Soft Slight Omega-Continuity and Soft Ultra-Separation Axioms

Abstract

The notions of continuity and separation axioms have significance in topological spaces. As a result, there has been a substantial amount of research on continuity and separation axioms, leading to the creation of several modifications of these axioms. In this paper, the concepts of soft slight $\omega$-continuity, soft ultra-Hausdorff, soft ultra-regular, and soft ultra-normal are initiated and investigated. Their characterizations and main features are determined. Also, the links between them and some other relevant concepts are obtained with the help of examples. Moreover, the equivalency between these notions and other related concepts is given under some necessary conditions. In addition, the inverse image of the introduced types of soft separation axioms under soft slight continuity and soft slight $\omega$-continuity is studied, and their reciprocal relationships with respect to their parametric topological spaces are investigated.

1. Introduction and Preliminaries

Real-world problems involve a lot of inherent uncertainty. Therefore, it is not possible to resolve these challenges utilizing traditional methods. For more than 30 years, fuzzy set theory, rough set theory, and ambiguous set theory have been instrumental in tackling these problems. According to Molodostov [1], each of these mathematical creations has its restrictions. Nowadays, most scientific fields use soft sets for their study. Soft set theory has been utilized in combination with game theory, function smoothness, operations research, measure theory, probability, and decision-making challenges to address issues [2,3,4,5,6].

Maji et al. [7] conducted the initial attempt to identify the key features of soft set theory. They introduced fundamental concepts of soft sets, including soft intersection and soft union operators, a soft set complement, and absolute and null soft sets. Numerous new mathematical structures, including soft group theory, soft ring theory, etc., have been studied since Maji’s contribution.

Soft sets, like fuzzy set topology, were utilized by topologists to construct the concept of soft topology by Shabir and Naz in [8]. Following that, topologists have defined various concepts related to soft topological spaces and their soft subsets.

The authors in [9] defined soft continuity. Then, different modifications of soft continuity and soft openness of functions appeared in the literature. For instance, soft $\alpha$-continuous functions [10], soft $\beta$-continuous functions [11], soft b-continuous functions [12], soft $\theta$-continuous functions [13], soft $\omega$-continuous functions [14], soft $\omega$-irresolute functions [15], Soft somewhat continuous functions [16], soft $b^{*}$-continuous functions [17], soft bi-continuous functions [18], weakly soft $\alpha$-continuous functions [19], weakly soft semi-continuous functions [20], soft somewhere dense open functions [21], and so on. Moreover, several important modifications of soft continuity appeared in [22,23,24].

Separation axioms divide spaces and mappings into families and serve as a starting point for investigating various aspects of compact and Lindelof spaces. As a result, topological researchers focused on soft separation axioms, displaying several sorts of soft separation axioms [25,26,27].

In this paper, we use the class of soft clopen sets to introduce soft slight $\omega$-continuity and several new soft separation axioms. We obtain several characterizations and relationships regarding each of these new concepts. Moreover, we study the relationships between these concepts and their analogous general topological concepts.

This work follows the concepts and terminology outlined in [28,29]. In this work, STS and TS will be used to represent soft topological space and topological space, respectively.

Let D be a collection of parameters and T be a general set. A function $H:D⟶\mathcal{P}\left(T\right)$ is a soft set over T with respect to D. $SS(T,D)$ stands for the family of all soft sets over T relative to D. The absolute soft set and the null soft set shall be identified in this research as $0_D$ and $1_D$, respectively. According to [5], STSs are a modern mathematical framework that goes as follows: An STS is a triplet $\left(T,\Gamma ,D\right)$, where $\left\{0_D,1_D\right\}\subseteq \Gamma \subseteq SS\left(T,D\right)$, and $\Gamma$ is closed under a finite soft intersection and closed under an arbitrary soft union. In an STS $\left(T,\Gamma ,D\right)$, the members of $\Gamma$ are called soft open sets and their soft complements are called soft closed sets.

Let $\left(T,\beta \right)$ and $\left(T,\Gamma ,D\right)$ be a TS and an STS, respectively. Let $W\subseteq T$ and $L\in SS(T,D)$. In this paper, the terms $In{t}_{\beta }\left(W\right)$, $C{l}_{\beta }\left(W\right)$, $In{t}_{\Gamma }\left(L\right)$, and $C{l}_{\Gamma }\left(L\right)$, respectively, will be used to refer to the interior of W in $\left(T,\beta \right)$, the closure of W in $\left(T,\beta \right)$, the soft interior of L in $\left(T,\Gamma ,D\right)$, and the soft closure of L in $\left(T,\Gamma ,D\right)$; while ${\beta }^c$, ${\Gamma }^c$, $CO\left(T,\beta \right)$, $CO\left(T,\Gamma ,D\right)$, and ${\Gamma }^{*}$ stand for the family of closed sets on $\left(T,\beta \right)$, the family of soft closed sets on $\left(T,\Gamma ,D\right)$, the family of clopen sets on $\left(T,\beta \right)$, the family of soft clopen sets on $\left(T,\Gamma ,D\right)$, and the soft topology on T relative to D having $CO\left(T,\Gamma ,D\right)$ as a soft base, respectively.

The following definitions are used in the sequel:

Definition 1.

A TS$\left(T,\beta \right)$ is called:

 (a) 

[30] zero-dimensional (0-dim, for short) if $CO\left(T,\beta \right)$ forms a base for $\left(T,\beta \right)$;

 (b) 

[31] ultra-Hasdorff if for any $x,y\in T$ such that $x\ne y$, we find V, $W\in CO\left(T,\beta \right)$ such that $x\in V$, $y\in W$, and $V\cap W=\varnothing$;

 (c) 

[31] ultra-regular if for any $x\in T$ and any $U\in T^c$ such that $x\in T-U$, we find V, $W\in CO\left(T,\beta \right)$ such that $x\in V$, $U\subseteq W$, and $V\cap W=\varnothing$;

 (d) 

[31] ultra-normal if for any $U,Z\in T^c$ such that $U\cap Z=\varnothing$, we find V, $W\in CO\left(T,\beta \right)$ such that $U\subseteq V$, $Z\subseteq W$, and $V\cap W=\varnothing$;

Definition 2.

An STS$\left(T,\Gamma ,D\right)$ is called:

 (a) 

[32] soft connected if $CO\left(T,\Gamma ,D\right)=\left\{0_D,1_D\right\}$ and called soft disconnected if it is not soft connected;

 (b) 

soft zero-dimensional (soft 0-dim, for short) if $CO\left(T,\Gamma ,D\right)$ forms a soft base for $\left(T,\Gamma ,D\right)$;

 (c) 

[33] soft Hausdorff space if for any two soft points $a_x,b_y\in SP(T,D)$, we find $L,M\in \Gamma$ such that $a_x\tilde{\in }L$, $b_y\tilde{\in }M$, and $L\tilde{\cap }M=0_D$;

 (d) 

[33] soft regular if for each $a_x\in SP(T,D)$ and each $G\in \Gamma$ such that $a_x\tilde{\in }G$, we find $L\in \Gamma$ such that $a_x\tilde{\in }L\tilde{\subseteq }C{l}_{\Gamma }\left(L\right)\tilde{\subseteq }G$;

 (e) 

[33] soft normal if for any $K,H\in {\Gamma }^c$ such that $K\tilde{\cap }H=0_D$, we find L, $M\in CO\left(T,\Gamma ,D\right)$ such that $K\tilde{\subseteq }L$, $H\tilde{\subseteq }M$, and $L\tilde{\cap }M=0_D$;

 (f) 

[34] soft Urysohn space if for any two soft points $a_x,b_y\in SP(T,D)$, we find $L,M\in \Gamma$ such that $a_x\tilde{\in }L$, $b_y\tilde{\in }M$, and $C{l}_{\Omega }\left(L\right)\tilde{\cap }C{l}_{\Omega }\left(M\right)=0_D$;

 (g) 

[35] soft locally indiscrete if $\Gamma \subseteq {\Gamma }^c$.

Definition 3.

Let $\left(X,\delta \right)$ and $\left(Y,\beta \right)$ be TSs. A function $g:\left(X,\delta \right)⟶\left(Y,\beta \right)$ is said to be the following:

 (a) 

[36] slightly continuous (s-c, for short) at $x\in X$ if for every $V\in CO\left(Y,\beta \right)$ such that $g\left(x\right)\in V$, we find $U\in \delta$ such that $x\in U$ and $g\left(U\right)\subseteq V$. If g is s-c at each $x\in X$, then g is called s-c.

 (b) 

[37] ω-continuous (ω-c, for short) at $x\in X$ if for every $V\in \beta$ such that $g\left(x\right)\in V$, we find $U\in {\delta }_{\omega }$ such that $x\in U$ and $g\left(U\right)\subseteq V$. If g is ω-c at each $x\in X$, then g is called ω-c.

 (c) 

[38] slightly ω-continuous (s-ω-c, for short) at $x\in X$ if for every $V\in CO\left(Y,\beta \right)$ such that $g\left(x\right)\in V$, we find $U\in {\delta }_{\omega }$ such that $x\in U$ and $g\left(U\right)\subseteq V$. If g is s-ω-c at each $x\in X$, then g is called s-ω-c.

Definition 4.

A soft set $K\in SS(X,M)$ defined by the following:

 (1) 

[28] $K\left(m\right)=\left\{\begin{array}{cc}V& \mathrm{if}m=b\\ \varnothing & \mathrm{if}m\ne b\end{array}\right.$ is denoted by $b_V$.

 (2) 

[39] $K\left(m\right)=V$ for all $m\in M$ is denoted by $C_V$.

 (3) 

[40] $K\left(m\right)=\left\{\begin{array}{cc}\left\{y\right\}& \mathrm{if}m=b\\ \varnothing & \mathrm{if}m\ne b\end{array}\right.$ is denoted by $b_y$ and is called a soft point. The set of all soft points in $SS(X,M)$ is denoted by $SP(X,M)$.

Definition 5 ([34]).

Let $K\in SS(X,M)$ and $b_y\in SP(X,M)$. Then, $b_y$ is said to belong to K (notation: $b_y\tilde{\in }K$) if $y\in K\left(b\right)$.

Definition 6.

Let $\left(X,\Omega ,M\right)$ and $\left(Y,{\rm Y},N\right)$ be STSs. A soft function $f_{pu}:\left(X,\Omega ,M\right)⟶\left(Y,{\rm Y},N\right)$ is said to be the following:

 (a) 

[41] soft slightly continuous (soft s-c, for short) at $m_x\in SP(X,M)$ if for every $G\in CO\left(Y,{\rm Y},N\right)$ such that $f_{pu}\left(m_x\right)\tilde{\in }G$, we find $H\in \Omega$ such that $m_x\tilde{\in }H$ and $f_{pu}\left(H\right)\tilde{\subseteq }G$. If $f_{pu}$ is soft s-c at each $m_x\in SP(X,M)$, then $f_{pu}$ is said to be soft s-c.

 (b) 

[14] soft ω-continuous (soft ω-c, for short) at $m_x\in SP(X,M)$ if for every $G\in {\rm Y}$ such that $f_{pu}\left(m_x\right)\tilde{\in }G$, we find $H\in {{\rm Y}}_{\omega }$ such that $m_x\tilde{\in }H$ and $f_{pu}\left(H\right)\tilde{\subseteq }G$. If $f_{pu}$ is soft ω-c at each $m_x\in SP(X,M)$, then $f_{pu}$ is said to be soft ω-c.

 (c) 

[15] soft ω-irresolute if for every $G\in {{\rm Y}}_{\omega }$ such that $f_{pu}^{-1}\left(G\right)\in \Omega$.

