Soft Weakly Quasi-Continuous Functions Between Soft Topological Spaces

Abstract

As an extension of quasi-continuity in general topology, we define soft quasi-continuity. We show that this notion is equivalent to the known notion of soft semi-continuity. Next, we define soft weak quasi-continuity. With the help of examples, we prove that soft weak quasi-continuity is strictly weaker than both soft semi-continuity and soft weak continuity. We introduce many characterizations of soft weak quasi-continuity. Moreover, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. Furthermore, we show that soft regularity of the co-domain of a soft function is a sufficient condition for equivalence between soft semi-continuity and soft weakly quasi-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and soft graph theorems in terms of soft weak quasi-continuity.

1. Introduction and Preliminaries

Finding solutions to difficult problems requires mathematical modeling of uncertainty across social sciences, engineering, health, environmental science, and economics. Despite their shortcomings, other theories, such as probability theory, fuzzy set theory [1], and rough set theory [2], may be useful in managing ambiguity and uncertainty. Scientific contributions using fuzzy sets are still a hot area for researchers [3,4,5,6,7,8,9,10,11].

One of the primary areas for improvement of this mathematical technique is the need for more parametrization tools.

In 1999, Molodtsov [12] developed the soft set theory in response to critiques of the uncertainty management techniques that had been previously discussed. the use of soft sets or parametrized universe possibilities was considered. Uncertainty in set modeling was first shown in [13] and improved in [14]. There are many useful applications for this uniform structure as well. Numerous studies (e.g., [15,16,17,18,19,20,21]) have effectively used set interpretation for modeling uncertainty in a variety of real-world contexts. These real-world applications have demonstrated the framework’s problem-solving capabilities and further supported its applicability and effectiveness. The main ideas and principles of soft set theory have been studied and investigated by several academics [21,22,23].

Shabir and Naz [24] established a soft topology across a family of soft sets in order to create one for a certain set of parameters. Their work elucidated the connections between notions in soft topology and classical topology, which stimulated more study in this area. Since the inception of soft topology, several contributions—such as those from [25,26,27,28,29,30,31,32,33,34,35]—have been made to the study of topological ideas in soft environments.

Majumdar and Samanta [36] looked at mappings on soft sets and how they may be used for medical diagnostics. Kharal and Ahmed [37] introduced the idea of soft mapping with characteristics and proposed soft continuity for soft mappings [38]. The concept of soft continuity, along with its many characterizations, is thoroughly examined in the literature reviews found in several publications, for instance, weakly soft $\alpha$-continuous [39], soft bi-continuity [40], weakly soft $\beta$-continuity [41], soft $\omega$-continuity [42], soft ${\omega }_p$-continuity [43], soft SD-continuity [44], soft b-continuity [45], soft $\beta$-continuity [46], and soft $\alpha$-continuity [47]. These works explore the intricacies of this mathematical concept, which characterizes the smooth transition of a function between its values at neighboring points.

In soft topology and other branches of mathematics, soft continuity has been the focus of much study. Soft continuity is widely used in many fields, such as data modeling, soft topological models, engineering, science, economics, and business. Scientists have demonstrated their interest in this subject.

Generalizing soft continuity is important because it enriches topological spaces and provides new tools for dealing with uncertainty; it has far-reaching implications for many fields, including computer science, physics, engineering, and economics; it provides a deeper understanding of mathematical structures and principles; and it can be used to unify different concepts and theories. Indeed, by generalizing soft continuity, researchers can create new mathematical frameworks, methodologies, and applications that can propel innovation and improvement in a variety of domains. Additionally, features of soft topological spaces including soft compactness, soft connectedness, and soft separation axioms can be studied using weaker versions of soft continuity. This motivated us to write this paper. We define and investigate soft quasi-continuity and soft weak quasi-continuity as two generalizations of soft continuity.

The first goal of this paper is to show how the definitions of quasi-continuity and weakly quasi-continuity can be modified in order to define soft quasi-continuity and soft weakly quasi-continuity. The second goal is to extend some known topological results to include soft topology.

Our research question is “What are the connections and characterizations of soft quasi-continuity, soft weak quasi-continuity, and their general topological equivalents, and how such concepts interact with soft semi-continuity and soft weak continuity in the setting of soft functions?”

This paper is organized as follows:

In Section 2, we define soft quasi-continuity and soft weakly quasi-continuity. We show that soft quasi-continuity is equivalent to soft semi-continuity. We offer many characterizations of soft weakly quasi-continuity. Also, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. With the help of examples, we prove that the class of soft weakly quasi-continuous functions contains strictly the classes of soft semi-continuous and soft weakly continuous functions. In addition, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology.

In Section 3, we present several results of soft composition, restrictions, preservation, and soft graph theorems. These results are presented in terms of soft weak quasi-continuity to illustrate its significance.

To clarify, we refer to notions and terminologies from [48,49] throughout this study. Topological space and soft topological space are abbreviated as TS and STS, respectively.

We let $\mathcal{L}$ and O be two non-empty sets where $\mathcal{L}$ is a set of parameters. A soft set over O relative to $\mathcal{L}$ is a function from $\mathcal{L}$ to the powerset of O. $SS(O,\mathcal{L})$ denotes the collection of all soft sets over O relative to $\mathcal{L}$. $0_{\mathcal{L}}$ and $1_{\mathcal{L}}$ denote the null and absolute soft sets, respectively. We let $H\in SS(O,\mathcal{L})$. If $H\left(e\right)=Y$ for all $e\in \mathcal{L}$, then H is denoted by $C_Y$. $0_{\mathcal{L}}$ and $1_{\mathcal{L}}$ denote $C_{\varnothing }$ and $C_O$, respectively. If $H\left(a\right)=Y$ and $H\left(e\right)=\varnothing$ for all $e\in \mathcal{L}-\left\{a\right\}$, then H is denoted by $a_Y$. For any $a\in \mathcal{L}$ and $x\in O$, $a_{\left\{x\right\}}$ is called a soft point and written as $a_x$, for simplicity. $SP\left(O,\mathcal{L}\right)$ denotes the collection of all soft points over O relative to $\mathcal{L}$. If $H\in SS(O,\mathcal{L})$ and $a_x\in SP\left(O,\mathcal{L}\right)$, then $a_x$ is said to belong to H (notation: $a_x\tilde{\in }H$) if $x\in H\left(a\right)$.

We now describe some key concepts that are used in the follow-up.

Definition 1 

([50]). Let $g:\left(O,\Im \right)⟶\left(T,\aleph \right)$ be a function and let $x\in O$. Then, g is called weakly quasi-continuous at x if for each $B\in \aleph$ such that $g\left(x\right)\in B$ and each $A\in \Im$ such that $x\in A$, there exists $N\in \Im -\left\{\varnothing \right\}$ such that $N\subseteq A$ and $g\left(N\right)\subseteq B$. If g is weakly quasi-continuous at every $x\in O$, then it is called weakly quasi-continuous.

Definition 2. 

We let $\left(X,\phi ,\mathcal{T}\right)$ be a STS and let $K\in SS\left(X,\mathcal{T}\right)$. Then,

(a) K is a soft semi-open [51] (resp. soft α-open [47], soft regular-open [52]) set in $\left(X,\phi ,\mathcal{T}\right)$ if $K\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(K\right)\right)$ (resp. $K\tilde{\subseteq }In{t}_{\phi }\left(C{l}_{\phi }\left(In{t}_{\phi }\left(K\right)\right)\right)$, $K=In{t}_{\phi }\left(C{l}_{\phi }\left(K\right)\right)$). The family of all soft semi-open sets (resp. soft α-open sets, soft regular-open sets) in $\left(X,\phi ,\mathcal{T}\right)$ is denoted by $SO\left(\phi \right)$ (resp. $\alpha \left(\phi \right)$, $RO\left(\phi \right)$).

(b) K is called a soft semi-closed [51] (resp. soft regular-closed [52]) set in $\left(X,\phi ,\mathcal{T}\right)$ if $1_{\mathcal{T}}-K\in$$SO\left(\phi \right)$ (resp. $1_{\mathcal{T}}-K\in RO\left(\phi \right)$). The family of all soft semi-closed sets (resp. soft regular-closed sets) in $\left(X,\phi ,\mathcal{T}\right)$ is denoted by $SC\left(\phi \right)$ (resp. $RC\left(\phi \right)$).

Definition 3 

([51]). We let $\left(O,\phi ,\mathcal{P}\right)$ be a STS and let $H\in SS\left(O,\mathcal{P}\right)$. Then,

(a) The soft semi-interior of H in $\left(O,\phi ,\mathcal{P}\right)$ is denoted by $sIn{t}_{\phi }\left(H\right)$ and defined by

$$sIn{t}_{\phi }\left(H\right)=\tilde{\cup }\left\{R:R\in SO\left(\phi \right)\mathit{and}R\tilde{\subseteq }H\right\}.$$

(b) The soft semi-closure of H in $\left(O,\phi ,\mathcal{P}\right)$ is denoted by $sC{l}_{\phi }\left(H\right)$ and defined by

$$sC{l}_{\phi }\left(H\right)=\tilde{\cap }\left\{L:L\in SC\left(\phi \right)\mathit{and}H\tilde{\subseteq }L\right\}.$$

Definition 4 

([53]). We let $\left(X,\phi ,\mathcal{T}\right)$ be a STS and let $K\in SS\left(X,\mathcal{T}\right)$. Then,

(a) The soft θ-closure of K in $\left(X,\phi ,\mathcal{T}\right)$ is denoted by $\theta C{l}_{\phi }\left(K\right)$, where $\theta C{l}_{\phi }\left(K\right)\in SS\left(X,\mathcal{T}\right)$ and defined as follows:

$a_x\tilde{\in }\theta C{l}_{\phi }\left(K\right)$ iff for each $H\in \phi$ such that $a_x\tilde{\in }H$, $K\tilde{\cap }C{l}_{\phi }\left(H\right)\ne 0_{\mathcal{T}}$.

(b) K is a soft θ-closed set in $\left(X,\phi ,\mathcal{T}\right)$ if $K=\theta C{l}_{\phi }\left(K\right)$.

(c) K is a soft θ-open set in $\left(X,\phi ,\mathcal{T}\right)$ if $1_{\mathcal{T}}-K$ is a soft θ-closed set in $\left(X,\phi ,\mathcal{T}\right)$. The family of all soft θ-open sets in $\left(X,\phi ,\mathcal{T}\right)$ is denoted by ${\phi }_{\theta }$.

Definition 5. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is said to be

(a) soft semi-continuous [54] if $f_{rw}^{-1}\left(H\right)\in SO\left(\phi \right)$ for every $H\in \sigma$;

(b) soft weakly continuous [55] if for each $a_x\in SP\left(O,\mathcal{P}\right)$ and each $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$, there exists $U\in \phi$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$;

(c) soft almost α-continuous [56] if for each $a_x\in SP\left(O,\mathcal{P}\right)$ and each $V\in RO\left(\sigma \right)$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$, there exists $U\in \alpha \left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }V$.

Definition 6. 

A STS $\left(O,\phi ,\mathcal{P}\right)$ is called

(a) [38] soft compact if for any $\mathcal{K}\subseteq \phi$ such that $1_{\mathcal{P}}={\tilde{\cup }}_{K\in \mathcal{K}}K$, there exists a finite subcollection ${\mathcal{K}}_1\subseteq \mathcal{K}$ such that $1_{\mathcal{P}}={\tilde{\cup }}_{K\in {\mathcal{K}}_1}K$;

(b) [57] soft connected if $\phi \cap {\phi }^c=\left\{0_{\mathcal{P}},1_{\mathcal{P}}\right\}$;

(c) [54] soft semi-compact if for any $\mathcal{K}\subseteq SO\left(\phi \right)$ such that $1_{\mathcal{P}}={\tilde{\cup }}_{K\in \mathcal{K}}K$, there exists a finite subcollection ${\mathcal{K}}_1\subseteq \mathcal{K}$ such that $1_{\mathcal{P}}={\tilde{\cup }}_{K\in {\mathcal{K}}_1}K$;

(d) [58] soft S-connected if $SO\left(\phi \right)\cap SC\left(\phi \right)=\left\{0_{\mathcal{P}},1_{\mathcal{P}}\right\}$;

(e) [59] soft hyperconnected if for any $A,B\in \phi -\left\{0_{\mathcal{P}}\right\}$, $A\tilde{\cap }B\ne 0_{\mathcal{P}}$;

(f) [60] soft Hausdorff if for every $a_x,b_y\in SP\left(O,\mathcal{P}\right)$ such that $a_x\ne b_y$, there exist $M,N\in \phi$ such that $a_x\tilde{\in }M$, $b_y\tilde{\in }N$, and $M\tilde{\cap }N=0_{\mathcal{P}}$;

(g) [60] soft regular if for every $a_x\in SP\left(O,\mathcal{P}\right)$ and every $M\in \phi$ such that $a_x\tilde{\in }M$, there exists $N\in \wp$ such that $a_x\tilde{\in }N\tilde{\subseteq }C{l}_{\phi }\left(N\right)\tilde{\subseteq }M$;

(h) [60] soft Urysohn if for every $a_x,b_y\in SP\left(O,\mathcal{P}\right)$ such that $a_x\ne b_y$, there exist $M,N\in \phi$ such that $a_x\tilde{\in }M$, $b_y\tilde{\in }N$, and $C{l}_{\phi }\left(M\right)\tilde{\cap }C{l}_{\phi }\left(N\right)=0_{\mathcal{P}}$;

(k) [61] soft semi-$T_2$ if for every $a_x,b_y\in SP\left(O,\mathcal{P}\right)$ such that $a_x\ne b_y$, there exist $M,N\in SO\left(\phi \right)$ such that $a_x\tilde{\in }M$, $b_y\tilde{\in }N$, and $M\tilde{\cap }N=0_{\mathcal{P}}$;

(l) [62] soft rim-compact if $\left(O,\phi ,\mathcal{P}\right)$ has a soft base $\mathcal{B}$ such that $B{d}_{\phi }\left(B\right)$ is soft compact for every $B\in \mathcal{B}$.

