Soft ω-θ-Continuous and Soft Weakly θ_ω-Continuous Mappings

Abstract

Soft $\omega$-$\theta$-continuity and soft weak-${\theta }_{\omega }$-continuity as two new concepts of continuity are presented and investigated. The investigation of the links between these forms of soft mappings and their general topological relatives is given. With the help of examples, it is investigated that soft $\omega$-$\theta$-continuity lies strictly between soft $\theta$-continuity and soft weak-continuity, while soft weak-${\theta }_{\omega }$-continuity lies strictly between soft continuity (i.e., soft ${\theta }_{\omega }$-continuity) and soft weak-continuity. A number of conditions for the equivalence between soft $\omega$-$\theta$-continuity and soft weak continuity (i.e., soft $\omega$-$\theta$-continuity and soft $\theta$-continuity, soft weak-${\theta }_{\omega }$-continuity and soft weak-continuity, soft weak-${\theta }_{\omega }$-continuity and soft continuity) are obtained. Additionally, soft $\theta$-closure and soft ${\theta }_{\omega }$-closure operators are used to characterize our new types of soft mappings.

1. Introduction

We encounter several uncertain situations in daily life that we are unable to solve using conventional mathematical techniques. It was suggested to use situations like fuzzy sets and soft sets to cope with these kinds of issues. In 1999, Molodtsov [1] introduced the idea of soft sets as a novel mathematical strategy for dealing with ambiguous circumstances and imprecise data. Molodtsov discussed the numerous sectors in which soft sets may be employed and the advantages they have over fuzzy sets. Following that, this method piqued the interest of a large number of academics and researchers interested in uncertainty in both theoretical and applied challenges. Maji et al. [2] employed soft sets to tackle decision-making problems in 2002, and they [3] provided the initial inspiration for a set of operations between soft sets in 2003. Several of these procedures, as demonstrated in the published literature, had flaws that caused certain writers to revise their definitions and adopt new types of them for a variety of reasons [4,5,6].

Shabir and Naz [7] constructed a soft topology over a fixed set of parameters by defining a topology over a family of soft sets. Shabir and Naz characterized soft topological notions as equal to their classical topological counterparts, which encourages and supports researchers in the area to continue down this line. Since the invention of soft topology, several contributions have been made to the discussion of topological ideas and concepts in soft contexts [8,9,10,11,12,13,14,15,16], and substantial contributions are yet possible.

Since they provide new insights into soft topological concepts such as soft compactness, soft separation axioms, new classes of soft mappings, etc., generalizations of soft open sets are important and attractive subjects for researchers. For instance, “soft semi-open sets” [17], “soft pre-open sets” [18], “soft $\beta$-open sets” [18], “soft $\alpha$-open sets” [19], “soft regular open sets” [20], “soft somewhere dense sets” [21], “weakly soft $\alpha$-open” [8], and so on. Al Ghour and Hamed [22] first suggested soft $\omega$-openness as an extension of soft openness, and numerous works on the topic have subsequently been published.

Soft continuity of mappings was defined by Nazmul and Samanta [23] in 2013. Then, many modifications of “soft continuity” appeared. For instance, “soft $\alpha$-continuous mappings” [19], “soft semicontinuous mappings” [24], “soft $\beta$-continuous mappings” [25], “soft $\omega$-continuous mappings” [13], “soft ${\omega }_s$-continuous mappings” [13], and so on.

In this work, soft $\omega$-$\theta$-continuous and soft weakly ${\theta }_{\omega }$-continuous classes of mappings are presented and investigated.

The arrangement of this article is as follows:

Section 2 introduces some fundamental concepts and outcomes that will be used in the next sections.

Section 3 defines soft $\omega$-$\theta$-continuous mappings. We study the features of these soft mappings and show how they relate to well-known soft continuity notions like soft weak continuity and soft $\theta$-continuity. Furthermore, we investigate the links between this class of soft mappings and its general topology analogs. We also provide many characterizations of soft $\omega$-$\theta$-continuous mappings.

In Section 4, we define soft weakly ${\theta }_{\omega }$-continuous mappings as a new class of soft mappings and investigate some of its properties. We give several characterizations of it. With the help of examples, we explain its connections with several types of soft continuity, such as soft continuous, soft weakly continuous, and soft ${\theta }_{\omega }$-continuous mappings. Moreover, we investigate the links between this class of soft mappings and its analogs in general topology.

Section 5 contains some findings and potential future studies.

2. Preliminaries

In this section, we introduce certain fundamental concepts and results that will be used in the paper.

For simplicity, throughout this paper, we shall use the concepts and terminologies from [22,26]. Topological space and soft topological space, respectively, shall be abbreviated as TS and STS.

Let $\left(W,ϝ,B\right)$ be a STS, $\left(W,\eta \right)$ be a TS, $H\in SS(W,B)$, and $V\subseteq W$. In this article, $C{l}_{ϝ}\left(H\right)$,$In{t}_{ϝ}\left(H\right)$, $C{l}_{\eta }\left(V\right)$, and $In{t}_{\eta }\left(V\right)$ will denote the soft closure of H in $\left(W,ϝ,B\right)$, the soft interior of H in $\left(W,ϝ,B\right)$, the closure of V in $\left(W,\eta \right)$, and the interior of V in $\left(W,\eta \right)$, respectively.

Now, we will go through a few concepts and results that will be used in the paper.

Definition 1.

A mapping $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ between the TSs $\left(W,\eta \right)$ and $\left(N,\xi \right)$ is said to be:

(1) 

θ-continuous (θ-c, for simplicity) if for any $w\in W$ and any $V\in \xi$ such that $q\left(T\right)\in V$, we find $L\in \eta$ such that $w\in L$ and $q\left(C{l}_{\eta }\left(L\right)\right)\subseteq C{l}_{\xi }\left(V\right)$ [27];

(2) 

Weakly continuous (w-c, for simplicity) if for any $w\in W$ and any $V\in \xi$ such that $q\left(T\right)\in V$, we find $L\in \eta$ such that $w\in L$ and $q\left(L\right)\subseteq C{l}_{\xi }\left(V\right)$ [28];

(3) 

Weakly ${\theta }_{\omega }$-continuous (w-${\theta }_{\omega }$-c, for simplicity) if for any $w\in W$ and any $V\in \xi$ such that $q\left(T\right)\in V$, we find $L\in \eta$ such that $w\in L$ and $q\left(L\right)\subseteq C{l}_{{\xi }_{\omega }}\left(V\right)$ [29];

(4) 

ω-θ-continuous (ω-θ-c, for simplicity) if for any $w\in W$ and any $V\in \xi$ such that $q\left(T\right)\in V$, we find $L\in \eta$ such that $w\in L$ and $q\left(C{l}_{{\eta }_{\omega }}\left(L\right)\right)\subseteq C{l}_{\xi }\left(V\right)$ [29].

Definition 2.

A soft set $K\in SS(W,B)$ defined by

(1) 

$K\left(b\right)=\left\{\begin{array}{cc}T& \mathrm{if}b=e\\ \varnothing & \mathrm{if}b\ne e\end{array}\right.$ is marked as $e_T$ [26];

(2) 

$K\left(b\right)=T$ for all $b\in B$ is marked as $C_T$ [26];

(3) 

$K\left(b\right)=\left\{\begin{array}{cc}\left\{w\right\}& \mathrm{if}b=a\\ \varnothing & \mathrm{if}b\ne a\end{array}\right.$ is marked as $a_w$ and said to be a soft point. $SP(W,B)$ will denote the collection of soft points in $SS(W,B)$ [30].

Definition 3

([30]). Let $H\in SS(W,B)$ and $b_w\in SP(W,B)$. Then, $b_w$ is considered to belong to H ($b_w\tilde{\in }H$) if $w\in H\left(b\right)$.

Definition 4

([22]). For any STS $\left(W,ϝ,E\right)$, the collection

$$\left\{H-S:H\in \tau \mathit{and}S\mathit{is}a\mathit{countable}\mathit{soft}\mathit{set}\right\}$$

forms a soft base for some soft topology. This soft topology is marked as ${ϝ}_{\omega }$.

Theorem 1

([26]). For a given TS $\left(W,\eta \right)$, the family

$$\left\{H\in SS\left(W,E\right):H\left(e\right)\in \eta \mathit{for}\mathit{every}e\in E\right\}$$

is a soft topology. This soft topology is marked as $\tau \left(\eta \right)$.

Theorem 2

([26]). For any collection of TSs $\left\{\left(W,{ϝ}_e\right):e\in E\right\}$, the collection

$$\left\{G\in SS\left(W,E\right):G\left(e\right)\in {ϝ}_e\mathit{for}\mathit{every}e\in E\right\}$$

forms a soft topology. This soft topology is denoted by ${\oplus }_{e\in E}{ϝ}_e$.

In Theorem 1, if we put ${ϝ}_e=\eta$ for every $e\in E$, then $\tau \left(\eta \right)={\oplus }_{e\in E}{ϝ}_e$. Therefore, Theorem 1 is a special case of one given in Theorem 2.

Definition 5.

