B}$, then $(\mathtt{F},\mathsf{B})$ is called an SS of $(\mathtt{G},\mathsf{B})$. Symbolically, we write $(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{G},\mathsf{B})$.

Definition 4 (see, [4]).

Let $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ be SSs over $\mathbf{O}$. Then:

(i) 

Soft union: $(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})=(\mathtt{H},\mathsf{B}),$ where $\mathtt{H}\left(\mathsf{b}\right)=\mathtt{F}\left(\mathsf{b}\right)\cup \mathtt{G}\left(\mathsf{b}\right)$ for all $\mathsf{b}\in \mathsf{B}$.

(ii) 

Soft intersection: $(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})=(\mathtt{H},\mathsf{B}),$ where $\mathtt{H}\left(\mathsf{b}\right)=\mathtt{F}\left(\mathsf{b}\right)\cap \mathtt{G}\left(\mathsf{b}\right)$ for all $\mathsf{b}\in \mathsf{B}$.

(iii) 

Soft difference: $(\mathtt{F},\mathsf{B})\backslash (\mathtt{G},\mathsf{B})=(\mathtt{H},\mathsf{B}),$ where $\mathtt{H}\left(\mathsf{b}\right)=\mathtt{F}\left(\mathsf{b}\right)\backslash \mathtt{G}\left(\mathsf{b}\right)$ for all $\mathsf{b}\in \mathsf{B}$.

(iv) 

Soft product: $(\mathtt{F},\mathsf{B})\times (\mathtt{G},\mathsf{B})=(\mathtt{H},\mathsf{B}),$ where $\mathtt{H}({\mathsf{b}}_1,{\mathsf{b}}_2)=\mathtt{F}\left({\mathsf{b}}_1\right)\times \mathtt{G}\left({\mathsf{b}}_2\right)$ for all $({\mathsf{b}}_1,{\mathsf{b}}_2)\in \mathsf{B}\times \mathsf{B}$.

The definition of soft functions was adjusted by Al-shami [18] as follows.

Definition 5 ([18]).

Let $\mathbb{I}:\mathbf{O}\to \mathbf{U}$ and $\mathbb{P}:\mathsf{B}\to \mathsf{E}$ be crisp functions. A soft function ${\mathbb{I}}_{\mathbb{P}}$ of $2^{{\mathbf{O}}_{\mathsf{B}}}$ into $2^{{\mathbf{U}}_{\mathsf{E}}}$ is a relation such that each ${\mathbf{\nu }}_{\mathsf{b}}\in 2^{{\mathbf{O}}_{\mathsf{B}}}$ is related to one and only one ${\mathbf{\mu }}_{\mathsf{e}}\in 2^{{\mathbf{U}}_{\mathsf{E}}}$ such that

${\mathbb{I}}_{\mathbb{P}}\left({\mathbf{\nu }}_{\mathsf{b}}\right)=\mathbb{I}{\left(\mathbf{\nu }\right)}_{\mathbb{P}\left(\mathsf{b}\right)}$ for all ${\mathbf{\nu }}_{\mathsf{b}}\in 2^{{\mathbf{O}}_{\mathsf{B}}}$.

In addition, ${\mathbb{I}}_{\mathbb{P}}^{-1}\left({\mathbf{\mu }}_{\mathsf{e}}\right)=\underset{\mathsf{b}\in {\mathbb{P}}^{-1}\left(\mathsf{e}\right)}{\underset{\mathbf{\nu }\in {\mathbb{I}}^{-1}\left(\mathbf{\mu }\right)}{\tilde{\cup }}}{\nu }_{\mathsf{b}}$ for each ${\mathbf{\mu }}_{\mathsf{e}}\in 2^{{\mathbf{U}}_{\mathsf{E}}}$.

That is, the image of $(\mathtt{F},\mathsf{B})$ and pre-image of $(\mathtt{G},\mathsf{E})$ under a soft function ${\mathbb{I}}_{\mathbb{P}}:2^{{\mathbf{O}}_{\mathsf{B}}}\to 2^{{\mathbf{U}}_{\mathsf{E}}}$ are, respectively, given by:

${\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})={\tilde{\cup }}_{{\mathbf{\nu }}_{\mathsf{b}}\in (\mathtt{F},\mathsf{B})}{\mathbb{I}}_{\mathbb{P}}\left({\mathbf{\nu }}_{\mathsf{b}}\right)$, and

${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{G},\mathsf{E})={\tilde{\cup }}_{{\mathbf{\mu }}_{\mathsf{e}}\in (\mathtt{G},\mathsf{E})}{\mathbb{I}}_{\mathbb{P}}^{-1}\left({\mathbf{\mu }}_{\mathsf{e}}\right)$.

A soft function is described as surjective (injective, bijective) if its two crisp functions satisfy this description.

Proposition 1 ([15,41]).

Let ${\mathbb{I}}_{\mathbb{P}}:2^{{\mathbf{O}}_{\mathsf{B}}}\to 2^{{\mathbf{U}}_{\mathsf{E}}}$ be a soft function and let $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ be SSs of $\tilde{\mathbf{O}}$ and $\tilde{\mathbf{U}}$, respectively. Then

(i) 

$(\mathtt{F},\mathsf{B})\tilde{\subseteq }{\mathbb{I}}_{\mathbb{P}}^{-1}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)$.

(ii) 

If ${\mathbb{I}}_{\mathbb{P}}$ is injective, then $(\mathtt{F},\mathsf{B})={\mathbb{I}}_{\mathbb{P}}^{-1}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)$.

(iii) 

${\mathbb{I}}_{\mathbb{P}}({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{G},\mathsf{E}))\tilde{\subseteq }(\mathtt{G},\mathsf{E})$.

(iv) 

If ${\mathbb{I}}_{\mathbb{P}}$ is surjective, then ${\mathbb{I}}_{\mathbb{P}}({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{G},\mathsf{E}))=(\mathtt{G},\mathsf{E})$

Definition 6 

([8]). A subfamily ℑ of $2^{{\mathbf{O}}_{\mathsf{B}}}$ is said to be an ST if the following terms are satisfied:

(i) 

$\tilde{\mathbf{O}}$ and ϕ are elements of ℑ.

(ii) 

ℑ is closed under the arbitrary unions.

(iii) 

ℑ is closed under the finite intersections.

We will call the triplet $(\mathbf{O},\Im ,\mathsf{B})$ an STS. Each element in ℑ is called soft open, and its complement is called soft closed.

Definition 7 ([8]).

Let $(\mathtt{F},\mathsf{B})$ be a soft subset of an STS $(\mathbf{O},\Im ,\mathsf{B})$. Then, the soft interior of $(\mathtt{F},\mathsf{B})$, denoted by $int(\mathtt{F},\mathsf{B})$, and the soft closure of $(\mathtt{F},\mathsf{B})$, denoted by $cl(\mathtt{F},\mathsf{B})$, are, respectively, given by

(i) 

The soft union of all soft open sets that are contained in $(\mathtt{F},\mathsf{B})$.

(ii) 

The soft intersection of all soft closed sets containing $(\mathtt{F},\mathsf{B})$.

Proposition 2 ([8]).

Let $(\mathbf{O},\Im ,\mathsf{B})$ be an STS. Then, ${\Im }_{\mathsf{b}}=\{\mathtt{F}\left(\mathsf{b}\right):(\mathtt{F},\mathsf{B})\in \Im \}$ institutes a topology on $\mathbf{O}$ for every $\mathsf{b}\in \mathsf{B}$. We will call this topology a parametric topology.

Definition 8 (see, [8]).

Let $(\mathtt{F},\mathsf{B})$ be an SS of an STS $(\mathbf{O},\Im ,\mathsf{B})$. Then $\left(int\right(\mathtt{F}),\mathsf{B})$ and $\left(cl\right(\mathtt{F}),\mathsf{B})$ are respectively defined by $int\left(\mathtt{F}\right)\left(\mathsf{b}\right)=int\left(\mathtt{F}\right(\mathsf{b}\left)\right)$ and $cl\left(\mathtt{F}\right)\left(\mathsf{b}\right)=cl\left(\mathtt{F}\right(\mathsf{b}\left)\right)$, where $int\left(\mathtt{F}\right(\mathsf{b}\left)\right)$ and $cl\left(\mathtt{F}\right(\mathsf{b}\left)\right)$ are, respectively, the interior and closure of $\mathtt{F}\left(\mathsf{b}\right)$ in $(\mathbf{O},{\Im }_{\mathsf{b}})$.

Definition 9 ([26]).

An STS $(\mathbf{O},\Im ,\mathsf{B})$ is called:

(i) 

Full provided in the case that every non-null soft open set has no empty component.

(ii) 

Hyperconnected provided in the case that the soft intersection of any two non-null soft open sets is non-null.

Definition 10 ([12,38]).

Let $(\mathbf{O},\Im ,\mathsf{B})$ be an STS.

(i) 

If all pseudo-constant SSs are elements of ℑ, then ℑ is called an enriched ST.

(ii) 

ℑ with the property “$(\mathtt{F},\mathsf{B})\in \Im$ iff $\mathtt{F}\left(\mathsf{b}\right)\in {\Im }_{\mathsf{b}}$ for each $\mathsf{b}\in \mathsf{B}$” is called an extended ST.