Definition 7 

([62]). We let $\left(O,\phi ,\mathcal{P}\right)$ be a STS and let $H\in SS\left(O,\mathcal{P}\right)$. Then, H is said to be a soft H-set if for every $\mathcal{A}\subseteq$ φ such that $H\tilde{\subseteq }{\tilde{\cup }}_{A\in \mathcal{A}}A$, there exists a finite sub-collection ${\mathcal{A}}_1$ ⊆ $\mathcal{A}$ such that $K\tilde{\subseteq }{\tilde{\cup }}_{A\in {\mathcal{A}}_1}C{l}_{\phi }\left(A\right)$.

Definition 8. 

A STS $\left(O,\phi ,\mathcal{P}\right)$ is said to be soft H-closed if $1_{\mathcal{P}}$ is a soft H-set.

Definition 9. 

For a given soft function $f_{rw}:SP\left(O,\mathcal{P}\right)⟶SP\left(T,\mathcal{S}\right)$, the soft set $\tilde{\cup }\left\{{\left(a,w\left(a\right)\right)}_{(x,r(x\left)\right)}:a\in \mathcal{P}\mathrm{and}x\in O\right\}$ is called the soft graph of $f_{rw}$ and is denoted by $G\left(f_{rw}\right)$. So, ${\left(a,b\right)}_{(x,y)}\tilde{\in }G\left(f_{rw}\right)$ iff $f_{rw}\left(a_x\right)=b_y$ iff $r\left(x\right)=y$ and $w\left(a\right)=b$.

Definition 10 

([62]). We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function. Then, $G\left(f_{rw}\right)$ is said to be soft strongly semi-closed with respect to $\left(O\times T,pr\left(\phi \times \supset \right),\mathcal{P}\times \mathcal{S}\right)$ if for each ${\left(a,b\right)}_{(x,y)}\tilde{\in }{1}_{\mathcal{P}\times \mathcal{S}}-G\left(f_{rw}\right)$, there exist $A\in \phi$ and $B\in \sigma$ such that $a_x\tilde{\in }A$, $b_y\tilde{\in }B$, and $\left(C{l}_{\phi }\left(A\right)\times C{l}_{\sigma }\left(B\right)\right)\tilde{\cap }G\left(f_{rw}\right)=0_{\mathcal{P}\times \mathcal{S}}$.

2. Characterizations

In this section, we define soft quasi-continuity and soft weakly quasi-continuity. We show that soft quasi-continuity is equivalent to soft semi-continuity. We offer many characterizations of soft weak quasi-continuity. Also, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. With the help of examples, we prove that the class of soft weakly quasi-continuous functions contains strictly the classes of soft semi-continuous and soft weakly continuous functions. In addition, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology.

Definition 11. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function and let $a_x\in SP\left(O,\mathcal{P}\right)$. Then, $f_{rw}$ is called soft quasi-continuous at $a_x$ if for each $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$ and each $U\in \phi$ such that $a_x\tilde{\in }U$, there exists $G\in \phi -\left\{0_{\mathcal{P}}\right\}$ such that $G\tilde{\subseteq }U$ and $f_{rw}\left(G\right)\tilde{\subseteq }V$. If $f_{rw}$ is soft quasi-continuous at every $a_x\in SP\left(O,\mathcal{P}\right)$, then it is called soft quasi-continuous.

Theorem 1. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft semi-continuous iff $f_{rw}$ is soft quasi-continuous.

Proof. 

Necessity. We suppose that $f_{rw}$ is soft semi-continuous. Let $a_t\tilde{\in }SP\left(O,\mathcal{P}\right)$. We let $H\in \phi$ and $K\in \sigma$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(a_t\right)\tilde{\in }K$. By soft semi-continuity of $f_{rw}$, $f_{rw}^{-1}\left(K\right)$ $\in SO\left(\phi \right)$ and so, $f_{rw}^{-1}\left(K\right)\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\right)$. We let $L=H\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$. Then, $L\in \phi$. Since $f_{rw}\left(a_t\right)\tilde{\in }K$, then $a_t\tilde{\in }$ $f_{rw}^{-1}\left(K\right)\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\right)$. Since $a_t\tilde{\in }H\in \phi$ and $a_t\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\right)$, then $L=H\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\ne 0_{\mathcal{P}}$. Now, we have $L\tilde{\subseteq }H$ and $f_{rw}\left(L\right)=f_{rw}\left(H\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\right)\tilde{\subseteq }{f}_{rw}\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\right)\tilde{\subseteq }{f}_{rw}\left(f_{rw}^{-1}\left(K\right)\right)\tilde{\subseteq }K$. This shows that $f_{rw}$ is soft quasi-continuous.

Sufficiency. We suppose that $f_{rw}$ is soft quasi-continuous. We let $K\in \sigma -\left\{0_{\mathcal{S}}\right\}$. We let $a_t\tilde{\in }{f}_{rw}^{-1}\left(K\right)$. To show that $a_t\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\right)$, we let $H\in \phi$ such that $a_t\tilde{\in }H$. By soft quasi-continuity of $f_{rw}$, there exists $L\in \phi -\left\{0_{\mathcal{P}}\right\}$ such that $L\tilde{\subseteq }H$ and $f_{rw}\left(L\right)\tilde{\subseteq }K$. Thus, $L\tilde{\subseteq }{f}_{rw}^{-1}\left(K\right)$, and so $In{t}_{\phi }\left(L\right)=L\tilde{\subseteq }In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$. So, $L\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)=L\ne 0_{\mathcal{P}}$. Hence, $H\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\ne 0_{\mathcal{P}}$. It follows that $a_t\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\right)$. This shows that $f_{rw}^{-1}\left(K\right)\in SO\left(\phi \right)$. □

Definition 12. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function and we let $a_x\in SP\left(O,\mathcal{P}\right)$. Then $f_{rw}$ is called soft weakly quasi-continuous at $a_x$ if for each $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$ and each $U\in \phi$ such that $a_x\tilde{\in }U$, there exists $G\in \phi -\left\{0_{\mathcal{P}}\right\}$ such that $G\tilde{\subseteq }U$ and $f_{rw}\left(G\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. If $f_{rw}$ is soft weakly quasi-continuous at every $a_x\in SP\left(O,\mathcal{P}\right)$, then it is called soft weakly quasi-continuous.

Lemma 1. 

We let $\left(O,\phi ,\mathcal{P}\right)$ be a STS and let $G\in SS\left(O,\mathcal{P}\right)$. Then, $G\in SO\left(\phi \right)$ iff $C{l}_{\phi }\left(G\right)=C{l}_{\phi }\left(In{t}_{\phi }\left(G\right)\right)$.

Proof. 

Necessity. We let $G\in SO\left(\phi \right)$. Then $G\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(G\right)\right)$, and so $C{l}_{\phi }\left(G\right)\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(G\right)\right)$. On the other hand, since $In{t}_{\phi }\left(G\right)\tilde{\subseteq }G\tilde{\subseteq }C{l}_{\phi }\left(G\right)$, then $C{l}_{\phi }\left(In{t}_{\phi }\left(G\right)\right)\tilde{\subseteq }C{l}_{\phi }\left(G\right)$. This shows that $C{l}_{\phi }\left(G\right)=C{l}_{\phi }\left(In{t}_{\phi }\left(G\right)\right)$.

Sufficiency. We let $C{l}_{\phi }\left(G\right)=C{l}_{\phi }\left(In{t}_{\phi }\left(G\right)\right)$. Then, $G\tilde{\subseteq }C{l}_{\phi }\left(G\right)=C{l}_{\phi }\left(In{t}_{\phi }\left(G\right)\right)$, and hence $G\in SO\left(\phi \right)$. □

Lemma 2. 

We let $\left(O,\phi ,\mathcal{P}\right)$ be a STS and $G\in SO\left(\phi \right)-\left\{0_{\mathcal{P}}\right\}$. Then, $In{t}_{\phi }\left(G\right)\ne 0_{\mathcal{P}}$.

Proof. 

We let $G\in SO\left(\phi \right)-\left\{0_{\mathcal{P}}\right\}$. We suppose to the contrary that $In{t}_{\phi }\left(G\right)=0_{\mathcal{P}}$. Then, by Lemma 1, $C{l}_{\phi }\left(G\right)=C{l}_{\phi }\left(0_{\mathcal{P}}\right)=0_{\mathcal{P}}$. Hence, $G=0_{\mathcal{P}}$, a contraction. □

Theorem 2. 

Soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft weakly quasi-continuous iff for each $a_x\in SP\left(O,\mathcal{P}\right)$ and each $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$, there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$.

Proof. 

Necessity. We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft weakly quasi-continuous. We let $a_x\in SP\left(O,\mathcal{P}\right)$ and $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$. We let $\psi =\left\{H\in \phi :a_x\tilde{\in }H\right\}$. For each $H\in \psi$, there exists $G\left(H\right)\in \phi -\left\{0_{\mathcal{P}}\right\}$ such that $G\left(H\right)\tilde{\subseteq }H$ and $f_{rw}\left(G\left(H\right)\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. We put $G={\tilde{\cup }}_{H\in \psi }G\left(H\right)$. Then, $G\in \phi$ and $a_x\tilde{\in }C{l}_{\phi }\left(G\right)$. We let $U=G\tilde{\cup }{a}_x$. Then, $a_x\tilde{\in }U\in SO\left(\phi \right)$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$.

Sufficiency. We let $a_x\in SP\left(O,\mathcal{P}\right)$. We let $V\in \sigma$ and $K\in \phi$ such that $a_x\tilde{\in }K$ and $f_{rw}\left(a_x\right)\tilde{\in }V$. By assumption, there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. We let $G=In{t}_{\phi }\left(K\tilde{\cap }U\right)$. Since $K\in \phi$ and $U\in SO\left(\phi \right)$, then $K\tilde{\cap }U\in SO\left(\phi \right)$. Since $a_x\tilde{\in }K\tilde{\cap }U$, then by Lemma 2, $G=In{t}_{\phi }\left(K\tilde{\cap }U\right)\ne 0_{\mathcal{P}}$. Now, we have

$$\begin{array}{ccc}G& =& In{t}_{\phi }\left(K\tilde{\cap }U\right)\\ & \tilde{\subseteq }& K\tilde{\cap }U\\ & \tilde{\subseteq }& U\end{array}$$

and

$$\begin{array}{ccc}{f}_{rw}\left(G\right)& =& f_{rw}\left(In{t}_{\phi }\left(K\tilde{\cap }U\right)\right)\\ & =& f_{rw}\left(In{t}_{\phi }\left(K\right)\tilde{\cap }In{t}_{\phi }\left(U\right)\right)\\ & =& f_{rw}\left(In{t}_{\phi }\left(K\right)\tilde{\cap }U\right)\\ & \tilde{\subseteq }& f_{rw}\left(U\right)\\ & \tilde{\subseteq }& C{l}_{\sigma }\left(V\right).\end{array}$$

Therefore, $f_{rw}$ is soft weakly quasi-continuous. □

Theorem 3. 

Soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft weakly quasi-continuous iff for every $V\in \sigma$, $f_{rw}^{-1}\left(V\right)\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)$.

Proof. 

Necessity. We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft weakly quasi-continuous. We let $V\in \sigma$. We let $a_x\tilde{\in }{f}_{rw}^{-1}\left(V\right)$. Then, $f_{rw}\left(a_x\right)\tilde{\in }V$. To see that $a_x\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)$, we let $U\in \phi$ such that $a_x\tilde{\in }U$. Then, there exists $G\in \phi -\left\{0_{\mathcal{P}}\right\}$ such that $G\tilde{\subseteq }U$ and $f_{rw}\left(G\right)\tilde{\subseteq }C{l}_{\phi }\left(V\right)$. Thus, it follows that $G\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$ and $G\tilde{\subseteq }In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$. Since $0_{\mathcal{P}}\ne G\tilde{\subseteq }U\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$, then $U\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\ne 0_{\mathcal{P}}$. Therefore, $a_x\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)$.

Sufficiency. We let $a_x\in SP\left(O,\mathcal{P}\right)$. We let $V\in \sigma$ and $U\in \phi$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(a_x\right)\tilde{\in }V$. By assumption, we have $a_x\tilde{\in }{f}_{rw}^{-1}\left(V\right)\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)$. Since $a_x\tilde{\in }U\in \phi$ and $a_x\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)$, then $U\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\ne 0_{\mathcal{P}}$. We put $G=U\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$. Then, $G\in \phi -\left\{0_{\mathcal{P}}\right\}$ such that $G\tilde{\subseteq }U$ and

$$\begin{array}{ccc}{f}_{rw}\left(G\right)& =& f_{rw}\left(U\tilde{\cap }In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)\\ & \tilde{\subseteq }& f_{rw}\left(In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)\\ & \tilde{\subseteq }& f_{rw}\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\\ & \tilde{\subseteq }& C{l}_{\sigma }\left(V\right).\end{array}$$

This shows that $f_{rw}$ is soft weakly quasi-continuous. □

Theorem 4. 

For soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$, the following are equivalent:

(a) $f_{rw}$ is soft weakly quasi-continuous;

(b) $sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(B\right)\right)$ for every $B\in SS\left(T,\mathcal{S}\right)$;

(c) $sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(R\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(R\right)$ for every $R\in RC\left(\sigma \right)$;

(d) $sC{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$ for every $V\in \sigma$;

(e) $f_{rw}^{-1}\left(V\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$ for every $V\in \sigma$.

Proof. 

(a) ⟶ (b): We let $B\in SS\left(T,\mathcal{S}\right)$. We suppose to the contrary that there exists $a_x\tilde{\in }sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\right)\right)-f_{rw}^{-1}\left(C{l}_{\sigma }\left(B\right)\right)$. Since $a_x\tilde{\notin }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(B\right)\right)$, then $f_{rw}\left(a_x\right)\tilde{\notin }C{l}_{\sigma }\left(B\right)$, and so, there exists $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$ and $V\tilde{\cap }B=0_{\mathcal{S}}$. □

Claim 1. 

$C{l}_{\sigma }\left(V\right)\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)=0_{\mathcal{S}}$.

Proof of Claim 1. 

We suppose to the contrary that there exists $d_z\tilde{\in }C{l}_{\sigma }\left(V\right)\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)$. Since $d_z\tilde{\in }In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\in \sigma$, then $V\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\ne 0_{\mathcal{S}}$. We choose $r_s\tilde{\in }V\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\tilde{\subseteq }V\tilde{\cap }C{l}_{\sigma }\left(B\right)$; then, we have $r_s\tilde{\in }V\in \sigma$ and $r_s\tilde{\in }C{l}_{\sigma }\left(B\right)$, which implies that $V\tilde{\cap }B\ne 0_{\mathcal{S}}$. This is a contradiction.

Now, by Theorem 2, there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. Since $a_x\tilde{\in }sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\right)\right)$, then $U\tilde{\cap }{f}_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\right)\ne 0_{\mathcal{P}}$. Since $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$ and by the above Claim $C{l}_{\sigma }\left(V\right)\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)=0_{\mathcal{S}}$, then $U\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\tilde{\subseteq }{1}_{\mathcal{P}}-f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\right)$ and so, $U\tilde{\cap }{f}_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(B\right)\right)\right)=1_{\mathcal{P}}$. This is a contradiction.

(b) ⟶ (c): We let $R\in RC\left(\sigma \right)$. Then, $R=C{l}_{\sigma }\left(In{t}_{\sigma }\left(R\right)\right)$ and by (b),

$$\begin{array}{ccc}sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(R\right)\right)\right)& =& sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(In{t}_{\sigma }\left(R\right)\right)\right)\right)\right)\\ & \tilde{\subseteq }& f_{rw}^{-1}\left(C{l}_{\sigma }\left(In{t}_{\sigma }\left(R\right)\right)\right)\\ & =& f_{rw}^{-1}\left(R\right).\end{array}$$

(c) ⟶ (d): We let $V\in \sigma$. □

Claim 2. 

$C{l}_{\sigma }\left(V\right)\in RC\left(\sigma \right)$.

Proof of Claim 2. 

Since $In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$, then $C{l}_{\sigma }\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. To show that $C{l}_{\sigma }\left(V\right)\tilde{\subseteq }C{l}_{\sigma }\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)$, we suppose to the contrary that there exists $a_x\tilde{\in }C{l}_{\sigma }\left(V\right)-C{l}_{\sigma }\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)$. Since $a_x\tilde{\notin }C{l}_{\sigma }\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)$, then there exists $K\in \sigma$ such that $a_x\tilde{\in }K$ and $In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\tilde{\cap }K=0_{\mathcal{P}}$. Since $a_x\tilde{\in }C{l}_{\sigma }\left(V\right)$, then $V\tilde{\cap }K\ne 0_{\mathcal{P}}$. Since $V\in \sigma$, then $V\tilde{\subseteq }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)$ and so $V\tilde{\cap }K\tilde{\subseteq }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\tilde{\cap }K$; hence, $In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\tilde{\cap }K\ne 0_{\mathcal{P}}$. This is a contradiction.

Since $V\in \sigma$, then $V\tilde{\subseteq }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)$ and so $sC{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\tilde{\subseteq }sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)$. On the other hand, since $C{l}_{\sigma }\left(V\right)\in RC\left(\sigma \right)$, then by (c), $sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$. Therefore, $sC{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$.

(d) ⟶ (e): We let $V\in \sigma$. Then,

$$\begin{array}{ccc}{1}_{\mathcal{P}}-sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)& =& sC{l}_{\phi }\left(1_{\mathcal{P}}-f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\\ & =& sC{l}_{\phi }\left(f_{rw}^{-1}\left(1_{\mathcal{S}}-C{l}_{\sigma }\left(V\right)\right)\right).\end{array}$$

Since $1_{\mathcal{S}}-C{l}_{\sigma }\left(V\right)\in \sigma$, then, by (d),

$$\begin{array}{ccc}sC{l}_{\phi }\left(f_{rw}^{-1}\left(1_{\mathcal{S}}-C{l}_{\sigma }\left(V\right)\right)\right)& \tilde{\subseteq }& f_{rw}^{-1}\left(C{l}_{\sigma }\left(1_{\mathcal{S}}-C{l}_{\sigma }\left(V\right)\right)\right)\\ & =& f_{rw}^{-1}\left(1_{\mathcal{S}}-In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right).\end{array}$$

Since $V\in \sigma$, then $V\tilde{\subseteq }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)$, and so $1_{\mathcal{S}}-In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\tilde{\subseteq }{1}_{\mathcal{S}}-V$. Thus,

$$\begin{array}{ccc}{f}_{rw}^{-1}\left(1_{\mathcal{S}}-In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)& \tilde{\subseteq }& f_{rw}^{-1}\left(1_{\mathcal{S}}-V\right)\\ & =& 1_{\mathcal{P}}-f_{rw}^{-1}\left(V\right).\end{array}$$

Therefore, $1_{\mathcal{P}}-sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\tilde{\subseteq }{1}_{\mathcal{P}}-f_{rw}^{-1}\left(V\right)$, and hence, $f_{rw}^{-1}\left(V\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$.

(e) ⟶ (a): We let $a_x\in SP\left(O,\mathcal{P}\right)$ and $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$. Then, $a_x\tilde{\in }{f}_{rw}^{-1}\left(V\right)$. So, by (e), $a_x\tilde{\in }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\in SO\left(\phi \right)$. We let $U=sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$. Then, we have $a_x\tilde{\in }U\in SO\left(\phi \right)$ and

$$\begin{array}{ccc}{f}_{rw}\left(U\right)& =& f_{rw}\left(sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)\\ & \tilde{\subseteq }& f_{rw}\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\\ & \tilde{\subseteq }& C{l}_{\sigma }\left(V\right).\end{array}$$

Therefore, by Theorem 2, $f_{rw}$ is soft weakly quasi-continuous. □

Lemma 3. 

We let $\left(O,\phi ,\mathcal{P}\right)$ be a STS and let $A\in SS\left(O,\mathcal{P}\right)$. Then,

$$sC{l}_{\phi }\left(A\right)=A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right).$$

Proof. 

Since $sC{l}_{\phi }\left(A\right)\in SC\left(\phi \right)$, then $In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\tilde{\subseteq }In{t}_{\phi }\left(C{l}_{\phi }\left(sC{l}_{\phi }\left(A\right)\right)\right)\tilde{\subseteq }sC{l}_{\phi }\left(A\right)$. Hence, $A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\tilde{\subseteq }sC{l}_{\phi }\left(A\right)$. □

Claim 3. 

$A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\in SC\left(\phi \right)$.

Proof of Claim 3. 

$$\begin{array}{ccc}C{l}_{\phi }\left(A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\right)& =& C{l}_{\phi }\left(A\right)\tilde{\cup }C{l}_{\phi }\left(In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\right)\\ & \tilde{\subseteq }& C{l}_{\phi }\left(A\right)\tilde{\cup }C{l}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\\ & =& C{l}_{\phi }\left(A\right)\tilde{\cup }C{l}_{\phi }\left(A\right)\\ & =& C{l}_{\phi }\left(A\right).\end{array}$$

So,

$$\begin{array}{ccc}In{t}_{\phi }\left(C{l}_{\phi }\left(A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\right)\right)& \tilde{\subseteq }& In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\\ & \tilde{\subseteq }& A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right).\end{array}$$

Therefore, $A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\in SC\left(\phi \right)$.

Since $A\tilde{\subseteq }A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)$, then, by the above claim, $sC{l}_{\phi }\left(A\right)\tilde{\subseteq }A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)$. □

Theorem 5. 

For a soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$, the following are equivalent:

(a) $f_{rw}$ is soft weakly quasi-continuous;

(b) $sC{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$ for every $B\in SS\left(T,\mathcal{S}\right)$;

(c) $f_{rw}\left(sC{l}_{\phi }\left(A\right)\right)\tilde{\subseteq }\theta C{l}_{\sigma }\left(f_{rw}\left(A\right)\right)$ for every $A\in SS\left(O,\mathcal{P}\right)$;

(d) $f_{rw}\left(In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\right)\tilde{\subseteq }\theta C{l}_{\sigma }\left(f_{rw}\left(A\right)\right)$ for every $A\in SS\left(O,\mathcal{P}\right)$;

(e) $In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$ for every $B\in SS\left(T,\mathcal{S}\right)$;

(f) $In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$ for every $V\in \sigma$.

Proof. 

(a) ⟶ (b): We let $B\in SS\left(T,\mathcal{S}\right)$. We suppose to the contrary that there exists $a_x\tilde{\in }sC{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)-f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$. Since $f_{rw}\left(a_x\right)\tilde{\notin }\theta C{l}_{\sigma }\left(B\right)$, then there exists $G\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }G$ and $C{l}_{\sigma }\left(G\right)\tilde{\cap }B=0_{\mathcal{S}}$. By (a) and Theorem 2, there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(G\right)$. Since $a_x\tilde{\in }sC{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)$, then $U\tilde{\cap }{f}_{rw}^{-1}\left(B\right)\ne 0_{\mathcal{P}}$. We choose $d_z\tilde{\in }U\tilde{\cap }{f}_{rw}^{-1}\left(B\right)$. Then, we have $f_{rw}\left(d_z\right)\tilde{\in }{f}_{rw}\left(U\right)\tilde{\cap }B\tilde{\subseteq }C{l}_{\sigma }\left(G\right)\tilde{\cap }B$. But $C{l}_{\sigma }\left(G\right)\tilde{\cap }B=0_{\mathcal{S}}$, a contradiction.

(b) ⟶ (c): We let $A\in SS\left(O,\mathcal{P}\right)$. By (b), $sC{l}_{\phi }\left(A\right)\tilde{\subseteq }sC{l}_{\phi }\left(f_{rw}^{-1}\left(f_{rw}\left(A\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(f_{rw}\left(A\right)\right)\right)$ and thus, $f_{rw}\left(sC{l}_{\phi }\left(A\right)\right)\tilde{\subseteq }{f}_{rw}\left(f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(f_{rw}\left(A\right)\right)\right)\right)\tilde{\subseteq }\theta C{l}_{\sigma }\left(f_{rw}\left(A\right)\right)$. This ends the proof.

(c) ⟶ (d): We let $A\in SS\left(O,\mathcal{P}\right)$. Then, by (c), $f_{rw}\left(sC{l}_{\phi }\left(A\right)\right)\tilde{\subseteq }\theta C{l}_{\sigma }\left(f_{rw}\left(A\right)\right)$. So, by Lemma 3,

$$\begin{array}{ccc}{f}_{rw}\left(In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\right)& \tilde{\subseteq }& f_{rw}\left(A\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(A\right)\right)\right)\\ & =& f_{rw}\left(sC{l}_{\phi }\left(A\right)\right)\\ & \tilde{\subseteq }& \theta C{l}_{\sigma }\left(f_{rw}\left(A\right)\right).\end{array}$$

(d) ⟶ (e): We let $B\in SS\left(T,\mathcal{S}\right)$. Then, $f_{rw}^{-1}\left(B\right)\in SS\left(O,\mathcal{P}\right)$ and by (d),

$$\begin{array}{ccc}{f}_{rw}\left(In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)\right)\right)& \tilde{\subseteq }& \theta C{l}_{\sigma }\left(f_{rw}\left(f_{rw}^{-1}\left(B\right)\right)\right)\\ & \tilde{\subseteq }& \theta C{l}_{\sigma }\left(B\right).\end{array}$$

Thus,

$$\begin{array}{ccc}In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)\right)& \tilde{\subseteq }& f_{rw}^{-1}\left(f_{rw}\left(In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)\right)\right)\right)\\ & \tilde{\subseteq }& f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right).\end{array}$$

(e) ⟶ (f): We let $V\in \sigma$. Then, by (e), $In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(V\right)\right)$. On the other hand, by Lemma 2.10 of [63], $\theta C{l}_{\sigma }\left(V\right)=C{l}_{\sigma }\left(V\right)$. Thus, we have $In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$.

(f) ⟶ (a): We apply Theorem 4 (d). We let $V\in \sigma$. Then, by (f), $In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$. On the other hand, we have $f_{rw}^{-1}\left(V\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$. Thus, $f_{rw}^{-1}\left(V\right)\tilde{\cup }In{t}_{\phi }\left(C{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$. Therefore, by Lemma 3, $sC{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$. □

Theorem 6. 

Soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft weakly quasi-continuous iff $sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$ for every $B\in SS\left(T,\mathcal{S}\right)$.

Proof. 