Definition  7 ([42]).

Let $\left(X,\Omega ,M\right)$ and $\left(Y,{\rm Y},N\right)$ be STSs. The soft topology over $X\times Y$ relative to $M\times N$ having $\left\{G\times H:G\in \Omega \mathrm{and}H\in {\rm Y}\right\}$ as a soft base will be denoted by $pr\left(\Omega \times {\rm Y}\right)$.

Definition 8 ([9]).

For any TS $\left(T,\mu \right)$, the soft topology $\left\{K\in SS\left(T,D\right):K\left(d\right)\in \mu \mathrm{for}\mathrm{every}d\in D\right\}$ on T relative to D will be denoted by $\tau \left(\mu \right)$.

Definition 9 ([28]).

For any collection of TSs $\left\{\left(T,{\psi }_d\right):d\in D\right\}$, the soft topology $\left\{H\in SS\left(T,D\right):H\left(d\right)\in {\psi }_d\mathrm{for}\mathrm{every}d\in D\right\}$ is denoted by ${\oplus }_{d\in D}{\psi }_d$.

2. Soft Slight $\mathbf{\omega }$-Continuity

Definition 10.

Let $f_{\mathit{pu}}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be a soft function. Then, it can be defined as follows:

 (a) 

$f_{\mathit{pu}}$ is called soft slightly ω-continuous (soft s-ω-c, for short) at a soft point $b_s\in SP\left(S,B\right)$ if for every $G\in CO\left(T,\Gamma ,D\right)$ such that $f_{\mathit{pu}}\left(b_s\right)\tilde{\in }G$, we find $H\in {\Gamma }_{\omega }$ such that $b_s\tilde{\in }H$ and $f_{\mathit{pu}}\left(H\right)\tilde{\subseteq }G$.

 (b) 

$f_{\mathit{pu}}$ is called soft s-ω-c if it is at soft s-ω-c at each soft point $b_s\in SP\left(S,B\right)$.

Theorem 1.

For any soft function $f_{\mathit{pu}}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$, the following are equivalent:

 (a) 

$f_{\mathit{pu}}$ is soft s-ω-c;

 (b) 

for every $G\in CO\left(T,\Gamma ,D\right)$, $f_{\mathit{pu}}^{-1}\left(G\right)\in {\Pi }_{\omega }$;

 (c) 

for every $G\in CO\left(T,\Gamma ,D\right)$, $f_{\mathit{pu}}^{-1}\left(G\right)\in {\left({\Pi }_{\omega }\right)}^c$;

 (d) 

for every $G\in CO\left(T,\Gamma ,D\right)$, $f_{\mathit{pu}}^{-1}\left(G\right)\in CO\left(S,{\Pi }_{\omega },B\right)$;

 (e) 

for every $G\in {\Gamma }^{*}$, $f_{\mathit{pu}}^{-1}\left(G\right)\in {\Pi }_{\omega }$;

 (f) 

for every $G\in {\left({\Gamma }^{*}\right)}^c$, $f_{\mathit{pu}}^{-1}\left(G\right)\in {\left({\Pi }_{\omega }\right)}^c$.

Proof. 

(a)

⟶ (b): Let $G\in CO\left(T,\Gamma ,D\right)$. Let $b_s\tilde{\in }{f}_{pu}^{-1}\left(G\right)$. Then, $f_{pu}\left(b_s\right)\tilde{\in }G$ and by (a), we find $H\in {\Pi }_{\omega }$ such that $b_s\tilde{\in }H$ and $f_{pu}\left(H\right)\tilde{\subseteq }G$. Thus, we have $b_s\tilde{\in }H\tilde{\subseteq }{f}_{pu}^{-1}$$\left(f_{pu}\left(H\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(G\right)$. Hence, $f_{pu}^{-1}\left(G\right)\in {\Pi }_{\omega }$.

(b)

⟶ (c): Let $G\in CO\left(T,\Gamma ,D\right)$. Then, $1_D-G\in CO\left(T,\Gamma ,D\right)$ and by (b), $f_{pu}^{-1}(1_D-G)=1_R-f_{pu}^{-1}\left(G\right)\in {\Gamma }_{\omega }$. Hence, $f_{pu}^{-1}\left(G\right)\in {\left({\Gamma }_{\omega }\right)}^c$.

(c)

⟶ (d): Let $G\in CO\left(T,\Gamma ,D\right)$. Then, $1_D-G\in CO\left(T,\Gamma ,D\right)$ and by (c), $f_{pu}^{-1}\left(G\right)\in {\left({\Pi }_{\omega }\right)}^c$ and $f_{pu}^{-1}(1_D-G)=1_R-f_{pu}^{-1}\left(G\right)\in {\left({\Pi }_{\omega }\right)}^c$. Hence, $f_{pu}^{-1}\left(G\right)\in CO\left(S,{\Pi }_{\omega },B\right)$.

(d)

⟶ (e): Let $G\in {\Gamma }^{*}$. Then, we find a subcollection $\mathcal{K}\subseteq CO\left(T,\Gamma ,D\right)$ such that $G={\tilde{\cup }}_{K\in \mathcal{K}}K$. Now, by (d), $f_{pu}^{-1}\left(K\right)\in CO\left(S,{\Pi }_{\omega },B\right)\subseteq {\Pi }_{\omega }$ for every $K\in \mathcal{K}$. Thus, $f_{pu}^{-1}\left(G\right)=f_{pu}^{-1}\left({\tilde{\cup }}_{K\in \mathcal{K}}K\right)={\tilde{\cup }}_{K\in \mathcal{K}}{f}_{pu}^{-1}\left(K\right)\in {\Pi }_{\omega }$.

(e)

⟶ (f): Let $G\in {\left({\Gamma }^{*}\right)}^c$. Then, $1_D-G\in {\Gamma }^{*}$ and by (e), $f_{pu}^{-1}(1_D-G)=1_R-f_{pu}^{-1}\left(G\right)\in {\Pi }_{\omega }$. Hence, $f_{pu}^{-1}\left(G\right)\in {\left({\Pi }_{\omega }\right)}^c$.

(f)

⟶ (a): Let $b_s\in SP\left(S,B\right)$ and $G\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(b_s\right)\tilde{\in }G$. Since $CO\left(T,\Gamma ,D\right)\subseteq {\Gamma }^{*}$, $1_D-G\in {\left({\Gamma }^{*}\right)}^c$ and by (f), $f_{pu}^{-1}(1_D-G)=1_R-f_{pu}^{-1}\left(G\right)\in {\left({\Pi }_{\omega }\right)}^c$. Put $H=f_{pu}^{-1}\left(G\right)$. Then, we have $b_s\tilde{\in }H\in {\Pi }_{\omega }$ and $f_{pu}\left(H\right)=f_{pu}\left(f_{pu}^{-1}\left(G\right)\right)\tilde{\subseteq }G$. Therefore, $f_{pu}$ is soft s-$\omega$-c.

□

Theorem 2.

Let$\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ and $\left\{\left(T,{\Gamma }_d\right):d\in D\right\}$ be two families of TSs. Let $p:S⟶T$ and $u:R⟶D$ be functions where u is bijective. Then, $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft s-ω-c if and only if $p:\left(S,{\Pi }_r\right)⟶\left(T,{\Gamma }_{u\left(r\right)}\right)$ is s-ω-c for all $r\in R$.

Proof. 

Necessity. Assume that $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft s-$\omega$-c. Let $t\in R$. Let $V\in CO\left(T,{\Gamma }_{u\left(t\right)}\right)$. Then, ${\left(u\left(t\right)\right)}_V\in CO\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$. Thus, by Theorem 1 (b), $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)\in {\left({\oplus }_{r\in R}{\Pi }_r\right)}_{\omega }$ and by Theorem 8 of [29], $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)\in {\oplus }_{r\in R}{\left({\Pi }_r\right)}_{\omega }$. Since $u:R⟶D$ is injective, $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)=t_{p^{-1}\left(V\right)}$. Thus, $\left(t_{p^{-1}\left(V\right)}\right)\left(t\right)=p^{-1}\left(V\right)\in {\left({\Pi }_t\right)}_{\omega }$. Therefore, $p:\left(S,{\Pi }_t\right)⟶\left(T,{\Gamma }_{u\left(t\right)}\right)$ is s-$\omega$-c.

Sufficiency. Assume that $p:\left(S,{\Pi }_r\right)⟶\left(T,{\Gamma }_{u\left(r\right)}\right)$ is s-$\omega$-c for all $r\in R$. Let $G\in CO\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$. Then, $G\left(d\right)\in CO\left(Y,{\Gamma }_d\right)$ for all $d\in D$. For every $d\in D$, $p:\left(S,{\Pi }_{u^{-1}\left(d\right)}\right)⟶\left(T,{\Gamma }_d\right)$ is s-$\omega$-c, and so $p^{-1}\left(G\left(d\right)\right)\in {\left({\Pi }_{u^{-1}\left(d\right)}\right)}_{\omega }$. Thus, for every $r\in R$, $\left(f_{pu}^{-1}\left(G\right)\right)\left(r\right)=p^{-1}(G\left(u\left(r\right)\right)\in {\Pi }_{{\left(u^{-1}\left(u\left(r\right)\right)\right)}_{\omega }}={\left({\Pi }_r\right)}_{\omega }$. Therefore, by Theorem 8 of [29], $f_{pu}^{-1}\left(G\right)\in {\left({\oplus }_{r\in R}{\Gamma }_r\right)}_{\omega }$. It follows that $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft s-$\omega$-c. □

Corollary 1.

Let $p:\left(S,\delta \right)⟶\left(T,\beta \right)$ and $u:R⟶D$ be two functions where u is a bijection. Then, $p:\left(S,\delta \right)⟶\left(T,\beta \right)$ is s-ω-c if and only if $f_{pu}:(S,\tau \left(\delta \right),R)⟶(T,\tau \left(\beta \right),D)$ is soft s-ω-c.

Proof. 

For every $r\in R$ and $d\in D$, let ${\Pi }_r=\delta$ and ${\Gamma }_d=\beta$. Then, $\tau \left(\delta \right)={\oplus }_{r\in R}{\Pi }_r$ and $\tau \left(\beta \right)={\oplus }_{d\in D}{\Gamma }_d$. We get the result as a consequence of Theorem 2. □

Theorem 3.

Let$\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ and $\left\{\left(T,{\Gamma }_d\right):d\in D\right\}$ be two families of TSs. Let $p:S⟶T$ and $u:R⟶D$ be functions where u is a bijection. Then, $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft s-c if and only if $p:\left(S,{\Pi }_r\right)⟶\left(T,{\Gamma }_{u\left(r\right)}\right)$ is s-c for all $r\in R$.

Proof. 

Necessity. Assume that $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft s-c. Let $t\in R$. Let $V\in CO\left(T,{\Gamma }_{u\left(t\right)}\right)$. Then, ${\left(u\left(t\right)\right)}_V\in CO\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$. Thus, $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)\in {\oplus }_{r\in R}{\Pi }_r$. Since $u:R⟶D$ is injective, $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)=t_{p^{-1}\left(V\right)}$. Thus, $\left(t_{p^{-1}\left(V\right)}\right)\left(t\right)=p^{-1}\left(V\right)\in {\Pi }_t$. Therefore, $p:\left(S,{\Pi }_t\right)⟶\left(T,{\Gamma }_{u\left(t\right)}\right)$ is s-c.