Let $\left(W,ϝ,B\right)$ be a STS and let $H\in SS\left(W,B\right)$. Then:

(1) H is a soft semi-open [17] (i.e., soft pre-open [18], soft β-open [18], soft regular closed [20]) set in $\left(W,ϝ,B\right)$ if we find $G\in ϝ$ such that $G\tilde{\subseteq }H\tilde{\subseteq }C{l}_{ϝ}\left(G\right)$ (resp. $H\tilde{\subseteq }In{t}_{ϝ}\left(C{l}_{ϝ}\left(H\right)\right)$, $H\tilde{\subseteq }C{l}_{ϝ}\left(In{t}_{ϝ}\left(C{l}_{ϝ}\left(H\right)\right)\right)$, $H=C{l}_{ϝ}\left(In{t}_{ϝ}\left(H\right)\right)$); $SO\left(W,ϝ,B\right)$ (i.e., $PO\left(W,ϝ,B\right)$, $\beta O\left(W,ϝ,\right.$$\left.B\right)$, $RC\left(W,ϝ,B\right)$) will denote the family of soft semi-open (i.e., pre-open, β-open, regular closed) sets in $\left(W,ϝ,B\right)$.

(2) The soft θ-closure of H in $\left(W,ϝ,B\right)$ is denoted by $C{l}_{\theta }^{ϝ}\left(H\right)$ and defined by $C{l}_{\theta }^{ϝ}\left(H\right)=$ $\tilde{\cup }\left\{b_w\in SP(W,B):C{l}_{ϝ}\left(G\right)\tilde{\cap }H\ne 0_B\mathit{for}\mathit{any}G\in ϝ\mathrm{with}{b}_w\tilde{\in }G\right\}$ [31].

(3) The soft ${\theta }_{\omega }$-closure of H in $\left(W,ϝ,B\right)$ is denoted by $C{l}_{{\theta }_{\omega }}^{ϝ}\left(H\right)$ and defined by $C{l}_{{\theta }_{\omega }}^{ϝ}\left(H\right)=$ $\tilde{\cup }\left\{b_w\in SP(W,B):C{l}_{{ϝ}_{\omega }}\left(G\right)\tilde{\cap }H\ne 0_B\mathit{for}\mathit{any}G\in ϝ\mathrm{with}{b}_w\tilde{\in }G\right\}$ [32].

Definition 6.

A soft mapping $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is called:

(1) Soft θ-continuous (soft θ-c, for simplicity) if for any $b_w$$\in SP(W,B)$ and any $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{ϝ}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$ [31].

(2) Soft weakly continuous (soft w-c, for simplicity) if for any $b_w$ $\in SP(W,B)$ and any $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$ [33].

(3) Soft ${\theta }_{\omega }$-continuous (soft ${\theta }_{\omega }$-c, for simplicity) if for any $b_w$$\in SP(W,B)$ and any $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{ϝ}\left(K\right)\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$ [32].

Definition 7.

A STS  $\left(W,ϝ,B\right)$ is called:

(a) Soft locally indiscrete (soft l-i, for simplicity) if $ϝ\subseteq {ϝ}^c$ [34];

(b) Soft locally countable (soft l-c, for simplicity) if for any $b_w\in SP(W,B)$, we find $G\in ϝ$ such that $b_w\tilde{\in }G\in CSS\left(W,B\right)$ [22];

(c) Soft anti-locally countable (soft anti-l-c, for simplicity) if for any $F\in ϝ-\left\{0_B\right\}$, $F\notin CSS\left(W,B\right)$ [22];

(d) Soft ω-locally indiscrete (soft ω-l-i, for simplicity) if $ϝ\subseteq {\left({ϝ}_{\omega }\right)}^c$ [35];

(e) Soft ω-regular if for any $b_w\in SP\left(M,B\right)$ and any $G\in ϝ$ such that $b_w\tilde{\in }G$, we find $K\in ϝ$ such that $b_w\tilde{\in }K\tilde{\subseteq }C{l}_{{ϝ}_{\omega }}\left(K\right)\tilde{\subseteq }G$ [35].

Lemma 1

([15]). $\left\{\left(W,{ϝ}_b\right):b\in B\right\}$ be a family of TSs and let $K\in SS\left(W,B\right)$. Then, $\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\left(K\right)\right)\left(a\right)=C{l}_{{\left({ϝ}_a\right)}_{\omega }}\left(K\left(a\right)\right)$ for every $a\in B$.

Lemma 2

([15]). Let $\left\{\left(W,{ϝ}_b\right):b\in B\right\}$ be a family of TSs and let $K\in SS\left(W,B\right)$. Then, $\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}\left(K\right)\right)\left(a\right)=C{l}_{{ϝ}_a}\left(K\left(a\right)\right)$ for every $a\in B$.

3. Soft $\mathit{\omega }$-$\mathit{\theta }$-Continuity

In this section, we define soft $\omega$-$\theta$-continuous mappings. We study the features of these soft mappings and show how they relate to well-known soft continuity notions like soft weak continuity and soft $\theta$-continuity. Furthermore, we investigate the links between this class of soft mappings and its general topology analogs. We also provide many characterizations of soft $\omega$-$\theta$-continuous mappings.

Definition 8.

A soft mapping $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is called soft ω-θ-continuous (soft ω-θ-c, for simplicity) if for any $b_w$$\in SP(W,B)$ and any $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$.

Theorem 3.

Let  $\left\{\left(W,{ϝ}_b\right):b\in B\right\}$ and $\left\{\left(N,{\Xi }_d\right):d\in D\right\}$ be two families of TSs. Let $q:W⟶N$ be a mapping and $v:B⟶D$ be an injective mapping. Then, $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft ω-θ-c if and only if $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is ω-θ-c for each $b\in B$.

Proof. 

Necessity. Suppose that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft $\omega$-$\theta$-c. Let $b\in B$. Let $w\in W$ and let $T\in {\Xi }_{v\left(b\right)}$ such that $q\left(T\right)\in T$. Then, we have $f_{qv}\left(b_w\right)={\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }{\left(v\left(b\right)\right)}_T\in {\oplus }_{d\in D}{\Xi }_d$. So, we find $K\in {\oplus }_{b\in B}{ϝ}_b$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}K\right)\tilde{\subseteq }C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)$, and so, $\left(f_{qv}\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}\left(K\right)\right)\right)\left(v\left(b\right)\right)\subseteq \left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\right.$$\left.\left({\left(v\left(b\right)\right)}_T\right)\right)\left(v\left(b\right)\right)$. By Theorem 8 of [22], ${\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }={\oplus }_{b\in B}{\left({ϝ}_b\right)}_{\omega }$ and so, $\left(f_{qv}\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}\right.\right.$$\left.\left.\left(K\right)\right)\right)\left(v\left(b\right)\right)=\left(f_{qv}\left(C{l}_{{\oplus }_{b\in B}{\left({ϝ}_b\right)}_{\omega }}\left(K\right)\right)\right)\left(v\left(b\right)\right)$. Since $v:B⟶D$ is injective, then, by Lemma 2, $\left(f_{qv}\left(C{l}_{{\oplus }_{b\in B}{\left({ϝ}_b\right)}_{\omega }}\left(K\right)\right)\right)\left(v\left(b\right)\right)=q\left(\left(C{l}_{{\oplus }_{b\in B}{\left({ϝ}_b\right)}_{\omega }}\left(K\right)\right)\left(b\right)\right)=$ $q\left(C{l}_{{\left({ϝ}_b\right)}_{\omega }}\left(K\left(b\right)\right)\right)$. Also, by Lemma 1, $\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)\right)\left(v\left(b\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}\left({\left(v\left(b\right)\right)}_T\left(v\left(b\right)\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}\left(T\right)$. Therefore, we have $w\in K\left(b\right)\in {ϝ}_b$ and $q\left(C{l}_{{\left({ϝ}_b\right)}_{\omega }}\left(K\left(b\right)\right)\right)\subseteq C{l}_{{\Xi }_{v\left(b\right)}}\left(T\right)$. It follows that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is soft $\omega$-$\theta$-c.

Sufficiency. Suppose that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is $\omega$-$\theta$-c for each $b\in B$. Let $b_w\in SP(W,B)$ and let $G\in {\oplus }_{d\in D}{\Xi }_d$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, we have ${\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }G$, and so $q\left(T\right)\in G\left(v\left(b\right)\right)\in {\Xi }_{v\left(b\right)}$. Since $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is $\omega$-$\theta$-c, then we find $X\in {ϝ}_b$ such that $w\in X$ and $q\left(C{l}_{{\left({ϝ}_b\right)}_{\omega }}\left(X\right)\right)\subseteq C{l}_{{\Xi }_{v\left(b\right)}}\left(G\left(v\left(b\right)\right)\right)$. Now, we have $b_w\tilde{\in }{b}_X\in {\oplus }_{b\in B}{ϝ}_b$. Also, since $v:B⟶D$ is injective, then, by Lemmas 1 and 2, we have $\left(f_{qv}\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}\left(b_X\right)\right)\right)\left(v\left(b\right)\right)=\left(f_{qv}\left(C{l}_{{\oplus }_{b\in B}{\left({ϝ}_b\right)}_{\omega }}\left(b_X\right)\right)\right)\left(v\left(b\right)\right)=p\left(\left(C{l}_{{\oplus }_{b\in B}{\left({ϝ}_b\right)}_{\omega }}\left(b_X\right)\right)\right.$$\left.\left(b\right)\right)=p\left(C{l}_{{\left({ϝ}_b\right)}_{\omega }}\left(X\right)\right)$, $\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)\right)\left(v\left(b\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}G\left(v\left(b\right)\right)$, and $\left(f_{qv}\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}\right.\right.$$\left.\left.\left(b_X\right)\right)\right)\left(s\right)=\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)\right)\left(s\right)=\varnothing$ for all $s\ne v\left(b\right)$. Hence, $f_{qv}\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}\left(b_X\right)\right)\tilde{\subseteq }C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)$. It follows that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft $\omega$-$\theta$-c. □

Corollary 1.