A deep investigation on the enriched and extended STs was conducted on [29]. The corresponding properties between these kinds of STs was one of the important and interesting results obtained in [29]. Henceforth, this type of soft topology will be called an extended soft topology. Under this soft topology, researchers proved several findings that associated soft topology with its parametric topologies. As a matter of fact, the following result represents a key point in the proof of many findings.

Theorem 1 ([29]).

An STS $(\mathbf{O},\Im ,\mathsf{B})$ is extended if $\left(int\right(\mathtt{F}),\mathsf{B})=int(\mathtt{F},\mathsf{B})$ and $\left(cl\right(\mathtt{F}),\mathsf{B})=cl(\mathtt{F},\mathsf{B})$ for any soft subset $(\mathtt{F},\mathsf{B})$.

Definition 11. 

A soft subset $(\mathtt{F},\mathsf{B})$ of $(\mathbf{O},\Im ,\mathsf{B})$ is termed as:

(i) 

Soft α-open [21] provided that $(\mathtt{F},\mathsf{B})\tilde{\subseteq }int\left(cl\left(int(\mathtt{F},\mathsf{B})\right)\right)$.

(ii) 

Soft semi-open [20] provided that $(\mathtt{F},\mathsf{B})\tilde{\subseteq }cl\left(int(\mathtt{F},\mathsf{B})\right)$.

(iii) 

Soft β-open [42] provided that $(\mathtt{F},\mathsf{B})\tilde{\subseteq }cl\left(int\left(cl(\mathtt{F},\mathsf{B})\right)\right)$.

(iv) 

Soft somewhat open set (briefly, soft $sw$-open) [26] provided that $(\mathtt{F},\mathsf{B})=\varphi$ or $int(\mathtt{F},\mathsf{B})\ne \varphi$.

Definition 12 ([17,20,21,26,42]).

A soft function ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is said to be:

(i) 

Soft open if ${\mathbb{I}}_{\mathbb{P}}(\mathtt{G},\mathsf{B})$ is a soft open set where $(\mathtt{G},\mathsf{B})$ is soft open.

(ii) 

Soft continuous (resp., soft α-continuous, soft semi-continuous, soft $sw$-continuous) if ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})$ is a soft open (resp., soft α-open, soft semi-open, soft $sw$-open) set where $(\mathtt{F},\mathsf{B})$ is soft open.

(iii) 

Soft bi-continuous (resp., soft homeomorphism) if it is soft open and continuous (resp., bijective and soft bi-continuous).

A topological property is a property of an STS that is preserved under any soft homeomorphism.

Theorem 2 ([29]).

If ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},\Im ,\mathsf{B})\to (\mathbf{U},\mathcal{S},\mathsf{E})$ is soft continuous, then $h:(\mathbf{O},{\Im }_{\mathsf{b}})\to (\mathbf{U},{\mathcal{S}}_{\mathbb{P}\left(\mathsf{b}\right)})$ is continuous for each $\mathsf{b}\in \mathsf{B}$.

3. On Weakly Soft Semi-Open Sets

Herein, we display the concept of “weakly soft semi-open sets”, which is the main idea of this document. It shows that the family of weakly soft semi-open sets is a proper generalization of soft open sets. We provide some interesting examples to prove that the extended soft topology guarantees that this family is a real superset of the families of soft semi-open sets and a genuine subset of the family of soft $sw$-open sets. In addition, we demonstrate some differences between this family and other generalizations of soft open sets such as this family is not closed under soft unions. In the end, we reveal how this family conducts itself with some topological principles such as functions and the product of soft spaces.

Definition 13. 

An SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$ is called a weakly soft semi-open set if it is a null soft set or there is a component of it which is a nonempty semi-open set. That is, $\mathtt{F}\left(\mathsf{b}\right)=\varnothing$ for all $\mathsf{b}\in \mathsf{B}$ or $\varnothing \ne \mathtt{F}\left(\mathsf{b}\right)\subseteq cl\left(int\right(\mathtt{F}\left(\mathsf{b}\right)\left)\right)$ for some $\mathsf{b}\in \mathsf{B}$.

We call $(\mathtt{F},\mathsf{B})$ a weakly soft semi-closed set if its complement is weakly soft semi-open.

Proposition 3. 

A subset $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$ is weakly soft semi-closed if $(\mathtt{F},\mathsf{B})=\tilde{\mathbf{O}}$ or $int\left(cl\right(\mathtt{F}\left(\mathsf{b}\right)\left)\right)\subseteq \mathtt{F}\left(\mathsf{b}\right)\ne \mathbf{O}$ for some $\mathsf{b}\in \mathsf{B}$.

Proof. 

$“\Rightarrow ”$: If $(\mathtt{F},\mathsf{B})$ is a weakly soft semi-closed set, then $({\mathtt{F}}^c,\mathsf{B})=\varphi$ or $\varnothing \ne F^c\left(\mathsf{b}\right)\subseteq cl\left(int\left({\mathtt{F}}^c\left(\mathsf{b}\right)\right)\right)$ for some $\mathsf{b}\in \mathsf{B}$. Therefore, $(\mathtt{F},\mathsf{B})=\tilde{\mathbf{O}}$ or $int\left(cl\right(\mathtt{F}\left(\mathsf{b}\right)\left)\right)\subseteq \mathtt{F}\left(\mathsf{b}\right)\ne \mathbf{O}$ for some $\mathsf{b}\in \mathsf{B}$, as demanded.

$“\Leftarrow ”$: Let $(\mathtt{F},\mathsf{B})$ be an SS such that $(\mathtt{F},\mathsf{B})=\tilde{\mathbf{O}}$ or $int\left(cl\right(\mathtt{F}\left(\mathsf{b}\right)\left)\right)\subseteq \mathtt{F}\left(\mathsf{b}\right)\ne \mathbf{O}$ for some $\mathsf{b}\in \mathsf{B}$. Then, $({\mathtt{F}}^c,\mathsf{B})=\varphi$ or $\varnothing \ne F^c\left(\mathsf{b}\right)\subseteq cl\left(int\left({\mathtt{F}}^c\left(\mathsf{b}\right)\right)\right)$ for some $\mathsf{b}\in \mathsf{B}$. This implies that $({\mathtt{F}}^c,\mathsf{B})$ is weakly soft semi-open. Hence, $(\mathtt{F},\mathsf{B})$ is weakly soft semi-closed, as demanded. □

The family of weakly soft semi-open (weakly soft semi-closed) subsets is not closed under soft union or soft intersection as the next example clarifies.

Example 1. 

Let the universe be the real numbers set $\mathcal{R}$ and a set of parameters be $\mathsf{B}=\{{\mathsf{b}}_1,{\mathsf{b}}_2\}$. Let ℑ be the ST on $\mathcal{R}$ generated by $\{({\mathsf{b}}_i,\mathtt{F}\left({\mathsf{b}}_i\right)):\mathtt{F}\left({\mathsf{b}}_i\right)=(a_i,b_i);a_i,b_i\in \mathcal{R};a_i\le b_i$ and $i=1,2\}$. Let $\mathcal{Q}$ be the set of rational numbers and take the next weakly soft semi-open sets:

$$({\mathtt{F}}_1,\mathsf{B})=\{({\mathsf{b}}_1,(0,1)\cup \left\{9\right\}),({\mathsf{b}}_2,[0,1])\}$$

$$({\mathtt{F}}_2,\mathsf{B})=\{({\mathsf{b}}_1,(0,1)),({\mathsf{b}}_2,[0,1]\cup \left\{9\right\})\}$$

$$({\mathtt{F}}_3,\mathsf{B})=\{({\mathsf{b}}_1,\mathcal{R}),({\mathsf{b}}_2,\mathcal{Q})\}$$

$$({\mathtt{F}}_4,\mathsf{B})=\{({\mathsf{b}}_1,\mathcal{Q}),({\mathsf{b}}_2,\mathcal{R})\}$$

Then:

(i) 

$({\mathtt{F}}_1,\mathsf{B})\tilde{\cup }({\mathtt{F}}_2,\mathsf{B})=\{({\mathsf{b}}_1,(0,1)\cup \left\{9\right\}),({\mathsf{b}}_2,[0,1]\cup \left\{9\right\})\}$ is not a weakly soft semi-open set.

(ii) 

$({\mathtt{F}}_3,\mathsf{B})\tilde{\cap }({\mathtt{F}}_4,\mathsf{B})=\{({\mathsf{b}}_1,\mathcal{Q}),({\mathsf{b}}_2,\mathcal{Q})\}$ is not a weakly soft semi-open set.

Take the next weakly soft semi-closed sets:

$$({\mathtt{G}}_1,\mathsf{B})=\{({\mathsf{b}}_1,\left\{9\right\}),({\mathsf{b}}_2,\mathcal{Q})\}$$

$$({\mathtt{G}}_2,\mathsf{B})=\{({\mathsf{b}}_1,\mathcal{Q}),({\mathsf{b}}_2,\left\{9\right\})\}$$

$$({\mathtt{G}}_3,\mathsf{B})=\{({\mathsf{b}}_1,(1,2)\cup (2,3)),({\mathsf{b}}_2,[1,3])\}$$

$$({\mathtt{G}}_4,\mathsf{B})=\{({\mathsf{b}}_1,[1,3]),({\mathsf{b}}_2,(1,2)\cup (2,3))\}$$

Then:

(i) 

$({\mathtt{G}}_1,\mathsf{B})\tilde{\cup }({\mathtt{G}}_2,\mathsf{B})=\{({\mathsf{b}}_1,\mathcal{Q}),({\mathsf{b}}_2,\mathcal{Q})\}$ is not a weakly soft semi-closed set.