Necessity. We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft weakly quasi-continuous. We let $B\in SS\left(T,\mathcal{S}\right)$. We assume to the contrary that there exists $a_x\tilde{\in }sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)\right)-f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$. Since $f_{rw}\left(a_x\right)\tilde{\notin }\theta C{l}_{\sigma }\left(B\right)$, then there exists $G\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }G$ and $C{l}_{\sigma }\left(G\right)\tilde{\cap }B=0_{\mathcal{S}}$. Thus, $\theta C{l}_{\sigma }\left(B\right)\tilde{\cap }G=0_{\mathcal{S}}$ and hence $G\tilde{\subseteq }{1}_{\mathcal{S}}-\theta C{l}_{\sigma }\left(B\right)$. So, $C{l}_{\sigma }\left(G\right)\tilde{\subseteq }C{l}_{\sigma }\left(1_{\mathcal{S}}-\theta C{l}_{\sigma }\left(B\right)\right)$. Since $f_{rw}$ is soft weakly quasi-continuous, then by Theorem 2, there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(G\right)\tilde{\subseteq }C{l}_{\sigma }\left(1_{\mathcal{S}}-\theta C{l}_{\sigma }\left(B\right)\right)$. Since $In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\tilde{\cap }C{l}_{\sigma }\left(1_{\mathcal{S}}-\theta C{l}_{\sigma }\left(B\right)\right)=0_{\mathcal{S}}$, then $In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\tilde{\cap }{f}_{rw}\left(U\right)=0_{\mathcal{S}}$. Thus,

$$\begin{array}{ccc}{f}_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\tilde{\cap }{f}_{rw}\left(U\right)\right)& =& f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)\tilde{\cap }{f}_{rw}^{-1}\left(f_{rw}\left(U\right)\right)\\ & =& 0_{\mathcal{P}}.\end{array}$$

Since $f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)\tilde{\cap }U\tilde{\subseteq }{f}_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)\tilde{\cap }{f}_{rw}^{-1}\left(f_{rw}\left(U\right)\right)$, then $f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)\tilde{\cap }U=0_{\mathcal{P}}$. This implies that $a_x\tilde{\notin }sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)\right)$, a contradiction.

Sufficiency. We apply Theorem 4 (d). We let $V\in \sigma$. Then $V\tilde{\subseteq }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)$. Also, by Lemma 2.10 of [63], $\theta C{l}_{\sigma }\left(V\right)=C{l}_{\sigma }\left(V\right)$. Since $V\tilde{\subseteq }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)$, then $sC{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\tilde{\subseteq }sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\right)\right)=\tilde{\subseteq }sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(V\right)\right)\right)\right)$. But by assumption, $sC{l}_{\phi }\left(f_{rw}^{-1}\left(In{t}_{\sigma }\left(\theta C{l}_{\sigma }\left(V\right)\right)\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(V\right)\right)=f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$. Therefore, $sC{l}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$. □

Theorem 7. 

We let $\left\{\left(O,{\phi }_a\right):a\in \mathcal{P}\right\}$ and $\left\{\left(T,{\sigma }_b\right):b\in \mathcal{S}\right\}$ be two collections of TSs. We consider the functions $r:O⟶T$ and $w:\mathcal{P}⟶\mathcal{S}$, where w is a bijection. Then, $f_{rw}:\left(O,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(T,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft weakly quasi-continuous iff $r:\left(O,{\phi }_a\right)⟶\left(T,{\sigma }_{w\left(a\right)}\right)$ is weakly quasi-continuous for all $a\in \mathcal{P}$.

Proof. 

Necessity. We let $f_{rw}:\left(O,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(T,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft weakly quasi-continuous. We let $a\in \mathcal{P}$. To show that $r:\left(O,{\phi }_a\right)⟶\left(T,{\sigma }_{w\left(a\right)}\right)$ is weakly quasi-continuous, we let $x\in O$ and let $V\in {\sigma }_{w\left(a\right)}$ such that $r\left(x\right)\in V$. Then, $f_{rw}\left(a_x\right)\tilde{\in }{\left(w\left(a\right)\right)}_V\in {\oplus }_{b\in \mathcal{S}}{\sigma }_b$. Since $f_{rw}$ is soft weakly continuous, then there exists $G\in SO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$ such that $a_x\tilde{\in }G$ and $f_{rw}\left(G\right)\tilde{\subseteq }C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left({\left(w\left(a\right)\right)}_V\right)$. So, $\left(f_{rw}\left(G\right)\right)\left(w\left(a\right)\right)\subseteq \left(C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left({\left(w\left(a\right)\right)}_V\right)\right)\left(w\left(a\right)\right)$. Since $G\in SO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$, then by Theorem 4.10 of [64], $G\left(a\right)\in SO\left({\phi }_a\right)$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, then $\left(f_{rw}\left(G\right)\right)\left(w\left(a\right)\right)=r\left(G\left(a\right)\right)$. On the other hand, by Lemma 4.9 of [64], $\left(C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left({\left(w\left(a\right)\right)}_V\right)\right)\left(w\left(a\right)\right)=C{l}_{{\sigma }_{w\left(a\right)}}\left({\left(w\left(a\right)\right)}_V\right)\left(w\left(a\right)\right)=C{l}_{{\sigma }_{w\left(a\right)}}\left(V\right)$. Therefore, we have $x\in G\left(a\right)\in SO\left({\phi }_a\right)$ and $r\left(G\left(a\right)\right)\subseteq C{l}_{{\sigma }_{w\left(a\right)}}\left(V\right)$. It follows that $r:\left(O,{\phi }_a\right)⟶\left(T,{\sigma }_{w\left(a\right)}\right)$ is weakly quasi-continuous.

Sufficiency. We let $r:\left(O,{\phi }_a\right)⟶\left(T,{\sigma }_{w\left(a\right)}\right)$ be weakly quasi-continuous for all $a\in \mathcal{P}$. We let $d_z\in SP\left(O,\mathcal{P}\right)$ and let $G\in {\oplus }_{a\in \mathcal{P}}{\phi }_a$ such that $f_{rw}\left(d_z\right)\tilde{\in }G$. Then, $r\left(z\right)\in G\left(w\left(d\right)\right)\in {\sigma }_{w\left(d\right)}$. Since $r:\left(O,{\phi }_d\right)⟶\left(T,{\sigma }_{w\left(d\right)}\right)$ is weakly quasi-continuous, then there exists $U\in SO\left({\phi }_d\right)$ such that $z\in U$ and $r\left(U\right)\subseteq C{l}_{{\sigma }_{w\left(d\right)}}\left(G\left(w\left(d\right)\right)\right)$. We let $H=d_U$. Then, $d_z\tilde{\in }H$, and by Theorem 4.10 of [64], $H\in SO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. □

Claim 4. 

${\left(w\left(d\right)\right)}_{r\left(U\right)}\tilde{\subseteq }C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left(G\right)$.

Proof of Claim 4. 

Let $b\in \mathcal{S}$. If $b=w\left(d\right)$, then $\left({\left(w\left(d\right)\right)}_{r\left(U\right)}\right)\left(b\right)=r\left(U\right)$ and by Lemma 4.9 of [64], $\left(C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left(G\right)\right)\left(b\right)=C{l}_{{\sigma }_b}\left(G\left(b\right)\right)$; hence, $\left({\left(w\left(d\right)\right)}_{r\left(U\right)}\right)\left(b\right)\subseteq \left(C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left(G\right)\right)\left(b\right)$. If $b\ne w\left(d\right)$, then $\left({\left(w\left(d\right)\right)}_{r\left(U\right)}\right)\left(b\right)=\varnothing$ and so, $\left({\left(w\left(d\right)\right)}_{r\left(U\right)}\right)\left(b\right)=\varnothing \subseteq \left(C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left(G\right)\right)\left(b\right)$.

Thus, by the above claim, we have $f_{rw}\left(H\right)={\left(w\left(d\right)\right)}_{r\left(U\right)}\tilde{\subseteq }C{l}_{{\oplus }_{b\in \mathcal{S}}{\sigma }_b}\left(G\right)$. This shows that $f_{rw}:\left(O,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$ $\left(T,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft weakly quasi-continuous. □

Corollary 1. 

We consider functions $r:\left(O,\Im \right)⟶\left(T,\aleph \right)$ and $w:\mathcal{P}⟶\mathcal{S}$, where w is a bijection. Then, $r:\left(O,\Im \right)⟶\left(T,\aleph \right)$ is weakly quasi-continuous iff $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(T,\tau \left(\aleph \right),\mathcal{S})$ is soft weakly quasi-continuous.

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we put ${\phi }_a=\Im$ and ${\sigma }_b=\aleph$. Then, $\tau \left(\Im \right)={\oplus }_{a\in \mathcal{P}}{\phi }_a$ and $\tau \left(\aleph \right)={\oplus }_{b\in \mathcal{S}}{\sigma }_b$. Theorem 7 ends the proof. □

Theorem 8. 

Every soft weakly continuous function is soft weakly quasi-continuous.

Proof. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft weakly continuous. We let $a_x\in SP\left(O,\mathcal{P}\right)$ and let $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$. Since $f_{rw}$ is soft weakly continuous, then there exists $U\in \phi \subseteq SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. This shows that $f_{rw}$ is soft weakly quasi-continuous. □

Theorem 9. 

Every soft semi-continuous function is soft weakly quasi-continuous.

Proof. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft semi-continuous. We let $a_x\in SP\left(O,\mathcal{P}\right)$ and let $V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$. Since $f_{rw}$ is soft semi-continuous, then there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }V\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. This shows that $f_{rw}$ is soft weakly quasi-continuous. □

The following example shows that the converse of Theorem 8 need not be true in general:

Example 1. 

We let $O=\left\{a,b,c\right\}$, $\mathcal{P}=\mathbb{R}$, $\Im =\left\{\varnothing ,O,\left\{a\right\},\left\{b\right\},\left\{a,b\right\}\right\}$, and $\aleph =\left\{\varnothing ,O,\left\{a\right\},\left\{b,c\right\}\right\}$. We consider the identity functions $r:\left(O,\Im \right)⟶$$\left(O,\aleph \right)$ and $w:\mathcal{P}⟶\mathcal{P}$. Since $r^{-1}\left(\left\{b,c\right\}\right)=\left\{b,c\right\}\in SO\left(\Im \right)$, then r is semi-continuous and hence it is weakly quasi-continuous. On the other hand, we suppose that r is weakly continuous. Since $r\left(c\right)=c\in \left\{b,c\right\}\in \aleph$, then there exists $L\in \Im$ such that $c\in L$ and $r\left(L\right)\subseteq C{l}_{\aleph }\left(\left\{b,c\right\}\right)=\left\{b,c\right\}$. So, we have $L=O$ and $r\left(O\right)=O\subseteq \left\{b,c\right\}$. Thus, r is not weakly continuous. Therefore, by Corollary 1 and Corollary 2 of [65], $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(O,\tau \left(\aleph \right),\mathcal{P})$ is soft weakly quasi-continuous but not soft weakly continuous.

The following example shows that the converse of Theorem 9 need not be true in general:

Example 2. 

We let $O=\left\{a,b,c,d\right\}$, $\mathcal{P}=\mathbb{R}$, and $\Im =\left\{\varnothing ,O,\left\{b\right\},\left\{c\right\},\left\{b,c\right\},\left\{a,b\right\},\left\{a,b,c\right\},\right.$$\left.\left\{b,c,d\right\}\right\}$. We define $r:\left(O,\Im \right)⟶$$\left(O,\Im \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ by $r\left(a\right)=c$, $r\left(b\right)=d$, $r\left(c\right)=b$, $r\left(d\right)=a$, and $w\left(t\right)=t$ for all $t\in \mathcal{P}$. Then, $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(O,\tau \left(\aleph \right),\mathcal{P})$ is soft weakly continuous (and by Theorem 8, it is soft weakly quasi-continuous) but not soft semi-continuous.

Theorem 10. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function such that $\left(T,\sigma ,\mathcal{S}\right)$ is soft regular. Then, the following are equivalent:

(a) $f_{rw}$ is soft quasi-continuous;

(b) $f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)\in SC\left(\phi \right)$ for every $B\in SS\left(T,\mathcal{S}\right)$;

(c) $f_{rw}$ is soft weakly quasi-continuous;

(d) $f_{rw}^{-1}\left(D\right)\in SC\left(\phi \right)$ for every $D\in {\left({\sigma }_{\theta }\right)}^c$;

(e) $f_{rw}^{-1}\left(V\right)\in SO\left(\phi \right)$ for every $V\in {\sigma }_{\theta }$.

Proof. 

(a) ⟶ (b): We let $B\in SS\left(T,\mathcal{S}\right)$. Then, $\theta C{l}_{\sigma }\left(B\right)\in {\sigma }^c$. So, by (a) and Theorem 1, $f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)\in SC\left(\phi \right)$.

(b) ⟶ (c): We apply Theorem 5 (b). We let $B\in SS\left(T,\mathcal{S}\right)$. Since $B\tilde{\subseteq }\theta C{l}_{\sigma }\left(B\right)$, $f_{rw}^{-1}\left(B\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$ and so, $sC{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)\tilde{\subseteq }sC{l}_{\phi }\left(f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)$. On the other hand, since by (b), $f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)\in SC\left(\phi \right)$, then $sC{l}_{\phi }\left(f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)\right)=f_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$. It follows that $sC{l}_{\phi }\left(f_{rw}^{-1}\left(B\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(B\right)\right)$.

