Three Weaker Forms of Soft Faint Continuity

Abstract

The authors of this paper introduce and discuss three weaker forms of soft faint continuity: soft faint semi-continuity, soft faint pre-continuity, and soft faint $\beta$-continuity. They characterize each of them in several ways. They also demonstrate how they are preserved under some restrictions. Moreover, they prove that a soft function is also soft faint semi-continuous (resp. soft faint pre-continuous, soft faint $\beta$-continuous) if its soft graph function is also soft faint semi-continuous (resp. soft faint pre-continuous, soft faint $\beta$-continuous). Moreover, they show that a soft function is soft faint semi-continuous (resp. soft faint pre-continuous, soft faint $\beta$-continuous) iff it is soft semi-continuous provided that it has a soft regular codomain. Finally, the symmetry between our new ideas and their analogous topological ones is investigated.

1. Introduction and Preliminaries

It is a common problem for scientists in many fields, such as economics, systems engineering, medicine, artificial intelligence, and others, to build complicated systems that involve uncertainty. Though widely used, “mathematical” approaches for handling such circumstances, conventional probability, fuzzy set [1], and rough set [2] theories, are unable to, due to parameter constraints, consistently generate satisfactory answers. Molodtsov [3] looked into soft set theory, a novel method for handling uncertainty in a way that surpasses the limitations of previous methods. The method uses soft sets, which lead to a universal set’s parameterized collection of subsets. Soft set theory, in contrast to earlier methods, does not impose any specific limitations on object lighting, and parameters can be selected in a number of ways, such as words, phrases, integers, and mappings. Because of this, the theory is highly flexible and simple to use in practical settings. Molodtsov demonstrated the versatility and wide applicability of soft set theory by extending it to a range of subjects, such as probability studies and game theory. Other scholars provide a variety of practical uses for soft sets and their expansions (see [4,5,6,7,8,9,10]).

General topology is a well-known and significant area of mathematics that deals with the application of set theory concepts and topological structures. Soft topology is a new topic of topology study established in 2011 [11], combining soft sets and topology concepts. The main concepts they introduced were soft open sets, soft neighborhoods, soft closure of a soft set, soft separation axioms, and soft regular and soft normal spaces. A few examples of classical topological ideas that have been extended and developed in soft set settings are presented in [12,13,14,15,16,17,18,19,20,21,22,23,24]. The notion of soft mapping with details was first presented in [25], and then, soft continuity for soft mappings was defined in [26]. Since then, researchers have focused on the study of soft continuity concepts (see [27,28,29,30,31,32]).

“Soft topology” is a useful extension of classical topology. There are several advantages of soft topology over standard topology, including the following: (i) Open set identification accuracy is increased by the soft topological structure. In contrast to soft topology, which permits the inclusion of intermediate degrees of openness, general topology works with binary ideas. This helps to provide more accurate information about the topological space’s characteristics. (ii) Classical topology is one instance of the broad area of soft topology. Since soft topology relaxes the rigid restrictions of classical topology, it broadens the range of topological structures and is therefore a valuable tool for comprehending a greater variety of complicated and varied settings. (iii) Since soft topology is the most efficient means of expressing uncertainty and imprecision, researchers have used it in computer science, image processing, fuzzy logic, and decision-making. (iv) When dealing with ambiguous or unclear data, soft topology is a useful mathematical modeling tool. It works especially well for encoding and assessing imprecise data, which is frequently encountered in real-world applications.

A lot of study has been done on soft continuity in soft topology and other math fields. Soft continuity is widely used in many fields, such as science, engineering, business, and economics. This encouraged us to write this work.

This work introduces and investigates three weak types of soft faint continuity: soft faint semi-continuity, soft faint pre-continuity, and soft faint $\beta$-continuity.

Let $\mathcal{L}$ be a set of parameters and let O be a non-empty set. A soft set over O relative to $\mathcal{L}$ is a function $G:O⟶\mathcal{P}\left(\mathcal{L}\right)$. $SS(O,\mathcal{L})$ represents the collection of all soft sets over O relative to $\mathcal{L}$. If $K\in SS(O,\mathcal{L})$ such that $K\left(b\right)=X$ for any $b\in \mathcal{L}$ (resp. $K\left(d\right)=X$ and $K\left(b\right)=\varnothing$ for each $b\in \mathcal{L}-\left\{d\right\}$), then K is represented by $C_X$ (resp. $d_X$). $1_{\mathcal{L}}$ and $0_{\mathcal{L}}$ will denote $C_O$ and $C_{\varnothing }$, respectively. If $K\in SS(O,\mathcal{L})$, then K is a soft point over O relative to $\mathcal{L}$ and represented as $d_y$ if $K\left(d\right)=\left\{y\right\}$ and $K\left(b\right)=\varnothing$ for any $b\in L-\left\{d\right\}$. $SP(O,L)$ represents the collection of all soft points over O with respect to $\mathcal{L}$. If $d_y\in SP\left(O,\mathcal{L}\right)$ and $K\in SS(O,\mathcal{L})$, then $d_y$ is said to belong to K (notation: $d_y\tilde{\in }K$) if $y\in K\left(d\right)$. Let $K,H\in SS(O,\mathcal{L})$. Then, K is a soft subset of H, denoted by $K\tilde{\subseteq }H$, if $K\left(a\right)\subseteq H\left(a\right)$ for each $a\in L$. The soft union (resp. intersection, difference) of K and H is denoted by $K\tilde{\cup }H$ (resp. $K\tilde{\cap }H$, $K-H$) and defined by $\left(K\tilde{\cup }H\right)\left(a\right)=K\left(a\right)\cup H\left(a\right)$ (resp. $\left(K\tilde{\cap }H\right)\left(a\right)=K\left(a\right)\cap H\left(a\right)$, $\left(K-H\right)\left(a\right)=K\left(a\right)-H\left(a\right)$) for each $a\in L$. For any sub-collection $\mathcal{K}\subseteq SS(O,\mathcal{L})$, the soft union (resp. intersection) of the members of $\mathcal{K}$ is denoted by ${\tilde{\cup }}_{K\in \mathcal{K}}K$ (resp. ${\tilde{\cap }}_{K\in \mathcal{K}}K$) and defined by $\left({\tilde{\cup }}_{K\in \mathcal{K}}K\right)\left(a\right)={\cup }_{K\in \mathcal{K}}K\left(a\right)$ (resp. ${\cap }_{K\in \mathcal{K}}K\left(a\right)$) for each $a\in L$. Let $SS\left(O,\mathcal{L}\right)$ and $SS\left(T,\mathcal{S}\right)$ be two families of soft sets, and $r:O⟶T$, $w:\mathcal{L}⟶S$ be two functions. Then, a soft mapping $f_{rw}:SS\left(O,\mathcal{L}\right)⟶SS\left(T,\mathcal{S}\right)$ is defined as follows: for each $H\in SS\left(O,\mathcal{L}\right)$ and $K\in SS\left(T,\mathcal{S}\right)$, $\left(f_{rw}\left(H\right)\right)\left(b\right)=\varnothing$ if $w^{-1}\left(b\right)=\varnothing$, $\left(f_{rw}\left(H\right)\right)\left(b\right)={\cup }_{a\in w^{-1}\left(b\right)}r\left(H\left(a\right)\right)$ if $w^{-1}\left(b\right)\ne \varnothing$, and $\left(f_{rw}^{-1}\left(K\right)\right)\left(a\right)=r^{-1}\left(K\left(w\left(a\right)\right)\right)$. A sub-collection $\sigma \subseteq SS\left(O,\mathcal{L}\right)$ is called a soft topology on O relative to $\mathcal{L}$, and the triplet $(O,\sigma ,\mathcal{L})$ is called a soft topological space if $\left\{0_{\mathcal{L}},1_{\mathcal{L}}\right\}\subseteq \sigma$, $K\tilde{\cap }H\in \sigma$ for any $\left\{K,H\right\}\subseteq \sigma$, and ${\tilde{\cup }}_{K\in \mathcal{K}}K$ for any $\mathcal{K}\subseteq \sigma$. Let $(O,\sigma ,\mathcal{L})$ be a soft topological space and let $K\in SS(O,\mathcal{L})$. Then, K is called a soft open set in $(O,\sigma ,\mathcal{L})$ if $K\in \sigma$ and K is called a soft closed set in $(O,\sigma ,\mathcal{L})$ if $1_{\mathcal{L}}-K\in \sigma$.

In this paper, we will follow the terminology and concepts from [15,33], and we will denote a topological space as TS and a soft topological space as STS.

Let $\left(O,\aleph \right)$ be a TS, $(O,\sigma ,\mathcal{L})$ be an STS, $X\subseteq O$, and $H\in S(O,\mathcal{L})$. Then, the closure of X in $\left(O,\aleph \right)$, the interior of X in $\left(O,\aleph \right)$, the soft closure of H in $(O,\sigma ,\mathcal{L})$, and the soft interior of H in $(O,\sigma ,\mathcal{L})$ will be denoted by $C{l}_{\aleph }\left(X\right)$, $In{t}_{\aleph }\left(X\right)$, $C{l}_{\sigma }\left(H\right)$, and $In{t}_{\sigma }\left(H\right)$, respectively, and the family of all closed sets in $\left(O,\aleph \right)$ (resp. soft closed sets in $(O,\sigma ,\mathcal{L})$) will be denoted by ${\aleph }^c$ (resp. ${\sigma }^c$).

We will use the following definitions and notations in the sequel.

Definition 1. 