Let $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ be a mapping between two TSs and let $v:B⟶D$ be an injective mapping. Then, $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ is ω-θ-c if and only if $f_{qv}:(M,\tau \left(\eta \right),B)⟶(N,\tau \left(\xi \right),D)$ is soft ω-θ-c.

Proof. 

For any $b\in B$ and $d\in D$, put ${ϝ}_b=\eta$ and ${\Xi }_d=\xi$. Then, $\tau \left(\eta \right)={\oplus }_{b\in B}{ϝ}_b$ and $\tau \left(\xi \right)={\oplus }_{d\in D}{\Xi }_d$. Thus, by Theorem 3, we get the result. □

Theorem 4.

Let  $\left\{\left(W,{ϝ}_b\right):b\in B\right\}$ and $\left\{\left(N,{\Xi }_d\right):d\in D\right\}$ be two families of TSs. Let $q:M⟶N$ be a mapping and $v:B⟶D$ be an injective mapping. Then, $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft w-c if and only if $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-c for each $b\in B$.

Proof. 

Necessity. Suppose that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft w-c. Let $b\in B$. Let $w\in W$ and let $T\in {\Xi }_{v\left(b\right)}$ such that $q\left(T\right)\in T$. Then, we have $f_{qv}\left(b_w\right)={\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }{\left(v\left(b\right)\right)}_T\in {\oplus }_{d\in D}{\Xi }_d$. Thus, we find $K\in {\oplus }_{b\in B}{ϝ}_b$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)$, and so $\left(f_{qv}\left(K\right)\right)\left(v\left(b\right)\right)\subseteq \left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)\right)\left(v\left(b\right)\right)$. Since $v:B⟶D$ is injective, then $\left(f_{qv}\left(K\right)\right)\left(v\left(b\right)\right)=q\left(K\left(b\right)\right)$. Also, by Lemma 1, $\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)\right)\left(v\left(b\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}(\left({\left(v\left(b\right)\right)}_T\right)\left(v\left(b\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}\left(T\right)$. Therefore, we have $w\in K\left(b\right)\in {ϝ}_b$ and $q\left(K\left(b\right)\right)\subseteq C{l}_{{\Xi }_{v\left(b\right)}}\left(T\right)$. It follows that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-c. □

Sufficiency. Suppose that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-c for each $b\in B$. Let $b_w\in SP(W,B)$ and let $G\in {\oplus }_{d\in D}{\Xi }_d$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, we have ${\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }G$, and so $q\left(w\right)\in G\left(v\left(b\right)\right)\in {\Xi }_{v\left(b\right)}$. Since $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-c, then we find $X\in {ϝ}_b$ such that $w\in X$ and $q\left(X\right)\subseteq C{l}_{{\Xi }_{v\left(b\right)}}\left(G\left(v\left(b\right)\right)\right)$. Now, we have $b_w\tilde{\in }{b}_X\in {\oplus }_{b\in B}{ϝ}_b$. Since $v:B⟶D$ is injective, then, by Lemma 1, $f_{qv}\left(C{l}_{{\left({\oplus }_{b\in B}{ϝ}_b\right)}_{\omega }}\left(b_X\right)\right)\tilde{\subseteq }C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)$. It follows that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft w-c.

Corollary 2.

Let $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ be a mapping between two TSs and let $v:B⟶D$ be an injective mapping. Then, $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ is w-c if and only if $f_{qv}:(M,\tau \left(\eta \right),B)⟶(N,\tau \left(\xi \right),D)$ is soft w-c.

Proof. 

For any $b\in B$ and $d\in D$, put ${ϝ}_b=\eta$ and ${\Xi }_d=\xi$. Then, $\tau \left(\eta \right)={\oplus }_{b\in B}{ϝ}_b$ and $\tau \left(\xi \right)={\oplus }_{d\in D}{\Xi }_d$. Thus, by Theorem 4, we get the result. □

Theorem 5.

Let  $\left\{\left(W,{ϝ}_b\right):b\in B\right\}$ and $\left\{\left(N,{\Xi }_d\right):d\in D\right\}$ be two families of TSs. Let $q:M⟶N$ be a mapping and $v:B⟶D$ be an injective mapping. Then, $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft θ-c if and only if $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is θ-c for each $b\in B$.

Proof. 

Necessity. Suppose that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft $\theta$-c. Let $b\in B$. Let $w\in W$ and let $T\in {\Xi }_{v\left(b\right)}$ such that $q\left(w\right)\in T$. Then, we have $f_{qv}\left(b_w\right)={\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }{\left(v\left(b\right)\right)}_T\in {\oplus }_{d\in D}{\Xi }_d$. Thus, we find $K\in {\oplus }_{b\in B}{ϝ}_b$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\right.$$\left.\left(K\right)\right)\tilde{\subseteq }C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)$, and so $\left(f_{qv}\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\left(K\right)\right)\right)\left(v\left(b\right)\right)\subseteq \left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)\right)\left(v\left(b\right)\right)$. Since $v:B⟶D$ is injective, then, by Lemma 1, $\left(f_{qv}\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\left(K\right)\right)\right)\left(v\left(b\right)\right)=q\left(C{l}_{{ϝ}_b}\left(K\left(b\right)\right)\right)$. Also, by Lemma 1, $\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left({\left(v\left(b\right)\right)}_T\right)\right)\left(v\left(b\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}\left(\left({\left(v\left(b\right)\right)}_T\right)\left(v\left(b\right)\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}\left(T\right)$. Therefore, we have $w\in K\left(b\right)\in {ϝ}_b$ and $q\left(C{l}_{{ϝ}_b}\left(K\left(b\right)\right)\right)\subseteq C{l}_{{\Xi }_{v\left(b\right)}}\left(T\right)$. It follows that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is $\theta$-c. □

Sufficiency 1.

Sufficiency. Suppose that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is θ-c for each $b\in B$. Let $b_w\in SP(W,B)$ and let $G\in {\oplus }_{d\in D}{\Xi }_d$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, we have ${\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }G$, and so $q\left(w\right)\in G\left(v\left(b\right)\right)\in {\Xi }_{v\left(b\right)}$. Since $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is θ-c, then we find $X\in {ϝ}_b$ such that $w\in X$ and $p\left(C{l}_{{ϝ}_b}\left(X\right)\right)\subseteq C{l}_{{\Xi }_{v\left(b\right)}}\left(G\left(v\left(b\right)\right)\right)$. Now, we have $b_w\tilde{\in }{b}_X\in {\oplus }_{b\in B}{ϝ}_b$. Since $v:B⟶D$ is injective, then, by Lemma 1, $\left(f_{qv}\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\left(b_X\right)\right)\right)\left(v\left(b\right)\right)=q\left(\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\left(b_X\right)\right)\left(b\right)\right)=q\left(C{l}_{{ϝ}_b}\left(X\right)\right)$, $\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)\right)\left(v\left(b\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}\left(G\left(v\left(b\right)\right)\right)$, and $\left(f_{qv}\right.$$\left.\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\left(b_X\right)\right)\right)\left(s\right)=\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)\right)\left(s\right)=\varnothing$ for all $s\ne v\left(b\right)$. Hence, $f_{qv}\left(C{l}_{{\oplus }_{b\in B}{ϝ}_b}\left(b_X\right)\right)\tilde{\subseteq }C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)$. It follows that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft θ-c.

Corollary 3.

Let $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ be a mapping between two TSs and let $v:B⟶D$ be an injective mapping. Then, $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ is θ-c if and only if $f_{qv}:(M,\tau \left(\eta \right),B)⟶(N,\tau \left(\xi \right),D)$ is soft θ-c.

Proof. 

For any $b\in B$ and $d\in D$, put ${ϝ}_b=\eta$ and ${\Xi }_d=\xi$. Then, $\tau \left(\eta \right)={\oplus }_{b\in B}{ϝ}_b$ and $\tau \left(\xi \right)={\oplus }_{d\in D}{\Xi }_d$. Thus, by Theorem 5, we get the result. □

Theorem 6.

Every soft ω-θ-c mapping is soft w-c.

Proof. 

Let $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ be soft $\omega$-$\theta$-c. Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, we find $K\in ϝ$ such that $b_w$$\tilde{\in }K$ and $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Since $f_{qv}\left(K\right)\tilde{\subseteq }{f}_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)$, then $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Hence, $f_{qv}$ is soft w-c. □

Theorem 7.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-c such that $\left(W,ϝ,B\right)$ is soft ω-l-i, then $f_{qv}$ is soft ω-θ-c.

Proof. 

Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft w-c, then we find $K\in ϝ$ such that $b_w$$\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Since $\left(W,ϝ,B\right)$ is soft $\omega$-l-i, then K is soft $\omega$-closed in $\left(W,ϝ,B\right)$, and so $C{l}_{{ϝ}_{\omega }}\left(K\right)=K$. Thus, we have $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)=f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Therefore, $f_{qv}$ is soft $\omega$-$\theta$-c. □

Corollary 4.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-c such that $\left(W,ϝ,B\right)$ is soft l-i, then $f_{qv}$ is soft ω-θ-c.

Proof. 