(ii) 

$({\mathtt{G}}_3,\mathsf{B})\tilde{\cap }({\mathtt{G}}_4,\mathsf{B})=\{({\mathsf{b}}_1,(1,2)\cup (2,3)),({\mathsf{b}}_2,(1,2)\cup (2,3))\}$ is not a weakly soft semi-closed set.

Proposition 4. 

Let $(\mathbf{O},\Im ,\mathsf{B})$ be a full and hyperconnected STS. Then, the soft intersection of soft semi-open and weakly soft semi-open subsets is weakly soft semi-open.

Proof. 

Assume that $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ are, respectively, soft semi-open and weakly soft semi-open sets. Then, there exists a non-null soft open set $(\mathtt{U},\mathsf{B})$ and $\mathsf{b}\in \mathsf{B}$ such that $(\mathtt{U},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$ and $\mathtt{G}\left(\mathsf{b}\right)$ is a nonempty semi-open subset of $(\mathbf{O},{\Im }_{\mathsf{b}})$. So, there exists a nonempty open subset $V_{\mathsf{b}}$ of $\mathtt{G}\left(\mathsf{b}\right)$. This means that ℑ contains a non-null soft open set $(\mathtt{V},\mathsf{B})$ with $\mathtt{V}\left(\mathsf{b}\right)=V_{\mathsf{b}}$. Since ℑ is soft hyperconnected and full, we obtain $V_{\mathsf{b}}\cap \mathtt{U}\left(\mathsf{b}\right)\ne \varnothing$. Therefore, $\mathtt{G}\left(\mathsf{b}\right)$ and $\mathtt{U}\left(\mathsf{b}\right)$ has a nonempty intersection. It follows from general topology that $\mathtt{G}\left(\mathsf{b}\right)\cap \mathtt{U}\left(\mathsf{b}\right)$ is a nonempty semi-open subset of $(\mathbf{O},{\Im }_{\mathsf{b}})$. Hence, $(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})$ is a weakly soft semi-open set. □

Corollary 1. 

Let $(\mathbf{O},\Im ,\mathsf{B})$ be a full and hyperconnected STS. Then, the soft intersection of soft open subsets and weakly soft semi-open is weakly soft semi-open.

Remark 1. 

(i)Every pseudo-constant soft subset $(\mathtt{F},\mathsf{B})$ is a weakly soft semi-subset because $\mathtt{F}\left(\mathsf{b}\right)=\varnothing$ for all $\mathsf{b}\in \mathsf{B}$ or $int\left(\mathtt{F}\right(\mathsf{b}\left)\right)=\mathbf{O}$ for some $\mathsf{b}\in \mathsf{B}$.

(ii) 

An SS $(\mathtt{F},\mathsf{B})$ of $(\mathbf{O},\Im ,\mathsf{B})$ with $\mathtt{F}\left(\mathsf{b}\right)=\mathbf{O}$ (resp. $\mathtt{F}\left(\mathsf{b}\right)=\varnothing$) is weakly soft semi-open (resp. weakly soft semi-closed).

The next proposition is straightforward.

Proposition 5. 

Every soft open set is weakly soft semi-open.

Now, we derive the conditions under which some relationships that associate weakly soft semi-open sets with some generalizations of soft open sets are obtained.

Proposition 6. 

Every soft α-open (soft semi-open) subset of an extended STS $(\mathbf{O},\Im ,\mathsf{B})$ is weakly soft semi-open.

Proof. 

Let $(\mathtt{F},\mathsf{B})$ be a non-null soft semi-open set. Then, $(\mathtt{F},\mathsf{B})\tilde{\subseteq }int\left(cl\left(int(\mathtt{F},\mathsf{B})\right)\right)$. Since ℑ is an extended soft topology, we obtain $\mathtt{F}\left(\mathsf{b}\right)\subseteq int\left(cl\right(int\left(\mathtt{F}\right(\mathsf{b}\left)\right)\left)\right)$ for each $\mathsf{b}\in \mathsf{B}$. This implies that there is a component of $(\mathtt{F},\mathsf{B})$ which is a nonempty $\alpha$-open set. As we know that any $\alpha$-open set is semi-open; hence, $(\mathtt{F},\mathsf{B})$ is weakly soft semi-open.

Following similar arguments, one can prove the case of a soft semi-open set. □

Proposition 7. 

If $(\mathbf{O},\Im ,\mathsf{B})$ is extended, then every weakly soft semi-open set is soft $sw$-open.

Proof. 

Let $(\mathtt{F},\mathsf{B})$ be a non-null weakly soft semi-open set. Then, there is a component of $(\mathtt{F},\mathsf{B})$ which is a nonempty semi-open set. So, $int\left(\mathtt{F}\right(\mathsf{b}\left)\right)\ne \varnothing$ for some $\mathsf{b}\in \mathsf{B}$. Since ℑ is extended, we obtain $int(\mathtt{F},\mathsf{B})=\left(int\right(\mathtt{F}),\mathsf{B})\ne \varphi$. This completes the proof. □

It cannot be dispensed of the stipulation of “extended soft topology” imposed in Proposition 6 and Proposition 7, as the following example elaborates.

Example 2. 

Let the universe be $\mathbf{O}=\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}$ and a parameters set be $\mathsf{B}=\{{\mathsf{b}}_1,{\mathsf{b}}_2\}$. Take the family ℑ consisting of ϕ, $\tilde{\mathbf{O}}$ and the following SSs over $\mathbf{O}$ with $\mathsf{B}$

$({\mathtt{F}}_1,\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\varnothing )\}$;

$({\mathtt{F}}_2,\mathsf{B})=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$({\mathtt{F}}_3,\mathsf{B})=\{({\mathsf{b}}_1,\mathbf{O}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$({\mathtt{F}}_4,\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\mathbf{O})\}$;

$({\mathtt{F}}_5,\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$({\mathtt{F}}_6,\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\})\}$; and

$({\mathtt{F}}_7,\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$.

Then, $(\mathbf{O},\Im ,\mathsf{B})$ is an STS. Remark that $(\mathtt{H},\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2\}),({\mathsf{b}}_2,\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2\})\}$ is soft semi-open because $cl\left(int(\mathtt{H},\mathsf{B})\right)=\tilde{\mathbf{O}}$. However, it is not a weakly soft semi-open set because $cl\left(int\left(\mathtt{H}\left({\mathsf{b}}_1\right)\right)\right)=cl\left(int\left(\mathtt{H}\left({\mathsf{b}}_2\right)\right)\right)=\left\{{\mathbf{\nu }}_1\right\}\supseteq \mathtt{H}\left({\mathsf{b}}_1\right)=\mathtt{H}\left({\mathsf{b}}_2\right)$. In addition, $(\mathtt{G},\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_3\right\})\}$ is a weakly soft semi-open set because $cl\left(int\left(\mathtt{G}\left({\mathsf{b}}_1\right)\right)\right)=\mathtt{G}\left({\mathsf{b}}_1\right)$. However, it is not a soft $sw$-open set because $int(\mathtt{G},\mathsf{B})=\varphi$.

The example below shows that Proposition 6 and Proposition 7 are irreversible.

Example 3. 

Let the universe be $\mathbf{O}=\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}$ and a parameters set be $\mathsf{B}=\{{\mathsf{b}}_1,{\mathsf{b}}_2\}$. Take the family ℑ consisting of ϕ, $\tilde{\mathbf{O}}$ and the following SSs over $\mathbf{O}$ with $\mathsf{B}$

$({\mathtt{F}}_1,\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\})\}$;

$({\mathtt{F}}_2,\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$({\mathtt{F}}_3,\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\varnothing )\}$;

$({\mathtt{F}}_4,\mathsf{B})=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$({\mathtt{F}}_5,\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\varnothing )\}$;

$({\mathtt{F}}_6,\mathsf{B})=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\})\}$;

$({\mathtt{F}}_7,\mathsf{B})=\{({\mathsf{b}}_1,\mathbf{O}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$({\mathtt{F}}_8,\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\mathbf{O})\}$;

$({\mathtt{F}}_9,\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\mathbf{O})\}$;

$({\mathtt{F}}_{10},\mathsf{B})=\{({\mathsf{b}}_1,\mathbf{O}),({\mathsf{b}}_2,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\})\}$;

$({\mathtt{F}}_{11},\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$({\mathtt{F}}_{12},\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\})\}$;

$({\mathtt{F}}_{13},\mathsf{B})=\{({\mathsf{b}}_1,\mathbf{O}),({\mathsf{b}}_2,\varnothing )\}$; and

$({\mathtt{F}}_{14},\mathsf{B})=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,\mathbf{O})\}$.