Let $\left(O,\Im \right)$ be a TS and let $A\subseteq O$. Then, A is called a semi-open [34] (resp. pre-open [35], β-open [36]) set in $\left(O,\Im \right)$ if $A\subseteq C{l}_{\Im }\left(In{t}_{\Im }\left(A\right)\right)$ (resp. $A\subseteq In{t}_{\Im }\left(C{l}_{\Im }\left(A\right)\right)$, $A\subseteq C{l}_{\Im }\left(In{t}_{\Im }\left(C{l}_{\Im }\left(A\right)\right)\right)$). The collection of all semi-open sets (resp. pre-open sets, β-open sets) in $\left(O,\Im \right)$ will be denoted by $SO\left(\Im \right)$ (resp. $PO\left(\Im \right)$, $\beta O\left(\Im \right)$).

Definition 2. 

A function $g:\left(O,\Im \right)⟶\left(T,\aleph \right)$ is called semi-continuous (S-C) [34] (resp. pre-continuous (P-C) [35], β-continuous (β-C) [36]) if $g^{-1}\left(V\right)\in SO\left(\Im \right)$ (resp. $g^{-1}\left(V\right)\in PO\left(\Im \right)$, $g^{-1}\left(V\right)\in \beta O\left(\Im \right)$) for every $V\in \aleph$.

Definition 3. 

A function $g:\left(O,\Im \right)⟶\left(T,\aleph \right)$ is called faintly continuous (F-C) [37] (resp. faintly semi-continuous (F-S-C) [38], faintly pre-continuous (F-P-C) [38], faintly β-continuous (F-β-C) [38]) if $g^{-1}\left(V\right)\in \Im$ (resp. $g^{-1}\left(V\right)\in SO\left(\Im \right)$, $g^{-1}\left(V\right)\in PO\left(\Im \right)$, $g^{-1}\left(V\right)\in \beta O\left(\Im \right)$) for every $V\in {\aleph }_{\theta }$.

Definition 4 

([38]). A function $g:\left(O,\Im \right)⟶\left(T,\aleph \right)$ is called quasi-θ-continuous if $g^{-1}\left(V\right)\in {\Im }_{\theta }$ for every $V\in {\aleph }_{\theta }$.

Definition 5. 

Let $\left(O,\phi ,\mathcal{P}\right)$ be an STS and let $H\in SS\left(O,\mathcal{P}\right)$. Then:

(a) 

H is called a soft semi-open [39] (resp. soft pre-open [40], soft β-open [40]) set in $\left(O,\phi ,\mathcal{P}\right)$ if $H\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(H\right)\right)$ (resp. $H\tilde{\subseteq }In{t}_{\phi }\left(C{l}_{\phi }\left(H\right)\right)$, $H\tilde{\subseteq }C{l}_{\phi }\left(In{t}_{\phi }\left(C{l}_{\phi }\left(H\right)\right)\right)$). The collection of all soft semi-open sets (resp. soft pre-open sets, soft β-open sets) in $\left(O,\phi ,\mathcal{P}\right)$ will be denoted by $SO\left(\phi \right)$ (resp. $PO\left(\phi \right)$, $\beta O\left(\phi \right)$).

(b) 

H is called a soft semi-closed [39] (resp. soft pre-closed [40], soft β-closed [40]) set in $\left(O,\phi ,\mathcal{P}\right)$ if $O-H\in$$SO\left(\phi \right)$ (resp. $O-H\in PO\left(\phi \right)$, $O-H\in \beta O\left(\phi \right)$). The collection of all soft semi-open sets (resp. soft pre-open sets, soft β-open sets) in $\left(O,\phi ,\mathcal{P}\right)$ will be denoted by $SC\left(\phi \right)$ (resp. $PC\left(\phi \right)$, $\beta C\left(\phi \right)$).

Definition 6. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is said to be called soft semi-continuous (soft S-C) [41] (resp. soft pre-continuous (soft P-C) [42], soft β-continuous (soft β-C) [43]) if $f_{rw}^{-1}\left(H\right)\in SO\left(\phi \right)$ (resp. $f_{rw}^{-1}\left(H\right)\in PO\left(\phi \right)$, $f_{rw}^{-1}\left(H\right)\in \beta O\left(\phi \right)$) for every $H\in \sigma$.

Definition 7 

([44]). Let $\left(O,\phi ,\mathcal{P}\right)$ be an STS and let $G\in SS(O,\mathcal{P})$. Then, G is called a soft θ-open set in $\left(O,\phi ,\mathcal{P}\right)$ if for every $a_x\tilde{\in }$ G, there exists H∈φ such that $a_x\tilde{\in }H\tilde{\subseteq }C{l}_{\phi }\left(H\right)\tilde{\subseteq }G$. The family of all soft θ-open sets in $\left(O,\phi ,\mathcal{P}\right)$ is denoted by ${\phi }_{\theta }$.

It is well known that ${\phi }_{\theta }\subseteq \phi$ and ${\phi }_{\theta }\ne \phi$ in general.

Definition 8 

([15]). A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is said to be soft faintly continuous (soft F-C) if, for every $a_t\in$ SP(O,$\mathcal{P})$ and $K\in {\sigma }_{\theta }$ with $f_{rw}\left(a_t\right)\tilde{\in }K$, we find $H\in \phi$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }K$.

Definition 9. 

Let $\left(O,\phi ,\mathcal{P}\right)$ be an STS and let $H\in SS\left(O,\mathcal{P}\right)$. Then, we obtain the following:

(a) 

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

(b) 

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

(c) 

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

(d) 

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

(e) 

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

(f) 

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

2. Soft Faint Semi-Continuity

Definition 10. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is said to be soft faintly semi-continuous (soft F-S-C, for simplicity) if, for every $a_t\in$ SP(O,$\mathcal{P})$ and $K\in {\sigma }_{\theta }$ with $f_{rw}\left(a_t\right)\tilde{\in }K$, we find $H\in SO\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }K$.

Theorem 1. 

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 F-S-C.

(b) 

$f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,{\sigma }_{\theta },\mathcal{S}\right)$ is soft S-C.

(c) 

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

(d) 

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

(e) 

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

(f) 

$f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(K\right)\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$ for every $K\in SS(T,\mathcal{S})$.

Proof. 

(a)

⟶ (b): Let $K\in {\sigma }_{\theta }$ and let $m_r\tilde{\in }{f}_{rw}^{-1}\left(K\right)$. Then, $f_{rw}\left(a_t\right)\tilde{\in }K$, and, by (a), we find $H\in SO\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }K$. Hence, $a_t\tilde{\in }H\tilde{\subseteq }{f}_{rw}^{-1}\left(f_{rw}\left(H\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(K\right)$. Therefore, $f_{rw}^{-1}\left(K\right)\in SO\left(\phi \right)$.

(b)

⟶ (c): Clear.

(c)

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

(d)

⟶ (e): Let $H\in SS(T,\mathcal{S})$. Then, $C{l}_{{\sigma }_{\theta }}\left(H\right)\in {\left({\sigma }_{\theta }\right)}^c$ and, by (d), $f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\in SC\left(\phi \right)$. Thus, $f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)=sC{l}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\right)$. Since $f_{rw}^{-1}\left(H\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$, then $sC{l}_{\phi }\left(f_{rw}^{-1}\left(H\right)\right)\tilde{\subseteq }sC{l}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\right)=f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$.

(e)

⟶ (f): Let $K\in SS(T,\mathcal{S})$. Then, by (e), $sC{l}_{\phi }\left(f_{rw}^{-1}(1_{\mathcal{S}}-K)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}(1_{\mathcal{S}}-K)\right)$. However,

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

and

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

Thus, $1_{\mathcal{P}}-sIn{t}_{\phi }\left(f_{rw}^{-1}\left(G\right)\right)\tilde{\subseteq }{1}_{\mathcal{P}}-f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(G\right)\right)$ and hence, $f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(G\right)\right)\tilde{\subseteq }sIn{t}_{\phi }$$\left(f_{rw}^{-1}\left(G\right)\right)$.

(f)

⟶ (a): Let $a_t\in SP(O,\mathcal{P})$ and $K\in {\sigma }_{\theta }$ such that $f_{rw}\left(a_t\right)\tilde{\in }K$. Then, $In{t}_{{\sigma }_{\theta }}\left(K\right)=K$. So, by (f), $f_{rw}^{-1}\left(K\right)=f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(K\right)\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$ and, thus, $f_{rw}^{-1}\left(K\right)\tilde{\subseteq }sIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$. Therefore, $f_{rw}^{-1}\left(K\right)=sIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$, and, hence, $f_{rw}^{-1}\left(K\right)\in SO\left(\phi \right)$. We set $H=f_{rw}^{-1}\left(K\right)$. Then, $H\in SO\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)=f_{rw}\left(f_{rw}^{-1}\left(K\right)\right)\tilde{\subseteq }K$. Therefore, $f_{rw}$ is soft F-S-C.

□

Theorem 2. 

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

Proof. 

Necessity. Let $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft F-S-C. Let $a\in \mathcal{P}$. Let $U\in {\left({\sigma }_{w\left(a\right)}\right)}_{\theta }$. Then, according to Theorem 2.21 of [45], ${\left(w\left(a\right)\right)}_U\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. So, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)\in SO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)=a_{r^{-1}\left(U\right)}$. Therefore, by Theorem 4.10 of [46], $\left(a_{r^{-1}\left(U\right)}\right)\left(a\right)=r^{-1}\left(U\right)\in SO\left({\phi }_a\right)$. This shows that $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is F-S-C.

