Connectedness and Stratification of Single-Valued Neutrosophic Topological Spaces

Abstract

This paper aims to introduce the notion of r-single-valued neutrosophic connected sets in single-valued neutrosophic topological spaces, which is considered as a generalization of r-connected sets in Šostak’s sense and r-connected sets in intuitionistic fuzzy topological spaces. In addition, it introduces the concept of r-single-valued neutrosophic separated and obtains some of its basic properties. It also tries to show that every r-single-valued neutrosophic component in single-valued neutrosophic topological spaces is an r-single-valued neutrosophic component in the stratification of it. Finally, for the purpose of symmetry, it defines the so-called single-valued neutrosophic relations.

1. Introduction

Under a neutrosophic environment, Smarandache had established a generalization of intuitionistic fuzzy sets. His neutrosophic framework has a very large impact of constant applications for different fields in applied and pure sciences. In 1965, Zadeh [1] defined the so-called fuzzy sets $(\mathcal{FS})$ and, later on, Atanassov [2] defined the intuitionistic fuzzy sets $(\mathcal{IFS})$ in 1983. Topology is, of course, a cornerstone notion of mathematics, especially for ordinary subjects. The main concept of fuzzy topology $(\mathcal{FT})$ was defined by Chang [3]. Moreover, Lowen [4] gave the introduction to the concept of stratified fuzzy topology in the sense of Chang’s fuzzy topology. Lee et al. and Liu et al. in their papers [5,6] investigated fuzzy connectedness $(\mathcal{F}$-$connected)$ in fuzzy topological spaces. Again, researchers in [7,8,9,10] have studied the concept of $(\mathcal{F}$-$connected)$. Sostak [11], however, also introduced the concept of smooth topology as an extension of Lowen and Chang’s work.

In his paper [12], Smarandache characterized the neutrosophic set into three segment neutrosophic sets (F-Falsehood, I-Indeterminacy, T-Truth), and neutrosophic topological spaces ($\mathcal{SVNT}$) presented by Salama et al. [13,14]. Single valued neutrosophic sets (in sort, $\mathcal{SVN}$) were proposed by Wang et al. [15]. Meanwhile, Kim et al. [16] inspected the single valued neutrosophic relations ($\mathcal{SVNR}s$ ) and symmetric closure of $\mathcal{SVNR}$, respectively. In recent times, Saber et al. [17] familiarized the concepts of single-valued neutrosophic ideal open local function and single-valued neutrosophic topological space.

In this paper, we introduce the concept of r-single-valued neutrosophic connected sets and r-single-valued neutrosophic component in single-valued neutrosophic topological spaces. We then define the stratification of the single-valued neutrosophic topological spaces and show that every r-single-valued neutrosophic component in a single-valued neutrosophic is an r-single-valued neutrosophic component in the stratification of it. We have performed distinguished definitions, theorems, and counterexamples in-depth analysis to investigate some of their significant properties and to find out the best results and consequences. It can be said that different crucial notions in single valued neutrosophic topology were developed and generalized in this article. Different attributes like connectedness and stratification which have a significant impact on the overall topology’s notions were also studied.

Innovative aspects and benefits of this article compared to relevant recent research on groups related to it are very useful. This paper studies connectedness and stratification of single-valued neutrosophic topological spaces. What makes this paper interesting is the introduction of the concept of r-single-valued neutrosophic separated. The authors obtain some of its basic properties. They show that every r-single-valued neutrosophic component in single-valued neutrosophic topological spaces is an r-single-valued neutrosophic component in the stratification of it.

A neutrosophic set is a power general formal framework, which generalizes the concept of the classic set, fuzzy set, interval valued fuzzy set, intuitionistic fuzzy set, and interval intuitionistic fuzzy set from a philosophical point of view. The applications aspects of these kinds of sets can be further noted. It can be seen In Geographical Information Systems (GIS) where there is a need to model spatial regions with indeterminate boundary and under indeterminacy (see [18]). In addition, possible applications to superstrings and ${\zeta }^{\infty }$ space–time are touched upon (see [19]). It can also be applicable to control engineering in average consensus in multi-agent systems with uncertain topologies, multiple time-varying delays, and emergence in random noisy environments (see [20]).

In this work, $\tilde{X}$ is assumed to be a nonempty set, $\zeta =[0,1]$ and ${\zeta }_0=(0,1]$. For $\alpha \in \zeta$, $\tilde{\alpha }(x)=\alpha$ for all $x\in \tilde{X}$. The family of all single-valued neutrosophic sets on $\tilde{X}$ is denoted by ${\zeta }^{\tilde{X}}$.

2. Preliminaries

This section is devoted to bring a complete survey and previous studies and important related notions and ideas.