Follows from Theorem 3.9 and Theorem 7 of [35]. □

Corollary 5.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-c such that $\left(W,ϝ,B\right)$ is soft l-c, then $f_{qv}$ is soft ω-θ-c.

Proof. 

Follows form Theorem 7 and Theorem 6 of [35]. □

Theorem 8.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-c such that $\left(W,ϝ,B\right)$ is soft ω-regular, then $f_{qv}$ is soft ω-θ-c.

Proof. 

Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft w-c, then we find $K\in ϝ$ such that $b_w$$\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Since $\left(W,ϝ,B\right)$ is soft $\omega$-regular, then we find $H\in ϝ$ such that $b_w$$\tilde{\in }H\tilde{\subseteq }C{l}_{{ϝ}_{\omega }}\left(H\right)\tilde{\subseteq }K$. Therefore, $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(H\right)\right)\tilde{\subseteq }{f}_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. It follows that $f_{qv}$ is soft $\omega$-$\theta$-c. □

Theorem 9.

Every soft θ-c mapping is soft ω-θ-c.

Proof. 

Let $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ be soft $\theta$-c. Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft $\theta$-c, then we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{ϝ}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Hence, $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }{f}_{qv}\left(C{l}_{ϝ}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Therefore, $f_{qv}$ is soft $\omega$-$\theta$-c. □

Theorem 10.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft ω-θ-c such that $\left(W,ϝ,B\right)$ is soft anti-l-c, then $f_{qv}$ is soft θ-c.

Proof. 

Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft $\omega$-$\theta$-c, then we find $K\in ϝ$ such that $b_w$$\tilde{\in }K$ and $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Since $\left(W,ϝ,B\right)$ is soft anti-l-c, then, by Theorem 14 of [22], $C{l}_{{ϝ}_{\omega }}\left(K\right)=C{l}_{ϝ}\left(K\right)$. Therefore, $f_{qv}\left(C{l}_{ϝ}\left(K\right)\right)=f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. It follows that $f_{qv}$ is soft $\theta$-c. □

From the above theorems, we have following implications. However, Examples 1 and 2 given below show that the converses of these implications are not true.

$$\begin{array}{ccccc}\mathrm{soft}\theta -\mathrm{c}& ⟶& \mathrm{soft}\omega -\theta -\mathrm{c}& ⟶& \mathrm{soft}\mathrm{w}-\mathrm{c}.\end{array}$$

The following example demonstrates that the contrary of Theorem 6 does not have to be true in general.

Example 1.

We utilize Example 3.3 of [36]. Let $W={\mathbb{R}}^2\cup \left\{s\right\}$, where  $s\notin {\mathbb{R}}^2$. Let α be the usual topology on ${\mathbb{R}}^2$. Let $\mathcal{B}=\left\{G\cap \left({\mathbb{R}}^2-\left\{\left(0,0\right)\right\}\right):G\in \alpha \right\}$ $\cup \left\{\left\{\left(x,y\right):x^2+y^2<\frac{1}{n^2},y>0\right\}\cup \right.$$\left.\left\{\left(0,0\right)\right\}:n\in \mathbb{N}\right\}\cup$ $\left\{\left\{\left(x,y\right):x^2+y^2<\frac{1}{n^2},y<0\right\}\cup \left\{s\right\}:n\in \mathbb{N}\right\}$. Let η be the topology on W having $\mathcal{B}$ as a base. Let $N=\left\{a,b,c\right\}$ and $\xi =\left\{\varnothing ,N,\left\{a\right\},\left\{c\right\},\left\{a,c\right\}\right\}$. Consider the mappings $q:W⟶N$ and $v:\mathbb{Z}⟶\mathbb{Z}$ defined as follows:

$q(x,y)=\left\{\begin{array}{ccc}a& \mathit{if}& y\ge 0\\ b& \mathit{if}& y<0\\ c& \mathit{if}& \left(x,y\right)=s\end{array}\right.$ and $v\left(z\right)=z$ for all $z\in \mathbb{Z}$.

It is proven in Example 3.3 of [36] that $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ is w-c but not $\theta$-c. So, by Corollaries 1 and 2, $f_{qv}:\left(W,\tau \left(\eta \right),\mathbb{Z}\right)⟶\left(N,\tau \left(\xi \right),\mathbb{Z}\right)$ is soft w-c but not soft $\theta$-c. Since $\left(W,\tau \left(\eta \right),\mathbb{Z}\right)$ is soft anti-l-c, then, by Theorem 10, $f_{qv}$ is not soft $\omega$-$\theta$-c.

The following example demonstrates that the opposite of Theorem 6 is not generally true.

Example 2.

We utilize Example 3.2 of [36]. Let $M=\left\{a,b,c,d\right\}$ and $\eta =\left\{\varnothing ,M,\left\{b\right\},\left\{c\right\},\left\{b,c\right\},\right.$$\left.\left\{a,b\right\},\left\{a,b,c\right\},\left\{b,c,d\right\}\right\}$. Define $q:M⟶M$ and $v:\mathbb{Z}⟶\mathbb{Z}$ by

$q\left(a\right)=c$, $q\left(b\right)=d$, $q\left(c\right)=b$, $q\left(d\right)=a$, and $u\left(z\right)=z$ for all $z\in \mathbb{Z}$. It is proven in Example 3.2 of [36] that $q:\left(W,\eta \right)⟶\left(W,\eta \right)$ is w-c but not θ-c. Since $\left(W,\eta \right)$ is l-c, then, by Corollaries 2 and 3, $f_{qv}:\left(W,\tau \left(\eta \right),\mathbb{Z}\right)⟶\left(W,\tau \left(\eta \right),\mathbb{Z}\right)$ is soft w-c but not soft θ-c. Since $\left(W,\tau \left(\eta \right),\mathbb{Z}\right)$ is soft l-c, then, by Corollary 5, $f_{qv}$ is soft ω-θ-c.

Theorem 11.

For a soft mapping  $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$, the following are equivalent:

(a) $f_{qv}$ is soft ω-θ-c;

(b) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(R\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(R\right)\right)$ for any $R\in SS(N,D)$;

(c) $f_{qv}\left(C{l}_{{\theta }_{\omega }}^{ϝ}\left(H\right)\right)\tilde{\subseteq }C{l}_{\theta }^{\Xi }\left(f_{qv}\left(H\right)\right)$ for any $H\in SS(W,B)$.

Proof. 

(a) ⟶ (b): Let $R\in SS(N,D)$. We show that $1_B-f_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(R\right)\right)\tilde{\subseteq }$

$1_B-C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(R\right)\right)$. Let $b_w$$\tilde{\in }{1}_B-f_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(R\right)\right)$. Then, $f_{qv}\left(b_w\right)\tilde{\notin }C{l}_{\theta }^{\Xi }\left(R\right)$, and so, we find $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$ and $C{l}_{\Xi }\left(G\right)\tilde{\cap }R=0_D$. By (a), we find $K\in ϝ$ such that $b_w$$\tilde{\in }K$ and $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Therefore, we have $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\cap }R=0_D$, and hence $C{l}_{{ϝ}_{\omega }}\left(K\right)\tilde{\cap }{f}_{qv}^{-1}\left(R\right)=0_B$. It follows that $a_m$$\tilde{\in }{1}_B-C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(R\right)\right)$.

(b) ⟶ (a): Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $C{l}_{\Xi }\left(G\right)\tilde{\cap }\left(1_D-\right.$$\left.C{l}_{\Xi }\left(G\right)\right)=0_D$, then $f_{qv}\left(b_w\right)\tilde{\notin }C{l}_{\theta }^{\Xi }\left(1_D-C{l}_{\Xi }\left(G\right)\right)$, and so $b_w$$\tilde{\notin }{f}_{qv}^{-1}\left(C{l}_{\theta }^{ϝ}\left(1_D-C{l}_{\Xi }\left(G\right)\right)\right)$. Thus, by (b), $b_w$$\tilde{\notin }C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(1_D-C{l}_{\Xi }\left(G\right)\right)\right)$. So, we find $K\in ϝ$ such that $b_w$$\tilde{\in }K$ and $C{l}_{{ϝ}_{\omega }}\left(K\right)\tilde{\cap }{f}_{qv}^{-1}\left(1_D-C{l}_{\Xi }\left(G\right)\right)$. Therefore, $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. It follows that $f_{qv}$ is soft $\omega$-$\theta$-c.

(b) ⟶ (c): Let $H\in SS(W,B)$. Then, by (b),

$C{l}_{{\theta }_{\omega }}^{ϝ}\left(H\right)\tilde{\subseteq }C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(f_{qv}\left(H\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(f_{qv}\left(H\right)\right)\right)$. Hence,

$f_{qv}\left(C{l}_{{\theta }_{\omega }}^{ϝ}\left(H\right)\right)\tilde{\subseteq }{f}_{qv}\left(f_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(f_{qv}\left(H\right)\right)\right)\right)\tilde{\subseteq }C{l}_{\theta }^{\Xi }\left(f_{qv}\left(H\right)\right)$.

(c) ⟶ (b): Let $R\in SS(N,D)$. Then, by (c),

$f_{qv}\left(C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}(R\right)\right))\tilde{\subseteq }C{l}_{\theta }^{\Xi }\left(f_{qv}\left(f_{qv}^{-1}(R\right)\right)\tilde{\subseteq }C{l}_{\theta }^{\Xi }\left(R\right)$. Hence,

$C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}(R\right))\tilde{\subseteq }{f}_{qv}^{-1}\left(f_{qv}\left(C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}(R\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(R\right)\right)$. □

Definition 9.