Sufficiency. Let $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ be F-S-C for all $a\in \mathcal{P}$. Let $G\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. Then, by Theorem 2.21 of [45], $G\left(b\right)\in {\left({\sigma }_b\right)}_{\theta }$ for all $b\in \mathcal{S}$. For every $b\in \mathcal{S}$, $r:\left(X,{\phi }_{w^{-1}\left(b\right)}\right)⟶\left(Y,{\sigma }_b\right)$ is F-S-C, and, so, $r^{-1}\left(G\left(b\right)\right)\in SO\left({\phi }_{r^{-1}\left(b\right)}\right)$. Hence, for each $a\in \mathcal{P}$, $\left(f_{rw}^{-1}\left(G\right)\right)\left(a\right)=r^{-1}(G\left(w\left(a\right)\right)\in SO\left({\phi }_{w^{-1}\left(w\left(a\right)\right)}\right)=SO\left({\phi }_a\right)$. Therefore, by Theorem 4.10 of [46], $f_{rw}^{-1}\left(G\right)\in SO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. This shows that $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft F-S-C. □

Corollary 1. 

We consider the 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 F-S-C iff $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(T,\tau \left(\aleph \right),\mathcal{S})$ is soft F-S-C.

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we set ${\phi }_a=\Im$ 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 2 ends the proof. □

Theorem 3. 

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

Proof. 

Necessity. Let $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft S-C. Let $a\in \mathcal{P}$. Let $U\in {\left({\sigma }_{w\left(a\right)}\right)}_{\theta }$. Then, ${\left(w\left(a\right)\right)}_U\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. So, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)\in SO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)=a_{r^{-1}\left(U\right)}$. Therefore, by Theorem 4.10 of [46], $\left(a_{r^{-1}\left(U\right)}\right)\left(a\right)=r^{-1}\left(U\right)\in SO\left({\phi }_a\right)$. This shows that $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is S-C.

Sufficiency. Let $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is S-C for all $a\in \mathcal{P}$. Let $G\in {\oplus }_{b\in \mathcal{S}}{\sigma }_b$. Then, $G\left(b\right)\in {\sigma }_b$ for all $b\in \mathcal{S}$. For every $b\in \mathcal{S}$, $r:\left(X,{\phi }_{w^{-1}\left(b\right)}\right)⟶\left(Y,{\sigma }_b\right)$ is S-C, and, so, $r^{-1}\left(G\left(b\right)\right)\in SO\left({\phi }_{r^{-1}\left(b\right)}\right)$. Hence, for each $a\in \mathcal{P}$, $\left(f_{rw}^{-1}\left(G\right)\right)\left(a\right)=r^{-1}(G\left(w\left(a\right)\right)\in SO\left({\phi }_{w^{-1}\left(w\left(a\right)\right)}\right)=SO\left({\phi }_a\right)$. Therefore, by Theorem 4.10 of [46], $f_{rw}^{-1}\left(G\right)\in SO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. This shows that $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft S-C. □

Corollary 2. 

We consider the 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 S-C iff $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(T,\tau \left(\aleph \right),\mathcal{S})$ is soft S-C.

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we set ${\phi }_a=\Im$ 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 3 ends the proof. □

Theorem 4. 

Every soft F-C function is soft F-S-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft F-C. Let $G\in {\sigma }_{\theta }$. Then, $f_{rw}^{-1}\left(G\right)\in \phi \subseteq SO\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-S-C. □

Theorem 5. 

Every soft S-C function is soft F-S-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft S-C. Let $G\in {\sigma }_{\theta }$. Then, $G\in \sigma$. So, $f_{rw}^{-1}\left(G\right)\in SO\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-S-C. □

The converse of Theorem 4 need not be true in general.

Example 1. 

Let $X=\left\{a,b,c\right\}$, $\mathcal{P}=\mathbb{R}$, $\Im =\left\{\varnothing ,X,\left\{a\right\},\left\{b\right\},\left\{a,b\right\}\right\}$, and $\aleph =\left\{\varnothing ,X,\left\{a\right\},\left\{b\right\}\right.$, $\left.\left\{a,b\right\},\left\{a,c\right\}\right\}$. We consider the identities functions $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ and $w:\mathcal{P}⟶\mathcal{P}$. Since $C{l}_{\Im }\left(\left\{a\right\}\right)=\left\{a,c\right\}$, then $r^{-1}\left(\left\{a,c\right\}\right)=\left\{a,c\right\}\in SO\left(\Im \right)$. Hence, $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ is S-C, and, by Theorem 4.1 (a) of [38], it is F-S-C. On the other hand, since $\left\{a,c\right\}\in {\aleph }_{\theta }$ while $r^{-1}\left(\left\{a,c\right\}\right)=\left\{a,c\right\}\notin \Im$, then $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ is not F-C.

Therefore, by Corollary 1 and Corollary 1 of [15], $f_{rw}:(X,\tau \left(\Im \right),\mathcal{P})⟶(X,\tau \left(\aleph \right),\mathcal{P})$ is soft F-S-C but not soft F-C.

The converse of Theorem 5 need not be true in general.

Example 2. 

Let $X=\left\{a,b\right\}$, $Y=\left\{1,2,3\right\}$, $\Im =\left\{\varnothing ,X,\left\{b\right\}\right\}$, $\aleph =\left\{\varnothing ,Y,\left\{1\right\},\left\{2\right\},\left\{1,2\right\}\right\}$, and $\mathcal{P}=\mathbb{N}$. We define $r:\left(X,\Im \right)⟶$$\left(Y,\aleph \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ by $r\left(a\right)=1$, $r\left(b\right)=2$, and $w\left(t\right)=t$ for all $t\in \mathcal{P}$. Since ${\aleph }_{\theta }=\left\{\varnothing ,Y\right\}$, then $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ is F-C. On the other hand, since $\left\{1\right\}\in \aleph$ while $r^{-1}\left(\left\{1\right\}\right)=\left\{a\right\}\notin SO\left(\Im \right)$, then $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ is not S-C.

Therefore, by Corollaries 1 and 2, $f_{rw}:(X,\tau \left(\Im \right),\mathcal{P})⟶(Y,\tau \left(\aleph \right),\mathcal{P})$ is soft F-C but not soft S-C. Finally, by Theorem 4, $f_{rw}$ is soft F-S-C.

Theorem 6. 

Let $\left(T,\sigma ,\mathcal{S}\right)$ be soft regular. The following are equivalent for a soft function$f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$:

(a) 

$f_{rw}$ is soft S-C.

(b) 

$f_{rw}$ is soft F-S-C.

Proof. 

(a)

⟶ (b): Follows from Theorem 5.

(b)

⟶ (c): Let $K\in \sigma$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft regular, then $\sigma ={\sigma }_{\theta }$. So, $K\in {\sigma }_{\theta }$ and, by (b), $f_{rw}^{-1}\left(K\right)\in SO\left(\phi \right)$. This shows that $f_{rw}$ is soft S-C.

□

Theorem 7. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function. If $f_{r^{\#}{w}^{\#}}:\left(O,\phi ,\mathcal{P}\right)⟶\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times \mathcal{S}\right)$ is soft F-S-C, then $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-S-C.

Proof. 

Let $H\in {\sigma }_{\theta }$. Since $1_{\mathcal{P}}\in {\phi }_{\theta }$, by Theorem 10 of [15], $1_{\mathcal{P}}\times H\in {\left(pr\left(\phi \times \sigma \right)\right)}_{\theta }$. Since $f_{p^{\#}{u}^{\#}}:\left(O,\phi ,\mathcal{P}\right)⟶\left(R\times L,pr\left(\phi \times \sigma \right),M\times N\right)$ is soft F-S-C, by Theorem 1b, $f_{r^{\#}{w}^{\#}}^{-1}\left(1_{\mathcal{P}}\times H\right)$$\in SO\left(\phi \right)$. By Lemma 2 of [15], $f_{r^{\#}{w}^{\#}}^{-1}\left(1_{\mathcal{P}}\times H\right)=1_{\mathcal{P}}\tilde{\cap }{f}_{rw}^{-1}\left(H\right)=f_{rw}^{-1}\left(H\right)$, and, hence, $f_{rw}^{-1}\left(H\right)\in SO\left(\phi \right)$. Thus, again, by Theorem 1b, $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-S-C. □

Lemma 1. 

Let $\left(O,\phi ,\mathcal{P}\right)$ be an STS. If $A\in \phi$ and $X$ is a non-empty subset of O such that $C_X\in PO\left(\phi \right)$, then $C{l}_{\phi }\left(A\right)\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{{\phi }_X}\left(A\tilde{\cap }{C}_X\right)$.

Proof. 