Sufficiency. Assume that $p:\left(S,{\Pi }_r\right)⟶\left(T,{\Gamma }_{u\left(r\right)}\right)$ is s-c for all $r\in R$. Let $G\in CO\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$. Then, $G\left(d\right)\in CO\left(Y,{\Gamma }_d\right)$ for all $d\in D$. For every $d\in D$, $p:\left(S,{\Pi }_{u^{-1}\left(d\right)}\right)⟶\left(T,{\Gamma }_d\right)$ is s-c, and so $p^{-1}\left(G\left(d\right)\right)\in {\Pi }_{u^{-1}\left(d\right)}$. Thus, for every $r\in R$, $\left(f_{pu}^{-1}\left(G\right)\right)\left(r\right)=p^{-1}(G\left(u\left(r\right)\right)\in {\Pi }_{u^{-1}\left(u\left(r\right)\right)}={\Pi }_r$. Therefore, $f_{pu}^{-1}\left(G\right)\in {\oplus }_{r\in R}{\Pi }_r$. It follows that $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft s-c. □

Corollary 2.

Let $p:\left(S,\delta \right)⟶\left(T,\beta \right)$ and $u:R⟶D$ be two functions where u is a bijection. Then, $p:\left(S,\delta \right)⟶\left(T,\beta \right)$ is s-c if and only if $f_{pu}:(S,\tau \left(\delta \right),R)⟶(T,\tau \left(\beta \right),D)$ is soft s-c.

Proof. 

For every $r\in R$ and $d\in D$, let ${\Pi }_r=\delta$ and ${\Gamma }_d=\beta$. Then, $\tau \left(\delta \right)={\oplus }_{r\in R}{\Pi }_r$ and $\tau \left(\beta \right)={\oplus }_{d\in D}{\Gamma }_d$. We get the result as a consequence of Theorem 3. □

Theorem 4.

Let$\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ and $\left\{\left(T,{\Gamma }_d\right):d\in D\right\}$ be two families of TSs. Let $p:S⟶T$ and $u:R⟶D$ be functions where u is a bijection. Then, $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft ω-c if and only if $p:\left(S,{\Pi }_r\right)⟶\left(T,{\Gamma }_{u\left(r\right)}\right)$ is ω-c for all $r\in R$.

Proof. 

Necessity. Assume that $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft $\omega$-c. Let $t\in R$. Let $V\in {\Gamma }_{u\left(t\right)}$. Then, ${\left(u\left(t\right)\right)}_V\in {\oplus }_{d\in D}{\Gamma }_d$. Thus, $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)\in {\left({\oplus }_{r\in R}{\Pi }_r\right)}_{\omega }$ and by Theorem 8 of [29], $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)\in {\oplus }_{r\in R}{\left({\Pi }_r\right)}_{\omega }$. Since $u:R⟶D$ is injective, $f_{pu}^{-1}\left({\left(u\left(t\right)\right)}_V\right)=t_{p^{-1}\left(V\right)}$. Thus, $\left(t_{p^{-1}\left(V\right)}\right)\left(t\right)=p^{-1}\left(V\right)\in {\left({\Pi }_t\right)}_{\omega }$. Therefore, $p:\left(S,{\Pi }_t\right)⟶\left(T,{\Gamma }_{u\left(t\right)}\right)$ is $\omega$-c.

Sufficiency. Assume that $p:\left(S,{\Pi }_r\right)⟶\left(T,{\Gamma }_{u\left(r\right)}\right)$ is $\omega$-c for all $r\in R$. Let $G\in {\oplus }_{d\in D}{\Gamma }_d$. Then, $G\left(d\right)\in {\Gamma }_d$ for all $d\in D$. For every $d\in D$, $p:\left(S,{\Pi }_{u^{-1}\left(d\right)}\right)⟶\left(T,{\Gamma }_d\right)$ is $\omega$-c, and so $p^{-1}\left(G\left(d\right)\right)\in {\left({\Pi }_{u^{-1}\left(d\right)}\right)}_{\omega }$. Thus, for every $r\in R$, $\left(f_{pu}^{-1}\left(G\right)\right)\left(r\right)=p^{-1}(G\left(u\left(r\right)\right)\in {\Pi }_{{\left(u^{-1}\left(u\left(r\right)\right)\right)}_{\omega }}={\left({\Pi }_r\right)}_{\omega }$. Therefore, by Theorem 8 of [29], $f_{pu}^{-1}\left(G\right)\in {\left({\oplus }_{r\in R}{\Pi }_r\right)}_{\omega }$. It follows that $f_{pu}:\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)⟶$$\left(T,{\oplus }_{d\in D}{\Gamma }_d,D\right)$ is soft $\omega$-c. □

Corollary 3 ([14]).

Let $p:\left(S,\delta \right)⟶\left(T,\beta \right)$ and $u:R⟶D$ be two functions where u is a bijection. Then, $p:\left(S,\delta \right)⟶\left(T,\beta \right)$ is ω-c if and only if $f_{pu}:(S,\tau \left(\delta \right),R)⟶(T,\tau \left(\beta \right),D)$ is soft ω-c.

Proof. 

For every $r\in R$ and $d\in D$, let ${\Pi }_r=\delta$ and ${\Gamma }_d=\beta$. Then, $\tau \left(\delta \right)={\oplus }_{r\in R}{\Pi }_r$ and $\tau \left(\beta \right)={\oplus }_{d\in D}{\Gamma }_d$. We get the result as a consequence of Theorem 4. □

Theorem 5.

If$f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft s-ω-c and $C_X\in {\Pi }_{\omega }$, then the soft restriction ${\left(f_{pu}\right)}_{\left|C_X\right.}:\left(X,{\Pi }_X,B\right)⟶\left(T,\Gamma ,D\right)$ is soft s-ω-c.

Proof. 

Let $G\in CO\left(T,\Gamma ,D\right)$. Since $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft s-$\omega$-c, $f_{pu}^{-1}\left(G\right)\in {\Pi }_{\omega }$ and so, ${\left({\left(f_{pu}\right)}_{\left|C_X\right.}\right)}^{-1}\left(G\right)=f_{pu}^{-1}\left(G\right)\tilde{\cap }{C}_X\in {\left({\Pi }_{\omega }\right)}_X$. Since by Theorem 15 of [29], ${\left({\Pi }_{\omega }\right)}_X={\left({\Pi }_X\right)}_{\omega }$, ${\left({\left(f_{pu}\right)}_{\left|C_X\right.}\right)}^{-1}\left(G\right)\in {\left({\Pi }_X\right)}_{\omega }$. This proves that ${\left(f_{pu}\right)}_{\left|C_X\right.}:\left(X,{\Pi }_X,B\right)⟶\left(T,\Gamma ,D\right)$ is soft s-$\omega$-c. □

Theorem 6.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is a soft function and $\left(S,\Pi ,R\right)$ is soft locally countable, then $f_{pu}$ is soft s-ω-c.

Proof. 

Since $\left(S,\Pi ,R\right)$ is soft locally countable, by Corollary 5 of [29], $\left(S,{\Pi }_{\omega },R\right)$ is soft discrete. This ends the proof. □

Theorem 7.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is a soft function and $\left(T,\Gamma ,D\right)$ is soft connected, then $f_{pu}$ is soft s-c.

Proof. 

Since $\left(T,\Gamma ,D\right)$ is soft connected, $CO\left(T,\Gamma ,D\right)=\left\{0_D,1_D\right\}$. This ends the proof. □

Theorem 8.

Each soft s-c function is soft s-ω-c.

Proof. 

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-c. Let $G\in CO\left(T,\Gamma ,D\right)$. Then, $f_{pu}^{-1}\left(G\right)\in \Pi \subseteq {\Pi }_{\omega }$. This proves that $f_{pu}$ is soft s-$\omega$-c. □

The following example shows that Theorem 8 is not reversible in general:

Example 1.

Let $X=\mathbb{N}$, $\delta =$$\left\{\varnothing ,X\right\}$, $\beta =\left\{\varnothing ,X,\left\{1\right\},\mathbb{N}-\left\{1\right\}\right\}$, and $D=\left\{a,b\right\}$. Consider the identities functions $p:\left(X,\delta \right)⟶\left(X,\beta \right)$ and $u:D⟶D$. Since $\left\{1\right\}\in CO\left(X,\beta \right)$ while $p^{-1}\left(\left\{1\right\}\right)=\left\{1\right\}\notin \delta$, $p:\left(X,\delta \right)⟶\left(X,\beta \right)$ is not soft s-c. Thus, by Corollary 2, $f_{pu}:(S,\tau \left(\delta \right),D)⟶(T,\tau \left(\beta \right),D)$ is not soft s-c. On the other hand, since $(S,\tau \left(\delta \right),D)$ is clearly soft locally countable, by Theorem 6, $f_{pu}:(S,\tau \left(\delta \right),D)⟶(T,\tau \left(\beta \right),D)$ is soft s-ω-c.

Theorem 9.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is a soft s-ω-c and $\left(S,\Pi ,R\right)$ is soft anti-locally countable, then $f_{pu}$ is soft s-c.

Proof. 

Let $G\in CO\left(T,\Gamma ,D\right)$. Then, by Theorem 1 (d), $f_{pu}^{-1}\left(G\right)\in CO\left(S,{\Pi }_{\omega },B\right)$. Since $f_{pu}^{-1}\left(G\right)\in {\Pi }_{\omega }$, $\left(1_R-f_{pu}^{-1}\left(G\right)\right)\in {\left({\Pi }_{\omega }\right)}^c$, and $\left(S,\Pi ,R\right)$ is soft anti-locally countable, by Theorem 14 of [29], $C{l}_{\Pi }\left(f_{pu}^{-1}\left(G\right)\right)=C{l}_{{\Pi }_{\omega }}\left(f_{pu}^{-1}\left(G\right)\right)=f_{pu}^{-1}\left(G\right)$ and $In{t}_{\Pi }(1_R-f_{pu}^{-1}\left(G\right))=In{t}_{{\Pi }_{\omega }}(1_R-f_{pu}^{-1}\left(G\right))=1_R-f_{pu}^{-1}\left(G\right)$. Therefore, we have $f_{pu}^{-1}\left(G\right)\in CO\left(S,\Pi ,R\right)$. This proves that $f_{pu}$ is soft s-c. □

Theorem 10.

Each soft ω-c function is soft s-ω-c.

Proof. 

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft $\omega$-c. Let $G\in CO\left(T,\Gamma ,D\right)\subseteq \Gamma$. Then, $G\in \Gamma$. Since $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft $\omega$-c, $f_{pu}^{-1}\left(G\right)\in {\Pi }_{\omega }$. This proves that $f_{pu}$ is soft s-$\omega$-c. □

The following example shows that Theorem 10 is not reversible in general:

Example 2.

Let $X=\mathbb{R}$, $\delta =$$\left\{\varnothing ,X\right\}$, $\beta =\left\{\varnothing ,X,\left\{1\right\}\right\}$, and $D=\left\{a\right\}$. Consider the functions $p:\left(X,\delta \right)⟶\left(X,\beta \right)$ and $u:D⟶D$ defined by:

$$p\left(x\right)=\left\{\begin{array}{cc}1& \mathrm{if}x\in \mathbb{Q}\\ 2& \mathrm{if}x\in \mathbb{R}-\mathbb{Q}\end{array}\right.$$

and $u\left(a\right)=a$. Since $(X,\tau \left(\beta \right),D)$ is clearly soft connected, by Theorems 7 and 8, $f_{pu}:(X,\tau \left(\delta \right),D)⟶(X,\tau \left(\beta \right),D)$ is soft s-ω-c. Since $\left\{1\right\}\in \beta$ while $p^{-1}\left(\left\{1\right\}\right)=\mathbb{Q}\notin {\delta }_{\omega }$, $p:\left(X,\delta \right)⟶\left(X,\beta \right)$ is not soft ω-c. Thus, by Corollary 3, $f_{pu}:(X,\tau \left(\delta \right),D)⟶(X,\tau \left(\beta \right),D)$ is soft ω-c.