Let $\left(W,ϝ,B\right)$ be a STS and let $h\in SS(W,B)$. The ${\theta }_{\omega }$-interior of H in $\left(W,ϝ,B\right)$ is denoted by $In{t}_{{\theta }_{\omega }}^{ϝ}\left(b\right)$ and defined as follows:

$$In{t}_{{\theta }_{\omega }}^{ϝ}\left(b\right)=\tilde{\cup }\left\{K\in ϝ:K\tilde{\subseteq }C{l}_{{ϝ}_{\omega }}\left(K\right)\tilde{\subseteq }H\right\}.$$

Theorem 12.

For a soft mapping  $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$, the following are equivalent:

(a) $f_{qv}$ is soft ω-θ-c;

(b) $f_{qv}^{-1}\left(G\right)\tilde{\subseteq }In{t}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)\right)$ for any $G\in \Xi$;

(c) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(G\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$ for any $G\in \Xi$.

Proof. 

(a) ⟶ (b): Let $G\in \Xi$ and let $b_w\tilde{\in }{f}_{qv}^{-1}\left(G\right)$. Then, $f_{qv}\left(b_w\right)\tilde{\in }G$ and by (a), we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. So, we have $b_w\tilde{\in }K\tilde{\subseteq }C{l}_{{ϝ}_{\omega }}$$\left(K\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$. Hence, $b_w\tilde{\in }In{t}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)\right)$.

(b) ⟶ (c): Let $G\in \Xi$. We will show that $1_B-f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)\tilde{\subseteq }{1}_B-C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(G\right)\right)$. Let $b_w$$\tilde{\in }{1}_B-f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)=f_{qv}^{-1}\left(1_D-C{l}_{\Xi }\left(G\right)\right)$. Then, $f_{qv}\left(b_w\right)\tilde{\in }{1}_D-C{l}_{\Xi }\left(G\right)$ and so, we find $H\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }H$ and $H\tilde{\cap }G=0_D$. Thus, $C{l}_{\Xi }\left(H\right)\tilde{\cap }G=0_D$ and hence $f_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)\tilde{\cap }{f}_{qv}^{-1}\left(G\right)=0_B$. Since $b_w\tilde{\in }{f}_{qv}^{-1}\left(H\right)$, then by (b), $b_w\tilde{\in }In{t}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)\right)$. So, we find $K\in ϝ$ such that $b_w\tilde{\in }K\tilde{\subseteq }C{l}_{{ϝ}_{\omega }}\left(K\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)$. Therefore, we have $C{l}_{{ϝ}_{\omega }}\left(K\right)\tilde{\cap }{f}_{qv}^{-1}\left(G\right)=0_B$ and hence, $b_w\tilde{\in }{1}_B-C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(G\right)\right)$.

(c) ⟶ (a): Let $G\in \Xi$ and let $b_w$$\in SP(W,B)$ such that $b_w\tilde{\in }{f}_{qv}^{-1}\left(G\right)$. Then, $f_{qv}\left(b_w\right)\tilde{\in }G$. Thus, $G\tilde{\cap }\left(1_D-C{l}_{\Xi }\left(G\right)\right)=0_D$ and $f_{qv}\left(b_w\right)\tilde{\notin }C{l}_{\Xi }\left(1_D-C{l}_{\Xi }\left(G\right)\right)$. So, $b_w\tilde{\notin }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(1_D-C{l}_{\Xi }\right.\right.$$\left.\left.\left(G\right)\right)\right)$ and by (c), $b_w\tilde{\notin }C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}(1_D-C{l}_{\Xi }\left(G\right))\right)$. Thus, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $C{l}_{{ϝ}_{\omega }}\left(K\right)\tilde{\cap }{f}_{qv}^{-1}(1_D-C{l}_{\Xi }\left(G\right))=0_B$. Therefore, we have $f_{qv}\left(C{l}_{{ϝ}_{\omega }}\left(K\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. □

Theorem 13.

For a soft mapping  $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$, the following are equivalent:

(a) $f_{qv}$ is soft ω-θ-c;

(b) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\theta }^{\Xi }\left(H\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)$ for any $H\in SS(N,D)$;

(c) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$ for any $G\in \Xi$;

(d) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(S\right))\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(S\right)$ for any $S\in {\Xi }^c$;

(e) $C{l}_{{\theta }_{\omega }}^{\Xi }\left(f_{qv}^{-1}\left(In{t}_{\Xi }(R\right))\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(R\right)$ for any $R\in RC\left(N,\Xi ,D\right)$.

Proof. 

(a) ⟶ (b): Let $H\in SS(N,D)$. Then, $In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)$$\in \Xi$ and by Theorem 12 (c), $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }$

$f_{qv}^{-1}\left(C{l}_{\Xi }\left(\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)\right)\right)=f_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)$.

(b) ⟶ (c): Let $G\in \Xi$. Then, $C{l}_{\Xi }\left(G\right)=C{l}_{\theta }^{\Xi }\left(G\right)$. Thus, by (b) we get the result.

(c) ⟶ (d): Let $S\in {\Xi }^c$. Then, $In{t}_{\Xi }\left(S\right)=In{t}_{\Xi }\left(C{l}_{\Xi }\left(In{t}_{\Xi }\left(S\right)\right)\right)$. So, by (c),

$C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(S\right))\right)=C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}(In{t}_{\Xi }\left(C{l}_{\Xi }\right(In{t}_{\Xi }\left(S\right)\right)\left)\right))\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(In{t}_{\Xi }\left(S\right)\right)\right)$

$\tilde{\subseteq }{f}_{qv}^{-1}\left(S\right)$.

(d) ⟶ (e): obvious.

(e) ⟶ (a): Let $K\in \Xi$. Then, $C{l}_{\Xi }\left(K\right)\in RC\left(N,\Xi ,D\right)$. So, by (e), $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(K\right)\right)\tilde{\subseteq }C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\right.$$\left.\left(In{t}_{\Xi }(C{l}_{\Xi }\left(K\right)\right))\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(K\right)\right)$. Therefore, by Theorem 12, it follows that $f_{qv}$ is soft $\omega$-$\theta$-c. □

Theorem 14.

For a soft mapping  $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$, the following are equivalent:

(a) $f_{qv}$ is soft ω-θ-c;

(b) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$ for any $G\in \beta O\left(N,\Xi ,D\right)$;

(c) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$ for any $G\in SO\left(N,\Xi ,D\right)$.

Proof. 

(a) ⟶ (b): Let $G\in \beta O\left(N,\Xi ,D\right)$. Then, $G\tilde{\subseteq }C{l}_{\Xi }(In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)$. So, $C{l}_{\Xi }\left(G\right)\tilde{\subseteq }C{l}_{\Xi }\left(C{l}_{\Xi }(In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)\right)=C{l}_{\Xi }(In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)$ and $C{l}_{\Xi }(In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)\tilde{\subseteq }C{l}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)=C{l}_{\Xi }\left(G\right)$. Therefore, $C{l}_{\Xi }\left(G\right)=C{l}_{\Xi }(In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)$. Thus, $C{l}_{\Xi }\left(G\right)\in RC\left(N,\Xi ,D\right)$. Hence, by (a) and Theorem 13 (e), $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\Xi }\left(G\right)\right))\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$.

(b) ⟶ (c): follows because $SO\left(N,\Xi ,D\right)\subseteq \beta O\left(N,\Xi ,D\right)$.

(c) ⟶ (a): Let $G\in \Xi$. Since $\Xi \subseteq SO\left(N,\Xi ,D\right)$, then $G\in SO\left(N,\Xi ,D\right)$. So, by (c), $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$. Therefore, by Theorem 13 (c), $f_{qv}$ is soft $\omega$-$\theta$-c. □

Lemma 3.

Let $\left(W,ϝ,B\right)$ be a STS and let $K\in SS(W,B)$. Then, $1_B-In{t}_{{\theta }_{\omega }}^{ϝ}\left(K\right)=C{l}_{{\theta }_{\omega }}^{ϝ}(1_B-K)$.

Proof. 

To see that $1_B-In{t}_{{\theta }_{\omega }}^{ϝ}\left(K\right)\tilde{\subseteq }C{l}_{{\theta }_{\omega }}^{ϝ}(1_B-K)$, let $b_w\tilde{\notin }C{l}_{{\theta }_{\omega }}^{ϝ}(1_B-K)$. Then, we find $T\in ϝ$ such that $b_w\tilde{\in }T$ and $C{l}_{{ϝ}_{\omega }}\left(T\right)\tilde{\cap }\left(1_B-K\right)=0_B$. Thus, $C{l}_{{ϝ}_{\omega }}\left(T\right)\tilde{\subseteq }K$ and hence, $b_w\tilde{\in }T\tilde{\subseteq }In{t}_{{\theta }_{\omega }}^{ϝ}\left(K\right)$. Therefore, $b_w\tilde{\notin }\left(1_B-In{t}_{{\theta }_{\omega }}^{ϝ}\left(K\right)\right)$. To see that $C{l}_{{\theta }_{\omega }}^{ϝ}(1_B-K)\tilde{\subseteq }{1}_B-In{t}_{{\theta }_{\omega }}^{ϝ}\left(K\right)$, let $b_w\tilde{\notin }\left(1_B-In{t}_{{\theta }_{\omega }}^{ϝ}\right.$$\left.\left(K\right)\right)$. Then, $b_w\tilde{\in }In{t}_{{\theta }_{\omega }}^{ϝ}\left(K\right)$. Thus, we find $T\in ϝ$ such that $b_w\tilde{\in }T\tilde{\subseteq }C{l}_{{ϝ}_{\omega }}\left(T\right)\tilde{\subseteq }K$ and so, $C{l}_{{ϝ}_{\omega }}\left(T\right)\tilde{\cap }\left(1_B-K\right)=0_B$. This implies that $b_w\tilde{\notin }C{l}_{{\theta }_{\omega }}^{ϝ}(1_B-K)$. □

Theorem 15.