Let $d_z\tilde{\in }C{l}_{\phi }\left(A\right)\tilde{\cap }{C}_X$. Let $G\in {\phi }_X$ such that $d_z\tilde{\in }G$. We choose $L\in \phi$ such that $G=L\tilde{\cap }{C}_X$. Since $C_X\in PO\left(\phi \right)$, then $C_X\tilde{\subseteq }In{t}_{\phi }\left(C{l}_{\phi }\left(C_X\right)\right)$. Since $d_z\tilde{\in }L\tilde{\cap }In{t}_{\phi }\left(C{l}_{\phi }\left(C_X\right)\right)\in \phi$ and $d_z\tilde{\in }C{l}_{\phi }\left(A\right)$, then $\left(L\tilde{\cap }In{t}_{\phi }\left(C{l}_{\phi }\left(C_X\right)\right)\right)\tilde{\cap }A\ne 0_{\mathcal{P}}$. Hence, $\left(L\tilde{\cap }C{l}_{\phi }\left(C_X\right)\right)\tilde{\cap }A\ne 0_{\mathcal{P}}$. We choose $b_y\tilde{\in }\left(L\tilde{\cap }C{l}_{\phi }\left(C_X\right)\right)\tilde{\cap }A$. Since $b_y\tilde{\in }L\tilde{\cap }A\in \phi$ and $b_y\tilde{\in }C{l}_{\phi }\left(C_X\right)$, then $\left(L\tilde{\cap }A\right)\tilde{\cap }{C}_X\ne 0_{\mathcal{P}}$. Hence, $G\tilde{\cap }\left(A\tilde{\cap }{C}_X\right)=\left(L\tilde{\cap }{C}_X\right)\tilde{\cap }\left(A\tilde{\cap }{C}_X\right)=\left(L\tilde{\cap }A\right)\tilde{\cap }{C}_X\ne 0_{\mathcal{P}}$. It follows that $d_z\tilde{\in }C{l}_{{\phi }_X}\left(A\tilde{\cap }{C}_X\right)$. □

Proposition 1. 

Let $\left(O,\phi ,\mathcal{P}\right)$ be an STS. If $A\in SO\left(\phi \right)$ and $X$ is a non-empty subset of O such that $C_X\in PO\left(\phi \right)$, then $A\tilde{\cap }{C}_X\in SO\left({\phi }_X\right)$.

Proof. 

Since $A\in SO\left(\phi \right)$, there exists $H\in \phi$ such that $H\tilde{\subseteq }A\tilde{\subseteq }C{l}_{\phi }\left(H\right)$, and, so, $H\tilde{\cap }{C}_X\tilde{\subseteq }$$A\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{\phi }\left(H\right)\tilde{\cap }{C}_X$. On the other hand, by Lemma 1, $C{l}_{\phi }\left(H\right)\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{{\phi }_X}\left(H\tilde{\cap }{C}_X\right)$. Thus, we have $H\tilde{\cap }{C}_X\in {\phi }_X$ and $H\tilde{\cap }{C}_X\tilde{\subseteq }A\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{{\phi }_X}\left(H\tilde{\cap }{C}_X\right)$. Hence, $A\tilde{\cap }{C}_X\in SO\left({\phi }_X\right)$. □

Theorem 8. 

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

Proof. 

Let $K\in {\sigma }_{\theta }$. Since $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-S-C, by Theorem 1b, $f_{rw}^{-1}\left(K\right)\in SO\left(\phi \right)$. So, by Proposition 1, ${\left({\left(f_{rw}\right)}_{\left|C_X\right.}\right)}^{-1}\left(H\right)=C_X\tilde{\cap }{f}_{rw}^{-1}\left(H\right)\in SO\left({\phi }_X\right)$. Hence, again, by Theorem 1b, ${\left(f_{rw}\right)}_{\left|C_X\right.}:\left(X,{\phi }_X,M\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-S-C. □

Definition 11. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is called soft quasi-θ-continuous if $f_{rw}^{-1}\left(K\right)\in {\phi }_{\theta }$ for every $K\in {\sigma }_{\theta }$.

Theorem 9. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft quasi-θ-continuous iff $f_{rw}^{-1}\left(K\right)\in {\left({\phi }_{\theta }\right)}^c$ for every $K\in {\left({\sigma }_{\theta }\right)}^c$.

Proof. 

Straightforward. □

Theorem 10. 

Let $\left\{\left(X,{\phi }_a\right):a\in \mathcal{P}\right\}$ and $\left\{\left(Y,{\sigma }_b\right):b\in \mathcal{S}\right\}$ be two collections of TSs. Let $r:X⟶Y$ and $w:\mathcal{P}⟶\mathcal{S}$ be two functions, with r being a bijection. Then, $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)$⟶$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft quasi-θ-continuous iff $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is quasi-θ-continuous for all $a\in \mathcal{P}$.

Proof. 

Necessity. Let $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft quasi-$\theta$-continuous. Let $a\in \mathcal{P}$. Let $U\in {\left({\sigma }_{w\left(a\right)}\right)}_{\theta }$. Then, according to Theorem 2.21 of [45], ${\left(w\left(a\right)\right)}_U\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. So, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)\in {\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)}_{\theta }$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)=a_{r^{-1}\left(U\right)}$. Therefore, by Theorem 2.21 of [45], $\left(a_{r^{-1}\left(U\right)}\right)\left(a\right)=r^{-1}\left(U\right)\in SO\left({\phi }_a\right)$. This shows that $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is quasi-$\theta$-continuous.

Sufficiency. $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is quasi-$\theta$-continuous for all $a\in \mathcal{P}$. Let $G\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. Then, by Theorem 2.21 of [45], $G\left(b\right)\in {\left({\sigma }_b\right)}_{\theta }$ for all $b\in \mathcal{S}$. For every $b\in \mathcal{S}$, $r:\left(X,{\phi }_{w^{-1}\left(b\right)}\right)⟶\left(Y,{\sigma }_b\right)$ is quasi-$\theta$-continuous, and, so, $r^{-1}\left(G\left(b\right)\right)\in SO\left({\phi }_{r^{-1}\left(b\right)}\right)$. Hence, for each $a\in \mathcal{P}$, $\left(f_{rw}^{-1}\left(G\right)\right)\left(a\right)=r^{-1}(G\left(w\left(a\right)\right)\in {\left({\phi }_{w^{-1}\left(w\left(a\right)\right)}\right)}_{\theta }={\left({\phi }_a\right)}_{\theta }$. Therefore, by Theorem 2.21 of [45], $f_{rw}^{-1}\left(G\right)\in {\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)}_{\theta }$. This shows that $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft quasi-$\theta$-continuous. □

Corollary 3. 

Let $r:\left(X,\Im \right)⟶\left(Y,\aleph \right)$ be a function between two TSs, and let $w:\mathcal{P}⟶\mathcal{S}$ be a bijective function. Then, $r:\left(X,\Im \right)⟶\left(Y,\aleph \right)$ is quasi-θ-continuous iff $f_{rw}:(X,\tau \left(\Im \right),\mathcal{P})⟶(Y,\tau \left(\aleph \right),\mathcal{S})$ is soft quasi-θ-continuous.

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we set ${\phi }_a=\Im$ 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 10 ends the proof. □

Theorem 11. 

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

Proof. 

The proof follows from Theorem 9 and Corollary 5.30 of [44]. □

It is not difficult to check that the soft function defined in Example 2 is soft quasi-$\theta$-continuous but not soft $\theta$-continuous.

Theorem 12. 

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

Proof. 

Let $K\in {\lambda }_{\theta }$. Then, $f_{r_2{w}_2}^{-1}\left(K\right)\in {\sigma }_{\theta }$ and, hence, $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}^{-1}\left(K\right)=f_{r_1{w}_1}^{-1}\left(f_{r_2{w}_2}^{-1}\left(K\right)\right)\in SO\left(\phi \right)$. Therefore, $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}$ is soft F-S-C. □

The following result follows from Theorems 11 and 12:

Corollary 4. 

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

Corollary 5. 

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

Proof. 

It follows from Corollary 4 and the fact that soft continuous functions are soft $\theta$-continuous. □

The subsequent illustrations demonstrate that, according to Corollary 5, the composition in reverse order may not possess the same quality.

Example 3. 

Let $X=\left\{a,b,c\right\}$, $\Im =\left\{\varnothing ,X,\left\{a\right\},\left\{b\right\},\left\{a,b\right\},\left\{b,c\right\}\right\}$, $\aleph =\left\{\varnothing ,X,\left\{a\right\},\left\{b\right\},\left\{a,b\right\}\right\}$, $\wp =\left\{\varnothing ,X,\left\{a\right\},\left\{b\right\},\left\{a,b\right\},\left\{c,c\right\}\right\}$, and $\mathcal{P}=\mathbb{N}$. Let $r_1:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$, $r_2:\left(X,\Im \right)⟶$$\left(X,\wp \right)$, and $w:\mathcal{P}⟶\mathcal{P}$ be the identity functions. Then, $f_{r_{1}w}:\left(X,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(X,\tau \left(\aleph \right),\mathcal{P}\right)$ is soft continuous and $f_{r_{2}w}:\left(X,\tau \left(\aleph \right),\mathcal{P}\right)⟶\left(X,\tau \left(\wp \right),\mathcal{P}\right)$ is soft S-C (and, hence, soft F-S-C), while $f_{\left(r_2\circ r_1\right)\left(w\circ w\right)}$ is not soft F-S-C.

3. Soft Faint Pre-Continuity

Definition 12. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is called soft faintly pre-continuous (soft F-P-C, for short) if, for each $a_t\in$ SP(O,$\mathcal{P})$ and $K\in {\sigma }_{\theta }$ such that $f_{rw}\left(a_t\right)\tilde{\in }K$, we find $H\in PO\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }K$.

Theorem 13. 

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 F-P-C.

(b) 

$f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,{\sigma }_{\theta },\mathcal{S}\right)$ is soft P-C.

(c) 

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

(d) 

$f_{rw}^{-1}\left(G\right)\in PC\left(\phi \right)$ for every $G\in {\left({\sigma }_{\theta }\right)}^c$.

(e) 

$pC{l}_{\phi }\left(f_{pu}^{-1}\left(H\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$ for every $H\in SS(T,\mathcal{S})$.

(f) 

$f_{pu}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(K\right)\right)\tilde{\subseteq }pIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$ for every $K\in SS(T,\mathcal{S})$.