Then, $(\mathbf{O},\Im ,\mathsf{B})$ is an extended STS. Now, $(\mathtt{H},\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_3\right\})\}$ is weakly soft semi-open because $\mathtt{H}\left({\mathsf{b}}_1\right)$ is a nonempty semi-open set. However, it is not a soft semi-open set because $cl\left(int(\mathtt{H},\mathsf{B})\right)=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\varnothing )\}\supseteq (\mathtt{H},\mathsf{B})$. In addition, $(\mathtt{G},\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2\})$, $({\mathsf{b}}_2,\varnothing )\}$ is soft $sw$-open because $int(\mathtt{G},\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\varnothing )\}\ne \varphi$. However, it is not a weakly soft semi-open set because $cl\left(int\left(\mathtt{G}\left({\mathsf{b}}_1\right)\right)\right)=\left\{{\mathbf{\nu }}_1\right\}\supseteq \mathtt{G}\left({\mathsf{b}}_1\right)$ and $\mathtt{G}\left({\mathsf{b}}_2\right)$ is empty.

Proposition 8. 

The image and pre-image of weakly soft semi-open set under a soft bi-continuous function is weakly soft semi-open.

Proof. 

To show the case of an image, let ${\mathbb{I}}_{\mathbb{P}}$ be a soft bi-continuous function from an STS $(\mathbf{O},\Im ,\mathsf{B})$ to an STS $(\mathbf{U},\mho ,\mathsf{E})$ and let $(\mathtt{F},\mathsf{B})$ be a weakly soft semi-subset of $(\mathbf{O},\Im ,\mathsf{B})$. Suppose that there exists $\mathsf{b}\in \mathsf{B}$ such that $\mathtt{F}\left(\mathsf{b}\right)$ is a nonempty semi-open subset and let $\mathbb{P}\left(\mathsf{b}\right)=\mathsf{e}$. According to Theorem 2, it follows from the soft bicontinuity of ${\mathbb{I}}_{\mathbb{P}}$ that $\mathbb{I}:(\mathbf{O},{\Im }_{\mathsf{b}})\to (\mathbf{U},{\mho }_{\mathbb{P}\left(\mathsf{b}\right)=\mathsf{e}})$ is a bi-continuous function. It is well known that a continuity of $\mathbb{I}$ implies that $\mathbb{I}\left(cl\right(U\left)\right)\subseteq cl\left(\mathbb{I}\right(U\left)\right)$, and an openness of $\mathbb{I}$ implies that $\mathbb{I}\left(int\right(U\left)\right)\subseteq int\left(\mathbb{I}\right(U\left)\right)$ for each subset U of $\mathbf{O}$. This implies that $\mathbb{I}\left(\mathtt{F}\left(\mathsf{b}\right)\right)\tilde{\subseteq }\mathbb{I}(cl\left(int\left(\mathtt{F}\left(\mathsf{b}\right)\right)\right)\tilde{\subseteq }cl\left(int\left(\mathbb{I}\left(\mathtt{F}\left(\mathsf{b}\right)\right)\right)\right)$. According to Definition 12, we find that $\mathbb{I}\left(\mathtt{F}\right(\mathsf{b}\left)\right)$ is a nonempty semi-open subset of ${\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})$; hence, ${\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})$ is a weakly soft semi-open subset of an STS $(\mathbf{U},\mho ,\mathsf{E})$. □

Corollary 2. 

The property of being a weakly soft semi-open set is a topological property.

Proposition 9. 

The product of two weakly soft semi-open sets is weakly soft semi-open.

Proof. 

Suppose that $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ are weakly soft semi-open subsets and let $(\mathtt{H},\mathsf{B}\times \mathsf{B})=(\mathtt{F},\mathsf{B})\times (\mathtt{G},\mathsf{B})$. Then, there are ${\mathsf{b}}_1,{\mathsf{b}}_2\in \mathsf{B}$ such that $\mathtt{F}\left({\mathsf{b}}_1\right)$ and $\mathtt{G}\left({\mathsf{b}}_2\right)$ are nonempty semi-open subsets. Now, $({\mathsf{b}}_1,{\mathsf{b}}_2)\in \mathsf{B}\times \mathsf{B}$ such that $\mathtt{H}({\mathsf{b}}_1,{\mathsf{b}}_2)=\mathtt{F}\left({\mathsf{b}}_1\right)\times \mathtt{G}\left({\mathsf{b}}_2\right)$. As we know from the classical topology that the product of two nonempty semi-open subsets is still a nonempty semi-open subset; therefore, $\mathtt{H}({\mathsf{b}}_1,{\mathsf{b}}_2)$ is a nonempty semi-open subset. Hence, $(\mathtt{H},\mathsf{B}\times \mathsf{B})$ is a weakly soft semi-open subset. □

4. Weakly Semi-Interior and Weakly Semi-Closure Soft Points

This part is dedicated to producing novel operators for each SS, namely, weakly semi-interior, weakly semi-closure, weakly semi-boundary, and weakly semi-limit soft points. We display their main characteristics and depict the relationships among them. By appropriate examples, we evince that the property that says that the “weakly semi-interior (resp., weakly semi-closure) of SS need not be weakly semi-open (resp., weakly semi-closed) sets” is false in general.

Definition 14. 

The weakly semi-interior soft points of an SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$, denoted by $in{t}_{ws}(\mathtt{F},\mathsf{B})$, is defined as the union of all weakly soft semi-open sets contained in $(\mathtt{F},\mathsf{B})$.

By Example 1, we remark that the weakly semi-interior soft points of an SS need not be a weakly semi-open set. That is, $in{t}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})$ does not imply that $(\mathtt{F},\mathsf{B})$ is a weakly semi-open set.

One can prove the next propositions.

Proposition 10. 

Let $(\mathtt{F},\mathsf{B})$ be an SS of an STS $(\mathbf{O},\Im ,\mathsf{B})$ and ${\mathbf{\nu }}_{\mathsf{b}}\in \tilde{\mathbf{O}}$. Then, ${\mathbf{\nu }}_{\mathsf{b}}\in in{t}_{ws}(\mathtt{F},\mathsf{B})$ if there is a weakly soft semi-open set $(\mathtt{G},\mathsf{B})$ that contains ${\mathbf{\nu }}_{\mathsf{b}}$ such that $(\mathtt{G},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$.

Proposition 11. 

Let $(\mathtt{F},\mathsf{B})$, $(\mathtt{G},\mathsf{B})$ be SSs of an STS $(\mathbf{O},\Im ,\mathsf{B})$. Then,

1. 

$in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$.

2. 

if $(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{G},\mathsf{B})$, then $in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }in{t}_{ws}(\mathtt{G},\mathsf{B})$.

Corollary 3. 

For any two subsets $(\mathtt{F},\mathsf{B})$, $(\mathtt{G},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$, we have the following results:

1. 

$in{t}_{ws}\left[(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\right]\tilde{\subseteq }in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }in{t}_{ws}(\mathtt{G},\mathsf{B})$.

2. 

$in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }in{t}_{ws}(\mathtt{G},\mathsf{B})\tilde{\subseteq }in{t}_{ws}\left[(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})\right]$.

Proof. 

It automatically comes from the following:

1. $(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$ and $(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\tilde{\subseteq }(\mathtt{G},\mathsf{B})$.

2. $(\mathtt{F},\mathsf{B})\tilde{\subseteq }\left[(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})\right]$ and $(\mathtt{G},\mathsf{B})\tilde{\subseteq }\left[(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})\right]$. □

In Proposition 11 and Corollary 3, the inclusion relations are proper. To demonstrate that, consider Example 1 and take the following SSs:

$(\mathtt{E},\mathsf{B})=\{({\mathsf{b}}_1,\left\{1\right\}),({\mathsf{b}}_2,\{2,3\})\}$, $(\mathtt{F},\mathsf{B})=\{({\mathsf{b}}_1,(1,2)),({\mathsf{b}}_2,[1,2])\}$,

$(\mathtt{G},\mathsf{B})=\{({\mathsf{b}}_1,(2,3)),({\mathsf{b}}_2,[2,3])\}$, $(\mathtt{H},\mathsf{B})=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,\mathcal{Q})\}$, and $(\mathtt{J},\mathsf{B})=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,{\mathcal{Q}}^c)\}$.

We remark on the following properties:

1.

$(\mathtt{E},\mathsf{B})\tilde{\subseteq }in{t}_{ws}(\mathtt{E},\mathsf{B})=\varphi$.

2.

$in{t}_{ws}(\mathtt{E},\mathsf{B})\tilde{\subseteq }in{t}_{ws}(\mathtt{F},\mathsf{B})$ whereas $(\mathtt{E},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$.

3.

$in{t}_{ws}\left[(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\right]=\varphi$ whereas $in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }in{t}_{ws}(\mathtt{G},\mathsf{B})=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,\left\{2\right\})\}$.

4.

$in{t}_{ws}(\mathtt{H},\mathsf{B})\tilde{\cup }in{t}_{ws}(\mathtt{J},\mathsf{B})=\varphi$ whereas $in{t}_{ws}\left[(\mathtt{H},\mathsf{B})\tilde{\cup }(\mathtt{J},\mathsf{B})\right]=\{({\mathsf{b}}_1,\varnothing ),({\mathsf{b}}_2,\mathcal{R})\}$.

Definition 15. 

The weakly semi-closure soft points of an SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$, denoted by $c{l}_{ws}(\mathtt{F},\mathsf{B})$, is defined as the intersection of all weakly soft semi-closed sets containing $(\mathtt{F},\mathsf{B})$.