(c) ⟶ (d): We let $D\in {\left({\sigma }_{\theta }\right)}^c$. Then, $\theta C{l}_{\sigma }\left(D\right)=D$. Now, by (c) and Theorem 5 (b), $sC{l}_{\phi }\left(f_{rw}^{-1}\left(D\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(\theta C{l}_{\sigma }\left(D\right)\right)=f_{rw}^{-1}\left(D\right)$. Therefore, $f_{rw}^{-1}\left(D\right)\in SC\left(\phi \right)$.

(d) ⟶ (e): We let $V\in {\sigma }_{\theta }$. Then, $1_{\mathcal{S}}-V\in {\left({\sigma }_{\theta }\right)}^c$ and by (d), $f_{rw}^{-1}\left(1_{\mathcal{S}}-V\right)=1_{\mathcal{P}}-f_{rw}^{-1}\left(V\right)\in SC\left(\phi \right)$. Thus, $f_{rw}^{-1}\left(V\right)\in SO\left(\phi \right)$.

(e) ⟶ (a): We apply Theorem 1. We let $V\in \sigma$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft regular, then ${\sigma }_{\theta }=\sigma$ and so $V\in {\sigma }_{\theta }$. Thus, by (e), $f_{rw}^{-1}\left(V\right)\in SO\left(\phi \right)$. □

3. Results

This section presents several results of soft composition, restrictions, preservation, and soft graph theorems. These results are presented regarding soft weak quasi-continuity to illustrate its significance.

Theorem 11. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft weakly quasi-continuous function such that $\left(T,\sigma ,\mathcal{S}\right)$ is soft rim-compact and $G\left(f_{rw}\right)$ is θ-closed with respect to $\left(O,\phi ,\mathcal{P}\right)$. Then, $f_{rw}$ is soft quasi-continuous.

Proof. 

We let $a_x\in SP\left(O,\mathcal{P}\right)$ and let $V\in \sigma$ such that $f_{pv}\left(a_x\right)\tilde{\in }V$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft rim-compact, there exists $K\in \sigma$ such that $f_{pv}\left(a_x\right)\tilde{\in }K\tilde{\subseteq }V$ and $B{d}_{\sigma }\left(K\right)$ is soft compact. For each $b_y\tilde{\in }B{d}_{\sigma }\left(K\right)$, ${\left(a,b\right)}_{\left(x,y\right)}\tilde{\in }{1}_{\mathcal{P}\times \mathcal{S}}-G\left(f_{rw}\right)$ and by assumption, there exist $A\left(b_y\right)\in \phi$ and $B\left(b_y\right)\in \sigma$ such that $a_x\tilde{\in }A\left(b_y\right)$, $b_y\tilde{\in }B\left(b_y\right)$, and $\left(C{l}_{\phi }\left(A\left(b_y\right)\right)\times B\left(b_y\right)\right)\tilde{\cap }G\left(f_{rw}\right)=0_{\mathcal{P}\times \mathcal{S}}$ or that $f_{rw}\left(C{l}_{\phi }\left(A\left(b_y\right)\right)\right)\tilde{\cap }B\left(b_y\right)=0_{\mathcal{S}}$. Since $B{d}_{\supset }\left(K\right)$ is soft compact and $B{d}_{\sigma }\left(K\right)\tilde{\subseteq }{\tilde{\cup }}_{b_y\tilde{\in }B{d}_{\sigma }\left(K\right)}B\left(b_y\right)$, there exists a finite subset $F\subseteq$$SP\left(T,\mathcal{S}\right)$ such that $d_z\tilde{\in }B{d}_{\sigma }\left(K\right)$ for all $d_z\in F$, and $B{d}_{\sigma }\left(K\right)\tilde{\subseteq }{\tilde{\cup }}_{d_z\in F}B\left(d_z\right)$. Since $f_{rw}$ is soft weakly quasi-continuous, then by Theorem 2, there exists $N\in SO\left(\phi \right)$ such that $a_x\tilde{\in }N$ and $f_{rw}\left(N\right)\tilde{\subseteq }C{l}_{\sigma }\left(K\right)$. We let $R=N\tilde{\cap }\left({\tilde{\cap }}_{d_z\in F}A\left(d_z\right)\right)$. Then, we have $a_x\tilde{\in }R\in SO\left(\phi \right)$ and $f_{rw}\left(R\right)\tilde{\cap }\left(1_{\mathcal{S}}-K\right)=0_{\mathcal{S}}$. Therefore, $f_{rw}\left(R\right)\tilde{\subseteq }K\tilde{\subseteq }G$. This shows that $f_{rw}$ is soft quasi-continuous. □

Theorem 12. 

Every soft rim-compact soft Hausdorff STS is soft regular.

Proof. 

We let $\left(O,\phi ,\mathcal{P}\right)$ be soft rim-compact and soft Hausdorff. We let $a_x\in SP\left(O,\mathcal{P}\right)$ and let $U\in \sigma$ such that $a_x\tilde{\in }U$. Since $\left(O,\phi ,\mathcal{P}\right)$ is soft rim-compact, there exists $G\in \phi$ such that $a_x\tilde{\in }G\tilde{\subseteq }U$ and $B{d}_{\phi }\left(G\right)$ is soft compact. For each $b_y\tilde{\in }B{d}_{\phi }\left(G\right)$, $a_x\ne b_y$, and by assumption that $\left(O,\phi ,\mathcal{P}\right)$ is soft Hausdorff, there exist $A\left(b_y\right)$, $B\left(b_y\right)\in \phi$ such that $a_x\tilde{\in }A\left(b_y\right)$, $b_y\tilde{\in }B\left(b_y\right)$, and $A\left(b_y\right)\tilde{\cap }B\left(b_y\right)=0_{\mathcal{P}}$ or $C{l}_{\phi }\left(A\left(b_y\right)\right)\tilde{\cap }B\left(b_y\right)=0_{\mathcal{P}}$. Since $B{d}_{\phi }\left(G\right)$ is soft compact and $B{d}_{\phi }\left(G\right)\tilde{\subseteq }{\tilde{\cup }}_{b_y\tilde{\in }B{d}_{\phi }\left(G\right)}B\left(b_y\right)$, there exists a finite subset $F\subseteq$$SP\left(O,\mathcal{P}\right)$ such that $d_z\tilde{\in }B{d}_{\phi }\left(G\right)$ for all $d_z\in F$, and $B{d}_{\phi }\left(G\right)\tilde{\subseteq }{\tilde{\cup }}_{d_z\in F}B\left(d_z\right)$. We let $V=G\tilde{\cap }\left({\tilde{\cap }}_{d_z\in F}A\left(d_z\right)\right)$. Then, $a_x\tilde{\in }V\in \phi$. Now,

$$\begin{array}{ccc}C{l}_{\phi }\left(V\right)\tilde{\cap }\left(1_{\mathcal{P}}-G\right)& =& C{l}_{\phi }\left(V\right)\tilde{\cap }B{d}_{\phi }\left(G\right)\\ & \tilde{\subseteq }& C{l}_{\phi }\left(V\right)\tilde{\cap }\left({\tilde{\cup }}_{d_z\in F}B\left(d_z\right)\right)\\ & =& \left({\tilde{\cup }}_{d_z\in F}\left(C{l}_{\phi }\left(A\left(b_y\right)\right)\tilde{\cap }B\left(d_z\right)\right)\right)\\ & =& 0_{\mathcal{P}}.\end{array}$$

This shows that $\left(O,\phi ,\mathcal{P}\right)$ is soft regular. □

Theorem 13. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft weakly quasi-continuous function such that $\left(T,\sigma ,\mathcal{S}\right)$ is soft rim-compact and soft Hausdorff. Then, $f_{rw}$ is soft quasi-continuous.

Proof. 

The proof follows from Theorems 10 and 12. □

Definition 13. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function. Then, $G\left(f_{rw}\right)$ is said to be soft strongly semi-closed with respect to $\left(O\times T,pr\left(\phi \times \supset \right),\mathcal{P}\times \mathcal{S}\right)$ if for each ${\left(a,b\right)}_{(x,y)}\tilde{\in }{1}_{\mathcal{P}\times \mathcal{S}}-G\left(f_{rw}\right)$, there exist $A\in SO\left(\phi \right)$ and $B\in SO\left(\sigma \right)$ such that $a_x\tilde{\in }A$, $b_y\tilde{\in }B$, and $\left(A\times sC{l}_{\sigma }\left(B\right)\right)\tilde{\cap }G\left(f_{rw}\right)=0_{\mathcal{P}\times \mathcal{S}}$.

Theorem 14. 

If $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is a soft weakly quasi-continuous function and $\left(T,\sigma ,\mathcal{S}\right)$ is soft Hausdorff, then $G\left(f_{rw}\right)$ is soft strongly semi-closed with respect to $\left(O\times T,pr\left(\phi \times \supset \right),\mathcal{P}\times \mathcal{S}\right)$.

Proof. 

We let ${\left(a,b\right)}_{\left(x,y\right)}\tilde{\in }{1}_{\mathcal{P}\times \mathcal{S}}-G\left(f_{rw}\right)$. Then, $f_{rw}\left(a_x\right)\ne b_y$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft Hausdorff, then there exist $U,V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }U$, $b_y\tilde{\in }V$, and $U\tilde{\cap }V=0_{\mathcal{S}}$. It is not difficult to check that $C{l}_{\sigma }\left(U\right)\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)=C{l}_{\sigma }\left(U\right)\tilde{\cap }sC{l}_{\sigma }\left(V\right)=0_{\mathcal{S}}$. Since $f_{rw}$ is soft weakly quasi-continuous, then there exists $W\in SO\left(\phi \right)$ such that $a_x\tilde{\in }W$ and $f_{rw}\left(W\right)\tilde{\subseteq }C{l}_{\sigma }\left(U\right)$. Therefore, $f_{rw}\left(W\right)\tilde{\cap }sC{l}_{\sigma }\left(V\right)=0_{\mathcal{S}}$ and hence, $\left(W\times sC{l}_{\sigma }\left(V\right)\right)\tilde{\cap }G\left(f_{rw}\right)=0_{\mathcal{P}\times \mathcal{S}}$. This shows that $G\left(f_{rw}\right)$ is soft strongly semi-closed with respect to $\left(O\times T,pr\left(\phi \times \supset \right),\mathcal{P}\times \mathcal{S}\right)$. □

Theorem 15. 

If $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is a soft weakly quasi-continuous injection and $\left(T,\sigma ,\mathcal{S}\right)$ is soft Urysohn, then $\left(O,\phi ,\mathcal{P}\right)$ is soft semi-$T_2$.

Proof. 

We let $a_x,b_y\in SP\left(O,\mathcal{P}\right)$ such that $a_x\ne b_y$. Since $f_{rw}$ is injective, then $f_{rw}\left(a_x\right)\ne f_{rw}\left(b_y\right)$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft Urysohn, then there exist $U,V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }U$, $f_{rw}\left(b_y\right)\tilde{\in }V$, and $C{l}_{\sigma }\left(U\right)\tilde{\cap }C{l}_{\sigma }\left(V\right)=0_{\mathcal{S}}$. Since $f_{rw}$ is soft weakly quasi-continuous, there are $A,B\in SO\left(\phi \right)$ such that $a_x\tilde{\in }A$, $b_y\tilde{\in }B$, $f_{rw}\left(A\right)\tilde{\subseteq }C{l}_{\sigma }\left(U\right)$, and $f_{rw}\left(B\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. Thus, $f_{rw}\left(A\right)\tilde{\cap }{f}_{rw}\left(B\right)\tilde{\subseteq }C{l}_{\sigma }\left(U\right)\tilde{\cap }C{l}_{\sigma }\left(V\right)=0_{\mathcal{S}}$ and hence, $f_{rw}\left(A\right)\tilde{\cap }{f}_{rw}\left(B\right)=0_{\mathcal{S}}$. Since $f_{rw}$ is injective, $f_{rw}\left(A\right)\tilde{\cap }{f}_{rw}\left(B\right)=f_{rw}\left(A\tilde{\cap }B\right)$. Therefore, $A\tilde{\cap }B=0_{\mathcal{P}}$. This shows that $\left(O,\phi ,\mathcal{P}\right)$ is soft semi-$T_2$. □

Theorem 16. 

We let $f_{rw},f_{pu}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft functions, and $\left(T,\sigma ,\mathcal{S}\right)$ is soft Urysohn. If $f_{rw}$ is soft weakly quasi-continuous and $f_{pu}$ is soft weakly continuous, then $\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}\in SC\left(\phi \right)$.

Proof. 

We let $G=\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}$. We show that $1_{\mathcal{P}}-G\tilde{\subseteq }{1}_{\mathcal{P}}-sC{l}_{\phi }\left(G\right)$. We let $a_x\tilde{\in }{1}_{\mathcal{P}}-G$. Then, $f_{rw}\left(a_x\right)\ne f_{pu}\left(a_x\right)$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft Urysohn, there exist $U,V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }U$, $f_{pu}\left(a_x\right)\tilde{\in }V$, and $C{l}_{\sigma }\left(U\right)\tilde{\cap }C{l}_{\sigma }\left(V\right)=0_{\mathcal{S}}$. Since $f_{rw}$ is soft weakly quasi-continuous, there exists $A\in SO\left(\phi \right)$ such that $a_x\tilde{\in }A$ and $f_{rw}\left(A\right)\tilde{\subseteq }C{l}_{\sigma }\left(U\right)$. Since $f_{pu}$ is soft weakly continuous, then there exists $B\in \phi$ such that $a_x\tilde{\in }B$ and $f_{pu}\left(B\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. We put $D=A\tilde{\cap }B$. Then, $a_x\tilde{\in }D\in SO\left(\phi \right)$. □

Claim 5. 