Proof. 

(a)

⟶ (b): Let $K\in {\sigma }_{\theta }$ and let $m_r\tilde{\in }{f}_{rw}^{-1}\left(K\right)$. Then, $f_{rw}\left(a_t\right)\tilde{\in }K$, and, by (a), we find $H\in PO\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }K$. Hence, $a_t\tilde{\in }H\tilde{\subseteq }{f}_{rw}^{-1}\left(f_{rw}\left(H\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(K\right)$. Therefore, $f_{rw}^{-1}\left(K\right)\in PO\left(\phi \right)$.

(b)

⟶ (c): Clear.

(c)

⟶ (d): Let $G\in {\left({\sigma }_{\theta }\right)}^c$. Then, $1_{\mathcal{S}}-G\in {\sigma }_{\theta }$, and, by (c), $f_{rw}^{-1}(1_{\mathcal{S}}-G)=1_{\mathcal{P}}-f_{rw}^{-1}\left(G\right)\in PO\left(\phi \right)$. Hence, $f_{rw}^{-1}\left(G\right)\in PC\left(\phi \right)$.

(d)

⟶ (e): Let $H\in SS(T,\mathcal{S})$. Then, $C{l}_{{\sigma }_{\theta }}\left(H\right)\in {\left({\sigma }_{\theta }\right)}^c$ and, by (d), $f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\in PC\left(\phi \right)$. Thus, $f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)=pC{l}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\right)$. Since $f_{rw}^{-1}\left(H\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$, then $pC{l}_{\phi }\left(f_{rw}^{-1}\left(H\right)\right)\tilde{\subseteq }pC{l}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\right)=f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$.

(e)

⟶ (f): Let $K\in SS(T,\mathcal{S})$. Then, by (e), $pC{l}_{\phi }\left(f_{rw}^{-1}(1_{\mathcal{S}}-K)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}(1_{\mathcal{S}}-K)\right)$. However,

$$\begin{array}{ccc}pC{l}_{\phi }\left(f_{rw}^{-1}(1_{\mathcal{S}}-K)\right)& =& pC{l}_{\phi }(1_{\mathcal{P}}-f_{rw}^{-1}\left(K\right))\\ & =& 1_{\mathcal{P}}-pIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\end{array}$$

and

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

Thus, $1_{\mathcal{P}}-pIn{t}_{\phi }\left(f_{rw}^{-1}\left(G\right)\right)\tilde{\subseteq }{1}_{\mathcal{P}}-f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(G\right)\right)$ and, hence, $f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(G\right)\right)\tilde{\subseteq }pIn{t}_{\phi }$$\left(f_{rw}^{-1}\left(G\right)\right)$.

(f)

⟶ (a): Let $a_t\in SP(O,\mathcal{P})$ and $K\in {\sigma }_{\theta }$ such that $f_{rw}\left(a_t\right)\tilde{\in }K$. Then, $In{t}_{{\sigma }_{\theta }}\left(K\right)=K$. So, by (f), $f_{rw}^{-1}\left(K\right)=f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(K\right)\right)\tilde{\subseteq }pIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$ and, thus, $f_{rw}^{-1}\left(K\right)\tilde{\subseteq }pIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$. Therefore, $f_{rw}^{-1}\left(K\right)=pIn{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$, and, hence, $f_{rw}^{-1}\left(K\right)\in PO\left(\phi \right)$. We set $H=f_{rw}^{-1}\left(K\right)$. Then, $H\in PO\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)=f_{rw}\left(f_{rw}^{-1}\left(K\right)\right)\tilde{\subseteq }K$. Therefore, $f_{rw}$ is soft F-P-C.

□

Theorem 14. 

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

Proof. 

Necessity. Let $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft F-P-C. Let $a\in \mathcal{P}$. Let $U\in {\left({\sigma }_{w\left(a\right)}\right)}_{\theta }$. Then, according to Theorem 2.21 of [45], ${\left(w\left(a\right)\right)}_U\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. So, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)\in PO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)=a_{r^{-1}\left(U\right)}$. Therefore, by Theorem 5.8 of [47], $\left(a_{r^{-1}\left(U\right)}\right)\left(a\right)=r^{-1}\left(U\right)\in PO\left({\phi }_a\right)$. This shows that $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is F-P-C.

Sufficiency. Let $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is F-P-C for all $a\in \mathcal{P}$. Let $G\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. Then, by Theorem 2.21 of [45], $G\left(b\right)\in {\left({\sigma }_b\right)}_{\theta }$ for all $b\in \mathcal{S}$. For every $b\in \mathcal{S}$, $r:\left(X,{\phi }_{w^{-1}\left(b\right)}\right)⟶\left(Y,{\sigma }_b\right)$ is F-P-C, and, so, $r^{-1}\left(G\left(b\right)\right)\in PO\left(\phi \right)$. Hence, for each $a\in \mathcal{P}$, $\left(f_{rw}^{-1}\left(G\right)\right)\left(a\right)=r^{-1}(G\left(w\left(a\right)\right)\in PO\left(\phi \right)$. Therefore, by Theorem 5.8 of [47], $f_{rw}^{-1}\left(G\right)\in PO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. This shows that $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft F-P-C. □

Corollary 6. 

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

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we set ${\phi }_a=\Im$ 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 14 ends the proof. □

Theorem 15. 

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

Proof. 

Necessity. Let $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft P-C. Let $a\in \mathcal{P}$. Let $U\in {\left({\sigma }_{w\left(a\right)}\right)}_{\theta }$. Then, ${\left(w\left(a\right)\right)}_U\in {\oplus }_{b\in \mathcal{S}}{\sigma }_b$. So, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)\in PO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)=a_{r^{-1}\left(U\right)}$. Therefore, by Theorem 5.8 of [47], $\left(a_{r^{-1}\left(U\right)}\right)\left(a\right)=r^{-1}\left(U\right)\in PO\left({\phi }_a\right)$. This shows that $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is P-C.

Sufficiency. $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is P-C for all $a\in \mathcal{P}$. Let $G\in {\oplus }_{b\in \mathcal{S}}{\sigma }_b$. Then, $G\left(b\right)\in {\sigma }_b$ for all $b\in \mathcal{S}$. For every $b\in \mathcal{S}$, $r:\left(X,{\phi }_{w^{-1}\left(b\right)}\right)⟶\left(Y,{\sigma }_b\right)$ is P-C, and, so, $r^{-1}\left(G\left(b\right)\right)\in PO\left(\phi \right)$. Hence, for each $a\in \mathcal{P}$, $\left(f_{rw}^{-1}\left(G\right)\right)\left(a\right)=r^{-1}(G\left(w\left(a\right)\right)\in PO\left(\phi \right)=PO\left(\phi \right)$. Therefore, by Theorem 5.8 of [47], $f_{rw}^{-1}\left(G\right)\in PO\left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. This shows that $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft P-C. □

Corollary 7. 

We consider the 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 P-C iff $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(T,\tau \left(\aleph \right),\mathcal{S})$ is soft P-C.

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we set ${\phi }_a=\Im$ 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 15 ends the proof. □

Theorem 16. 

Every soft F-C function is soft F-P-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft F-C. Let $G\in {\sigma }_{\theta }$. Then, $f_{rw}^{-1}\left(G\right)\in \phi \subseteq PO\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-P-C. □

Theorem 17. 

Every soft P-C function is soft F-P-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft P-C. Let $G\in {\sigma }_{\theta }$. Then, $G\in \sigma$. So, $f_{rw}^{-1}\left(G\right)\in PO\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-P-C. □

The converse of Theorem 16 need not be true in general.

Example 4. 

Let $X=\mathbb{R}$, $\mathcal{P}=\left\{a,b\right\}$, and ℑ the usual topology on X. We define the functions $r:\left(X,\Im \right)⟶$$\left(X,\Im \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ by

$$r\left(x\right)=\left\{\begin{array}{cc}1& \mathit{if}x\in \mathbb{Q}\\ -1& \mathit{if}x\in \mathbb{R}-\mathbb{Q}\end{array}\right.,w\left(a\right)=a,\mathit{and}w\left(b\right)=b.$$

Then, $f_{rw}:(X,\tau \left(\Im \right),\mathcal{P})⟶(X,\tau \left(\Im \right),\mathcal{P})$ is soft F-P-C but not soft F-C.

The converse of Theorem 17 need not be true in general.

Example 5. 

Let $X=\left\{a,b\right\}$, $Y=\left\{1,2,3\right\}$, $\Im =\left\{\varnothing ,X,\left\{b\right\}\right\}$, $\aleph =\left\{\varnothing ,Y,\left\{1\right\},\left\{2\right\},\left\{1,2\right\}\right\}$, and $\mathcal{P}=\mathbb{N}$. We define $r:\left(X,\Im \right)⟶$$\left(Y,\aleph \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ by $r\left(a\right)=1$, $r\left(b\right)=2$, and $w\left(t\right)=t$ for all $t\in \mathcal{P}$. Since ${\aleph }_{\theta }=\left\{\varnothing ,Y\right\}$, then $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ is F-P-C. On the other hand, since $\left\{1\right\}\in \aleph$ while $r^{-1}\left(\left\{1\right\}\right)=\left\{a\right\}\notin PO\left(\Im \right)$, then $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ is not P-C.

Therefore, by Corollaries 6 and 7, $f_{rw}:(X,\tau \left(\Im \right),\mathcal{P})⟶(Y,\tau \left(\aleph \right),\mathcal{P})$ is soft F-P-C but not soft P-C.