By Example 1, we remark that the weakly semi-closure points of an SS need not be a weakly semi-closed set. That is, $c{l}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})$ does not imply that $(\mathtt{F},\mathsf{B})$ is a weakly semi-closed set.

Proposition 12. 

Let $(\mathtt{F},\mathsf{B})$ be an SS of an STS $(\mathbf{O},\Im ,\mathsf{B})$ and ${\mathbf{\nu }}_{\mathsf{b}}\in \tilde{\mathbf{O}}$. Then, ${\mathbf{\nu }}_{\mathsf{b}}\in c{l}_{ws}(\mathtt{F},\mathsf{B})$ if $(\mathtt{G},\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})\ne \varphi$ for each weakly soft semi-open set $(\mathtt{G},\mathsf{B})$ contains ${\mathbf{\nu }}_{\mathsf{b}}$.

Proof. 

$[\Rightarrow ]$ Let ${\mathbf{\nu }}_{\mathsf{b}}\in c{l}_{ws}(\mathtt{F},\mathsf{B})$. Suppose that there is weakly soft semi-open set $(\mathtt{G},\mathsf{B})$ containing ${\mathbf{\nu }}_{\mathsf{b}}$ with $(\mathtt{G},\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})=\varphi$. Then, $(\mathtt{F},\mathsf{B})\tilde{\subseteq }({\mathtt{G}}^c,\mathsf{B})$. Therefore, $c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }({\mathtt{G}}^c,\mathsf{B})$. Thus, ${\mathbf{\nu }}_{\mathsf{b}}\notin c{l}_{ws}(\mathtt{F},\mathsf{B})$. This is a contradiction, which means that $(\mathtt{G},\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})\ne \varphi$, as demanded.

$[\Leftarrow ]$ Suppose that $(\mathtt{G},\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})\ne \varphi$ for each weakly soft semi-open set $(\mathtt{G},\mathsf{B})$ contains ${\mathbf{\nu }}_{\mathsf{b}}$. Let ${\mathbf{\nu }}_{\mathsf{b}}\notin c{l}_{ws}(\mathtt{F},\mathsf{B})$. Then, there is a weakly soft semi-closed set $(\mathtt{H},\mathsf{B})$ containing $(\mathtt{F},\mathsf{B})$ with ${\mathbf{\nu }}_{\mathsf{b}}\notin (\mathtt{H},\mathsf{B})$. So, ${\mathbf{\nu }}_{\mathsf{b}}\in ({\mathtt{H}}^c,\mathsf{B})$ and $({\mathtt{H}}^c,\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})=\varphi$, which contradicts the assumption. Hence, we obtain the desired result. □

Corollary 4. 

If $(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})=\varphi$ such that $(\mathtt{F},\mathsf{B})$ is a weakly soft semi-open set and $(\mathtt{G},\mathsf{B})$ is an SS in $(\mathbf{O},\Im ,\mathsf{B})$, then $(\mathtt{F},\mathsf{B})\tilde{\cap }c{l}_{ws}(\mathtt{G},\mathsf{B})=\varphi$.

Proposition 13. 

The following properties hold for an SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$.

(1)

${\left[in{t}_{ws}(\mathtt{F},\mathsf{B})\right]}^c=c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})$.

(2)

${\left[c{l}_{ws}(\mathtt{F},\mathsf{B})\right]}^c=in{t}_{ws}({\mathtt{F}}^c,\mathsf{B})$.

Proof. 

1. If ${\mathbf{\nu }}_{\mathsf{b}}\notin {\left[in{t}_{ws}(\mathtt{F},\mathsf{B})\right]}^c$, then there is a weakly soft semi-open set $(\mathtt{G},\mathsf{B})$ with ${\mathbf{\nu }}_{\mathsf{b}}\in (\mathtt{G},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$. Therefore, $({\mathtt{F}}^c,\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})=\varphi$ and hence, ${\mathbf{\nu }}_{\mathsf{b}}\notin c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})$. Conversely, if ${\mathbf{\nu }}_{\mathsf{b}}\notin c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})$, we can follow the previous steps to verify ${\mathbf{\nu }}_{\mathsf{b}}\notin {\left[in{t}_{ws}(\mathtt{F},\mathsf{B})\right]}^c$.

2. Following similar approach given in 2. □

The proof of the next proposition is following from Definition 15.

Proposition 14. 

Let $(\mathtt{F},\mathsf{B})$, $(\mathtt{G},\mathsf{B})$ be SSs of an STS $(\mathbf{O},\Im ,\mathsf{B})$. Then

1. 

$(\mathtt{F},\mathsf{B})\tilde{\subseteq }c{l}_{ws}(\mathtt{F},\mathsf{B})$.

2. 

if $(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{G},\mathsf{B})$, then $c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }c{l}_{ws}(\mathtt{G},\mathsf{B})$.

Corollary 5. 

The following results hold for any subsets $(\mathtt{F},\mathsf{B})$, $(\mathtt{G},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$.

1. 

$c{l}_{ws}\left[(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\right]\tilde{\subseteq }c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }c{l}_{ws}(\mathtt{G},\mathsf{B})$.

2. 

$c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }c{l}_{ws}(\mathtt{G},\mathsf{B})\tilde{\subseteq }c{l}_{ws}\left[(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})\right]$.

Proof. 

It automatically comes from the following:

1. $(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$ and $(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\tilde{\subseteq }(\mathtt{G},\mathsf{B})$.

2. $(\mathtt{F},\mathsf{B})\tilde{\subseteq }\left[(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})\right]$ and $(\mathtt{G},\mathsf{B})\tilde{\subseteq }\left[(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})\right]$. □

In Proposition 14 and Corollary 5, the inclusion relations are proper. To demonstrate that, consider Example 1 and take the following SSs:

Let $(\mathtt{E},\mathsf{B})=\{({\mathsf{b}}_1,\mathcal{R}),({\mathsf{b}}_2,\mathcal{Q})\}$, $(\mathtt{F},\mathsf{B})=\{({\mathsf{b}}_1,(1,2)),({\mathsf{b}}_2,[1,2])\}$, and $(\mathtt{G},\mathsf{B})=\{({\mathsf{b}}_1,\mathcal{R})$, $({\mathsf{b}}_2,{\mathcal{Q}}^c)\}$.

We remark on the following properties:

1.

$c{l}_{ws}(\mathtt{E},\mathsf{B})=\tilde{\mathcal{R}}\tilde{\subset }(\mathtt{E},\mathsf{B})$.

2.

$c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }c{l}_{ws}(\mathtt{E},\mathsf{B})$ whereas $(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{E},\mathsf{B})$.

3.

$c{l}_{ws}\left[(\mathtt{E},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\right]=\{({\mathsf{b}}_1,\mathcal{R}),({\mathsf{b}}_2,\varnothing )\}$ whereas $c{l}_{ws}(\mathtt{E},\mathsf{B})\tilde{\cap }c{l}_{ws}(\mathtt{G},\mathsf{B})=\tilde{\mathcal{R}}$.

Definition 16. 

A soft point ${\mathbf{\nu }}_{\mathsf{b}}$ is said to be a weakly semi-boundary soft point of an SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$ if ${\mathbf{\nu }}_{\mathsf{b}}$ belongs to the complement of $in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }in{t}_{ws}({\mathtt{F}}^c,\mathsf{B})$.

All weakly semi-boundary soft points of $(\mathtt{F},\mathsf{B})$ are called a weakly semi-boundary set and denoted by $b_{ws}(\mathtt{F},\mathsf{B})$.

Proposition 15. 

$b_{ws}(\mathtt{F},\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})$ for every SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$.

Proof. 

$b_{ws}(\mathtt{F},\mathsf{B})={\left[in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }in{t}_{ws}({\mathtt{F}}^c,\mathsf{B})\right]}^c$

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\left[in{t}_{ws}(\mathtt{F},\mathsf{B})\right]}^c\tilde{\cap }{\left[in{t}_{ws}({\mathtt{F}}^c,\mathsf{B})\right]}^c$ (De Morgan’s law);

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})\tilde{\cap }c{l}_{ws}(\mathtt{F},\mathsf{B})$ (Proposition 13(2)). □

Corollary 6. 

For every SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$, the following properties hold.

1. 

$b_{ws}(\mathtt{F},\mathsf{B})=b_{ws}({\mathtt{F}}^c,\mathsf{B})$.

2. 

$b_{ws}(\mathtt{F},\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})\backslash in{t}_{ws}(\mathtt{F},\mathsf{B})$.

3. 

$c{l}_{ws}(\mathtt{F},\mathsf{B})=in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }{b}_{ws}(\mathtt{F},\mathsf{B})$.

4. 

$in{t}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})\backslash b_{ws}(\mathtt{F},\mathsf{B})$.

Proof. 

1. Obvious.

2.

$b_{ws}(\mathtt{F},\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})\setminus {\left[c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})\right]}^c$. By 2 of Proposition 13, we obtain the required relation.

3.

$in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }{b}_{ws}(\mathtt{F},\mathsf{B})=in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }[c{l}_{ws}(\mathtt{F},\mathsf{B})\setminus in{t}_{ws}(\mathtt{F},\mathsf{B})]=c{l}_{ws}(\mathtt{F},\mathsf{B})$.

4.