Theorem 11.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is a soft s-ω-c and $\left(T,\Gamma ,D\right)$ is soft locally indiscrete, then $f_{pu}$ is soft ω-c.

Proof. 

Let $G\in \Gamma$. Since $\left(T,\Gamma ,D\right)$ is soft locally indiscrete, $G\in CO\left(T,\Gamma ,D\right)$. Since $f_{pu}$ is soft s-$\omega$-c, $f_{pu}^{-1}\left(G\right)\in {\Pi }_{\omega }$. This proves that $f_{pu}$ is soft $\omega$-c. □

Theorem 12.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is a soft s-ω-c surjection and $\left(S,{\Pi }_{\omega },B\right)$ is soft connected, then $\left(T,\Gamma ,D\right)$ is soft connected.

Proof. 

Assume, however, that $\left(T,\Gamma ,D\right)$ is soft disconnected. Then, we find $K\in CO\left(T,\Gamma ,D\right)-\left\{0_D,1_D\right\}$. Since $f_{pu}$ is soft s-$\omega$-c, by Theorem 1 (d), $f_{pu}^{-1}\left(K\right)\in CO\left(S,{\Pi }_{\omega },B\right)$. Since $K\ne 0_D$ and $f_{pu}$ is surjective, $f_{pu}^{-1}\left(K\right)\ne 0_R$. If $f_{pu}^{-1}\left(K\right)=1_R$, then $f_{pu}\left(f_{pu}^{-1}\left(K\right)\right)=f_{pu}\left(1_R\right)=1_D\tilde{\subseteq }K$ and so $K=1_D$. Therefore, $f_{pu}^{-1}\left(K\right)\ne 1_R$. This proves that $\left(S,{\Pi }_{\omega },B\right)$ is soft $\omega$-disconnected, a contradiction. □

Theorem 13.

If $f_{p_1{u}_1}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft ω-irresolute and $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ is soft s-ω-c, then $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}:\left(S,\Pi ,R\right)⟶\left(R,\Phi ,E\right)$ is soft s-ω-c.

Proof. 

Let $G\in CO\left(R,\Phi ,E\right)$. Since $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ is soft s-$\omega$-c, $f_{p_2{u}_2}^{-1}\left(G\right)\in {\Gamma }_{\omega }$. Since $f_{p_1{u}_1}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft $\omega$-irresolute, $f_{p_1{u}_1}^{-1}\left(f_{p_2{u}_2}^{-1}\left(G\right)\right)=f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}^{-1}\left(G\right)\in {\Pi }_{\omega }$. This proves that $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}:\left(S,\Pi ,R\right)⟶\left(R,\Phi ,E\right)$ is soft s-$\omega$-c. □

Theorem 14.

If $f_{p_1{u}_1}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft s-ω-c and $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ is soft s-c, then $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}:\left(S,\Pi ,R\right)⟶\left(R,\Phi ,E\right)$ is soft s-ω-c.

Proof. 

Let $G\in CO\left(R,\Phi ,E\right)$. Since $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ is soft s-c, $f_{p_2{u}_2}^{-1}\left(G\right)\in CO\left(T,\Gamma ,D\right)$. Since $f_{p_1{u}_1}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft s-$\omega$-c, $f_{p_1{u}_1}^{-1}\left(f_{p_2{u}_2}^{-1}\left(G\right)\right)=f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}^{-1}\left(G\right)\in {\Pi }_{\omega }$. This proves that $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}:\left(S,\Pi ,R\right)⟶\left(R,\Phi ,E\right)$ is soft s-$\omega$-c. □

Theorem 15.

Let $f_{p_1{u}_1}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be a soft ω-irresolute surjection such that $f_{p_1{u}_1}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,{\Gamma }_{\omega },D\right)$ is soft open and let $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ be a soft function. Then, $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ is soft s-ω-c if and only if $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}:\left(S,\Pi ,R\right)⟶\left(R,\Phi ,E\right)$ is soft s-ω-c.

Proof. 

Necessity. Assume that $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ is soft s-$\omega$-c. Then, by Theorem 13, $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}:\left(S,\Pi ,R\right)⟶\left(R,\Phi ,E\right)$ is soft s-$\omega$-c.

Sufficiency. Assume that $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}:\left(S,\Pi ,R\right)⟶\left(R,\Phi ,E\right)$ is soft s-$\omega$-c. Let $G\in CO\left(R,\Phi ,E\right)$. Then, $f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}^{-1}\left(G\right)\in {\Pi }_{\omega }$. Since $f_{p_1{u}_1}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,{\Gamma }_{\omega },D\right)$ is soft open, $f_{p_1{u}_1}\left(f_{\left(p_2\circ p_1\right)\left(u_2\circ u_1\right)}^{-1}\left(G\right)\right)=f_{p_1{u}_1}\left(f_{f_{p_1{u}_1}}^{-1}\left(f_{f_{p_2{u}_2}}^{-1}\left(G\right)\right)\right)\in {\Gamma }_{\omega }$. Since $f_{p_1{u}_1}$ is surjective, $f_{p_1{u}_1}\left(f_{f_{p_1{u}_1}}^{-1}\left(f_{f_{p_2{u}_2}}^{-1}\left(G\right)\right)\right)=f_{f_{p_2{u}_2}}^{-1}\left(G\right)$. This proves that $f_{p_2{u}_2}:\left(T,\Gamma ,D\right)⟶\left(R,\Phi ,E\right)$ is soft s-$\omega$-c.

For any function $g:Z⟶W$, the function $w:Z⟶Z\times W$ defined by $w\left(z\right)=\left(z,w\left(z\right)\right)$ will be denoted by $g^{\#}$. □

Theorem 16.

A soft function $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft s-ω-c if $f_{p^{\#}{u}^{\#}}:\left(S,\Pi ,R\right)⟶\left(S\times T,pr\left(\Pi \times \Gamma \right),R\times D\right)$ is soft s-ω-c.

Proof. 

Assume that $f_{p^{\#}{u}^{\#}}:\left(S,\Pi ,R\right)⟶\left(S\times T,pr\left(\Pi \times \Gamma \right),R\times D\right)$ is soft s-$\omega$-c. Let $G\in CO\left(T,\Gamma ,D\right)$. Then, $1_R\times G\in CO\left(S\times T,pr\left(\Pi \times \Gamma \right),R\times D\right)$ and so $f_{p^{\#}{u}^{\#}}^{-1}\left(1_R\times G\right)=f_{pu}^{-1}\left(G\right)\in {\Pi }_{\omega }$. This proves that $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft s-$\omega$-c. □

3. Soft Ultra-Separation Axioms

Definition 11.

An STS $\left(S,\Pi ,R\right)$ is called:

 (a) 

soft ultra-Hausdorff if for any two soft points $a_x$, $b_y\in SP\left(S,B\right)$ with $a_x\ne b_y$, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\in }G$, $b_y\tilde{\in }H$, and $G\tilde{\cap }H=0_R$.

 (b) 

soft ultra-$T_1$ if for any two soft points $a_x$, $b_y\in SP\left(S,B\right)$ with $a_x\ne b_y$, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\in }G-H$ and $b_y\tilde{\in }H-G$.

 (c) 

soft ultra-$T_0$ if for any two soft points $a_x$, $b_y\in SP\left(S,B\right)$ with $a_x\ne b_y$, we find $G\in CO\left(S,\Pi ,R\right)$ such that $\left(a_x\tilde{\in }G\mathrm{but}{b}_y\tilde{\notin }G\right)$ or $\left(b_y\tilde{\in }G\mathrm{but}{a}_x\tilde{\notin }G\right)$.

Theorem 17.

For an STS $\left(S,\Pi ,R\right)$, the following are equivalent:

 (a) 

$\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff;

 (b) 

$\left(S,\Pi ,R\right)$ is soft ultra-$T_1$;

 (c) 

$\left(S,\Pi ,R\right)$ is soft ultra-$T_0$;

Proof. 

(a) ⟶ (b) and (b) ⟶ (c) are obvious.

(c) ⟶ (a): Let $a_x$, $b_y\in SP\left(S,B\right)$ with $a_x\ne b_y$. Then, by (a), we find $G\in CO\left(S,\Pi ,R\right)$ such that $\left(a_x\tilde{\in }G\mathrm{but}{b}_y\tilde{\notin }G\right)$ or $\left(b_y\tilde{\in }G\mathrm{but}{a}_x\tilde{\notin }G\right)$. Put $H=1_R-G$. Then, we have $G,H\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\in }G$, $b_y\tilde{\in }H$, and $G\tilde{\cap }H=0_R$. This proves that $\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff. □

Theorem 18.

If $\left(S,\Pi ,D\right)$ is soft ultra-Hausdorff, then $\left(S,{\Pi }_d\right)$ is ultra-Hausdorff for all $d\in D$.

Proof. 

Assume that $\left(S,\Pi ,D\right)$ is soft ultra-Hausdorff and let $d\in D$. Let $x,y\in S$ such that $x\ne y$. Then, $d_x,d_y\in SP\left(S,D\right)$ such that $d_x\ne d_y$. Since $\left(S,\Pi ,D\right)$ is soft ultra-Hausdorff, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $d_x\tilde{\in }G$, $d_y\tilde{\in }H$, and $G\tilde{\cap }H=0_R$. Then, $x\in G\left(d\right)\in CO\left(S,{\Pi }_d\right)$, $y\in H\left(d\right)\in CO\left(S,{\Pi }_d\right)$, and $G\left(d\right)\cap H\left(d\right)=\left(G\tilde{\cap }H\right)\left(d\right)=\varnothing$. This proves that $\left(S,{\Pi }_d\right)$ is ultra-Hausdorff. □

Lemma 1.

Let $\left\{\left(S,{\Pi }_d\right):d\in D\right\}$ be a family of TSs. Then, $G\in CO\left(S,{\oplus }_{d\in D}{\Pi }_d,D\right)$ if and only if $G\left(d\right)\in CO\left(S,{\Pi }_d\right)$ for all $d\in D$.

Proof. 

Straightforward. □

Theorem 19.

Let $\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ be a family of TSs. Then, $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft ultra-Hausdorff if and only if $\left(S,{\Pi }_r\right)$ is ultra-Hausdorff for every $r\in R$.

Proof. 

Necessity. Assume that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft ultra-Hausdorff and let $a\in R$. Then, by Theorem 18, $\left(S,{\left({\oplus }_{r\in R}{\Pi }_r\right)}_a\right)$ is ultra-Hausdorff. On the other hand, by Theorem 3.7 of [28], ${\left({\oplus }_{r\in R}{\Pi }_r\right)}_a={\Pi }_a$.

Sufficiency. Assume that $\left(S,{\Pi }_r\right)$ is ultra-Hausdorff for every $r\in R$. Let $a_x,b_y\in SP\left(S,B\right)$ such that $a_x\ne b_y$.

Case 1. $a\ne b$. Then, $a_x\tilde{\in }{a}_S\in CO\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$, $b_x\tilde{\in }{b}_S\in CO\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$, and $a_S\tilde{\cap }{b}_S=0_R$.

Case 2. $a=b$. Then, $x\ne y$. Since $\left(S,{\Pi }_a\right)$ is ultra-Hausdorff, we find $V,W\in CO\left(S,{\Pi }_a\right)$ such that $x\in V$, $y\in W$, and $V\cap W=\varnothing$. Then, we have $a_x\tilde{\in }{a}_V\in CO\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$, $b_y\tilde{\in }{b}_W\in CO\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ and $a_V\tilde{\cap }{b}_W=a_V\tilde{\cap }{a}_W=0_R$. □

Corollary 4.