Theorem 18. 

Let $\left(T,\sigma ,\mathcal{S}\right)$ be soft regular. The following are equivalent for a soft function  $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$:

(a) 

$f_{rw}$ is soft P-C.

(b) 

$f_{rw}$ is soft F-P-C.

Proof. 

(a)

⟶ (b): Follows from Theorem 17.

(b)

⟶ (c): Let $K\in \sigma$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft regular, then $\sigma ={\sigma }_{\theta }$. So, $K\in {\sigma }_{\theta }$ and, by (b), $f_{rw}^{-1}\left(K\right)\in PO\left(\phi \right)$. This shows that $f_{rw}$ is soft P-C.

□

The following two examples show that “soft F-S-C” and “soft F-P-C” are independent concepts:

Example 6. 

Let $X=\mathbb{R}$ and $\mathcal{P}=\mathbb{N}$. Let ℑ and ℵ be the indiscrete and the discrete topologies on X, respectively. Let $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ be the identity functions. Then, $f_{rw}:\left(X,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(X,\tau \left(\aleph \right),\mathcal{P}\right)$ is soft F-P-C but it is not soft F-S-C.

Example 7. 

Let $X=\mathbb{R}$, $Y=\left\{a,b\right\}$, $\mathcal{P}=\mathbb{N}$, ℑ be the usual typology on X, and $\aleph =\left\{\varnothing ,Y,\left\{a\right\},\left\{b\right\}\right\}$. We define $r:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ as follows:

$$r\left(x\right)=\left\{\begin{array}{cc}a& \mathit{if}x\ge 0\\ b& \mathit{if}x<0\end{array}\right.\mathit{and}w\left(t\right)=t\mathit{for}\mathit{all}t\in \mathcal{P}.$$

Then, $f_{rw}:\left(X,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(X,\tau \left(\aleph \right),\mathcal{P}\right)$ is soft F-S-C but it is not soft F-P-C.

Theorem 19. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function. If $f_{r^{\#}{w}^{\#}}:\left(O,\phi ,\mathcal{P}\right)⟶\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times \mathcal{S}\right)$ is soft F-P-C, then $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-P-C.

Proof. 

Let $H\in {\sigma }_{\theta }$. Since $1_{\mathcal{P}}\in {\phi }_{\theta }$, by Theorem 10 of [15], $1_{\mathcal{P}}\times H\in {\left(pr\left(\phi \times \sigma \right)\right)}_{\theta }$. Since $f_{p^{\#}{u}^{\#}}:\left(O,\phi ,\mathcal{P}\right)⟶\left(R\times L,pr\left(\phi \times \sigma \right),M\times N\right)$ is soft F-P-C, by Theorem 13b, $f_{r^{\#}{w}^{\#}}^{-1}\left(1_{\mathcal{P}}\times H\right)\in PO\left(\phi \right)$. By Lemma 2 of [15], $f_{r^{\#}{w}^{\#}}^{-1}\left(1_{\mathcal{P}}\times H\right)=1_{\mathcal{P}}\tilde{\cap }{f}_{rw}^{-1}\left(H\right)=f_{rw}^{-1}\left(H\right)$, and, hence, $f_{rw}^{-1}\left(H\right)\in PO\left(\phi \right)$. Thus, again, by Theorem 13b, $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-P-C. □

Proposition 2. 

Let $\left(O,\phi ,\mathcal{P}\right)$ be an STS. If $A\in PO\left(\phi \right)$ and $X$ is a non-empty subset of O such that $C_X\in SO\left(\phi \right)$, then $A\tilde{\cap }{C}_X\in PO\left({\phi }_X\right)$.

Proof. 

Since $A\in PO\left(\phi \right)$ and $C_X\in SO\left(\phi \right)$, there exist $U,V\in \phi$ such that $A\tilde{\subseteq }U\tilde{\subseteq }C{l}_{\phi }\left(A\right)$ and $V\tilde{\subseteq }{C}_X\tilde{\subseteq }C{l}_{\phi }\left(V\right)$. We have $U\tilde{\cap }{C}_X\in {\phi }_X$ and $A\tilde{\cap }{C}_X\tilde{\subseteq }U\tilde{\cap }{C}_X$. □

Claim 1. 

$U\tilde{\cap }{C}_X\tilde{\subseteq }C{l}_{{\phi }_X}\left(A\tilde{\cap }{C}_X\right)$ and, hence, $A\tilde{\cap }{C}_X\in PO\left({\phi }_X\right)$.

Proof of Claim 1. 

Let $b_y\tilde{\in }U\tilde{\cap }{C}_X$. To see that $b_y\tilde{\in }C{l}_{{\phi }_X}\left(A\tilde{\cap }{C}_X\right)$, let $G\in {\phi }_X$ such that $b_y\tilde{\in }G$. We choose $H\in \phi$ such that $G=H\tilde{\cap }{C}_X$. Since $b_y\tilde{\in }{C}_X\tilde{\subseteq }C{l}_{\phi }\left(V\right)$ and $b_y\tilde{\in }U\tilde{\cap }H\in \phi$, $U\tilde{\cap }H\tilde{\cap }V\ne 0_{\mathcal{P}}$. We choose $d_z\tilde{\in }U\tilde{\cap }H\tilde{\cap }V$. Since $d_z\tilde{\in }U\tilde{\subseteq }C{l}_{\phi }\left(A\right)$ and $d_z\tilde{\in }H\tilde{\cap }V\in \phi$, $A\tilde{\cap }H\tilde{\cap }V\ne 0_{\mathcal{P}}$. Since $A\tilde{\cap }H\tilde{\cap }V\tilde{\subseteq }A\tilde{\cap }H\tilde{\cap }{C}_X=\left(H\tilde{\cap }{C}_X\right)\tilde{\cap }\left(A\tilde{\cap }{C}_X\right)=G\tilde{\cap }\left(A\tilde{\cap }{C}_X\right)$, then $G\tilde{\cap }\left(A\tilde{\cap }{C}_X\right)\ne 0_{\mathcal{P}}$. Therefore, $b_y\tilde{\in }C{l}_{{\phi }_X}\left(A\tilde{\cap }{C}_X\right)$. □

Theorem 20. 

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

Proof. 

Let $K\in {\sigma }_{\theta }$. Since $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-P-C, by Theorem 13b, $f_{rw}^{-1}\left(K\right)\in PO\left(\phi \right)$. So, by Proposition 2, ${\left({\left(f_{rw}\right)}_{\left|C_X\right.}\right)}^{-1}\left(H\right)=C_X\tilde{\cap }{f}_{rw}^{-1}\left(H\right)\in PO\left(\phi \right)$. Hence, again, by Theorem 13b, ${\left(f_{rw}\right)}_{\left|C_X\right.}:\left(X,{\phi }_X,M\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-P-C. □

Theorem 21. 

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

Proof. 

Let $K\in {\lambda }_{\theta }$. Then, $f_{r_2{w}_2}^{-1}\left(K\right)\in {\sigma }_{\theta }$ and, hence, $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}^{-1}\left(K\right)=f_{r_1{w}_1}^{-1}\left(f_{r_2{w}_2}^{-1}\left(K\right)\right)\in PO\left(\phi \right)$. Therefore, $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}$ is soft F-P-C. □

The following result follows from Theorems 11 and 21.

Corollary 8. 

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

Corollary 9. 

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

Proof. 

It follows from Corollary 8 and the fact that soft continuous functions are soft $\theta$-continuous. □

The subsequent illustrations demonstrate that, according to Corollary 9, the composition in reverse order may not possess the same quality.

Example 8. 

Let $X=\mathbb{R}$ and $\mathcal{P}=\mathbb{N}$. Let ℑ, ℵ, and ℘ be the usual, indiscrete, and discrete topologies on X, respectively. Let $r_1:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$, $r_2:\left(X,\Im \right)⟶$$\left(X,\wp \right)$, and $w:\mathcal{P}⟶\mathcal{P}$ be the identity functions. Then, $f_{r_{1}w}:\left(X,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(X,\tau \left(\aleph \right),\mathcal{P}\right)$ is soft continuous and $f_{r_{2}w}:\left(X,\tau \left(\aleph \right),\mathcal{P}\right)⟶\left(X,\tau \left(\wp \right),\mathcal{P}\right)$ is soft P-C (and, hence, soft F-P-C) while $f_{\left(r_2\circ r_1\right)\left(w\circ w\right)}$ is not soft F-P-C.

4. Soft Faint $\mathit{\beta }$-Continuity

Definition 13. 

A soft function $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is called soft faintly β-continuous (soft F-β-C, for short) if, for each $a_t\in$ SP(O,$\mathcal{P})$ and $K\in {\sigma }_{\theta }$ such that $f_{rw}\left(a_t\right)\tilde{\in }K$, we find $H\in \beta O\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }K$.

Theorem 22. 

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 F-β-C.

(b) 

$f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,{\sigma }_{\theta },\mathcal{S}\right)$ is soft β-C.

(c) 

$f_{rw}^{-1}\left(K\right)\in \beta O\left(\phi \right)$ for every $K\in {\sigma }_{\theta }$.

(d) 

$f_{rw}^{-1}\left(G\right)\in \beta C\left(\phi \right)$ for every $G\in {\left({\sigma }_{\theta }\right)}^c$.

(e) 

$\beta C{l}_{\phi }\left(f_{pu}^{-1}\left(H\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$ for every $H\in SS(T,\mathcal{S})$.