$(\mathtt{F},\mathsf{B})\setminus b_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})\setminus [c{l}_{ws}(\mathtt{F},\mathsf{B})\setminus in{t}_{ws}(\mathtt{F},\mathsf{B})]$;

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=(\mathtt{F},\mathsf{B})\tilde{\cap }{\left[c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }{\left(in{t}_{ws}(\mathtt{F},\mathsf{B})\right)}^c\right]}^c$;

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=(\mathtt{F},\mathsf{B})\tilde{\cap }\left[{\left(c{l}_{ws}(\mathtt{F},\mathsf{B})\right)}^c\tilde{\cup }in{t}_{ws}(\mathtt{F},\mathsf{B})\right]$;

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left[(\mathtt{F},\mathsf{B})\tilde{\cap }{\left(c{l}_{ws}(\mathtt{F},\mathsf{B})\right)}^c\right]\tilde{\cup }\left[(\mathtt{F},\mathsf{B})\tilde{\cap }in{t}_{ws}(\mathtt{F},\mathsf{B})\right];$

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=in{t}_{ws}(\mathtt{F},\mathsf{B})$. □

Proposition 16. 

Let $(\mathtt{F},\mathsf{B}),(\mathtt{G},\mathsf{B})$ be SSs of an STS $(\mathbf{O},\Im ,\mathsf{B})$; then, the following properties hold.

1. 

$b_{ws}\left(in{t}_{ws}(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }{b}_{ws}(\mathtt{F},\mathsf{B})$.

2. 

$b_{ws}\left(c{l}_{ws}(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }{b}_{ws}(\mathtt{F},\mathsf{B})$.

Proof. 

By substituting in Formula 3 from Corollary 6, the proof follows. □

Proposition 17. 

Let $(\mathtt{F},\mathsf{B})$ be an SS of an STS $(\mathbf{O},\Im ,\mathsf{B})$. Then

1. 

$(\mathtt{F},\mathsf{B})=in{t}_{ws}(\mathtt{F},\mathsf{B})$ if $b_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})=\varphi$.

2. 

$(\mathtt{F},\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})$ if $b_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$.

Proof. 

1. Suppose that $(\mathtt{F},\mathsf{B})=in{t}_{ws}(\mathtt{F},\mathsf{B})$. Then, by (4) from Corollary 6, $(\mathtt{F},\mathsf{B})=in{t}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})\backslash b_{ws}(\mathtt{F},\mathsf{B})$ and hence $b_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})=\varphi$. Conversely, let ${\mathbf{\nu }}_{\mathsf{b}}\in (\mathtt{F},\mathsf{B})$. Since ${\mathbf{\nu }}_{\mathsf{b}}\notin b_{ws}(\mathtt{F},\mathsf{B})$ and ${\mathbf{\nu }}_{\mathsf{b}}\in c{l}_{ws}(\mathtt{F},\mathsf{B})$, by (3) from Corollary 6, ${\mathbf{\nu }}_{\mathsf{b}}\in in{t}_{ws}(\mathtt{F},\mathsf{B})$. Therefore, $in{t}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})$, as demanded.

2.

Assume that $(\mathtt{F},\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})$. Then, $b_{ws}(\mathtt{F},\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }c{l}_{ws}({\mathtt{F}}^c,\mathsf{B})$$\tilde{\subseteq }$$c{l}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})$, as demanded. Conversely, if $b_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$, then by (3) from Corollary 6, $c{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }in{t}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})$ and hence $c{l}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})$, as demanded. □

Corollary 7. 

Let $(\mathtt{F},\mathsf{B})$ be an SS of an STS $(\mathbf{O},\Im ,\mathsf{B})$. Then, $in{t}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})=c{l}_{ws}(\mathtt{F},\mathsf{B})$ if $b_{ws}(\mathtt{F},\mathsf{B})=\varphi$.

Definition 17. 

A soft point ${\mathbf{\nu }}_{\mathsf{b}}$ is said to be a weakly semi-limit soft point of an SS $(\mathtt{F},\mathsf{B})$ of an STS $(\mathbf{O},\Im ,\mathsf{B})$ if $[(\mathtt{G},\mathsf{B})\backslash {\mathbf{\nu }}_{\mathsf{b}}]\cap (\mathtt{F},\mathsf{B})\ne \varphi$ for each weakly soft semi-open set $(\mathtt{G},\mathsf{B})$ containing ${\mathbf{\nu }}_{\mathsf{b}}$.

All weakly semi-limit points of $(\mathtt{F},\mathsf{B})$ are called a weakly semi-derived set and denoted by $l_{ws}(\mathtt{F},\mathsf{B})$.

Proposition 18. 

Let $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ be subsets of an STS $(\mathbf{O},\Im ,\mathsf{B})$. If $(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{G},\mathsf{B})$, then $l_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }{l}_{ws}(\mathtt{G},\mathsf{B})$.

Proof. 

Straightforward by Definition 17. □

Corollary 8. 

Consider $(\mathtt{F},\mathsf{B})$ and $(\mathtt{G},\mathsf{B})$ are subsets of an STS $(\mathbf{O},\Im ,\mathsf{B})$. Then:

1. 

$l_{ws}\left[(\mathtt{F},\mathsf{B})\tilde{\cap }(\mathtt{G},\mathsf{B})\right]\tilde{\subseteq }{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\cap }{l}_{ws}(\mathtt{G},\mathsf{B})$.

2. 

$l_{ws}(\mathtt{F},\mathsf{B})\tilde{\cup }{l}_{ws}(\mathtt{G},\mathsf{B})\tilde{\subseteq }{l}_{ws}\left[(\mathtt{F},\mathsf{B})\tilde{\cup }(\mathtt{G},\mathsf{B})\right]$.

Theorem 3. 

Let $(\mathtt{F},\mathsf{B})$ be an SS of an STS $(\mathbf{O},\Im ,\mathsf{B})$; then, $c{l}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})\tilde{\cup }{l}_{ws}(\mathtt{F},\mathsf{B})$.

Proof. 

The side $(\mathtt{F},\mathsf{B})\tilde{\cup }{l}_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }c{l}_{ws}(\mathtt{F},\mathsf{B})$ is obvious. To prove the other side, let ${\mathbf{\nu }}_{\mathsf{b}}\notin \left[(\mathtt{F},\mathsf{B})\tilde{\cup }{l}_{ws}(\mathtt{F},\mathsf{B})\right]$. Then, ${\mathbf{\nu }}_{\mathsf{b}}\notin (\mathtt{F},\mathsf{B})$ and ${\mathbf{\nu }}_{\mathsf{b}}\notin l_{ws}(\mathtt{F},\mathsf{B})$. Therefore, there is a weakly soft semi-open $(\mathtt{G},\mathsf{B})$ containing ${\mathbf{\nu }}_{\mathsf{b}}$ with $(\mathtt{G},\mathsf{B})\tilde{\cap }(\mathtt{F},\mathsf{B})=\varphi$. Thus, ${\mathbf{\nu }}_{\mathsf{b}}\notin c{l}_{ws}(\mathtt{F},\mathsf{B})$. Hence, we find that $c{l}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})\tilde{\cup }{l}_{ws}(\mathtt{F},\mathsf{B})$. □

Corollary 9. 

Let $(\mathtt{F},\mathsf{B})$ be a weakly soft semi-closed subset of an STS $(\mathbf{O},\Im ,\mathsf{B})$; then, $l_{ws}(\mathtt{F},\mathsf{B})\tilde{\subseteq }(\mathtt{F},\mathsf{B})$.

5. Continuity via Weakly Soft Semi-Open Sets

In this section, we tackle the concept of soft continuity via weakly soft semi-open sets. We derive its main characterizations and point out that the loss of the property says that “the weakly semi-interior of an SS is weakly soft semi-open” results to evaporating some descriptions of this type of soft continuity. An elucidative counterexample is supplied.

Definition 18. 

A soft function ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is said to be weakly soft semi-continuous if the inverse image of each soft open set is weakly soft semi-open.

It is straightforward to prove the next result, so we omit its proof.

Proposition 19. 

If ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is a weakly soft semi-continuous function and ${\mathbb{N}}_{\mathbb{K}}:(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})\to (\mathbf{Z},{\Im }_{\mathbf{Z}},\mathsf{B})$ is a soft continuous function, then ${\mathbb{N}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-continuous.

The proof of the next proposition follows from Proposition 5.

Proposition 20. 

Every soft continuous function is weakly soft semi-continuous.

Proposition 21. 

Let ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ be a soft function such that ${\Im }_{\mathbf{O}}$ is extended. Then

1. 

If ${\mathbb{I}}_{\mathbb{P}}$ is soft α-continuous (soft semi-continuous), then ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-continuous.

2. 

If ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-continuous, then ${\mathbb{I}}_{\mathbb{P}}$ is soft $sw$-continuous.

Proof. 

It, respectively, follows from Proposition 6 and Proposition 7. □

Proposition 22. 

A soft function ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is weakly soft semi-continuous if the inverse image of every soft closed subset is weakly soft semi-closed.

Proof. 

Necessity: Suppose that $(\mathtt{F},\mathsf{B})$ is a soft closed subset of $(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$. Then, $({\mathtt{F}}^c,\mathsf{B})$ is soft open. Therefore, ${\mathbb{I}}_{\mathbb{P}}^{-1}({\mathtt{F}}^c,\mathsf{B})=\tilde{\mathbf{O}}-{\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})$ is weakly soft semi-open. Thus, ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})$ is a weakly soft semi-closed set.