Let $\left(S,\beta \right)$ be a TS and D be a set of parameters. $\left(S,\tau \left(\beta \right),D\right)$ is soft ultra-Hausdorff if and only if $\left(S,\beta \right)$ is ultra-Hausdorff.

Proof. 

For every $d\in D$, let ${\Pi }_d=\beta$. Then, $\tau \left(\beta \right)={\oplus }_{d\in D}{\Pi }_d$. We get the result as a consequence of Theorem 19. □

Theorem 20.

If $\left(S,\Pi ,R\right)$ is a soft connected STS where S contains at least two distinct points, then $\left(S,\Pi ,R\right)$ is not soft ultra-Hausdorff.

Proof. 

Choose $r\in R$ and $x,y\in S$ such that $x\ne y$. Then, $b_x,b_y\in SP\left(S,B\right)$. If $\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff, we find $G,H\in CO\left(S,\Pi ,R\right)-\left\{0_R,1_R\right\}$ such that $b_x\tilde{\in }G$, $b_y\tilde{\in }H$, and $G\tilde{\cap }H=0_R$. Since $\left(S,\Pi ,R\right)$ is soft connected, $CO\left(S,\Pi ,R\right)-\left\{0_R\right\}=1_R$ and so $G\tilde{\cap }H=1_R\ne 0_R$. This proves that $\left(S,\Pi ,R\right)$ is not soft ultra-Hausdorff. □

Theorem 21.

Each soft ultra-Hausdorff STS is soft Urysohn.

Proof. 

Let $\left(S,\Pi ,R\right)$ be soft ultra-Hausdorff. Let $a_x$, $b_y\in SP\left(S,B\right)$. Then, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\in }G$, $b_y\tilde{\in }H$, and $G\tilde{\cap }H=0_R$. Since $G,H\in CO\left(S,\Pi ,R\right)$, then $G,H\in \Pi$, $C{l}_{\Pi }\left(G\right)\tilde{\cap }C{l}_{\Pi }\left(H\right)=G\tilde{\cap }H=0_R$. This ends the proof. □

Corollary 5.

Each soft ultra-Hausdorff STS is soft Hausdorff.

The opposite of Theorem 21 does not have to be true in all cases:

Example 3.

If ν is the usual topology $\mathbb{R}$, then $\left(\mathbb{R},\tau \left(\upsilon \right),\left\{a\right\}\right)$ is soft Urysohn. On the other hand, since $\left(\mathbb{R},\tau \left(\upsilon \right),\left\{a\right\}\right)$ is soft connected, by Theorem 20, it is not soft ultra-Hausdorff.

Theorem 22.

If $\left(S,\Pi ,R\right)$ is soft 0-dim, then $\left(S,{\Pi }_d\right)$ is 0-dim for every $d\in D$.

Proof. 

Assume that $\left(S,\Pi ,D\right)$ is soft 0-dim and let $d\in D$. Since $\left(S,\Pi ,R\right)$ is soft 0-dim, then we find a soft base $\mathcal{H}$ of $\left(S,\Pi ,D\right)$ such that $\mathcal{H}\subseteq CO\left(S,\Pi ,D\right)$. Let ${\mathcal{H}}_d=\left\{H\left(d\right):H\in \mathcal{H}\right\}$. Then, ${\mathcal{H}}_d\subseteq CO\left(S,{\Pi }_d\right)$. To show that ${\mathcal{H}}_d$ is a base for $\left(S,{\Pi }_d\right)$, let $W\in {\Pi }_d$ and let $x\in W$. Choose $K\in \Pi$ such that $U=K\left(d\right)$. Then, we have $d_x\tilde{\in }K$. Since $\mathcal{H}$ is a soft base for $\left(S,\Pi ,D\right)$, then we find $H\in \mathcal{H}$ such that $d_x\tilde{\in }H\tilde{\subseteq }K$. So, we have $H\left(d\right)\in {\mathcal{H}}_d$ and $x\in H\left(d\right)\subseteq K\left(d\right)=W$. This proves that ${\mathcal{H}}_d$ is a base for $\left(S,{\Pi }_d\right)$. It follows that $\left(S,{\Pi }_d\right)$ is 0-dim. □

Theorem 23.

Let $\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ be a family of TSs. Then, $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft 0-dim if and only if $\left(S,{\Pi }_r\right)$ is 0-dim for every $r\in R$.

Proof. 

Necessity. Assume that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft 0-dim and let $a\in R$. Then, by Theorem 22, $\left(S,{\left({\oplus }_{r\in R}{\Pi }_r\right)}_a\right)$ is 0-dim. By Theorem 3.7 of [28], ${\left({\oplus }_{r\in R}{\Pi }_r\right)}_a={\Pi }_a$. This proves that $\left(S,{\Pi }_a\right)$ is 0-dim.

Sufficiency. Assume that $\left(S,{\Pi }_r\right)$ is 0-dim for every $r\in R$. Then, for every $r\in R$, we find a base ${\mathcal{H}}_b$ is a base for $\left(S,{\Pi }_r\right)$.

Let $\mathcal{H}=\left\{b_U:r\in R\mathrm{and}U\in {\mathcal{H}}_b\right\}$. Then, by Lemma 1, $\mathcal{H}\subseteq CO\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$. On the other hand, by Corollary 3.6 of [28], $\mathcal{H}$ is a soft base of $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$. This proves that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft 0-dim. □

Corollary 6.

Let $\left(S,\beta \right)$ be a TS and D be a set of parameters. Then, $\left(S,\tau \left(\beta \right),D\right)$ is soft 0-dim if and only if $\left(S,\beta \right)$ is 0-dim.

Proof. 

For every $d\in D$, let ${\Pi }_d=\beta$. Then, $\tau \left(\beta \right)={\oplus }_{d\in D}{\Pi }_d$. We get the result as a consequence of Theorem 23. □

Theorem 24.

For an STS soft 0-dim $\left(S,\Pi ,R\right)$, the following are equivalent:

 (a) 

$\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff;

 (b) 

$\left(S,\Pi ,R\right)$ is soft Urysohn;

 (c) 

$\left(S,\Pi ,R\right)$ is soft Hausdorff;

Proof. 

(a)

⟶ (b): Theorem 21 leads to the proof.

(b)

⟶ (c): Obvious.

(c)

⟶ (a): Let $a_x$, $b_y\in SP\left(S,B\right)$ such that $a_x\ne b_y$. Then, by (c), we find $R,W\in \Pi$ such that $a_x\tilde{\in }R$, $b_y\tilde{\in }W$, and $R\tilde{\cap }W=0_R$. Since $\left(S,\Pi ,R\right)$ is soft 0-dim, we find a soft base $\mathcal{H}$ of $\left(S,\Pi ,R\right)$ such that $\mathcal{H}\subseteq CO\left(S,\Pi ,R\right)$. Choose $G,H\in \mathcal{H}$ such that $a_x\tilde{\in }G\tilde{\subseteq }R$, $b_y\tilde{\in }H\tilde{\subseteq }W$. Then, we have $G,H\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\in }G$, $b_y\tilde{\in }H$, and $G\tilde{\cap }H\tilde{\subseteq }R\tilde{\cap }W=0_R$. This proves that $\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff.

□

Theorem 25.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is soft s-c and $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, then:

 (a) 

$\tilde{\cup }\left\{{\left(m,u\left(m\right)\right)}_{\left(s,p\left(s\right)\right)}:m_s\in SP(S,R)\right\}$$\in {\left({\left(pr\left(\Pi \times \Gamma \right)\right)}^{*}\right)}^c$.

 (b) 

$\tilde{\cup }\left\{{\left(m,n\right)}_{\left(s,t\right)}:m_s,n_t\in SP(S,R)\mathrm{and}{f}_{pu}\left(m_s\right)=f_{pu}\left(n_t\right)\right\}$$\in {\left({\left(pr\left(\Pi \times \Pi \right)\right)}^{*}\right)}^c$.

Proof. 

(a)

Put $K=\tilde{\cup }\left\{{\left(m,u\left(m\right)\right)}_{\left(s,p\left(s\right)\right)}:m_s\in SP(S,R)\right\}$. We will show that $1_{R\times D}-K\in {\left(pr\left(\Pi \times \Gamma \right)\right)}^{*}$. Let ${\left(a,b\right)}_{\left(x,y\right)}\tilde{\in }{1}_{R\times D}-K$. Then, $f_{pu}\left(a_x\right)\ne b_y$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, we find $G,H\in CO\left(T,\Gamma ,D\right)\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }G$, $b_y\tilde{\in }H$, and $G\tilde{\cap }H=0_D$. Since $f_{pu}$ is soft s-c, $f_{pu}^{-1}\left(G\right)\in CO\left(S,\Pi ,R\right)$. Therefore, we have, ${\left(a,b\right)}_{\left(x,y\right)}\tilde{\in }{f}_{pu}^{-1}\left(G\right)\times H\in CO\left(S\times T,pr\left(\Pi \times \Gamma \right),R\times D\right)$ and $\left(f_{pu}^{-1}\left(G\right)\times H\right)\tilde{\cap }$$K=0_{R\times D}$. This proves that $1_{R\times D}-K\in {\left(pr\left(\Pi \times \Gamma \right)\right)}^{*}$.

(b)

Put $L=\tilde{\cup }\left\{{\left(m,n\right)}_{\left(s,t\right)}:m_s,n_t\in SP(S,R)\mathrm{and}{f}_{pu}\left(m_s\right)=f_{pu}\left(n_t\right)\right\}$. We will show that $1_{R\times R}-L\in {\left(pr\left(\Pi \times \Pi \right)\right)}^{*}$. Let ${\left(m,n\right)}_{\left(s,t\right)}\tilde{\in }{1}_{R\times R}-L$. Then, $f_{pu}\left(m_s\right)=f_{pu}\left(n_t\right)$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, we find $G,H\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(m_s\right)\tilde{\in }G$, $f_{pu}\left(n_t\right)\tilde{\in }H$, and $G\tilde{\cap }H=0_D$. Since $f_{pu}$ is soft s-c, $f_{pu}^{-1}\left(G\right),f_{pu}^{-1}\left(H\right)\in CO\left(S,\Pi ,R\right)$. Therefore, we have ${\left(m,n\right)}_{\left(s,t\right)}\tilde{\in }{f}_{pu}^{-1}\left(G\right)\times f_{pu}^{-1}\left(H\right)\in CO\left(S\times T,pr\left(\Pi \times \Pi \right),R\times R\right)$ and $\left(f_{pu}^{-1}\left(G\right)\times f_{pu}^{-1}\left(H\right)\right)\tilde{\cap }L=0_{R\times R}$. This proves that $1_{R\times R}-L\in {\left(pr\left(\Pi \times \Pi \right)\right)}^{*}$.

□

Theorem 26.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-ω-c such that $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff and $\left\{G\times H:G\in CO\left(S,{\Pi }_{\omega },B\right)\mathrm{and}H\in CO\left(T,\Gamma ,D\right)\right\}\subseteq$$CO\left(S\times T,{\left(pr\left(\Pi \times \Gamma \right)\right)}_{\omega },R\times D\right)$, then $\tilde{\cup }\left\{{\left(m,u\left(m\right)\right)}_{\left(s,p\left(s\right)\right)}:m_s\in SP(S,R)\right\}$$\in {\left({\left({\left(pr\left(\Pi \times \Gamma \right)\right)}_{\omega }\right)}^{*}\right)}^c$.

Proof. 