(f) 

$f_{pu}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(K\right)\right)\tilde{\subseteq }\beta In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$ for every $K\in SS(T,\mathcal{S})$.

Proof. 

(a)

⟶ (b): Let $K\in {\sigma }_{\theta }$ and let $m_r\tilde{\in }{f}_{rw}^{-1}\left(K\right)$. Then, $f_{rw}\left(a_t\right)\tilde{\in }K$, and, by (a), we find $H\in \beta O\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)\tilde{\subseteq }K$. Hence, $a_t\tilde{\in }H\tilde{\subseteq }{f}_{rw}^{-1}\left(f_{rw}\left(H\right)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(K\right)$. Therefore, $f_{rw}^{-1}\left(K\right)\in \beta O\left(\phi \right)$.

(b)

⟶ (c): Clear.

(c)

⟶ (d): Let $G\in {\left({\sigma }_{\theta }\right)}^c$. Then, $1_{\mathcal{S}}-G\in {\sigma }_{\theta }$, and, by (c), $f_{rw}^{-1}(1_{\mathcal{S}}-G)=1_{\mathcal{P}}-f_{rw}^{-1}\left(G\right)\in \beta O\left(\phi \right)$. Hence, $f_{rw}^{-1}\left(G\right)\in \beta C\left(\phi \right)$.

(d)

⟶ (e): Let $H\in SS(T,\mathcal{S})$. Then, $C{l}_{{\sigma }_{\theta }}\left(H\right)\in {\left({\sigma }_{\theta }\right)}^c$ and, by (d), $f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\in \beta C\left(\phi \right)$. Thus, $f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)=\beta C{l}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\right)$. Since $f_{rw}^{-1}\left(H\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$, then $\beta C{l}_{\phi }\left(f_{rw}^{-1}\left(H\right)\right)\tilde{\subseteq }\beta C{l}_{\phi }\left(f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)\right)=f_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}\left(H\right)\right)$.

(e)

⟶ (f): Let $K\in SS(T,\mathcal{S})$. Then, by (e), $\beta C{l}_{\phi }\left(f_{rw}^{-1}(1_{\mathcal{S}}-K)\right)\tilde{\subseteq }{f}_{rw}^{-1}\left(C{l}_{{\sigma }_{\theta }}(1_{\mathcal{S}}-K)\right)$. However,

$$\begin{array}{ccc}\beta C{l}_{\phi }\left(f_{rw}^{-1}(1_{\mathcal{S}}-K)\right)& =& \beta C{l}_{\phi }(1_{\mathcal{P}}-f_{rw}^{-1}\left(K\right))\\ & =& 1_{\mathcal{P}}-\beta In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)\end{array}$$

and

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

Thus, $1_{\mathcal{P}}-\beta In{t}_{\phi }\left(f_{rw}^{-1}\left(G\right)\right)\tilde{\subseteq }{1}_{\mathcal{P}}-f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(G\right)\right)$ and, hence, $f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(G\right)\right)\tilde{\subseteq }\beta In{t}_{\phi }$$\left(f_{rw}^{-1}\left(G\right)\right)$.

(f)

⟶ (a): Let $a_t\in SP(O,\mathcal{P})$ and $K\in {\sigma }_{\theta }$ such that $f_{rw}\left(a_t\right)\tilde{\in }K$. Then, $In{t}_{{\sigma }_{\theta }}\left(K\right)=K$. So, by (f), $f_{rw}^{-1}\left(K\right)=f_{rw}^{-1}\left(In{t}_{{\sigma }_{\theta }}\left(K\right)\right)\tilde{\subseteq }\beta In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$ and, thus, $f_{rw}^{-1}\left(K\right)\tilde{\subseteq }\beta In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$. Therefore, $f_{rw}^{-1}\left(K\right)=\beta In{t}_{\phi }\left(f_{rw}^{-1}\left(K\right)\right)$, and, hence, $f_{rw}^{-1}\left(K\right)\in \beta O\left(\phi \right)$. We set $H=f_{rw}^{-1}\left(K\right)$. Then, $H\in \beta O\left(\phi \right)$ such that $a_t\tilde{\in }H$ and $f_{rw}\left(H\right)=f_{rw}\left(f_{rw}^{-1}\left(K\right)\right)\tilde{\subseteq }K$. Therefore, $f_{rw}$ is soft F-$\beta$-C.

□

Theorem 23. 

Let $\left\{\left(X,{\phi }_a\right):a\in \mathcal{P}\right\}$ and $\left\{\left(Y,{\sigma }_b\right):b\in \mathcal{S}\right\}$ be two collections of TSs. We consider the functions $r:X⟶Y$ and $w:\mathcal{P}⟶\mathcal{S}$, where w is a bijection. Then, $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft F-β-C iff $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is F-β-C for all $a\in \mathcal{P}$.

Proof. 

Necessity. Let $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft F-$\beta$-C. Let $a\in \mathcal{P}$. Let $U\in {\left({\sigma }_{w\left(a\right)}\right)}_{\theta }$. Then, according to Theorem 2.21 of [45], ${\left(w\left(a\right)\right)}_U\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. So, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)\in \beta \left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)=a_{r^{-1}\left(U\right)}$. Therefore, by Theorem 1 of [13], $\left(a_{r^{-1}\left(U\right)}\right)\left(a\right)=r^{-1}\left(U\right)\in \beta \left({\phi }_a\right)$. This shows that $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is F-$\beta$-C.

Sufficiency. Let $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is F-$\beta$-C for all $a\in \mathcal{P}$. Let $G\in {\left({\oplus }_{b\in \mathcal{S}}{\sigma }_b\right)}_{\theta }$. Then, by Theorem 2.21 of [45], $G\left(b\right)\in {\left({\sigma }_b\right)}_{\theta }$ for all $b\in \mathcal{S}$. For every $b\in \mathcal{S}$, $r:\left(X,{\phi }_{w^{-1}\left(b\right)}\right)⟶\left(Y,{\sigma }_b\right)$ is F-$\beta$-C, and, so, $r^{-1}\left(G\left(b\right)\right)\in \beta O\left(\phi \right)$. Hence, for each $a\in \mathcal{P}$, $\left(f_{rw}^{-1}\left(G\right)\right)\left(a\right)=r^{-1}(G\left(w\left(a\right)\right)\in \beta O\left(\phi \right)$. Therefore, by Theorem 1 of [13], $f_{rw}^{-1}\left(G\right)\in \beta \left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. This shows that $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft F-$\beta$-C. □

Corollary 10. 

We consider the 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 F-β-C iff $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(T,\tau \left(\aleph \right),\mathcal{S})$ is soft F-β-C.

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we set ${\phi }_a=\Im$ 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 23 ends the proof. □

Theorem 24. 

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

Proof. 

Necessity. Let $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ be soft $\beta$-C. Let $a\in \mathcal{P}$. Let $U\in {\left({\sigma }_{w\left(a\right)}\right)}_{\theta }$. Then, ${\left(w\left(a\right)\right)}_U\in {\oplus }_{b\in \mathcal{S}}{\sigma }_b$. So, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)\in \beta \left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. Since $w:\mathcal{P}⟶\mathcal{S}$ is injective, $f_{rw}^{-1}\left({\left(w\left(a\right)\right)}_U\right)=a_{r^{-1}\left(U\right)}$. Therefore, by Theorem 1 of [13], $\left(a_{r^{-1}\left(U\right)}\right)\left(a\right)=r^{-1}\left(U\right)\in \beta \left({\phi }_a\right)$. This shows that $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is $\beta$-C.

Sufficiency. $r:\left(X,{\phi }_a\right)⟶\left(Y,{\sigma }_{w\left(a\right)}\right)$ is $\beta$-C for all $a\in \mathcal{P}$. Let $G\in {\oplus }_{b\in \mathcal{S}}{\sigma }_b$. Then, $G\left(b\right)\in {\sigma }_b$ for all $b\in \mathcal{S}$. For every $b\in \mathcal{S}$, $r:\left(X,{\phi }_{w^{-1}\left(b\right)}\right)⟶\left(Y,{\sigma }_b\right)$ is $\beta$-C, and, so, $r^{-1}\left(G\left(b\right)\right)\in \beta O\left(\phi \right)$. Hence, for each $a\in \mathcal{P}$, $\left(f_{rw}^{-1}\left(G\right)\right)\left(a\right)=r^{-1}(G\left(w\left(a\right)\right)\in \beta O\left(\phi \right)=\beta O\left(\phi \right)$. Therefore, by Theorem 1 of [13], $f_{rw}^{-1}\left(G\right)\in \beta \left({\oplus }_{a\in \mathcal{P}}{\phi }_a\right)$. This shows that $f_{rw}:\left(X,{\oplus }_{a\in \mathcal{P}}{\phi }_a,\mathcal{P}\right)⟶$$\left(Y,{\oplus }_{b\in \mathcal{S}}{\sigma }_b,\mathcal{S}\right)$ is soft $\beta$-C. □

Corollary 11. 

We consider the 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 β-C iff $f_{rw}:(O,\tau \left(\Im \right),\mathcal{P})⟶(T,\tau \left(\aleph \right),\mathcal{S})$ is soft β-C.

Proof. 

For each $a\in \mathcal{P}$ and $b\in \mathcal{S}$, we set ${\phi }_a=\Im$ 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 24 ends the proof. □

Theorem 25. 

Every soft F-C function is soft F-β-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft F-C. Let $G\in {\sigma }_{\theta }$. Then, $f_{rw}^{-1}\left(G\right)\in \phi \subseteq \beta O\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-$\beta$-C. □

Theorem 26. 