Following a similar argument, one can prove the sufficient part. □

Theorem 4. 

If ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is weakly soft semi-continuous, then the next properties are equivalent.

1. 

For each soft open subset $(\mathtt{F},\mathsf{B})$ of $(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$, we have ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})=in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$.

2. 

For each soft closed subset $(\mathtt{F},\mathsf{B})$ of $(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$, we have ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})=c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$.

3. 

$c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }{\mathbb{I}}_{\mathbb{P}}^{-1}\left(cl(\mathtt{F},\mathsf{B})\right)$ for each $(\mathtt{F},\mathsf{B})\tilde{\subseteq }\tilde{\mathbf{U}}$.

4. 

${\mathbb{I}}_{\mathbb{P}}\left(c{l}_{ws}(\mathtt{G},\mathsf{B})\right)\tilde{\subseteq }cl\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{G},\mathsf{B})\right)$ for each $(\mathtt{G},\mathsf{B})\tilde{\subseteq }\tilde{\mathbf{O}}$.

5. 

${\mathbb{I}}_{\mathbb{P}}^{-1}\left(int(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$ for each $(\mathtt{F},\mathsf{B})\tilde{\subseteq }\tilde{\mathbf{U}}$.

Proof. 

(1→2): Suppose that $(\mathtt{F},\mathsf{B})$ is a soft closed subset of $(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$. Then, $({\mathtt{F}}^c,\mathsf{B})$ is soft open. Therefore, ${\mathbb{I}}_{\mathbb{P}}^{-1}({\mathtt{F}}^c,\mathsf{B})=in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}({\mathtt{F}}^c,\mathsf{B})\right)$. According to Proposition 13, we obtain ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})=c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$.

$(2\to 3)$: For any soft set $(\mathtt{F},\mathsf{B})\tilde{\subseteq }\tilde{\mathbf{U}}$, we have ${\mathbb{I}}_{\mathbb{P}}^{-1}\left(cl(\mathtt{F},\mathsf{B})\right)=c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}\left(cl(\mathtt{F},\mathsf{B})\right)\right)$. Then, $c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}\left(cl(\mathtt{F},\mathsf{B})\right)\right)={\mathbb{I}}_{\mathbb{P}}^{-1}\left(cl(\mathtt{F},\mathsf{B})\right)$.

$(3\to 4)$: It is obvious that $c{l}_{ws}(\mathtt{G},\mathsf{B})\tilde{\subseteq }c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{G},\mathsf{B})\right)\right)$ for each $(\mathtt{G},\mathsf{B})\tilde{\subseteq }\tilde{\mathbf{O}}$. By 3, we obtain $c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{G},\mathsf{B})\right)\right)\tilde{\subseteq }$ ${\mathbb{I}}_{\mathbb{P}}^{-1}\left(cl\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{G},\mathsf{B})\right)\right)$. Therefore, ${\mathbb{I}}_{\mathbb{P}}(c{l}_{ws}(\mathtt{G},\mathsf{B})\tilde{\subseteq }$${\mathbb{I}}_{\mathbb{P}}\left({\mathbb{I}}_{\mathbb{P}}^{-1}\left(cl\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{G},\mathsf{B})\right)\right)\right)$ $\tilde{\subseteq }$$cl\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{G},\mathsf{B})\right)$.

$(4\to 5)$: Let $(\mathtt{F},\mathsf{B})$ be an arbitrary soft set in $(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$. Then, ${\mathbb{I}}_{\mathbb{P}}(c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}({\mathtt{F}}^c,\mathsf{B})\right)$ $\tilde{\subseteq }cl\left({\mathbb{I}}_{\mathbb{P}}\left({\mathbb{I}}_{\mathbb{P}}^{-1}({\mathtt{F}}^c,\mathsf{B})\right)\right)$ $\tilde{\subseteq }cl({\mathtt{F}}^c,\mathsf{B})$. So that, $c{l}_{ws}\left({\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)}^c\right)\tilde{\subseteq }{\mathbb{I}}_{\mathbb{P}}^{-1}({\left(int(\mathtt{F},\mathsf{B})\right)}^c$. Hence, ${\mathbb{I}}_{\mathbb{P}}^{-1}\left(int(\mathtt{F},\mathsf{B})\right)$$\tilde{\subseteq }in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$.

$(5\to 1)$: Suppose that $(\mathtt{F},\mathsf{B})$ is a soft open subset in $(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$. By 5, we obtain ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})={\mathbb{I}}_{\mathbb{P}}^{-1}\left(int(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }$$in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$. However, $in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }{\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})$, so ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})=in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$, as demanded. □

To prove that the converse of the above theorem is generally false, the next example is shown.

Example 4. 

Let $\mathbf{O}=\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}$ and $\mathbf{U}=\{{\mathbf{\mu }}_1,{\mathbf{\mu }}_2\}$ with $\mathsf{B}=\{{\mathsf{b}}_1,{\mathsf{b}}_2\}$. Let ${\Im }_{\mathbf{O}}=\{\varphi ,\tilde{\mathbf{O}},(\mathtt{F},\mathsf{B}),(\mathtt{G},\mathsf{B})\}$ and ${\Im }_{\mathbf{U}}=\{\varphi ,\tilde{\mathbf{U}},(\mathtt{H},\mathsf{B})\}$ be two STs defined on $\mathbf{O}$ and $\mathbf{U}$, respectively, with the same set of parameters $\mathsf{B}$, where

$(\mathtt{F},\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\nu }}_1\right\}),({\mathsf{b}}_2,\left\{{\mathbf{\nu }}_1\right\})\}$;

$(\mathtt{G},\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\}),({\mathsf{b}}_2,\{{\mathbf{\nu }}_2,{\mathbf{\nu }}_3\})\}$; and

$(\mathtt{H},\mathsf{B})=\{({\mathsf{b}}_1,\left\{{\mathbf{\mu }}_1\right\}),({\mathsf{b}}_2,\left\{{\mathbf{\mu }}_1\right\})\}$.

Let ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ be a soft function, where $\mathbb{P}:\mathsf{B}\to \mathsf{B}$ is the identity function and $\mathbb{I}:\mathbf{O}\to \mathbf{U}$ is defined as follows

$\mathbb{I}\left({\mathbf{\nu }}_1\right)=\mathbb{I}\left({\mathbf{\nu }}_2\right)={\mathbf{\mu }}_1$ and $\mathbb{I}\left({\mathbf{\nu }}_3\right)={\mathbf{\mu }}_2$.

Now, ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{H},\mathsf{B})=\{({\mathsf{b}}_1,\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2\}),({\mathsf{b}}_2,\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2\})\}$ is not a weakly soft semi-open subset of $(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})$ because $cl\left(int\left(\{{\mathbf{\nu }}_1,{\mathbf{\nu }}_2\}\right)\right)=\left\{{\mathbf{\nu }}_1\right\}$. Then, ${\mathbb{I}}_{\mathbb{P}}$ is not weakly soft semi-continuous. On the other hand, ${\mathbb{I}}_{\mathbb{P}}^{-1}\left(\varphi \right)=in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}\left(\varphi \right)\right)$, ${\mathbb{I}}_{\mathbb{P}}^{-1}\left(\tilde{\mathbf{U}}\right)=in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}\left(\tilde{\mathbf{U}}\right)\right)$ and ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{H},\mathsf{B})=in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{H},\mathsf{B})\right)$, which means that the all properties given in Theorem 4 hold true.

Now, we introduce the concepts of weakly soft semi-open, weakly soft semi-closed and weakly soft semi-homeomorphism functions.

Definition 19. 

A soft function ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is said to be weakly soft semi-open (resp., weakly soft semi-closed) provided that the image of each soft open (resp., soft closed) set is weakly soft semi-open (resp., weakly soft semi-closed).

Proposition 23. 

Let ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ be a soft function and $(\mathtt{F},\mathsf{B})$ be any SS of $\tilde{\mathbf{O}}$.

1. 

If ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-open, then ${\mathbb{I}}_{\mathbb{P}}\left(int(\mathtt{F},\mathsf{B})\right)\tilde{\subseteq }in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)$.

2. 

If ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-closed, then $c{l}_{ws}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)$ $\tilde{\subseteq }{\mathbb{I}}_{\mathbb{P}}\left(cl(\mathtt{F},\mathsf{B})\right)$.

Proof. 

1. Let $(\mathtt{F},\mathsf{B})$ be an SS of $\tilde{\mathbf{O}}$. Then, ${\mathbb{I}}_{\mathbb{P}}\left(int(\mathtt{F},\mathsf{B})\right)$ is a weakly soft semi-open subset of $(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ and so ${\mathbb{I}}_{\mathbb{P}}\left(int(\mathtt{F},\mathsf{B})\right)=in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}\left(int(\mathtt{F},\mathsf{B})\right)\right)\tilde{\subseteq }in{t}_{ws}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)$.

2.

The proof is similar to that of 1. □

Proposition 24. 

A bijective soft function ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is weakly soft semi-open if it is weakly soft semi-closed.

Proof. 