$D\tilde{\cap }G=0_{\mathcal{P}}$.

Proof of Claim 5. 

We suppose to the contrary that there exists $b_y\tilde{\in }D\tilde{\cap }G$. Then $b_y\tilde{\in }A$, $b_y\tilde{\in }B$, and $f_{rw}\left(b_y\right)=f_{pu}\left(b_y\right)$. So, we have $f_{rw}\left(b_y\right)\tilde{\in }{f}_{rw}\left(A\right)\tilde{\subseteq }C{l}_{\sigma }\left(U\right)$, $f_{pu}\left(b_y\right)=f_{rw}\left(b_y\right)\tilde{\in }{f}_{pu}\left(B\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. Thus, $f_{rw}\left(b_y\right)\tilde{\in }C{l}_{\sigma }\left(U\right)\tilde{\cap }C{l}_{\sigma }\left(V\right)=0_{\mathcal{S}}$, a contradiction.

Therefore, $a_x\tilde{\in }\left(1_{\mathcal{P}}-sC{l}_{\phi }\left(G\right)\right)\in SC\left(\phi \right)$. □

Theorem 17. 

We let $\left(O,\phi ,\mathcal{P}\right)$ and let $D\in SS\left(O,\mathcal{P}\right)$. If $C{l}_{\phi }\left(D\right)=1_{\mathcal{P}}$, then $sC{l}_{\phi }\left(D\right)=1_{\mathcal{P}}$.

Proof. 

We suppose to the contrary that there exists $b_y\tilde{\in }{1}_{\mathcal{P}}-sC{l}_{\phi }\left(D\right)$. Then, there exists $G\in SO\left(\phi \right)$ such that $b_y\tilde{\in }G$ and $G\tilde{\cap }D=0_{\mathcal{P}}$. We choose $V\in \phi$ such that $V\tilde{\subseteq }G\tilde{\subseteq }C{l}_{\phi }\left(V\right)$. Since $b_y\tilde{\in }G\tilde{\subseteq }C{l}_{\phi }\left(V\right)$, $C{l}_{\phi }\left(V\right)\ne 0_{\mathcal{P}}$, and so $V\ne 0_{\mathcal{P}}$. Since $C{l}_{\phi }\left(D\right)=1_{\mathcal{P}}$, $D\tilde{\cap }V\ne 0_{\mathcal{P}}$. But $D\tilde{\cap }V\tilde{\subseteq }D\tilde{\cap }G=0_{\mathcal{P}}$, a contradiction. □

Theorem 18. 

We let $f_{rw},f_{pu}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft functions and $\left(T,\sigma ,\mathcal{S}\right)$ is soft Urysohn, and we et D be a soft dense set in $\left(O,\phi ,\mathcal{P}\right)$. If $f_{rw}$ is soft weakly quasi-continuous, $f_{pu}$ is soft weakly continuous, and $f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)$ for all $a_x\tilde{\in }D$, then $f_{rw}=f_{pu}$.

Proof. 

We let $G=\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}$. Then, $D\tilde{\subseteq }G$, and so $sC{l}_{\phi }\left(D\right)\tilde{\subseteq }sC{l}_{\phi }\left(G\right)$. By Theorem 17, $sC{l}_{\phi }\left(D\right)=1_{\mathcal{P}}$ and by Theorem 16, $sC{l}_{\phi }\left(G\right)=G$. Therefore, $G=1_{\mathcal{P}}$. Hence, $f_{rw}=f_{pu}$. □

Theorem 19. 

We let $\left(O,\phi ,\mathcal{P}\right)$ be a STS, $A\in \alpha \left(\phi \right)$, and $B\in SO\left(\phi \right)$. Then, $A\tilde{\cap }B\in SO\left(\phi \right)$.

Proof. 

We suppose that $A\in \alpha \left(\phi \right)$ and $B\in SO\left(\phi \right)$. Then, $A\tilde{\subseteq }In{t}_{\phi }\left(C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)$ and $B\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(B\right)\right)$. We show that $A\tilde{\cap }B\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(A\tilde{\cap }B\right)\right)$. We let $a_x\tilde{\in }A\tilde{\cap }B$ and we let $H\in \phi$ such that $a_x\tilde{\in }H$. Since $a_x\tilde{\in }A\tilde{\subseteq }In{t}_{\phi }\left(C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)\in \phi$, we have $a_x\tilde{\in }H\tilde{\cap }\left(In{t}_{\phi }\left(C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)\right)\in \phi$. Since $a_x\tilde{\in }B\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(B\right)\right)$, $a_x\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(B\right)\right)$ and so $\left(H\tilde{\cap }\left(In{t}_{\phi }\left(C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)\right)\right)\tilde{\cap }In{t}_{\phi }\left(B\right)\ne 0_{\mathcal{P}}$. Since $In{t}_{\phi }\left(C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)$, $0_{\mathcal{P}}\ne \left(H\tilde{\cap }\left(In{t}_{\phi }\left(C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)\right)\right)\tilde{\cap }In{t}_{\phi }\left(B\right)\tilde{\subseteq }\left(H\tilde{\cap }C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)\tilde{\cap }In{t}_{\phi }\left(B\right)$. We choose $d_z\tilde{\in }\left(H\tilde{\cap }C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)\right)\tilde{\cap }In{t}_{\phi }\left(B\right)$. Then, we have $d_z\tilde{\in }H\tilde{\cap }In{t}_{\phi }\left(B\right)\in \phi$ and $d_z\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(A\right)\right)$, which implies that $\left(H\tilde{\cap }In{t}_{\phi }\left(B\right)\right)\tilde{\cap }In{t}_{\phi }\left(A\right)=H\tilde{\cap }\left(In{t}_{\phi }\left(B\right)\tilde{\cap }In{t}_{\phi }\left(A\right)\right)=H\tilde{\cap }\left(In{t}_{\phi }\left(A\tilde{\cap }B\right)\right)\ne 0_{\mathcal{P}}$. This shows that $a_x\tilde{\in }C{l}_{\phi }\left(In{t}_{\phi }\left(A\tilde{\cap }B\right)\right)$. □

Theorem 20. 

We let $f_{rw},f_{pu}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft functions, and $\left(T,\sigma ,\mathcal{S}\right)$ is soft Hausdorff. If $f_{rw}$ is soft weakly quasi-continuous, $f_{pu}$ is soft almost α-continuous, and $G=\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}$. Then, $G\in SC\left(\phi \right)$.

Proof. 

We show that $1_{\mathcal{P}}-G\tilde{\subseteq }{1}_{\mathcal{P}}-sC{l}_{\phi }\left(G\right)$. We let $a_x\tilde{\in }{1}_{\mathcal{P}}-G$. Then, $f_{rw}\left(a_x\right)\ne f_{pu}\left(a_x\right)$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft Hausdorff, then there exist $U,V\in \sigma$ such that $f_{rw}\left(a_x\right)\tilde{\in }U$, $f_{pu}\left(a_x\right)\tilde{\in }V$, and $U\tilde{\cap }V=0_{\mathcal{S}}$; hence, $C{l}_{\sigma }\left(U\right)\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)=0_{\mathcal{S}}$. Since $f_{rw}$ is soft weakly quasi-continuous, there exists $A\in SO\left(\phi \right)$ such that $a_x\tilde{\in }A$ and $f_{rw}\left(A\right)\tilde{\subseteq }C{l}_{\sigma }\left(U\right)$. Since $V\in \sigma$, $In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\in RO\left(\sigma \right)$. Since $f_{pu}$ is soft almost $\alpha$-continuous and $In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)\in RO\left(\sigma \right)$, there exists $B\in \alpha \left(\phi \right)$ such that $a_x\tilde{\in }B$ and $f_{pu}\left(B\right)\tilde{\subseteq }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)$. Since we have $f_{rw}\left(A\right)\tilde{\cap }{f}_{pu}\left(B\right)\tilde{\subseteq }C{l}_{\sigma }\left(U\right)\tilde{\cap }In{t}_{\sigma }\left(C{l}_{\sigma }\left(V\right)\right)=0_{\mathcal{S}}$, $f_{rw}\left(A\right)\tilde{\cap }{f}_{pu}\left(B\right)=0_{\mathcal{S}}$ and hence $\left(A\tilde{\cap }B\right)\tilde{\cap }G=0_{\mathcal{P}}$. Since $a_x\tilde{\in }A\tilde{\cap }B$ and by Theorem 19, $A\tilde{\cap }B\in SO\left(\phi \right)$, $a_x\tilde{\in }{1}_{\mathcal{P}}-sC{l}_{\phi }\left(G\right)$. □

Corollary 2. 

We let $f_{rw},f_{pu}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft functions and $\left(T,\sigma ,\mathcal{S}\right)$ is soft Hausdorff, and let D be a soft dense set in $\left(O,\phi ,\mathcal{P}\right)$. If $f_{rw}$ is soft weakly quasi-continuous, $f_{pu}$ is soft almost α-continuous, and $f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)$ for all $a_x\tilde{\in }D$, then $f_{rw}=f_{pu}$.

Proof. 

We let $G=\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}$. Then, by Theorem 20, $G\in SC\left(\phi \right)$. By assumption, we have $D\tilde{\subseteq }G$, and so $sC{l}_{\phi }\left(D\right)\tilde{\subseteq }sC{l}_{\phi }\left(G\right)=G$. On the other hand, by Theorem 17, $sC{l}_{\phi }\left(D\right)=1_{\mathcal{P}}$. Therefore, $G=1_{\mathcal{P}}$, which ends the proof. □

Theorem 21. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function such that $G\left(f_{rw}\right)$ is θ-closed with respect to $\left(O,\phi ,\mathcal{P}\right)$ and we let $f_{pu}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft weakly quasi-continuous. Then, $\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}\in SC\left(\phi \right)$.

Proof. 

We let $G=\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}$. We show that $1_{\mathcal{P}}-G\tilde{\subseteq }{1}_{\mathcal{P}}-sC{l}_{\phi }\left(G\right)$. We let $a_x\tilde{\in }{1}_{\mathcal{P}}-G$. Then, $f_{rw}\left(a_x\right)\ne f_{pu}\left(a_x\right)$. So, ${\left(a,u\left(a\right)\right)}_{\left(x,p\left(x\right)\right)}\tilde{\in }{1}_{\mathcal{P}\times \mathcal{S}}-G\left(f_{rw}\right)$. Since $G\left(f_{rw}\right)$ is $\theta$-closed with respect to $\left(O,\phi ,\mathcal{P}\right)$, there exist $A\in \phi$ and $B\in \sigma$ such that $a_x\tilde{\in }A$, $f_{pu}\left(a_x\right)\tilde{\in }B$, and $\left(C{l}_{\phi }\left(A\right)\times C{l}_{\phi }\left(B\right)\right)\tilde{\cap }G\left(f_{rw}\right)=0_{\mathcal{P}\times \mathcal{S}}$ or that $f_{rw}\left(C{l}_{\phi }\left(A\right)\right)\tilde{\cap }C{l}_{\sigma }\left(B\right)=0_{\mathcal{S}}$. Now, since $f_{pu}$ is soft weakly quasi-continuous, there exists $H\in SO\left(\phi \right)$ such that $a_x\tilde{\in }H$ and $f_{pu}\left(H\right)\tilde{\subseteq }C{l}_{\phi }\left(B\right)$. So, we have $a_x\tilde{\in }A\tilde{\cap }H\in SO\left(\phi \right)$ and $\left(A\tilde{\cap }H\right)\tilde{\cap }G=0_{\mathcal{S}}$, and hence $a_x\tilde{\in }{1}_{\mathcal{P}}-sC{l}_{\phi }\left(G\right)$. □

Corollary 3. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function such that $G\left(f_{rw}\right)$ is θ-closed with respect to $\left(O,\phi ,\mathcal{P}\right)$ and we let $f_{pu}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft weakly quasi-continuous. If D is a soft dense set in $\left(O,\phi ,\mathcal{P}\right)$ such that $f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)$ for all $a_x\tilde{\in }D$, then $f_{rw}=f_{pu}$.

Proof. 

We let $G=\tilde{\cup }\left\{a_x\in SP\left(O,\mathcal{P}\right):f_{rw}\left(a_x\right)=f_{pu}\left(a_x\right)\right\}$. Then, by Theorem 21, $G\in SC\left(\phi \right)$. By assumption, we have $D\tilde{\subseteq }G$, and so $sC{l}_{\phi }\left(D\right)\tilde{\subseteq }sC{l}_{\phi }\left(G\right)=G$. On the other hand, by Theorem 17, $sC{l}_{\phi }\left(D\right)=1_{\mathcal{P}}$. Therefore, $G=1_{\mathcal{P}}$, which ends the proof. □

Theorem 22. 

We let $\left(O,\phi ,\mathcal{P}\right)$ be soft Hausdorff, and we let Y be a non-empty subset of O. If there is a soft weakly quasi-continuous function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(Y,{\phi }_Y,\mathcal{P}\right)$ such that $f_{rw}\left(a_x\right)=a_x$ for all $a_x\in SP\left(Y,\mathcal{P}\right)$. Then, $C_Y\in SC\left(\phi \right)$.

Proof. 