Put $K=\tilde{\cup }\left\{{\left(m,u\left(m\right)\right)}_{\left(s,p\left(s\right)\right)}:m_s\in SP(S,R)\right\}$. We will show that $1_{R\times D}-K\in {\left({\left(pr\left(\Pi \times \Gamma \right)\right)}_{\omega }\right)}^{*}$. Let ${\left(a,b\right)}_{\left(x,y\right)}\tilde{\in }{1}_{R\times D}-K$. Then, $f_{pu}\left(a_x\right)\ne b_y$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, we find $L,M\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }L$, $b_y\tilde{\in }M$, and $L\tilde{\cap }M=0_D$. Since $f_{pu}$ is soft s-$\omega$-c, $f_{pu}^{-1}\left(L\right)\in CO\left(S,{\Pi }_{\omega },B\right)$. Since $\left\{G\times H:G\in \left(S,{\Pi }_{\omega },B\right)\mathrm{and}H\in CO\left(T,\Gamma ,D\right)\right\}\subseteq$$CO\left(S\times T,{\left(pr\left(\Pi \times \Gamma \right)\right)}_{\omega },R\times D\right)$, $f_{pu}^{-1}\left(L\right)\times M\in CO\left(S\times T,{\left(pr\left(\Pi \times \Gamma \right)\right)}_{\omega },R\times D\right)$. Therefore, we have ${\left(a,b\right)}_{\left(x,y\right)}\tilde{\in }{f}_{pu}^{-1}\left(L\right)\times M\in CO\left(S\times T,{\left(pr\left(\Pi \times \Gamma \right)\right)}_{\omega },R\times D\right)$ and $\left(f_{pu}^{-1}\left(L\right)\times M\right)\tilde{\cap }K=0_{R\times D}$. This proves that $1_{R\times D}-K\in {\left({\left(pr\left(\Pi \times \Gamma \right)\right)}_{\omega }\right)}^{*}$. □

Theorem 27.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-ω-c such that $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff and $\left\{G\times H:G,H\in CO\left(S,{\Pi }_{\omega },B\right)\right\}\subseteq CO\left(S\times S,{\left(pr\left(\Pi \times \Pi \right)\right)}_{\omega },R\times R\right)$, then $\tilde{\cup }\left\{{\left(m,n\right)}_{\left(s,t\right)}:m_s,n_t\in SP(S,R)\mathrm{and}{f}_{pu}\left(m_s\right)=f_{pu}\left(n_t\right)\right\}$$\in {\left({\left({\left(pr\left(\Pi \times \Pi \right)\right)}_{\omega }\right)}^{*}\right)}^c$.

Proof. 

Put $L=\tilde{\cup }\left\{{\left(m,n\right)}_{\left(s,t\right)}:m_s,n_t\in SP(S,R)\mathrm{and}{f}_{pu}\left(m_s\right)=f_{pu}\left(n_t\right)\right\}$. We will show that $1_{R\times R}-L\in {\left({\left(pr\left(\Pi \times \Pi \right)\right)}_{\omega }\right)}^{*}$. Let ${\left(m,n\right)}_{\left(s,t\right)}\tilde{\in }{1}_{R\times R}-L$. Then, $f_{pu}\left(m_s\right)=f_{pu}\left(n_t\right)$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, we find $F,W\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(m_s\right)\tilde{\in }F$, $f_{pu}\left(n_t\right)\tilde{\in }W$, and $F\tilde{\cap }W=0_D$. Since $f_{pu}$ is soft s-$\omega$-c, $f_{pu}^{-1}\left(F\right),f_{pu}^{-1}\left(W\right)\in CO\left(S,{\Pi }_{\omega },B\right)$. Since $\left\{G\times H:G,H\in CO\left(S,{\Pi }_{\omega },B\right)\right\}\subseteq CO\left(S\times S,{\left(pr\left(\Pi \times \Pi \right)\right)}_{\omega },R\times R\right)$, $f_{pu}^{-1}\left(F\right)\times f_{pu}^{-1}\left(W\right)\in CO\left(S\times S,{\left(pr\left(\Pi \times \Pi \right)\right)}_{\omega },R\times R\right)$. Therefore, we have ${\left(m,n\right)}_{\left(s,t\right)}\tilde{\in }{f}_{pu}^{-1}\left(F\right)\times f_{pu}^{-1}\left(W\right)\in$$CO\left(S\times T,{\left(pr\left(\Pi \times \Pi \right)\right)}_{\omega },R\times R\right)$ and $\left(f_{pu}^{-1}\left(F\right)\times f_{pu}^{-1}\left(W\right)\right)\tilde{\cap }L=0_{R\times R}$. This proves that $1_{R\times R}-L\in {\left({\left(pr\left(\Pi \times \Pi \right)\right)}_{\omega }\right)}^{*}$. □

Theorem 28.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is a soft s-c and injection, and $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, then $\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff.

Proof. 

Let $a_x$, $b_y\in SP\left(S,B\right)$ such that $a_x\ne b_y$. Since $f_{pu}$ is injection, $f_{pu}\left(a_x\right)\ne f_{pu}\left(b_y\right)$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, we find $G,H\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }G$, $f_{pu}\left(b_y\right)\tilde{\in }H$, and $G\tilde{\cap }H=0_D$. Since $f_{pu}$ is a soft s-c, $f_{pu}^{-1}\left(G\right)$, $f_{pu}^{-1}\left(H\right)\in CO\left(S,\Pi ,R\right)$. Also, we have $a_x\tilde{\in }$$f_{pu}^{-1}\left(G\right)$, $b_y\tilde{\in }$$f_{pu}^{-1}\left(H\right)$, and $f_{pu}^{-1}\left(G\right)\tilde{\cap }{f}_{pu}^{-1}\left(H\right)=f_{pu}^{-1}\left(G\tilde{\cap }H\right)=f_{pu}^{-1}\left(0_D\right)=0_R$. This proves that $\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff. □

Corollary 7.

Let $\left(T,\Gamma ,D\right)$ be soft Hausdorff and soft 0-dim, and $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be a soft s-c and injection. Then, $\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff.

Proof. 

The proof follows from Theorems 24 and 28. □

Theorem 29.

If $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ is a soft s-ω-c and injection, and $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, then $\left(S,{\Pi }_{\omega },B\right)$ is soft ultra-Hausdorff.

Proof. 

Let $a_x$, $b_y\in SP\left(S,B\right)$ such that $a_x\ne b_y$. Since $f_{pu}$ is injection, $f_{pu}\left(a_x\right)\ne f_{pu}\left(b_y\right)$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-Hausdorff, we find $G,H\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }G$, $f_{pu}\left(b_y\right)\tilde{\in }H$, and $G\tilde{\cap }H=0_D$. Since $f_{pu}$ is a soft s-$\omega$-c, $f_{pu}^{-1}\left(G\right)$, $f_{pu}^{-1}\left(H\right)\in CO\left(S,{\Pi }_{\omega },B\right)$. Also, we have $a_x\tilde{\in }$$f_{pu}^{-1}\left(G\right)$, $b_y\tilde{\in }$$f_{pu}^{-1}\left(H\right)$, and $f_{pu}^{-1}\left(G\right)\tilde{\cap }{f}_{pu}^{-1}\left(H\right)=f_{pu}^{-1}\left(G\tilde{\cap }H\right)=f_{pu}^{-1}\left(0_D\right)=0_R$. This proves that $\left(S,{\Pi }_{\omega },B\right)$ is soft ultra-Hausdorff. □

Corollary 8.

Let $\left(T,\Gamma ,D\right)$ be soft Hausdorff and soft 0-dim, and $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be a soft s-ω-c and injection. Then, $\left(S,{\Pi }_{\omega },B\right)$ is soft ultra-Hausdorff.

Proof. 

The proof follows from Theorems 24 and 29. □

Definition 12.

An STS $\left(S,\Pi ,R\right)$ is said to be soft ultra-regular if for any $a_x\in SP\left(S,B\right)$ and any $F\in {\Pi }^c$ with $a_x\tilde{\in }{1}_R-F$, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\in }G$, $F\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_R$.

Theorem 30.

For any STS $\left(S,\Pi ,D\right)$, the following are equivalent:

 (a) 

$\left(S,\Pi ,D\right)$ is soft ultra-regular;

 (b) 

for any $a_x\in SP\left(S,D\right)$ and any $T\in \Pi$ such that $a_x\tilde{\in }T$, we find $G\in CO\left(S,\Pi ,D\right)$ such that $a_x\tilde{\in }G\tilde{\subseteq }T$;

 (c) 

$\Pi ={\Pi }^{*}$.

 (d) 

$\left(S,\Pi ,D\right)$ is soft 0-dim.

Proof. 

(a)

⟶ (b): Let $a_x\in SP\left(S,D\right)$ and $T\in \Pi$ such that $a_x\tilde{\in }T$. Then, we have $1_D-T\in {\Pi }^c$ with $a_x\tilde{\in }{1}_D-\left(1_D-T\right)$. So, by (a), we find $G,H\in CO\left(S,\Pi ,D\right)$ such that $a_x\tilde{\in }G$, $1_D-T\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_D$. Thus, we have $a_x\tilde{\in }G\tilde{\subseteq }{1}_D-H\tilde{\subseteq }T$.

(b)

⟶ (c): We only need to show that $\Pi \subseteq {\Pi }^{*}$. Let $T\in \Pi -\left\{0_D\right\}$ and let $a_x\tilde{\in }T$. Then, by (b), we find, $G\in CO\left(S,\Pi ,D\right)$ such that $a_x\tilde{\in }G\tilde{\subseteq }T$. This proves that $T\in {\Pi }^{*}$.

(c)

⟶ (d): Since $CO\left(S,\Pi ,D\right)$ is a soft base for $\left(S,{\Pi }^{*},D\right)$, by (c), $CO\left(S,\Pi ,D\right)$ is a soft base for $\left(S,\Pi ,D\right)$. This proves that $\left(S,\Pi ,D\right)$ is soft 0-dim.

(d)

⟶ (a): Let $a_x\in SP\left(S,D\right)$ and $F\in {\Pi }^c$ with $a_x\tilde{\in }{1}_D-F$. By (d), we find a soft base $\mathcal{G}$ for $\left(S,\Pi ,D\right)$ such that $\mathcal{G}\subseteq CO\left(S,\Pi ,D\right)$. So, we find $G\in \mathcal{G}$ such that $a_x\tilde{\in }G\tilde{\subseteq }{1}_D-F$. Let $H=1_D-G$. Then, we have G, $H\in CO\left(S,\Pi ,D\right)$, $a_x\tilde{\in }G$, $F\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_D$. This proves that $\left(S,\Pi ,D\right)$ is soft ultra-regular.

□

Theorem 31.

Soft ultra-regularity implies soft regularity.

Proof. 

Let $\left(S,\Pi ,R\right)$ be soft ultra-regular. Let $a_x\in SP\left(S,B\right)$ and $T\in \Pi$ such that $a_x\tilde{\in }T$. Then, by Theorem 30, we find $G\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\in }G\tilde{\subseteq }T$. Thus, we have $a_x\tilde{\in }G=C{l}_{\Pi }\left(G\right)\tilde{\subseteq }T$. This proves that $\left(S,\Pi ,R\right)$ is soft regular. □

In Example 3, the STS is soft regular, but not soft ultra-regular.

The example below demonstrates that soft ultra-regular STSs are not often soft ultra-Hausdorff:

Example 4.