For a soft mapping  $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$, the following are equivalent:

(a) $f_{qv}$ is soft ω-θ-c;

(b) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$ for any $G\in PO\left(N,\Xi ,D\right)$;

(c) $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(G\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$ for any $G\in PO\left(N,\Xi ,D\right)$;

(d) $f_{qv}^{-1}\left(G\right)\tilde{\subseteq }In{t}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)\right)$ for any $G\in PO\left(N,\Xi ,D\right)$.

Proof. 

(a) ⟶ (b): Let $G\in PO\left(N,\Xi ,D\right)$. Since $PO\left(N,\Xi ,D\right)\subseteq \beta O\left(N,\Xi ,D\right)$, then $G\in \beta O\left(N,\Xi ,D\right)$. Thus, by Theorem 14 (b),

$C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$.

(b) ⟶ (c): Let $G\in PO\left(N,\Xi ,D\right)$. Then, $G\tilde{\subseteq }In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)$, and so $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(G\right)\right)\tilde{\subseteq }C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right)\right)\right)$. Since by (b),

$C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }(C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$, then $C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(G\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)$.

(c) ⟶ (d): Let $G\in PO\left(N,\Xi ,D\right)$. We will show that $1_B-In{t}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)\right)$

$\tilde{\subseteq }{1}_B-f_{qv}^{-1}\left(G\right)$. By (c) and Lemma 3, we have $1_B-In{t}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)\right)=C{l}_{{\theta }_{\omega }}^{ϝ}(1_B-f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right))=C{l}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(1_D-C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }(1_D-C{l}_{\Xi }\left(G\right))\right)$

$=f_{qv}^{-1}(1_D-In{t}_{\Xi }\left(C{l}_{\Xi }\left(G\right)\right))\tilde{\subseteq }{f}_{qv}^{-1}(1_D-G)=1_B-f_{qv}^{-1}\left(G\right)$.

(d) ⟶ (a): We will apply Theorem 12 (b). Let $G\in \Xi$. Then, $G\in PO\left(N,\Xi ,D\right)$. So, by (d), $f_{qv}^{-1}\left(G\right)\tilde{\subseteq }In{t}_{{\theta }_{\omega }}^{ϝ}\left(f_{qv}^{-1}\left(C{l}_{\Xi }\left(G\right)\right)\right)$. □

4. Weak-${\mathit{\theta }}_{\mathit{\omega }}$-Continuity

In this section, we define soft weakly ${\theta }_{\omega }$-continuous mappings as a new class of soft mappings and investigate some of its properties. We give several characterizations of it. With the help of examples, we explain its connections with several types of soft continuity such as soft continuous, soft weakly continuous, and soft ${\theta }_{\omega }$-continuous mappings. Moreover, we investigate the links between this class of soft mappings and its analogs in general topology.

Definition 10.

A soft mapping $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is called soft weakly ${\theta }_{\omega }$-continuous (w-${\theta }_{\omega }$-c, for simplicity) if for any $b_w$$\in SP(W,B)$ and any $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$.

Theorem 16.

Let  $\left\{\left(W,{ϝ}_b\right):b\in B\right\}$ and $\left\{\left(N,{\Xi }_d\right):d\in D\right\}$ be two families of TSs. Let $q:M⟶N$ be a mapping and $v:B⟶D$ be an injective mapping. Then, $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft w-${\theta }_{\omega }$-c if and only if $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-${\theta }_{\omega }$-c for each $b\in B$.

Proof. 

Necessity. Suppose that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft w-${\theta }_{\omega }$-c. Let $b\in B$. Let $w\in W$ and let $V\in {\Xi }_{v\left(b\right)}$ such that $q\left(w\right)\in V$. Then we have $f_{qv}\left(b_w\right)={\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }{\left(v\left(b\right)\right)}_V\in {\oplus }_{d\in D}{\Xi }_d$. Thus, we find $K\in {\oplus }_{b\in B}{ϝ}_b$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\left({\oplus }_{d\in D}{\Xi }_d\right)}_{\omega }}\left({\left(v\left(b\right)\right)}_V\right)$, and so $\left(f_{qv}\left(K\right)\right)\left(v\left(b\right)\right)\subseteq \left(C{l}_{{\left({\oplus }_{d\in D}{\Xi }_d\right)}_{\omega }}\left({\left(v\left(b\right)\right)}_V\right)\right)\left(v\left(b\right)\right)$. By Theorem 8 of [22], ${\left({\oplus }_{d\in D}{\Xi }_d\right)}_{\omega }={\oplus }_{d\in D}{\left({\Xi }_d\right)}_{\omega }$. Since $v:B⟶D$ is injective, then by Lemma 2, $\left(C{l}_{{\left({\oplus }_{d\in D}{\Xi }_d\right)}_{\omega }}\left({\left(v\left(b\right)\right)}_V\right)\right)\left(v\left(b\right)\right)=C{l}_{{\Xi }_{v\left(b\right)}}(\left({\left(v\left(b\right)\right)}_V\right)\left(v\left(b\right)\right)=C{l}_{{\left({\Xi }_{v\left(b\right)}\right)}_{\omega }}\left(V\right)$. Therefore, we have $w\in K\left(b\right)\in {ϝ}_b$ and $q\left(K\left(b\right)\right)\subseteq C{l}_{{\left({\Xi }_{v\left(b\right)}\right)}_{\omega }}\left(V\right)$. It follows that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-${\theta }_{\omega }$-c. □

Sufficiency. Suppose that $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-${\theta }_{\omega }$-c for each $b\in B$. Let $b_w\in SP(W,B)$ and let $G\in {\oplus }_{d\in D}{\Xi }_d$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, we have ${\left(v\left(b\right)\right)}_{q\left(w\right)}\tilde{\in }G$ and $q\left(w\right)\in G\left(v\left(b\right)\right)\in {\Xi }_{v\left(b\right)}$. Since $q:\left(W,{ϝ}_b\right)⟶\left(N,{\Xi }_{v\left(b\right)}\right)$ is w-${\theta }_{\omega }$-c, then we find $X\in {ϝ}_b$ such that $w\in X$ and $q\left(X\right)\subseteq C{l}_{{\left({\Xi }_{v\left(b\right)}\right)}_{\omega }}\left(G\left(v\left(b\right)\right)\right)$. Now, we have $b_w\tilde{\in }{b}_X\in {\oplus }_{b\in B}{ϝ}_b$. Also, since $v:B⟶D$ is injective, then, by Lemmas 1 and 2,we have $\left(f_{qv}\left(b_X\right)\right)\left(v\left(b\right)\right)=q\left(X\right)$ and $\left(f_{qv}\left(\left(b_X\right)\right)\right)\left(s\right)=\left(C{l}_{{\oplus }_{d\in D}{\Xi }_d}\left(G\right)\right)\left(s\right)=\varnothing$ for all $s\ne v\left(b\right)$. Moreover, by Lemma 2, $\left(C{l}_{{\left({\oplus }_{d\in D}{\Xi }_d\right)}_{\omega }}\left(G\right)\right)\left(v\left(b\right)\right)=C{l}_{{\left({\Xi }_{v\left(b\right)}\right)}_{\omega }}\left(G\left(v\left(b\right)\right)\right)$, and $\left(C{l}_{{\left({\oplus }_{d\in D}{\Xi }_d\right)}_{\omega }}\right.$$\left.\left(G\right)\right)\left(s\right)=\varnothing$ for all $s\ne v\left(b\right)$. Hence, $f_{qv}\left(b_X\right)\tilde{\subseteq }C{l}_{{\left({\oplus }_{d\in D}{\Xi }_d\right)}_{\omega }}\left(G\right)$. It follows that $f_{qv}:\left(W,{\oplus }_{b\in B}{ϝ}_b,B\right)⟶$$\left(N,{\oplus }_{d\in D}{\Xi }_d,D\right)$ is soft w-${\theta }_{\omega }$-c.

Corollary 6.

Let $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ be a mapping between two TSs and let $v:B⟶D$ be an injective mapping. Then, $q:\left(W,\eta \right)⟶\left(N,\xi \right)$ is w-${\theta }_{\omega }$-c if and only if $f_{qv}:(M,\tau \left(\eta \right),B)⟶(N,\tau \left(\xi \right),D)$ is soft w-${\theta }_{\omega }$-c.

Proof. 

For any $b\in B$ and $d\in D$, put ${ϝ}_b=\eta$ and ${\Xi }_d=\xi$. Then, $\tau \left(\eta \right)={\oplus }_{b\in B}{ϝ}_b$ and $\tau \left(\xi \right)={\oplus }_{d\in D}{\Xi }_d$. Thus, by Theorem 16, we get the result. □

Theorem 17.