We suppose to the contrary that there is $a_x\tilde{\in }sC{l}_{\phi }\left(C_Y\right)-C_Y$. Then, $f_{rw}\left(a_x\right)\ne a_x$. Since $\left(O,\phi ,\mathcal{P}\right)$ is soft Hausdorff, there exist $U,V\in \phi$ such that $a_x\tilde{\in }U$, $f_{rw}\left(a_x\right)\tilde{\in }V$, and $U\tilde{\cap }V=0_{\mathcal{P}}$; hence, $U\tilde{\cap }C{l}_{\phi }\left(V\right)=0_{\mathcal{P}}$. Since $f_{rw}$ is soft weakly quasi-continuous, then there exists $W\in SO\left(\phi \right)$ such that $a_x\tilde{\in }W$ and $f_{rw}\left(W\right)\tilde{\subseteq }C{l}_{\phi }\left(V\right)$. Since $a_x\tilde{\in }U\tilde{\cap }W\in SO\left(\phi \right)$ and $a_x\tilde{\in }sC{l}_{\phi }\left(C_Y\right)$, then $\left(U\tilde{\cap }W\right)\tilde{\cap }{C}_Y\ne 0_{\mathcal{P}}$. We choose $b_y\tilde{\in }\left(U\tilde{\cap }W\right)\tilde{\cap }{C}_Y$. Since $b_y\tilde{\in }{C}_Y$, then $f_{rw}\left(b_y\right)=b_y$. Since $b_y\tilde{\in }W$, then $f_{rw}\left(b_y\right)=b_y\in f_{rw}\left(W\right)\tilde{\subseteq }C{l}_{\phi }\left(V\right)$. Since $U\tilde{\cap }C{l}_{\phi }\left(V\right)=0_{\mathcal{P}}$, then $b_y\tilde{\notin }U$. This is a contradiction. □

Theorem 23. 

If $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is a soft weakly quasi-continuous surjection and $\left(O,\phi ,\mathcal{P}\right)$ is soft S-connected, then $\left(T,\sigma ,\mathcal{S}\right)$ is soft connected.

Proof. 

We suppose to the contrary that $\left(T,\sigma ,\mathcal{S}\right)$ is not soft connected. Then, there are $U,V\in \sigma -\left\{0_{\mathcal{S}}\right\}$ such that $U\tilde{\cup }V=1_{\mathcal{S}}$ and $U\tilde{\cap }V=0_{\mathcal{S}}$. Since $U\tilde{\cup }V=1_{\mathcal{S}}$, $f_{rw}^{-1}\left(U\right)\tilde{\cup }{f}_{rw}^{-1}\left(V\right)=f_{rw}^{-1}\left(U\tilde{\cup }V\right)=f_{rw}^{-1}\left(1_{\mathcal{S}}\right)=1_{\mathcal{P}}$. Since $U\tilde{\cap }V=0_{\mathcal{S}}$, $f_{rw}^{-1}\left(U\right)\tilde{\cap }{f}_{rw}^{-1}\left(V\right)=f_{rw}^{-1}\left(U\tilde{\cap }V\right)=f_{rw}^{-1}\left(0_{\mathcal{S}}\right)=0_{\mathcal{P}}$. Since $f_{rw}$ is surjective, $f_{rw}^{-1}\left(U\right)\ne 0_{\mathcal{P}}$ and $f_{rw}^{-1}\left(V\right)\ne 0_{\mathcal{P}}$. Since $f_{rw}$ is soft weakly quasi-continuous, by Theorem 4 (e), we have $f_{rw}^{-1}\left(U\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(U\right)\right)\right)$ and $f_{rw}^{-1}\left(V\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$. Since $\left\{U,V\right\}\subseteq \sigma \cap {\sigma }^c$, then $C{l}_{\sigma }\left(U\right)=U$ and $C{l}_{\sigma }\left(V\right)=V$. Hence, $f_{rw}^{-1}\left(U\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(U\right)\right)$ and $f_{rw}^{-1}\left(V\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(V\right)\right)$. This implies that $\left\{f_{rw}^{-1}\left(U\right),f_{rw}^{-1}\left(V\right)\right\}\subseteq SO\left(\phi \right)$. It follows that $\left(O,\phi ,\mathcal{P}\right)$ is not soft S-connected. This is a contradiction. □

Definition 14. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function. Then, $G\left(f_{pv}\right)$ is said to be soft strongly closed with respect to $\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times S\right)$ if for each ${\left(a,b\right)}_{(x,y)}\tilde{\in }{1}_{\mathcal{P}\times S}-G\left(f_{rw}\right)$, there exist $A\in \phi$ and $B\in \sigma$ such that $a_x\tilde{\in }A$, $b_y\tilde{\in }B$, and $\left(A\times C{l}_{\sigma }\left(B\right)\right)\tilde{\cap }G\left(f_{rw}\right)=0_{\mathcal{P}\times S}$.

Theorem 24. 

A STS $\left(O,\phi ,\mathcal{P}\right)$ is soft S-connected iff it is soft hyperconnected.

Proof. 

Necessity. We suppose that $\left(O,\phi ,\mathcal{P}\right)$ is soft S-connected. We suppose to the contrary that $\left(O,\phi ,\mathcal{P}\right)$ is not soft hyperconnected. Then, there are $A,B\in \phi -\left\{0_{\mathcal{P}}\right\}$ such that $A\tilde{\cap }B=0_{\mathcal{P}}$. So, $C{l}_{\phi }\left(A\right)\tilde{\cap }B=0_{\mathcal{P}}$ and hence $0_{\mathcal{P}}\ne B\tilde{\subseteq }{1}_{\mathcal{P}}-C{l}_{\phi }\left(A\right)$. Therefore, we have $\left\{C{l}_{\phi }\left(A\right),1_{\mathcal{P}}-C{l}_{\phi }\left(A\right)\right\}\subseteq SO\left(\phi \right)-\left\{0_{\mathcal{P}}\right\}$, $C{l}_{\phi }\left(A\right)\tilde{\cap }\left(1_{\mathcal{P}}-C{l}_{\phi }\left(A\right)\right)=0_{\mathcal{P}}$ and $C{l}_{\phi }\left(A\right)\tilde{\cup }\left(1_{\mathcal{P}}-C{l}_{\phi }\left(A\right)\right)=1_{\mathcal{P}}$. This implies that $\left(O,\phi ,\mathcal{P}\right)$ is not soft S-connected, which is a contradiction.

Sufficiency. We suppose that $\left(O,\phi ,\mathcal{P}\right)$ is soft hyperconnected. We suppose to the contrary that $\left(O,\phi ,\mathcal{P}\right)$ is not soft S-connected. Then, there are $A,B\in SO\left(\phi \right)-\left\{0_{\mathcal{P}}\right\}$ such that $A\tilde{\cap }B=0_{\mathcal{P}}$ and $A\tilde{\cup }B=0_{\mathcal{P}}$. This contradicts Theorem 3 of [66]. □

Theorem 25. 

If $\left(O,\phi ,\mathcal{P}\right)$ is soft S-connected and $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is a soft weakly quasi-continuous function such that $G\left(f_{pv}\right)$ is soft strongly closed with respect to $\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times S\right)$, then $f_{rw}$ is constant.

Proof. 

We suppose to the contrary that $f_{rw}$ is not constant. Then, there exists $a_x,b_y\in SP\left(O,\mathcal{P}\right)$ such that $f_{rw}\left(a_x\right)\ne f_{rw}\left(b_y\right)$. So, ${\left(a,w\left(b\right)\right)}_{(x,r\left(y\right))}\tilde{\in }{1}_{\mathcal{P}\times S}-G\left(f_{rw}\right)$. Since $G\left(f_{pv}\right)$ is soft strongly closed with respect to $\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times S\right)$, there exist $U\in \phi$ and $V\in \sigma$ such that $a_x\tilde{\in }U$, $f_{rw}\left(b_y\right)\tilde{\in }V$, and $\left(U\times C{l}_{\sigma }\left(V\right)\right)\tilde{\cap }G\left(f_{rw}\right)=0_{\mathcal{P}\times S}$. So, $f_{rw}\left(U\right)\tilde{\cap }C{l}_{\sigma }\left(V\right)=0_S$. Since $f_{rw}$ is soft weakly quasi-continuous, there exists $H\in SO\left(\phi \right)$ such that $b_y\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. So, we have $\left\{U,H\right\}\subseteq SO\left(\phi \right)-\left\{0_{\mathcal{P}}\right\}$ and $U\tilde{\cap }H=0_{\mathcal{P}}$. Thus, by Theorem 3 of [66] and Theorem 24, we conclude that $\left(O,\phi ,\mathcal{P}\right)$ is not soft S-connected, a contradiction. □

Theorem 26. 

If $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is a soft weakly quasi-continuous surjection and $\left(O,\phi ,\mathcal{P}\right)$ is soft semi-compact, $\left(T,\sigma ,\mathcal{S}\right)$ is soft quasi H-closed.

Proof. 

We let $\mathcal{V}\subseteq$$\sigma$ such that $1_{\mathcal{S}}={\tilde{\cup }}_{V\in \mathcal{V}}V$. Then, $1_{\mathcal{P}}=f_{rw}^{-1}\left(1_{\mathcal{S}}\right)=f_{rw}^{-1}\left({\tilde{\cup }}_{V\in \mathcal{V}}V\right)={\tilde{\cup }}_{V\in \mathcal{V}}{f}_{rw}^{-1}\left(V\right)$. Since $f_{rw}$ is soft weakly quasi-continuous, by Theorem 4 (e), $f_{rw}^{-1}\left(V\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)$ for every $V\in \sigma$. Thus, we have $\left\{sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right):V\in \mathcal{V}\right\}\subseteq SO\left(\phi \right)$ and $1_{\mathcal{P}}={\tilde{\cup }}_{V\in \mathcal{V}}{f}_{rw}^{-1}\left(V\right)\tilde{\subseteq }{\tilde{\cup }}_{V\in \mathcal{V}}s$$In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\tilde{\subseteq }{1}_{\mathcal{P}}$. Since $\left(O,\phi ,\mathcal{P}\right)$ is soft semi-compact, there is a finite sub-collection ${\mathcal{V}}_1$ of $\mathcal{V}$ such that ${\tilde{\cup }}_{V\in {\mathcal{V}}_1}sIn{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)=1_{\mathcal{P}}$. Since ${\tilde{\cup }}_{V\in {\mathcal{V}}_1}s$$In{t}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)\right)\tilde{\subseteq }{\tilde{\cup }}_{V\in {\mathcal{V}}_1}{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)$, $f_{rw}^{-1}\left({\tilde{\cup }}_{V\in {\mathcal{V}}_1}C{l}_{\sigma }\left(V\right)\right)={\tilde{\cup }}_{V\in {\mathcal{V}}_1}{f}_{rw}^{-1}\left(C{l}_{\sigma }\left(V\right)\right)=1_{\mathcal{P}}$ and so $f_{rw}\left(1_{\mathcal{P}}\right)=f_{rw}\left(f_{rw}^{-1}\left({\tilde{\cup }}_{V\in {\mathcal{V}}_1}C{l}_{\sigma }\left(V\right)\right)\right)\tilde{\subseteq }{\tilde{\cup }}_{V\in {\mathcal{V}}_1}C{l}_{\sigma }\left(V\right)$. On the other hand, since $f_{rw}$ is surjective, $f_{rw}\left(1_{\mathcal{P}}\right)=1_{\mathcal{S}}$. It follows that ${\tilde{\cup }}_{V\in {\mathcal{V}}_1}C{l}_{\sigma }\left(V\right)=1_{\mathcal{S}}$. This shows that $\left(T,\sigma ,\mathcal{S}\right)$ is soft quasi H-closed. □

The soft composition of two soft weakly quasi-continuous functions need not be soft weakly quasi-continuous:

Example 3. 

We let $O=\left\{a,b\right\}$, $T=M=\left\{a,b,c\right\}$, $\mathcal{P}=\left\{1,2\right\}$, $\Im =\left\{\varnothing ,O,\left\{b\right\}\right\}$, $\aleph =\left\{\varnothing ,T,\left\{a,c\right\},\left\{b,c\right\},\left\{c\right\}\right\}$, and $\wp =\left\{\varnothing ,M,\left\{a\right\},\left\{b\right\},\left\{a,b\right\}\right\}$. We let $r_1:O⟶T$, $r_2:T⟶M$, and $w_1,w_2:\mathcal{P}⟶\mathcal{P}$ be the inclusion functions. Then, $f_{r_1{w}_1}:\left(O,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(T,\tau \left(\aleph \right),\mathcal{P}\right)$ and $f_{r_2{w}_2}:\left(T,\tau \left(\aleph \right),\mathcal{P}\right)⟶\left(M,\tau \left(\wp \right),\mathcal{P}\right)$ are soft weakly quasi-continuous functions, while $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}:\left(O,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(M,\tau \left(\wp \right),\mathcal{P}\right)$ is not soft weakly quasi-continuous.

Theorem 27. 

If $f_{r_1{w}_1}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft weakly quasi-continuous and $f_{r_2{w}_2}:\left(T,\sigma ,\mathcal{S}\right)⟶\left(M,\rho ,\mathcal{N}\right)$ is soft continuous, then $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}:\left(O,\phi ,\mathcal{P}\right)⟶\left(M,\rho ,\mathcal{N}\right)$ is soft weakly quasi-continuous.

Proof. 