Consider the topology $\sigma =\left\{\varnothing ,\mathbb{R},\left(-\infty ,0\right),\left[0,\infty \right)\right\}$ on $\mathbb{R}$. Consider the STS $\left(\mathbb{R},\tau \left(\sigma \right),\mathbb{Z}\right)$. Since $\left(\mathbb{R},\sigma \right)$ is 0-dim, by Corollary 6, $\left(\mathbb{R},\tau \left(\sigma \right),\mathbb{Z}\right)$ is soft 0-dim. Thus, by Theorem 30, $\left(\mathbb{R},\tau \left(\sigma \right),\mathbb{Z}\right)$ is soft ultra-regular. On the other hand, since $\left(\mathbb{R},\sigma \right)$ is clearly not ultra-Hausdorff, then by Corollary 4, it is not soft ultra-Hausdorff.

Definition 13.

If an STS $\left(S,\Pi ,R\right)$ is both soft ultra-regular and soft $T_1$, it is referred to be soft ultra-$T_3$.

Theorem 32.

Each soft ultra-$T_3$ STS is soft $T_3$.

Proof. 

The proof follows from the definitions and Theorem 31. □

In Example 3, the STS is soft $T_3$ but not soft ultra-$T_3$.

Theorem 33.

Each soft ultra-$T_3$ STS is soft ultra-Hausdorff.

Proof. 

Let $\left(S,\Pi ,R\right)$ be soft ultra-$T_3$. Then, by Theorem 32, $\left(S,\Pi ,R\right)$ is soft $T_3$. Thus, $\left(S,\Pi ,R\right)$ is soft Hausdorff. Hence, by Theorem 24, $\left(S,\Pi ,R\right)$ is soft ultra-Hausdorff. □

Question 1.

Is it true that every soft ultra-Hausdorff is soft ultra-$T_3$?

Theorem 34.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-c and soft open. If $\left(T,\Gamma ,D\right)$ is soft ultra-regular, then $\left(S,\Pi ,R\right)$ is soft ultra-regular.

Proof. 

Let $a_x\in SP\left(S,B\right)$ and $T\in \Pi$ such that $a_x\tilde{\in }T$. Since $f_{pu}$ is soft open, $f_{pu}\left(a_x\right)\tilde{\in }{f}_{pu}\left(T\right)\in \Gamma$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-regular, by Theorem 30, we find $G\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }G\tilde{\subseteq }{f}_{pu}\left(T\right)$ and so $a_x\tilde{\in }{f}_{pu}^{-1}\left(G\right)\tilde{\subseteq }T$. Since $f_{pu}$ is soft s-c, $f_{pu}^{-1}\left(G\right)\in CO\left(S,\Pi ,R\right)$. Thus, again by Theorem 30, $\left(S,\Pi ,R\right)$ is soft ultra-regular. □

Theorem 35.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-ω-c such that $f_{pu}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,\Gamma ,D\right)$ is soft open. If $\left(T,\Gamma ,D\right)$ is soft ultra-regular, then $\left(S,{\Pi }_{\omega },B\right)$ is soft ultra-regular.

Proof. 

Let $a_x\in SP\left(S,B\right)$ and $T\in {\Pi }_{\omega }$ such that $a_x\tilde{\in }T$. Since $f_{pu}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,\Gamma ,D\right)$ is soft open, $f_{pu}\left(a_x\right)\tilde{\in }{f}_{pu}\left(T\right)\in \Gamma$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-regular, by Theorem 30, we find $G\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }G\tilde{\subseteq }{f}_{pu}\left(T\right)$ and so $a_x\tilde{\in }{f}_{pu}^{-1}\left(G\right)\tilde{\subseteq }T$. Since $f_{pu}$ is soft s-$\omega$-c, $f_{pu}^{-1}\left(G\right)\in CO\left(S,{\Pi }_{\omega },B\right)$. Thus, again by Theorem 30, $\left(S,{\Pi }_{\omega },B\right)$ is soft ultra-regular. □

Theorem 36.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-ω-c, soft closed,and injective. If $\left(T,\Gamma ,D\right)$ is soft ultra-regular, then $\left(S,\Pi ,R\right)$ is soft ultra-regular.

Proof. 

Let $F\in {\Pi }^c$ and let $a_x\tilde{\in }{1}_R-F$. Then, $f_{pu}\left(a_x\right)\tilde{\in }{f}_{pu}\left(1_R-F\right)$. Since $f_{pu}$ is soft closed, $f_{pu}\left(F\right)\in {\Gamma }^c$. Since $f_{pu}$ is injective, $f_{pu}\left(1_R-F\right)=f_{pu}\left(1_R\right)-f_{pu}\left(F\right)$ and so $f_{pu}\left(a_x\right)\tilde{\in }{1}_D-f_{pu}\left(F\right)$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-regular, we find $G,H\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }G$, $f_{pu}\left(F\right)\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_D$. So, we have $a_x\tilde{\in }{f}_{pu}^{-1}\left(G\right)$, $F\tilde{\subseteq }{f}_{pu}^{-1}\left(f_{pu}\left(F\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(H\right)$ and $f_{pu}^{-1}\left(G\right)\tilde{\cap }{f}_{pu}^{-1}\left(H\right)=f_{pu}^{-1}\left(G\tilde{\cap }H\right)=f_{pu}^{-1}\left(0_D\right)=0_R$. Since $f_{pu}$ is s-c, $f_{pu}^{-1}\left(G\right)$, $f_{pu}^{-1}\left(H\right)\in CO\left(S,\Pi ,R\right)$. This proves that $\left(S,\Pi ,R\right)$ is soft ultra-regular. □

Theorem 37.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-ω-c and injective with $f_{pu}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,\Gamma ,D\right)$ is soft closed. If $\left(T,\Gamma ,D\right)$ is soft ultra-regular, then $\left(S,{\Pi }_{\omega },B\right)$ is soft ultra-regular.

Proof. 

Let $F\in {\left({\Pi }_{\omega }\right)}^c$ and let $a_x\tilde{\in }{1}_R-F$. Then, $f_{pu}\left(a_x\right)\tilde{\in }{f}_{pu}\left(1_R-F\right)$. Since $f_{pu}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,\Gamma ,D\right)$ is soft closed, $f_{pu}\left(F\right)\in {\Gamma }^c$. Since $f_{pu}$ is injective, $f_{pu}\left(1_R-F\right)=f_{pu}\left(1_R\right)-f_{pu}\left(F\right)$ and so $f_{pu}\left(a_x\right)\tilde{\in }{1}_D-f_{pu}\left(F\right)$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-regular, we find $G,H\in CO\left(T,\Gamma ,D\right)$ such that $f_{pu}\left(a_x\right)\tilde{\in }G$, $f_{pu}\left(F\right)\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_D$. So, we have $a_x\tilde{\in }{f}_{pu}^{-1}\left(G\right)$, $F\tilde{\subseteq }{f}_{pu}^{-1}\left(f_{pu}\left(F\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(H\right)$, and $f_{pu}^{-1}\left(G\right)\tilde{\cap }{f}_{pu}^{-1}\left(H\right)=f_{pu}^{-1}\left(G\tilde{\cap }H\right)=f_{pu}^{-1}\left(0_D\right)=0_R$. Since $f_{pu}$ is s-$\omega$-c, $f_{pu}^{-1}\left(G\right)$, $f_{pu}^{-1}\left(H\right)\in CO\left(S,\Pi ,R\right)$. This proves that $\left(S,\Pi ,R\right)$ is soft ultra-regular. □

Definition 14.

An STS $\left(S,\Pi ,R\right)$ is said to be the following:

 (a) 

soft strongly zero-dimensional (soft strongly 0-dim, for short) if $F,L\in {\Pi }^c$ such that $F\tilde{\cap }L=0_R$, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $F\tilde{\subseteq }G$, $L\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_R$.

 (b) 

soft ultra-normal if for any $F\in {\Pi }^c$ and any $T\in \Pi$ such that $F\tilde{\subseteq }T$, we find $G\in CO\left(S,\Pi ,R\right)$ such that $F\tilde{\subseteq }G\tilde{\subseteq }T$.

Theorem 38.

An STS $\left(S,\Pi ,R\right)$ is soft ultra-normal if and only if it is soft strongly 0-dim.

Proof. 

Necessity. Suppose that $\left(S,\Pi ,R\right)$ is soft ultra-normal. Let $F\in {\Pi }^c$ and $T\in \Pi$ such that $F\tilde{\subseteq }T$. Then, $F,1_R-T\in {\Pi }^c$ such that $F\tilde{\cap }\left(1_R-T\right)=0_R$. So, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $F\tilde{\subseteq }G$, $1_R-T\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_R$. Thus, we have $G\in CO\left(S,\Pi ,R\right)$ and $F\tilde{\subseteq }G\tilde{\subseteq }{1}_R-H\tilde{\subseteq }T$.

Sufficiency. Suppose that $\left(S,\Pi ,R\right)$ is soft strongly 0-dim. Let $F,L\in {\Pi }^c$ such that $F\tilde{\cap }L=0_R$. Then, we have $F\tilde{\subseteq }{1}_R-L$$\in \Pi$. So, we find $G\in CO\left(S,\Pi ,R\right)$ such that $F\tilde{\subseteq }G\tilde{\subseteq }{1}_R-L$. Thus, we have G, $1_R-G\in CO\left(S,\Pi ,R\right)$, $F\tilde{\subseteq }G$, $L\tilde{\subseteq }{1}_R-G$, and $G\tilde{\cap }\left(1_R-G\right)=0_R$. □

Theorem 39.

If $\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ is a family of TSs such that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft ultra-normal, then $\left(S,{\Pi }_r\right)$ is ultra-normal for every $r\in R$.

Proof. 

Assume that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft ultra-normal and let $a\in R$. Let $C\in {\left({\Pi }_a\right)}^c$ and $U\in {\Pi }_a$ such that $C\subseteq U$. Then, $a_C\in {\Pi }^c$, $a_U\in \Pi$, and $a_C\tilde{\subseteq }{a}_U$. Thus, by soft ultra-normality of $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$, we find $H\in CO\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ such that $a_C\tilde{\subseteq }H\tilde{\subseteq }{a}_U$. Thus, we have $C\subseteq H\left(a\right)\subseteq U$ and by Lemma 1, $H\left(a\right)\in CO\left(S,{\Pi }_a\right)$. This proves that $\left(S,{\Pi }_a\right)$ is ultra-normal. □

Question 2.

Let $\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ be a family of ultra-normal TSs. Is it true that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft ultra-normal.

Question 2’s partial solution is provided by the following result:

Theorem 40.

Let $\left(S,\beta \right)$ be a TS and B be a set of parameters. Then, $\left(S,\tau \left(\beta \right),B\right)$ is soft ultra-normal if and only if $\left(S,\beta \right)$ is ultra-normal.

Proof. 

Necessity. Suppose that $\left(S,\tau \left(\beta \right),B\right)$ is soft ultra-normal. For every $r\in R$, let ${\Pi }_r=\beta$. Then, $\tau \left(\beta \right)={\oplus }_{r\in R}{\Pi }_r$. As a consequence of Theorem 39, we get that $\left(S,\beta \right)$ is ultra-normal.

Sufficiency. Suppose that $\left(S,\beta \right)$ is ultra-normal. Let $F\in {\left(\tau \left(\beta \right)\right)}^c$ and $T\in \tau \left(\beta \right)$ such that $F\tilde{\subseteq }T$. For each $r\in R$, we have $F\left(b\right)\in {\beta }^c$, $T\left(b\right)\in \beta$, and $F\left(b\right)\subseteq T\left(r\right)$ and thus we find $H_b\in CO\left(S,\beta \right)$ such that $F\left(b\right)\subseteq H_b\subseteq T\left(r\right)$. Consider $H\in SS(S,B)$ where $H\left(b\right)=H_b$ for every $r\in R$. Then, $H\in CO\left(S,\tau \left(\beta \right),B\right)$ and $F\tilde{\subseteq }H\tilde{\subseteq }T$. This proves that $\left(S,\tau \left(\beta \right),B\right)$ is soft ultra-normal. □

Theorem 41.