Every soft w-${\theta }_{\omega }$-c mapping is soft w-c.

Proof. 

Let $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ be soft w-${\theta }_{\omega }$-c. Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft w-${\theta }_{\omega }$-c, then we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Therefore, $f_{qv}$ is soft w-c. □

Theorem 17’s converse is not often true, as shown by the example below.

Example 3.

Let  $W=\mathbb{Z}$, $\eta =\left\{\varnothing ,W\right\}$, $\xi =\left\{\varnothing ,W,\left\{2\right\}\right\}$, and $B=\left[0,1\right]$. Consider the identity mappings $q:W⟶W$ and $v:B⟶B$.

Claim 1.

$q:\left(W,\eta \right)⟶\left(M,\xi \right)$ is w-c.

Claim 2.

$q:\left(W,\eta \right)⟶\left(M,\xi \right)$ is not w-${\theta }_{\omega }$-c.

Proof. 

Proof of Claim 1. Let $w\in W$ and $V\in \xi$ such that $q\left(w\right)=w\in V$. Then, $V=W$ or $V=\left\{2\right\}$, and in both cases $C{l}_{\xi }\left(V\right)=W$. Choose $U=W$. Then, $w\in U\in \eta$ and $q\left(w\right)=W\subseteq W=C{l}_{\xi }\left(V\right)$.

2. Suppose to the contrary that $q:\left(W,\eta \right)⟶\left(W,\xi \right)$ is w-${\theta }_{\omega }$-c. Let $w=2$ and take $V=\left\{2\right\}$. Then, we find $U\in \eta$ such that $2\in U$ and $q\left(U\right)=U\subseteq C{l}_{{\xi }_{\omega }}\left(\left\{2\right\}\right)=\left\{2\right\}$. But $U=W$, a contradiction.

Therefore, by Corollaries 2 and 6, $f_{qv}:\left(W,\tau \left(\eta \right),B\right)⟶\left(N,\tau \left(\xi \right),B\right)$ is soft w-c but not soft w-${\theta }_{\omega }$-c. □

Theorem 18.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-c such that $\left(N,\Xi ,D\right)$ is soft anti-l-c, then $f_{qv}$ is soft w-${\theta }_{\omega }$-c.

Proof. 

Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft w-c, then we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Since $\left(N,\Xi ,D\right)$ is soft anti-l-c, then, by Theorem 14 of [22], $C{l}_{{\Xi }_{\omega }}\left(G\right)=C{l}_{\Xi }\left(G\right)$. Therefore, $f_{qv}$ is soft w-${\theta }_{\omega }$-c. □

Theorem 19.

Every soft continuous mapping is soft w-${\theta }_{\omega }$-c.

Proof. 

Let $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ be soft continuous. Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft continuous, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }G\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. Hence, $f_{qv}$ is soft w-${\theta }_{\omega }$-c. □

The converse of Theorem 19 is not always true, as shown by the following example.

Example 4.

Let  $W=\mathbb{R}$, η be the usual topology on W, $\xi$ be the co-countable topology on W, and $B=\left[0,1\right]$. Consider the identity mappings $q:W⟶W$ and $v:B⟶B$. Then, clearly $q:\left(W,\eta \right)⟶\left(W,\xi \right)$ is not continuous. To show that $q:\left(W,\eta \right)⟶\left(W,\xi \right)$ is w-${\theta }_{\omega }$-c, let $w\in W$ and $V\in \xi$ such that $q\left(w\right)=w\in V$. Since $\left(W,\xi \right)$ is anti-l-c, then $C{l}_{{\xi }_{\omega }}\left(V\right)=C{l}_{\xi }\left(V\right)=W$. Choose $U=W$. Then, $w\in U\in \eta$ and $q\left(U\right)=U\subseteq C{l}_{{\xi }_{\omega }}\left(V\right)$. Therefore, by Theorem 5.31 of [26] and Corollary 6, $f_{qv}:\left(W,\tau \left(\eta \right),B\right)⟶\left(N,\tau \left(\xi \right),B\right)$ is not soft continuous and soft w-${\theta }_{\omega }$-c.

Theorem 20.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-${\theta }_{\omega }$-c such that $\left(N,\Xi ,D\right)$ is soft ω-l-i, then $f_{qv}$ is soft continuous.

Proof. 

Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft w-${\theta }_{\omega }$-c, then we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$. Since $\left(N,\Xi ,D\right)$ is soft $\omega$-l-i, then G is soft $\omega$-closed and $C{l}_{{\Xi }_{\omega }}\left(G\right)=G$. Thus, $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)=G$. Hence, $f_{qv}$ is soft continuous. □

Corollary 7.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-${\theta }_{\omega }$-c such that $\left(N,\Xi ,D\right)$ is soft l-i, then $f_{qv}$ is soft continuous.

Proof. 

Follows from Theorem 7 of [35] and Theorem 4.9. □

Corollary 8.

If $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-${\theta }_{\omega }$-c such that $\left(N,\Xi ,D\right)$ is soft l-c, then $f_{qv}$ is soft continuous.

Proof. 

Follows from Theorem 6 of [35] and Theorem 4.9. □

Theorem 21.

Every soft ${\theta }_{\omega }$-c mapping is soft w-${\theta }_{\omega }$-c.

Proof. 

Let $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ be soft ${\theta }_{\omega }$-c. Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Since $f_{qv}$ is soft ${\theta }_{\omega }$-continuous, then we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(C{l}_{ϝ}\left(K\right)\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$. Thus, $f_{qv}\left(K\right)\tilde{\subseteq }{f}_{qv}\left(C{l}_{ϝ}\left(K\right)\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$. Hence, $f_{qv}$ is soft w-${\theta }_{\omega }$-c. □

The opposite of Theorem 21 is not necessarily true, as shown by the following example.

Example 5.

Take $f_{qv}$ as in Example 3.10 of [32]. Since $f_{qv}$ is soft continuous, then, by Theorem 19, it is soft w-${\theta }_{\omega }$-c. On the other hand, it is proved in Example 3.10 of [32] that $f_{qv}$ is not soft ${\theta }_{\omega }$-c.

From Theorems 17, 19, and 21 we have following implications; however, Examples 3–5 show that the converses of these implications are not true.

$$\begin{array}{ccccc}& & \mathrm{soft}{\theta }_{\omega }-\mathrm{c}& & \\ & & \downarrow & & \\ \mathrm{soft}\mathrm{continuity}& ⟶& \mathrm{soft}\mathrm{w}-{\theta }_{\omega }-\mathrm{c}& ⟶& \mathrm{soft}\mathrm{w}-\mathrm{c}\end{array}$$

Theorem 22.

A soft mapping $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$ is soft w-${\theta }_{\omega }$-c if and only if for any $G\in \Xi$, $f_{qv}^{-1}\left(G\right)\tilde{\subseteq }In{t}_{ϝ}\left(f_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$.

Proof. 

Necessity. Suppose that $f_{qv}$ is soft w-${\theta }_{\omega }$-c. Let $G\in \Xi$ and $b_w\tilde{\in }{f}_{qv}^{-1}\left(G\right)$. Then, $f_{qv}\left(b_w\right)\tilde{\in }G$ and we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$. Hence,

$b_w\tilde{\in }K$$\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)$ and $b_w\tilde{\in }In{t}_{ϝ}\left(f_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$.

Sufficiency. Suppose that for any $G\in \Xi$, $f_{qv}^{-1}\left(G\right)\tilde{\subseteq }In{t}_{ϝ}\left(f_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$. Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, by assumption, we have $b_w\tilde{\in }{f}_{qv}^{-1}\left(G\right)\tilde{\subseteq }In{t}_{ϝ}\left(f_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$. Let $K=In{t}_{ϝ}\left(f_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$. Then, $K\in ϝ$ and $f_{qv}\left(K\right)=f_{qv}\left(In{t}_{ϝ}\left(f_{qv}^{-1}\right.\right.$$\left.\left.\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}\left(f_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$ $\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$. □

Theorem 23.

For a soft mapping  $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$, the following are equivalent:

(a) $f_{qv}$ is soft w-c;

(b) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(S\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(S\right)\right)$ for any $S\in SS(N,D)$;

(c) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)$ for any $H\in \Xi$;

(d) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(R\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(R\right)$ for any $R\in RC\left(N,\Xi ,D\right)$;

(e) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(H\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)$ for any $H\in \Xi$.

Proof. 

(a) ⟶ (b): Let $S\in SS(N,D)$ and let $b_w\tilde{\in }{1}_B-f_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(S\right)\right)$. Then, $f_{qv}\left(b_w\right)\tilde{\notin }C{l}_{\theta }^{\Xi }\left(S\right)$ and so, we find $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$ and $C{l}_{\Xi }\left(G\right)\tilde{\cap }S=0_D$. Thus, $G\tilde{\cap }C{l}_{\theta }^{\Xi }\left(S\right)=0_D$ and hence $G\tilde{\subseteq }{1}_D-C{l}_{\theta }^{\Xi }\left(S\right)$. On the other hand, by (a), we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)\tilde{\subseteq }$ $C{l}_{\Xi }(1_D-C{l}_{\theta }^{\Xi }\left(S\right))=1_D-In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(S\right)\right)$. This gives that $In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(S\right)\right)\tilde{\cap }{f}_{qv}\left(K\right)=0_D$, and hence $f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(S\right)\right)\right)\tilde{\cap }K=0_B$. This implies that $b_w\tilde{\in }{1}_B-C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(S\right)\right)\right)\right)$.