Definition 1

([21]). Let $\tilde{X}$ be a non-empty set. A neutrosophic set (briefly, $\mathcal{NS}$) in $\tilde{X}$ is an object having the form

$$\begin{array}{l}\mathcal{S}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\},\end{array}$$

where ${\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}$ and the degree of membership (namely ${\tilde{\gamma }}_{\mathcal{S}}(x)$), the degree of indeterminacy (namely ${\tilde{\eta }}_{\mathcal{S}}(x)$), and the degree of non-membership (namely ${\tilde{\mu }}_{\mathcal{S}}(x)$); for all $x\in \tilde{X}$ to the set $\mathcal{S}$. A neutrosophic set $\mathcal{S}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\}$ can be identified as ${\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}$ in ${\rfloor }^{-}0,1^{+}\lfloor$ in $\tilde{X}$.

Definition 2

([22]). Suppose that $\mathcal{S}$ and $\mathcal{E}$ are ${\mathcal{NS}}^{\prime }s$ of the form $\mathcal{S}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\}$ and $\mathcal{E}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\}$ Then, $\mathcal{S}\subseteq \mathcal{E}$, iff for every $x\in \tilde{X}$,

$$\begin{array}{l}inf{\tilde{\gamma }}_{\mathcal{S}}(x)\le inf{\tilde{\gamma }}_{\mathcal{E}}(x),inf{\tilde{\eta }}_{\mathcal{S}}(x)\ge inf{\tilde{\eta }}_{\mathcal{E}}(x),inf{\tilde{\mu }}_{\mathcal{S}}(x)\ge inf{\tilde{\mu }}_{\mathcal{E}}(x),\end{array}$$

$$\begin{array}{l}sup{\tilde{\gamma }}_{\mathcal{S}}(x)\le sup{\tilde{\gamma }}_{\mathcal{E}}(x),sup{\tilde{\eta }}_{\mathcal{S}}(x)\ge sup{\tilde{\eta }}_{\mathcal{E}}(x),sup{\tilde{\mu }}_{\mathcal{S}}(x)\ge sup{\tilde{\mu }}_{\mathcal{E}}(x).\end{array}$$

Definition 3

([15]). Let $\tilde{X}$ be a space of points (objects), with a generic element in $\tilde{X}$ denoted by x. Then, $\mathcal{S}$ is called a single valued neutrosophic set (briefly, $\mathcal{SVNS}$) in $\tilde{X}$, if $\mathcal{S}$ has the form $\mathcal{S}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\}$, where ${\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}:\tilde{X}\to [0,1]$.

In this case, ${\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}$ are called truth-membership, indeterminacy-membership, falsify-membership mappings, respectively, and we will denote the set of all ${\mathcal{SVNS}}^{\prime }s$ in $\tilde{X}$ as $I^{\tilde{X}}$. Moreover, we will refer to the Null (empty) $\mathcal{SVNS}$ (resp. the absolute (universe) $\mathcal{SVNS}$) in $\tilde{X}$ as $\tilde{0}$ (resp. $\tilde{1}$) and defined by $\tilde{0}=\langle 0,1,1\rangle$ (resp. $\tilde{1}=\langle 1,0,0\rangle$) for each $x\in \tilde{X}$.

Definition 4

([15]). Let $\mathcal{S}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\}$ be an $\mathcal{SVNS}$ on $\tilde{X}$. The complement of the set $\mathcal{S}$ (briefly ${\mathcal{S}}^c$) is defined as follows:

$$\begin{array}{l}{\tilde{\gamma }}_{{\mathcal{S}}^c}(x)={\tilde{\mu }}_{\mathcal{S}}(x),{\tilde{\eta }}_{{\mathcal{S}}^c}(x)=1-{\tilde{\eta }}_{\mathcal{S}}(x),{\tilde{\mu }}_{{\mathcal{S}}^c}(x)={\tilde{\gamma }}_{\mathcal{S}}(x),\end{array}$$

for every $x\in \tilde{X}$.

Definition 5

([23]). Let $\mathcal{S}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\}$ and $\mathcal{E}=\{\langle x,{\tilde{\gamma }}_{\mathcal{E}},{\tilde{\eta }}_{\mathcal{E}},{\tilde{\mu }}_{\mathcal{E}}\rangle :x\in \tilde{X}\}$ be an $\mathcal{SVNS}$. Then,

(i) 

A $\mathcal{SVNS}$$\mathcal{S}$ is contained in the other $\mathcal{SVNS}$$\mathcal{E}$ (briefly, $\mathcal{S}\subseteq \mathcal{E}$), if and only if

$$\begin{array}{l}{\tilde{\gamma }}_{\mathcal{S}}(x)\le {\tilde{\gamma }}_{\mathcal{E}}(x),{\tilde{\eta }}_{\mathcal{S}}(x)\ge {\tilde{\eta }}_{\mathcal{E}}(x),{\tilde{\mu }}_{\mathcal{S}}(x)\ge {\tilde{\mu }}_{\mathcal{E}}(x)\end{array}$$

for every $\omega \in \tilde{X}$,

(ii) 

we say that $\mathcal{S}$ is equal to $\mathcal{E}$, denoted by $\mathcal{S}=\mathcal{E}$, if $\mathcal{S}\subseteq \mathcal{E}$ and $\mathcal{S}\supseteq \mathcal{E}$.