If $\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ is a family of TSs such that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft normal, then $\left(S,{\Pi }_r\right)$ is normal for every $r\in R$.

Proof. 

Assume that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft normal and let $a\in R$. Let $C\in {\left({\Pi }_a\right)}^c$ and $U\in {\Pi }_a$ such that $C\subseteq U$. Then, $a_C\in {\Pi }^c$, $a_U\in \Pi$, and $a_C\tilde{\subseteq }{a}_U$. Thus, by soft normality of $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$, we find $H\in {\oplus }_{r\in R}{\Pi }_r$ such that $a_C\tilde{\subseteq }H\tilde{\subseteq }C{l}_{{\oplus }_{r\in R}{\Pi }_r}\left(H\right)\tilde{\subseteq }{a}_U$. Thus, we have $C\subseteq H\left(a\right)\subseteq \left(C{l}_{{\oplus }_{r\in R}{\Pi }_r}\left(H\right)\right)\left(a\right)\subseteq U$. Now, by Lemma 4.9 of [43], $\left(C{l}_{{\oplus }_{r\in R}{\Pi }_r}\left(H\right)\right)\left(a\right)=C{l}_{{\Pi }_r}\left(H(a\right))$. This proves that $\left(S,{\Pi }_a\right)$ is normal. □

Question 3.

Let $\left\{\left(S,{\Pi }_r\right):r\in R\right\}$ be a family of normal TSs. Is it true that $\left(S,{\oplus }_{r\in R}{\Pi }_r,R\right)$ is soft normal.

Question 3’s partial solution is provided by the following result:

Theorem 42.

Let $\left(S,\beta \right)$ be a TS and R be a set of parameters. Then, $\left(S,\tau \left(\beta \right),R\right)$ is soft normal if and only if $\left(S,\beta \right)$ is normal.

Proof. 

Necessity. Suppose that $\left(S,\tau \left(\beta \right),R\right)$ is soft normal. For every $r\in R$, let ${\Pi }_r=\beta$. Then, $\tau \left(\beta \right)={\oplus }_{r\in R}{\Pi }_r$. As a consequence of Theorem 41, we get that $\left(S,\beta \right)$ is normal.

Sufficiency. Suppose that $\left(S,\beta \right)$ is normal. Let $F\in {\left(\tau \left(\beta \right)\right)}^c$ and $T\in \tau \left(\beta \right)$ such that $F\tilde{\subseteq }T$. For each $r\in R$, we have $F\left(r\right)\in {\beta }^c$, $T\left(r\right)\in \beta$, and $F\left(r\right)\subseteq T\left(r\right)$, and thus we find $H_r\in \beta$ such that $F\left(r\right)\subseteq H_r\subseteq C{l}_{\beta }\left(H_r\right)\subseteq T\left(r\right)$. Consider $H\in SS(S,R)$ where $H\left(r\right)=H_r$ for every $r\in R$. Then, $H\in \tau \left(\beta \right)$ and $C{l}_{\tau \left(\beta \right)}\left(\left(H\right)\right)\tilde{\subseteq }T$. Therefore, we have $F\tilde{\subseteq }H\tilde{\subseteq }C{l}_{\tau \left(\beta \right)}\left(\left(H\right)\right)\tilde{\subseteq }T$. This proves that $\left(S,\tau \left(\beta \right),R\right)$ is soft normal. □

Theorem 43.

Each soft ultra-normal STS is soft normal.

Proof. 

Let $\left(S,\Pi ,R\right)$ be soft ultra-normal. Let $F\in {\Pi }^c$ and $T\in \Pi$ such that $F\tilde{\subseteq }T$. Then, by Theorem 38, we find $G\in CO\left(S,\Pi ,R\right)$ such that $F\tilde{\subseteq }G\tilde{\subseteq }T$. Thus, we have $F\tilde{\subseteq }G=C{l}_{\Pi }\left(G\right)\tilde{\subseteq }T$. This proves that $\left(S,\Pi ,R\right)$ is soft normal. □

In Example 3, the STS is soft normal but not soft ultra-normal.

Theorem 44.

Each soft locally indiscrete STS is soft ultra-normal.

Proof. 

Let $\left(S,\Pi ,R\right)$ be soft locally indiscrete. Let $F\in {\Pi }^c$ and $T\in \Pi$ such that $F\tilde{\subseteq }T$. Then, we have $T\in CO\left(S,\Pi ,R\right)$ and $F\tilde{\subseteq }T\tilde{\subseteq }T$. □

The following example shows that soft ultra-normal STSs are not soft ultra-Hausdorff in general:

Example 5.

Consider the topology $\sigma =\left\{\varnothing ,\mathbb{R},\left(-\infty ,0\right),\left[0,\infty \right)\right\}$ on $\mathbb{R}$. It is clear that $\left(\mathbb{R},\tau \left(\sigma \right),\mathbb{Z}\right)$ is soft locally indiscrete. On the other hand, since $\left(\mathbb{R},\sigma \right)$ is clearly not ultra-Hausdorff, then by Corollary 4, it is not soft ultra-Hausdorff.

Definition 15.

An STS $\left(S,\Pi ,R\right)$ is said to be soft ultra-$T_4$ if it is soft ultra-normal and soft $T_1$.

Theorem 45.

Each soft ultra-$T_4$ STS is soft $T_4$.

Proof. 

The proof follows from the definitions and Theorem 43. □

In Example 3, the STS is soft T4 but not soft ultra-T4.

Theorem 46.

Each soft ultra-$T_4$ STS is soft ultra-$T_3$.

Proof. 

Let $\left(S,\Pi ,R\right)$ be soft ultra-$T_4$. We will apply Theorem 30 (b) to show that show that $\left(S,\Pi ,R\right)$ is soft ultra-regular. Let $a_x\in SP\left(S,B\right)$ and $T\in \Pi$ such that $a_x\tilde{\in }T$. Since $\left(S,\Pi ,R\right)$ is soft $T_1$, $a_x\in {\Pi }^c$. Since $a_x\tilde{\subseteq }T$ and $\left(S,\Pi ,R\right)$ is soft ultra-normal, we find $G\in CO\left(S,\Pi ,R\right)$ such that $a_x\tilde{\subseteq }G\tilde{\subseteq }T$ and, hence, $a_x\tilde{\in }G\tilde{\subseteq }T$. □

The following example shows that the converse of Theorem 46 need not be true in general:

Example 6.

Let $\mathcal{H}=\left\{\left[a,b\right)\times \left[c,d\right):a,b,c,d\in \mathbb{R},a<b\mathit{and}c<d\right\}$. Consider the topology σ on ${\mathbb{R}}^2$ having $\mathcal{H}$ as a base. It is known $\left({\mathbb{R}}^2,\sigma \right)$ is 0-dim but not normal. Thus, by Corollary 6 and Theorem 42, $\left(\mathbb{R},\tau \left(\sigma \right),\mathbb{Z}\right)$ is soft 0-dim but not soft normal. On the other hand, it is clear that $\left(\mathbb{R},\tau \left(\sigma \right),\mathbb{Z}\right)$ is soft $T_1$.

Theorem 47.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-c, soft closed, and injective. If $\left(T,\Gamma ,D\right)$ is soft ultra-normal, then $\left(S,\Pi ,R\right)$ is soft ultra-normal.

Proof. 

Let $F,L\in {\Pi }^c$ such that $F\tilde{\cap }L=0_R$. Since $f_{pu}$ is soft closed, $f_{pu}\left(F\right)$, $f_{pu}\left(L\right)\in {\Gamma }^c$. Since $f_{pu}$ is injective, $f_{pu}\left(F\right)\tilde{\cap }{f}_{pu}\left(L\right)=f_{pu}\left(F\tilde{\cap }L\right)=f_{pu}\left(0_R\right)=0_D$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-normal, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $f_{pu}\left(F\right)\tilde{\subseteq }G$, $f_{pu}\left(L\right)\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_D$. Thus, we have $F\tilde{\subseteq }{f}_{pu}^{-1}\left(f_{pu}\left(F\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(G\right)$, $L\tilde{\subseteq }{f}_{pu}^{-1}\left(f_{pu}\left(L\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(H\right)$, and $f_{pu}^{-1}\left(G\right)\tilde{\cap }{f}_{pu}^{-1}\left(H\right)=f_{pu}^{-1}\left(G\tilde{\cap }H\right)=f_{pu}^{-1}\left(0_D\right)=0_R$. Since $f_{pu}$ is soft s-c, $f_{pu}^{-1}\left(G\right)$, $f_{pu}^{-1}\left(H\right)\in CO\left(S,\Pi ,R\right)$. This proves that $\left(S,\Pi ,R\right)$ is soft ultra-normal. □

Theorem 48.

Let $f_{pu}:\left(S,\Pi ,R\right)⟶\left(T,\Gamma ,D\right)$ be soft s-ω-c and injective such that $f_{pu}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,\Gamma ,D\right)$ is soft closed. If $\left(T,\Gamma ,D\right)$ is soft ultra-normal, then $\left(S,{\Pi }_{\omega },B\right)$ is soft ultra-normal.

Proof. 

Let $F,L\in {\left({\Pi }_{\omega }\right)}^c$ such that $F\tilde{\cap }L=0_R$. Since $f_{pu}:\left(S,{\Pi }_{\omega },B\right)⟶\left(T,\Gamma ,D\right)$ is soft closed, $f_{pu}\left(F\right)$, $f_{pu}\left(L\right)\in {\Gamma }^c$. Since $f_{pu}$ is injective, $f_{pu}\left(F\right)\tilde{\cap }{f}_{pu}\left(L\right)=f_{pu}\left(F\tilde{\cap }L\right)=f_{pu}\left(0_R\right)=0_D$. Since $\left(T,\Gamma ,D\right)$ is soft ultra-normal, we find $G,H\in CO\left(S,\Pi ,R\right)$ such that $f_{pu}\left(F\right)\tilde{\subseteq }G$, $f_{pu}\left(L\right)\tilde{\subseteq }H$, and $G\tilde{\cap }H=0_D$. Thus, we have $F\tilde{\subseteq }{f}_{pu}^{-1}\left(f_{pu}\left(F\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(G\right)$, $L\tilde{\subseteq }{f}_{pu}^{-1}\left(f_{pu}\left(L\right)\right)\tilde{\subseteq }{f}_{pu}^{-1}\left(H\right)$, and $f_{pu}^{-1}\left(G\right)\tilde{\cap }{f}_{pu}^{-1}\left(H\right)=f_{pu}^{-1}\left(G\tilde{\cap }H\right)=f_{pu}^{-1}\left(0_D\right)=0_R$. Since $f_{pu}$ is soft s-$\omega$-c, $f_{pu}^{-1}\left(G\right)$, $f_{pu}^{-1}\left(H\right)\in CO\left(S,{\Pi }_{\omega },B\right)$. This proves that $\left(S,\Pi ,R\right)$ is soft ultra-normal. □

4. Conclusions

We define the concepts of soft slight $\omega$-continuity, soft ultra-Hausdorff, soft ultra-regular, and soft ultra-normal. Related to them, we introduce several characterizations, relationships, and examples. Moreover, via soft slight continuity and soft slight $\omega$-continuity, we obtain several soft inverse mapping theorems of the newly defined soft separation axioms. In addition to these, we raise three open questions.

Future research might look into the following topics: (1) To solve some of the three questions raised in this paper; (2) To investigate the soft product of soft ultra-normal STSs.