We let $f_{r_1{w}_1}$ be soft weakly quasi-continuous and $f_{r_2{w}_2}$ be soft continuous. We let $a_x\in SP\left(O,\mathcal{P}\right)$ and we let $G\in \rho$ such that $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}\left(a_x\right)\tilde{\in }G$. Since $f_{r_2{w}_2}$ is soft continuous, $f_{r_2{w}_2}^{-1}\left(G\right)\in \sigma$. Since $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}\left(a_x\right)=f_{r_2{w}_2}\left(f_{r_1{w}_1}\left(a_x\right)\right)\in G$, $f_{r_1{w}_1}\left(a_x\right)\tilde{\in }{f}_{r_2{w}_2}^{-1}\left(G\right)$. Since $f_{r_1{w}_1}$ is soft weakly quasi-continuous, there exists $H\in SO\left(\phi \right)$ such that $a_x\tilde{\in }H$ and $f_{r_1{w}_1}\left(H\right)\tilde{\subseteq }C{l}_{\sigma }\left(f_{r_2{w}_2}^{-1}\left(G\right)\right)$; hence, $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}\left(H\right)=f_{r_2{w}_2}\left(f_{r_1{w}_1}\left(H\right)\right)\tilde{\subseteq }{f}_{r_2{w}_2}\left(C{l}_{\sigma }\left(f_{r_2{w}_2}^{-1}\left(G\right)\right)\right)$. On the other hand, since $f_{r_2{w}_2}$ is soft continuous, $f_{r_2{w}_2}\left(C{l}_{\sigma }\left(f_{r_2{w}_2}^{-1}\left(G\right)\right)\right)\tilde{\subseteq }C{l}_{\rho }\left(f_{r_2{w}_2}\left(\left(f_{r_2{w}_2}^{-1}\left(G\right)\right)\right)\right)\tilde{\subseteq }C{l}_{\rho }\left(G\right)$. Thus, we have $a_x\tilde{\in }H\in SO\left(\phi \right)$ and $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}\left(H\right)\tilde{\subseteq }C{l}_{\rho }\left(G\right)$. This shows that $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}$ is soft weakly quasi-continuous. □

Composition $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}$ of a soft continuous function $f_{r_1{w}_1}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ and a soft semi-continuous function $f_{r_2{w}_2}:\left(T,\sigma ,\mathcal{S}\right)⟶\left(M,\rho ,\mathcal{N}\right)$ is not necessarily soft weakly quasi-continuous:

Example 4. 

We let $O=T=M=\left\{a,b,c,d\right\}$, $\mathcal{P}=\left\{1,2\right\}$, $\Im =\left\{\varnothing ,O,\left\{a\right\},\left\{b\right\},\left\{a,b\right\},\left\{a,c,d\right\}\right\}$, $\aleph =\left\{\varnothing ,T,\left\{a\right\},\left\{b\right\},\left\{a,b\right\}\right\}$, and $\wp =\left\{\varnothing ,M,\left\{a\right\},\left\{b\right\},\left\{a,b\right\},\left\{b,c,d\right\}\right\}$. We let $r_1:O⟶T$, $r_2:T⟶M$, and $w_1,w_2:\mathcal{P}⟶\mathcal{P}$ be the identity functions. Then, $f_{r_1{w}_1}:\left(O,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(T,\tau \left(\aleph \right),\mathcal{P}\right)$ is soft continuous and $f_{r_2{w}_2}:\left(T,\tau \left(\aleph \right),\mathcal{P}\right)⟶\left(M,\tau \left(\wp \right),\mathcal{P}\right)$ is soft semi-continuous while $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}:\left(O,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(M,\tau \left(\wp \right),\mathcal{P}\right)$ is not soft weakly quasi-continuous.

Lemma 4. 

We let $\left(O,\phi ,\mathcal{P}\right)$ be a STS and we let X be a non-empty subset of O such that $C_X\in \phi$; then, for every $A\in SO\left(\phi \right)$, $A\tilde{\cap }{C}_X\in SO\left({\phi }_X\right)$.

Proof. 

We let $A\in SO\left(\phi \right)$. Then, there exist $U\in \phi$ such that $U\tilde{\subseteq }A\tilde{\subseteq }C{l}_{\phi }\left(U\right)$. So, $U\tilde{\cap }{C}_X\tilde{\subseteq }A\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{\phi }\left(U\right)\tilde{\cap }{C}_X$. □

Claim 6. 

$C{l}_{\phi }\left(U\right)\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{{\phi }_X}\left(U\tilde{\cap }{C}_X\right)$.

Proof of Claim 6. 

We let $b_y\tilde{\in }C{l}_{\phi }\left(U\right)\tilde{\cap }{C}_X$. To see that $b_y\tilde{\in }C{l}_{{\phi }_X}\left(U\tilde{\cap }{C}_X\right)$, we let $G\in {\phi }_X$ such that $b_y\tilde{\in }G$. We choose $H\in \phi$ such that $G=H\tilde{\cap }{C}_X$. Then, $G\in \phi$. Since $b_y\tilde{\in }G\tilde{\cap }{C}_X\in \phi$ and $b_y\tilde{\in }C{l}_{\phi }\left(U\right)$, then $\left(G\tilde{\cap }{C}_X\right)\tilde{\cap }U=G\tilde{\cap }\left(U\tilde{\cap }{C}_X\right)\ne 0_{\mathcal{P}}$. Thus, $b_y\tilde{\in }C{l}_{{\phi }_X}\left(U\tilde{\cap }{C}_X\right)$.

Since $U\tilde{\cap }{C}_X\in {\phi }_X$ and by the above claim $U\tilde{\cap }{C}_X\tilde{\subseteq }A\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{\phi }\left(U\right)\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{{\phi }_X}\left(U\tilde{\cap }{C}_X\right)$, then $A\tilde{\cap }{C}_X\in SO\left({\phi }_X\right)$. □

Theorem 28. 

If $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is a soft weakly quasi-continuous function and $X\subseteq O$ such that $C_X\in \phi -\left\{0_{\mathcal{P}}\right\}$, then ${\left(f_{rw}\right)}_{\left|C_X\right.}:\left(X,{\phi }_X,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft weakly quasi-continuous.

Proof. 

We let $a_x\in SP\left(X,\mathcal{P}\right)$ and let $G\in \sigma$ such that ${\left(f_{rw}\right)}_{\left|C_X\right.}\left(a_x\right)\tilde{\in }G$. Since $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft weakly quasi-continuous, there exists $H\in SO\left(\phi \right)$ such that $a_x\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }C{l}_{\sigma }\left(G\right)$. Since $C_X\in \phi$ and $H\in SO\left(\phi \right)$, by Lemma 4, $C_X\tilde{\cap }H\in SO\left({\phi }_X\right)$. Thus, we have $a_x\tilde{\in }{C}_X\tilde{\cap }H\in SO\left({\phi }_X\right)$ and ${\left(f_{rw}\right)}_{\left|C_X\right.}\left(C_X\tilde{\cap }H\right)=f_{rw}\left(C_X\tilde{\cap }H\right)\tilde{\subseteq }{f}_{rw}\left(H\right)\tilde{\subseteq }C{l}_{\sigma }\left(G\right)$. Therefore, ${\left(f_{rw}\right)}_{\left|C_X\right.}$ is soft weakly quasi-continuous. □

The restriction of a soft semi-continuous function to a soft regular closed subset is not necessarily soft weakly quasi-continuous:

Example 5. 

We let $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(O,\tau \left(\aleph \right),\mathcal{P})$ be as in Example 1. Then, $f_{rw}$ is soft semi-continuous. We let $X=\left\{a,c\right\}$. Then, $C_X\in SC\left(\phi \right)$ while ${\left(f_{rw}\right)}_{\left|C_X\right.}:\left(X,{\left(\tau \left(\Im \right)\right)}_X,\mathcal{P}\right)⟶\left(O,\tau \left(\aleph \right),\mathcal{P}\right)$ is not soft weakly quasi-continuous.

For any two non-empty sets O and T, the projection functions $h:O\times T⟶O$ and $g:O\times T⟶T$ defined by $h\left(x,y\right)=x$ and $g\left(x,y\right)=y$ for all $\left(x,y\right)\in O\times T$ are denoted by ${\pi }_O$ and ${\pi }_T$, respectively.

Theorem 29. 

We let $\left(O,\phi ,\mathcal{P}\right)$, $\left(T,\sigma ,\mathcal{S}\right)$, and $\left(M,\lambda ,\mathcal{N}\right)$ be three STSs. If $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T\times M,pr\left(\sigma \times \lambda \right),\mathcal{S}\times \mathcal{N}\right)$ is soft weakly quasi-continuous, then $f_{\left({\pi }_T\circ r\right)\left({\pi }_{\mathcal{S}}\circ w\right)}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ and $f_{\left({\pi }_M\circ r\right)\left({\pi }_{\mathcal{N}}\circ w\right)}:\left(O,\phi ,\mathcal{P}\right)⟶\left(M,\lambda ,\mathcal{N}\right)$ are soft weakly quasi-continuous.

Proof. 

We let $f_{rw}$ be soft weakly quasi-continuous. Since $f_{\left({\pi }_T\right)\left({\pi }_{\mathcal{S}}\right)}:\left(T\times M,pr\left(\sigma \times \lambda \right),\mathcal{S}\times \mathcal{N}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ and $f_{\left({\pi }_M\right)\left({\pi }_{\mathcal{N}}\right)}:\left(T\times M,pr\left(\sigma \times \lambda \right),\mathcal{S}\times \mathcal{N}\right)⟶\left(M,\lambda ,\mathcal{N}\right)$ are always soft continuous, then by Theorem 27, $f_{\left({\pi }_T\circ r\right)\left({\pi }_{\mathcal{S}}\circ w\right)}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ and $f_{\left({\pi }_M\circ r\right)\left({\pi }_{\mathcal{N}}\circ w\right)}:\left(O,\phi ,\mathcal{P}\right)⟶\left(M,\lambda ,\mathcal{N}\right)$ are soft weakly quasi-continuous. □

For any function $g:O⟶T$, function $h:O⟶O\times T$ defined by $h\left(x\right)=\left(x,g\left(x\right)\right)$ is denoted by $g^{\#}$.

Lemma 5. 

For a soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$, the following are equivalent:

(a) $f_{rw}$ is soft weakly quasi-continuous.

(b) For a soft base $\mathcal{B}$ of σ, we have, for every $a_x\in SP\left(O,\mathcal{P}\right)$ and every $B\in \mathcal{B}$ such that $f_{rw}\left(a_x\right)\tilde{\in }B$, there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(B\right)$.

Proof. 

(a) ⟶ (b) is obvious;

(b) ⟶ (a): We let $a_x\in SP\left(O,\mathcal{P}\right)$ and let $V\in \supset$ such that $f_{rw}\left(a_x\right)\tilde{\in }V$. Since $\mathcal{B}$ is a soft base of $\sigma$, there exists $B\in \mathcal{B}$ such that $f_{rw}\left(a_x\right)\tilde{\in }B\tilde{\subseteq }V$. By (b), there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(B\right)\tilde{\subseteq }C{l}_{\sigma }\left(V\right)$. □

Theorem 30. 

We let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function. Then, $f_{r^{\#}{w}^{\#}}:\left(O,\phi ,\mathcal{P}\right)⟶\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times \mathcal{S}\right)$ is soft weakly quasi-continuous iff $f_{rw}$ is soft weakly quasi-continuous.

Proof. 

Necessity. We let $f_{r^{\#}{w}^{\#}}$ be soft weakly quasi-continuous. Then, by Theorem 29, $f_{rw}=f_{\left({\pi }_T\circ r^{\#}\right)\left({\pi }_S\circ w^{\#}\right)}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft weakly quasi-continuous.

Sufficiency. We let $f_{rw}$ be soft weakly quasi-continuous. We apply Lemma 5. We consider the soft base $\mathcal{B}=\left\{A\times B:A\in \phi \mathrm{and}B\in \sigma \right\}$ of $\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times \mathcal{S}\right)$. We let $a_x\in SP\left(O,\mathcal{P}\right)$ and we let $A\times B\in \mathcal{B}$ such that $f_{r^{\#}{w}^{\#}}\left(a_x\right)={\left(a,w\left(a\right)\right)}_{\left(x,r\left(x\right)\right)}\tilde{\in }A\times B$. Since $f_{rw}$ is soft weakly quasi-continuous and $f_{rw}\left(a_x\right)={\left(w\left(a\right)\right)}_{r\left(x\right)}\tilde{\in }B\in \sigma$, then there exists $U\in SO\left(\phi \right)$ such that $a_x\tilde{\in }U$ and $f_{rw}\left(U\right)\tilde{\subseteq }C{l}_{\sigma }\left(B\right)$. We put $G=A\tilde{\cap }U$. Then, we have $a_x\tilde{\in }G$ and $f_{r^{\#}{w}^{\#}}\left(G\right)\tilde{\subseteq }C{l}_{pr\left(\phi \times \sigma \right)}\left(A\times B\right)$. It follows that $f_{r^{\#}{w}^{\#}}$ is soft weakly quasi-continuous. □

4. Conclusions

In the setting of soft topological spaces, soft continuity is an extension of classical continuity. Weaker versions of soft continuity enable a more in-depth examination of the characteristics of soft functions and a more sophisticated comprehension of the connections between soft topological spaces.

In this paper, we define soft quasi-continuity in soft topological spaces which is an extension of quasi-continuity in general topology. We prove that the concepts of soft quasi-continuity and soft semi-continuity are equivalent. Also, we define soft weak quasi-continuity as a weak form of both soft semi-continuity and soft weak continuity. Several characterizations of soft weak quasi-continuity are given. Also, the relationships between these functions and their topological equivalents are studied. Moreover, a sufficient condition for soft semi-continuity and soft weakly quasi-continuity to be equivalent is given. In addition, many results about soft composition, preservation, restriction, and soft graph theorems of soft weakly quasi-continuity are introduced.

Soft weak quasi-continuity is not presverved under both soft composition and soft restrictions in general, which are significant limitations.

Future research might look into the following topics: (1) defining soft almost weakly continuous functions; (2) defining soft weakly closed continuous functions; (3) finding a use for these new concepts in a decision making problem.