Every soft β-C function is soft F-β-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft $\beta$-C. Let $G\in {\sigma }_{\theta }$. Then, $G\in \sigma$. So, $f_{rw}^{-1}\left(G\right)\in \beta O\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-$\beta$-C. □

Theorem 27. 

Every soft F-S-C function is soft F-β-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft F-S-C. Let $G\in {\sigma }_{\theta }$. Then, $f_{rw}^{-1}\left(G\right)\in SO\left(\phi \right)\subseteq \beta O\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-$\beta$-C. □

Theorem 28. 

Every soft F-P-C function is soft F-β-C.

Proof. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be soft F-P-C. Let $G\in {\sigma }_{\theta }$. Then, $f_{rw}^{-1}\left(G\right)\in PO\left(\phi \right)\subseteq \beta O\left(\phi \right)$. Consequently, $f_{rw}$ is soft F-$\beta$-C. □

The converse of Theorem 25 need not be true in general.

Example 9. 

Let $X=\mathbb{R}$, $\mathcal{P}=\left\{a,b\right\}$, and ℑ the usual topology on X. We define the functions $r:\left(X,\Im \right)⟶$$\left(X,\Im \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ by

$$r\left(x\right)=\left\{\begin{array}{cc}1& \mathit{if}x\in \mathbb{Q}\\ -1& \mathit{if}x\in \mathbb{R}-\mathbb{Q}\end{array}\right.,w\left(a\right)=a,\mathit{and}w\left(b\right)=b.$$

Then, $f_{rw}:(X,\tau \left(\Im \right),\mathcal{P})⟶(X,\tau \left(\Im \right),\mathcal{P})$ is soft F-β-C but not soft F-C.

The converse of Theorem 26 need not be true in general.

Example 10. 

Let $X=\left\{1,2,3,4\right\}$, $\Im =\left\{\varnothing ,X,\left\{2\right\},\left\{3\right\},\left\{2,3\right\},\left\{1,2\right\},\left\{1,2,3\right\},\left\{2,3,4\right\}\right\}$, and $\mathcal{P}=\mathbb{N}$. We define $r:\left(X,\Im \right)⟶$$\left(X,\Im \right)$ and $w:\mathcal{P}⟶\mathcal{P}$ by $r\left(1\right)=3$, $r\left(2\right)=4$, $r\left(3\right)=2$, $r\left(4\right)=1$, and $w\left(t\right)=t$ for all $t\in \mathcal{P}$. Then, $f_{rw}:(X,\tau \left(\Im \right),\mathcal{P})⟶(Y,\tau \left(\aleph \right),\mathcal{P})$ is soft F-β-C but not soft β-C.

The soft function in Example 6 is soft F-$\beta$-C but not soft F-S-C. Furthermore, the soft function in Example 7 is soft F-$\beta$-C but not soft F-P-C. Therefore, Theorems 27 and 28 are not reversible, in general.

Theorem 29. 

Let $\left(T,\sigma ,\mathcal{S}\right)$ be soft regular. The following are equivalent for a soft function$f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$:

(a) 

$f_{rw}$ is soft β-C.

(b) 

$f_{rw}$ is soft F-β-C.

Proof. 

(a)

⟶ (b): Follows from Theorem 26.

(b)

⟶ (c): Let $K\in \sigma$. Since $\left(T,\sigma ,\mathcal{S}\right)$ is soft regular, then $\sigma ={\sigma }_{\theta }$. So, $K\in {\sigma }_{\theta }$ and, by (b), $f_{rw}^{-1}\left(K\right)\in \beta O\left(\phi \right)$. This shows that $f_{rw}$ is soft $\beta$-C.

□

Theorem 30. 

Let $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ be a soft function. If $f_{r^{\#}{w}^{\#}}:\left(O,\phi ,\mathcal{P}\right)⟶\left(O\times T,pr\left(\phi \times \sigma \right),\mathcal{P}\times \mathcal{S}\right)$ is soft F-β-C, then $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-β-C.

Proof. 

Let $H\in {\sigma }_{\theta }$. Since $1_{\mathcal{P}}\in {\phi }_{\theta }$, by Theorem 10 of [15], $1_{\mathcal{P}}\times H\in {\left(pr\left(\phi \times \sigma \right)\right)}_{\theta }$. Since $f_{p^{\#}{u}^{\#}}:\left(O,\phi ,\mathcal{P}\right)⟶\left(R\times L,pr\left(\phi \times \sigma \right),M\times N\right)$ is soft F-$\beta$-C, by Theorem 22b, $f_{r^{\#}{w}^{\#}}^{-1}\left(1_{\mathcal{P}}\times H\right)\in \beta O\left(\phi \right)$. By Lemma 2 of [15], $f_{r^{\#}{w}^{\#}}^{-1}\left(1_{\mathcal{P}}\times H\right)=1_{\mathcal{P}}\tilde{\cap }{f}_{rw}^{-1}\left(H\right)=f_{rw}^{-1}\left(H\right)$, and, hence, $f_{rw}^{-1}\left(H\right)\in \beta O\left(\phi \right)$. Thus, again, by Theorem 22b, $f_{rw}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-$\beta$-C. □

Theorem 31. 

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

Proof. 

Let $K\in {\lambda }_{\theta }$. Then, $f_{r_2{w}_2}^{-1}\left(K\right)\in {\sigma }_{\theta }$ and, hence, $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}^{-1}\left(K\right)=f_{r_1{w}_1}^{-1}\left(f_{r_2{w}_2}^{-1}\left(K\right)\right)\in \beta O\left(\phi \right)$. Therefore, $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}$ is soft F-$\beta$-C. □

The following result follows from Theorems 11 and 31.

Corollary 12. 

If $f_{r_1{w}_1}:\left(O,\phi ,\mathcal{P}\right)⟶\left(T,\sigma ,\mathcal{S}\right)$ is soft F-β-C and $f_{r_2{w}_2}:\left(T,\sigma ,\mathcal{S}\right)⟶\left(N,\lambda ,\mathcal{L}\right)$ is soft θ-continuous, then $f_{\left(r_2\circ r_1\right)\left(w_2\circ w_1\right)}$ is soft F-β-C.

Corollary 13. 

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

Proof. 

It follows from Corollary 12 and the fact that soft continuous functions are soft $\theta$-continuous. □

The subsequent illustrations demonstrate that, according to Corollary 13, the composition in reverse order may not possess the same quality.

Example 11. 

Let $X=\mathbb{R}$ and $\mathcal{P}=\mathbb{N}$. Let ℑ, ℵ, and ℘ be the usual, indiscrete, and discrete topologies on X, respectively. Let $r_1:\left(X,\Im \right)⟶$$\left(X,\aleph \right)$, $r_2:\left(X,\Im \right)⟶$$\left(X,\wp \right)$, and $w:\mathcal{P}⟶\mathcal{P}$ be the identity functions. Then, $f_{r_{1}w}:\left(X,\tau \left(\Im \right),\mathcal{P}\right)⟶\left(X,\tau \left(\aleph \right),\mathcal{P}\right)$ is soft continuous and $f_{r_{2}w}:\left(X,\tau \left(\aleph \right),\mathcal{P}\right)⟶\left(X,\tau \left(\wp \right),\mathcal{P}\right)$ is soft β-C (and, hence, soft F-β-C), while $f_{\left(r_2\circ r_1\right)\left(w\circ w\right)}$ is not soft F-β-C.

5. Conclusions

Many of the things we deal with daily include ambiguity. Soft set theory and its associated notions are one of the most significant theories for addressing uncertainty. One of the most significant frameworks to come out of soft set theory is soft topology. One of the most crucial ideas in soft topology is that of soft continuity, which is the subject of this paper.

In this paper, three generalizations of soft continuity—soft faint semi-continuity, soft faint pre-continuity, and soft faint $\beta$-continuity—were defined and explored. We characterized each of them (Theorems 1, 13, and 22). We also investigated the correspondence between each of them and their analog concept in general topology (Theorems 2, 14, and 23, and Corollaries 1, 6, and 10). Moreover, we proved that soft faint semi-continuity is strictly weaker than both soft faint continuity (Theorem 4 and Example 1) and soft semi-continuity (Theorem 5 and Example 2), and we proved that soft faint pre-continuity is strictly weaker than both soft faint continuity (Theorem 16 and Example 4) and soft pre-continuity (Theorem 17 and Example 5). We also proved that the concepts soft faint semi-continuity and soft faint pre-continuity are independent (Examples 6 and 7). In addition, we proved that soft faint $\beta$-continuity is a strictly weaker form of each of soft $\beta$-continuity (Theorem 26 and Example 10), soft faint semi-continuity (Theorem 27 and Example 6), and soft faint pre-continuity (Theorem 28 and Example 7). Soft regularity, on the codomain of a soft function, is given as a sufficient condition for the equivalence between soft faint semi-continuity (resp. soft faint pre-continuity, soft faint $\beta$-continuity) and soft semi-continuity (resp. soft pre-continuity, soft $\beta$-continuity). In addition to these, we provided several results on soft restriction (Theorems 8 and 20), soft composition (Theorems 12, 21, and 31 and Corollaries 4, 5, 8, 9, 12, and 13), and soft graph (Theorems 7, 19, and 30).

Future research might look into the following topics: (1) defining soft weakly quasi-continuous functions; (2) defining soft almost weakly continuous functions; (3) finding a use for these new concepts in a “decision making problem”; and (4) extending the concept of $\theta$-Menger spaces [48] to include STSs.