(b) ⟶ (c): Let $H\in \Xi$. Then, by (b),

$C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\theta }^{\Xi }\left(H\right)\right)$. Since $H\in \Xi$, then $C{l}_{\theta }^{\Xi }\left(H\right)=C{l}_{\Xi }\left(H\right)$ and so $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)$.

(c) ⟶ (d): Let $R\in RC\left(N,\Xi ,D\right)$. Then, $In{t}_{\Xi }\left(R\right)\in \Xi$ and by (c), $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(R\right)\right)\right)=C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\Xi }\left(In{t}_{\Xi }\left(R\right)\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(In{t}_{\Xi }\left(R\right)\right)\right)=f_{qv}^{-1}\left(R\right)$.

(d) ⟶ (e): Let $H\in \Xi$. Then $C{l}_{\Xi }\left(H\right)\in RC\left(N,\Xi ,D\right)$. So, by (d), $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)$. Since $H\in \Xi$, then $H\tilde{\subseteq }In{t}_{\Xi }\left(C{l}_{\Xi }\left(H\right)\right)$ and so $C{l}_{ϝ}\left(f_{qv}^{-1}\left(H\right)\right)\tilde{\subseteq }C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\right.\right.$$\left.\left.\left(C{l}_{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(H\right)\right)$.

(e) ⟶ (a): Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, $f_{qv}\left(b_w\right)\tilde{\notin }C{l}_{\Xi }\left(1_D\right.$$\left.-C{l}_{\Xi }\left(G\right)\right)$ and $b_w\tilde{\notin }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(1_D-C{l}_{\Xi }\left(G\right)\right)\right)$. Since $1_D-C{l}_{\Xi }\left(G\right)\in \Xi$, then, by (e), $C{l}_{ϝ}\left(f_{qv}^{-1}\right.$$\left.\left(1_D-C{l}_{\Xi }\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{\Xi }\left(1_D-C{l}_{\Xi }\left(G\right)\right)\right)$. Hence, $b_w\tilde{\notin }C{l}_{ϝ}\left(f_{qv}^{-1}\left(1_D-C{l}_{\Xi }\left(G\right)\right)\right)$. So, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $K\tilde{\cap }{f}_{qv}^{-1}\left(1_D-C{l}_{\Xi }\left(G\right)\right)=0_B$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{\Xi }\left(G\right)$. □

Theorem 24.

For a soft mapping  $f_{qv}:\left(W,ϝ,B\right)⟶\left(N,\Xi ,D\right)$, the following are equivalent:

(a) $f_{qv}$ is soft w-${\theta }_{\omega }$-c;

(b) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)\right)$ for any $S\in SS(N,D)$;

(c) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)$ for any $H\in \Xi$;

(d) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(R\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(R\right)$ for any $R\in RC\left(N,{\Xi }_{\omega }D\right)$;

(e) $C{l}_{ϝ}\left(f_{qv}^{-1}\left(H\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)$ for any $H\in {\Xi }_{\omega }$.

Proof. 

(a) ⟶ (b): Let $S\in SS(N,D)$ and let $b_w\tilde{\in }{1}_B-f_{qv}^{-1}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)\right)$. Then, $f_{qv}\left(b_w\right)\tilde{\notin }C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)$, and so, we find $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$ and $C{l}_{{\Xi }_{\omega }}\left(G\right)\tilde{\cap }S=0_D$. Thus, $G\tilde{\cap }C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)=0_D$ and hence $G\tilde{\subseteq }{1}_D-C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)$. On the other hand, by (a), we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $f_{qv}\left(K\right)\tilde{\subseteq }$ $C{l}_{{\Xi }_{\omega }}\left(G\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}(1_D-C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right))=1_D-In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)\right)$. This gives that $In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)\right)\tilde{\cap }{f}_{qv}\left(K\right)=0_D$, and hence $f_{qv}^{-1}\left(In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)\right)\right)\tilde{\cap }K=0_B$. This implies that $b_w\tilde{\in }{1}_B-C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(S\right)\right)\right)\right)$.

(b) ⟶ (c): Let $H\in \Xi$. Then, by (b), $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(H\right)\right)\right)\right)\tilde{\subseteq }$ $f_{qv}^{-1}\left(C{l}_{{\theta }_{\omega }}^{\Xi }\left(H\right)\right)$. Since $H\in \Xi$, then by Theorem 1 (e) of [32], $C{l}_{{\theta }_{\omega }}^{\Xi }\left(H\right)=C{l}_{{\Xi }_{\omega }}\left(H\right)$ and so $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\Xi }_{\omega }}\right.\right.\right.$$\left.\left.\left.\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)$.

(c) ⟶ (d): Let $R\in RC\left(N,{\Xi }_{\omega }D\right)$. Then, $In{t}_{\Xi }\left(R\right)\in \Xi$ and by (c), $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(R\right)\right)\right)=C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{{\Xi }_{\omega }}\left(In{t}_{{\Xi }_{\omega }}\left(R\right)\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(In{t}_{{\Xi }_{\omega }}\left(R\right)\right)\right)$ $=f_{qv}^{-1}\left(R\right)$.

(d) ⟶ (e): Let $H\in {\Xi }_{\omega }$. Then, $C{l}_{{\Xi }_{\omega }}\left(H\right)\in RC\left(N,{\Xi }_{\omega }D\right)$. So, by (d), $C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{\Xi }\left(C{l}_{{\Xi }_{\omega }}\right.\right.\right.$$\left.\left.\left.\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)$. Since $H\in {\Xi }_{\omega }$, then

$H\tilde{\subseteq }In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)$, and so

$C{l}_{ϝ}\left(f_{qv}^{-1}\left(H\right)\right)\tilde{\subseteq }C{l}_{ϝ}\left(f_{qv}^{-1}\left(In{t}_{{\Xi }_{\omega }}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(H\right)\right)$.

(e) ⟶ (a): Let $b_w$$\in SP(W,B)$ and let $G\in \Xi$ such that $f_{qv}\left(b_w\right)\tilde{\in }G$. Then, $f_{qv}\left(b_w\right)\tilde{\notin }C{l}_{{\Xi }_{\omega }}\left(1_D\right.$$\left.-C{l}_{{\Xi }_{\omega }}\left(G\right)\right)$ and $b_w\tilde{\notin }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(1_D-C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$. Since $1_D-C{l}_{{\Xi }_{\omega }}\left(G\right)\in {\Xi }_{\omega }$, then, by (e), $C{l}_{ϝ}\left(f_{qv}^{-1}\left(1_D-C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)\tilde{\subseteq }{f}_{qv}^{-1}\left(C{l}_{{\Xi }_{\omega }}\left(1_D-C{l}_{{\Xi }_{\omega }}\left(G\right)\right)\right)$. Hence, $b_w\tilde{\notin }C{l}_{ϝ}\left(f_{qv}^{-1}\left(1_D-C{l}_{{\Xi }_{\omega }}\right.\right.$$\left.\left.\left(G\right)\right)\right)$. So, we find $K\in ϝ$ such that $b_w\tilde{\in }K$ and $K\tilde{\cap }{f}_{qv}^{-1}\left(1_D-C{l}_{{\Xi }_{\omega }}\left(G\right)\right)=0_B$ and $f_{qv}\left(K\right)\tilde{\subseteq }C{l}_{{\Xi }_{\omega }}\left(G\right)$. □

5. Conclusions

A lot of facets of daily life are unpredictable. One of the concepts proposed to cope with uncertainty is the soft set theory. Soft topology, a unique mathematical framework constructed by topologists employing soft sets, is the topic of this study.

This study is in the category of soft topology. In soft topological spaces, the notions of $\omega$-$\theta$-continuity and soft weak ${\theta }_{\omega }$-continuity are introduced and their features are examined. In particular, the relationships between these classes of soft mappings with their analogs in general topology are examined (Theorems 3, 16 and Corollaries 1, 6). Also, the relationships between the soft weak continuity and soft $\theta$-continuity as two known classes of soft mappings with their analogs in general topology are examined (Theorems 4, 5 and Corollaries 2, 3). The soft $\theta$-closure and soft ${\theta }_{\omega }$-closure operators are employed to characterize soft $\omega$-$\theta$-continuity (Theorems 11–15) and soft weak ${\theta }_{\omega }$-continuity (Theorem 24). It is also shown that soft $\omega$-$\theta$-continuity is strictly between soft $\theta$-continuity and soft weak continuity, while soft weak ${\theta }_{\omega }$-continuity is strictly between soft continuity (i.e., soft $\theta$-continuity) and soft weak continuity. For instance, Theorem 6 and Example 1 show that soft $\omega$-$\theta$-continuity strictly implies soft weak continuity, Theorem 6 and Example 2 show that soft $\theta$-continuity strictly implies soft $\omega$-$\theta$-continuity, Theorem 17 and Example 3 show that soft weak ${\theta }_{\omega }$-continuity strictly implies soft weak continuity, Theorem 19 and Example 4 show that soft continuity strictly implies soft weak ${\theta }_{\omega }$-continuity, and Theorem 21 and Example 5 show that soft ${\theta }_{\omega }$-continuity strictly implies soft weak ${\theta }_{\omega }$-continuity.

The following topics could be considered in future studies: (1) to define soft faintly ${\theta }_{\omega }$-continuous; and (2) to define soft define several soft covering properties via soft ${\theta }_{\omega }$-open sets.