Necessity: Let $(\mathtt{F},\mathsf{B})$ be a weakly soft semi-closed subset of $(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})$. Since ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-open, ${\mathbb{I}}_{\mathbb{P}}({\mathtt{F}}^c,\mathsf{B})$ is weakly soft semi-open. By the bijectiveness of ${\mathbb{I}}_{\mathbb{P}}$, we obtain ${\mathbb{I}}_{\mathbb{P}}({\mathtt{F}}^c,\mathsf{B})={\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)}^c$. So that, ${\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})$ is a weakly soft semi-closed set. Hence, ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-closed. To prove the sufficient, we follow a similar approach. □

Proposition 25. 

Let ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ be a weakly soft semi-closed function and $\tilde{\mathbf{Z}}$ be a soft closed subset of $\tilde{\mathbf{O}}$. Then, ${\mathbb{I}}_{\mathbb{P}}{\mid }_{\mathbf{Z}}:(\mathbf{Z},{\Im }_{\mathbf{Z}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ is weakly soft semi-closed.

Proof. 

Suppose that $(\mathtt{F},\mathsf{B})$ is a soft closed subset of $(\mathbf{Z},{\Im }_{\mathbf{Z}},\mathsf{B})$. Then, there is a soft closed subset $(\mathtt{G},\mathsf{B})$ of $(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})$ with $(\mathtt{F},\mathsf{B})=(\mathtt{G},\mathsf{B})\tilde{\cap }\tilde{\mathbf{Z}}$. Since $\tilde{\mathbf{Z}}$ is a soft closed subset of $(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})$, then $(\mathtt{F},\mathsf{B})$ is also a soft closed subset of $(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})$. Since ${\mathbb{I}}_{\mathbb{P}}{\mid }_{\mathbf{Z}}(\mathtt{F},\mathsf{B})={\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})$, then ${\mathbb{I}}_{\mathbb{P}}{\mid }_{\mathbf{Z}}(\mathtt{F},\mathsf{B})$ is a weakly soft semi-closed set. Thus, ${\mathbb{I}}_{\mathbb{P}}{\mid }_{\mathbf{Z}}$ is a weakly soft semi-closed. □

Proposition 26. 

The next three statements hold for soft functions ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ and ${\mathbb{J}}_{\mathbb{K}}:(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})\to (\mathbf{Z},{\Im }_{\mathbf{Z}},\mathsf{B})$.

1. 

If ${\mathbb{I}}_{\mathbb{P}}$ is soft open and ${\mathbb{J}}_{\mathbb{K}}$ is soft j-open such that ${\Im }_{\mathbf{Z}}$ is extended, then ${\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-open, where $j\in \{\alpha ,semi\}$.

2. 

If ${\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-open and ${\mathbb{I}}_{\mathbb{P}}$ is soft continuous surjective, then ${\mathbb{J}}_{\mathbb{K}}$ is weakly soft semi-open.

3. 

If ${\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is soft open and ${\mathbb{J}}_{\mathbb{K}}$ is weakly soft semi-continuous injective, then ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-open.

Proof. 

1. Without loss of generality, let $j=\alpha$. Then, consider $(\mathtt{F},\mathsf{B})\ne \varphi$ as a soft open subset of $\tilde{\mathbf{O}}$. So, ${\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\ne \varphi$ is a soft open subset of $\tilde{\mathbf{U}}$. Thus, ${\mathbb{J}}_{\mathbb{K}}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)$ is a soft $\alpha$-open subset. According to Proposition 6, ${\mathbb{J}}_{\mathbb{K}}\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)$ is a weakly soft semi-open subset. Hence, ${\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-open.

2. Suppose that $(\mathtt{F},\mathsf{B})\ne \varphi$ is a soft open subset of $\tilde{\mathbf{U}}$. Then, ${\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\ne \varphi$ is a soft open subset of $\tilde{\mathbf{O}}$. Therefore, $({\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}})\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)$ is a weakly soft semi-open subset of $\tilde{\mathbf{Z}}$. Since ${\mathbb{I}}_{\mathbb{P}}$ is surjective, then $({\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}})\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)={\mathbb{J}}_{\mathbb{K}}\left({\mathbb{I}}_{\mathbb{P}}\left({\mathbb{I}}_{\mathbb{P}}^{-1}(\mathtt{F},\mathsf{B})\right)\right)={\mathbb{J}}_{\mathbb{K}}(\mathtt{F},\mathsf{B})$. Thus, ${\mathbb{J}}_{\mathbb{K}}$ is weakly soft semi-open.

3. Let $(\mathtt{F},\mathsf{B})\ne \varphi$ be a soft open subset of $\tilde{\mathbf{O}}$. Then, $({\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}})(\mathtt{F},\mathsf{B})\ne \varphi$ is a soft open subset of $\tilde{\mathbf{Z}}$. Therefore, ${\mathbb{J}}_{\mathbb{K}}^{-1}({\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B}))$ is a weakly soft semi-open subset of $\tilde{\mathbf{U}}$. Since ${\mathbb{J}}_{\mathbb{K}}$ is injective, ${\mathbb{J}}_{\mathbb{K}}^{-1}({\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B}))=\left({\mathbb{J}}_{\mathbb{K}}^{-1}{\mathbb{J}}_{\mathbb{K}}\right)\left({\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})\right)={\mathbb{I}}_{\mathbb{P}}(\mathtt{F},\mathsf{B})$. Thus, ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-open. □

We cancel the proof of the next finding because it can be obtained following a similar approach to the above proposition.

Proposition 27. 

The next three statements hold for soft functions ${\mathbb{I}}_{\mathbb{P}}:(\mathbf{O},{\Im }_{\mathbf{O}},\mathsf{B})\to (\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})$ and ${\mathbb{J}}_{\mathbb{K}}:(\mathbf{U},{\Im }_{\mathbf{U}},\mathsf{B})\to (\mathbf{Z},{\Im }_{\mathbf{Z}},\mathsf{B})$.

1. 

If ${\mathbb{I}}_{\mathbb{P}}$ is soft closed and ${\mathbb{J}}_{\mathbb{K}}$ is soft j-closed, then ${\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-closed, where $i,j\in \{\alpha ,semi\}$.

2. 

If ${\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-closed and ${\mathbb{I}}_{\mathbb{P}}$ is soft continuous surjective, then ${\mathbb{J}}_{\mathbb{K}}$ is weakly soft semi-closed.

3. 

If ${\mathbb{J}}_{\mathbb{K}}\circ {\mathbb{I}}_{\mathbb{P}}$ is soft closed and ${\mathbb{J}}_{\mathbb{K}}$ is weakly soft semi-continuous injective, then ${\mathbb{I}}_{\mathbb{P}}$ is weakly soft semi-closed.

Definition 20. 

A bijective soft function ${\mathbb{I}}_{\mathbb{P}}$ in which is weakly soft semi-continuous and weakly soft semi-open is called a weakly soft semi-homeomorphism.

6. Conclusions

In 2011, the structure of soft topology was invented as a novel branch of topology and has gained extreme interest among researchers in recent years. This structure is influenced by the basic tenets of classical topological space because it is a combination of soft set theory and topology.

In this article, we have introduced the concept of “weakly soft semi-open sets” which broadens the structure of soft open sets. This concept has been built by utilizing its classical counterpart “semi-open sets” obtained from any parametric topology. The essential properties of this concept have been studied, and its relationship with some extensions of soft open sets has been revealed under specific sorts of STs with the assistance of suitable counterexamples. Next, new types of soft interior, closure, boundary, and limit operators were defined with respect to weakly soft semi-open and weakly soft semi-closed sets. Some mathematical formulas that are connected among these operators have been inferred. This work has been completed by presenting the concept of weakly soft semi-continuity and examining its main characteristics.

It is important to refer to the major divergences between this class and the other generalizations of soft open sets that were defined by the soft operators of interior and closure. First, the class of weakly soft semi-open sets does not institute a supra soft topology because it is not closed under arbitrary unions, whereas the classes of SSs such as soft semi-open or soft pre-open sets form a supra soft topology. This abolishes the way of determining whether the soft subset is weakly soft semi-open or not by its weakly semi-interior points. That is, the property “$in{t}_{ws}(\mathtt{F},\mathsf{B})=(\mathtt{F},\mathsf{B})$” does not imply that $(\mathtt{F},\mathsf{B})$ is weakly soft semi-open. These results contradict some well-known descriptions of soft continuity for this type of continuity. In addition, the property reports that “the boundary soft points of a soft subset $(\mathtt{F},\mathsf{B})$ is soft closed” need not be true for a weakly semi-boundary set $b_{ws}(\mathtt{F},\mathsf{B})$ because $c{l}_{ws}(\mathtt{F},\mathsf{B})$ is not a weakly soft semi-closed set in general.

The methodology proposed herein to define this category of extension of soft open sets is completely different than the existing popular one in the published literature, which is based on the soft interior and closure operators, whereas the current approach is inspired by the extension of open sets on the classical (parametric) topological spaces. This means that we have successfully employed classical ideas to produce new notions via soft structures, which will open the door for more studies in this branch of research. Of course, it will be interesting and useful to restate the soft topological concepts with respect to this class of SSs. Another proposal for upcoming work is formulating another type of soft continuity inspired by weakly soft semi-open sets in a way that helps to obviate the irregular behavior of the current soft continuity. Furthermore, one can develop and discuss these concepts via the generalized structures of STs.