Definition 6

([22]). Let $\mathcal{S}=\{\langle x,{\tilde{\gamma }}_{\mathcal{S}},{\tilde{\eta }}_{\mathcal{S}},{\tilde{\mu }}_{\mathcal{S}}\rangle :x\in \tilde{X}\}$ and $\mathcal{E}=\{\langle x,{\tilde{\gamma }}_{\mathcal{E}},{\tilde{\eta }}_{\mathcal{E}},{\tilde{\mu }}_{\mathcal{E}}\rangle :x\in \tilde{X}\}$ be an $\mathcal{SVNS}$. Then,

(i) 

the intersection of $\mathcal{S}$ and $\mathcal{E}$ (briefly, $\mathcal{S}\cap \mathcal{E}$) is a $\mathcal{SVNS}$ in $\tilde{X}$ defined as:

$$\begin{array}{l}\mathcal{S}\cap \mathcal{E}=({\tilde{\gamma }}_{\mathcal{S}}\cap {\tilde{\gamma }}_{\mathcal{E}},{\tilde{\eta }}_{\mathcal{S}}\cup {\tilde{\eta }}_{\mathcal{E}},{\tilde{\mu }}_{\mathcal{S}}\cup {\tilde{\mu }}_{\mathcal{E}})\end{array}$$

where $({\tilde{\mu }}_{\mathcal{S}}\cup {\tilde{\mu }}_{\mathcal{E}})(x)={\tilde{\mu }}_{\mathcal{S}}(x)\cup {\tilde{\mu }}_{\mathcal{E}}(x)$ and $({\tilde{\gamma }}_{\mathcal{S}}\cap {\tilde{\gamma }}_{\mathcal{E}})(x)={\tilde{\gamma }}_{\mathcal{S}}(x)\cap {\tilde{\gamma }}_{\mathcal{E}}(x)$, for all $x\in \tilde{X}$,

(ii) 

the union of $\mathcal{S}$ and $\mathcal{E}$ (briefly, $\mathcal{S}\cup \mathcal{E}$) is an $\mathcal{SVNS}$ on $\tilde{X}$ defined as:

$$\begin{array}{l}\mathcal{S}\cup \mathcal{E}=({\tilde{\gamma }}_{\mathcal{S}}\cup {\tilde{\gamma }}_{\mathcal{E}},{\tilde{\eta }}_{\mathcal{S}}\cap {\tilde{\eta }}_{\mathcal{E}},{\tilde{\mu }}_{\mathcal{S}}\cap {\tilde{\mu }}_{\mathcal{E}}).\end{array}$$

Definition 7

([13]). Let $\{{\mathcal{S}}_j,j\in \mathsf{\Gamma }\}$ be an arbitrary family of ${\mathcal{SVNS}}^{\prime }s$ on $\tilde{X}$. Then,

(i) 

the intersection of $\{{\mathcal{S}}_j,j\in \mathsf{\Gamma }\}$ (briefly, ${\bigcap }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j$) is $\mathcal{SVNS}$ over $\tilde{X}$ defined as:

$$\begin{array}{l}({\displaystyle \bigcap _{J\in \mathsf{\Gamma }}}{\mathcal{S}}_j)(x)=({\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\gamma }}_{{\mathcal{S}}_j}(x),{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\eta }}_{{\mathcal{S}}_j}(x),{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\mu }}_{{\mathcal{S}}_j}(x)),\end{array}$$

for all $x\in \tilde{X}$,

(ii) 

the union of $\{{\mathcal{S}}_j,j\in \mathsf{\Gamma }\}$ (briefly, ${\bigcup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j)$ is $\mathcal{SVNS}$ over $\tilde{X}$ defined as:

$$\begin{array}{l}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{S}}_j)(x)=({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\gamma }}_{{\mathcal{S}}_j}(x),{\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\eta }}_{{\mathcal{S}}_j}(x),{\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\mu }}_{{\mathcal{S}}_j}(x)),\end{array}$$

for all $x\in \tilde{X}$.

Definition 8

([24]). A single-valued neutrosophic topology $(\mathcal{SVNT})$ on $\tilde{X}$ is an ordered triple (${\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}$) as mappings from ${\zeta }^{\tilde{X}}$ to ζ such that:

(SVNT1) 

${\tilde{\tau }}^{\tilde{\gamma }}(\tilde{0})={\tilde{\tau }}^{\tilde{\gamma }}(\tilde{1})=1$ and ${\tilde{\tau }}^{\tilde{\eta }}(\tilde{0})={\tilde{\tau }}^{\tilde{\eta }}(\tilde{1})={\tilde{\tau }}^{\tilde{\mu }}(\tilde{0})={\tilde{\tau }}^{\tilde{\mu }}(\tilde{1})=0$,

(SVNT2) 

${\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S}\cap \mathcal{E})\ge {\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})\cap {\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{E})$,    ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S}\cap \mathcal{E})\le {\tau }^{\tilde{\eta }}(\mathcal{S})\cup {\tilde{\tau }}^{\tilde{\eta }}(\mathcal{E})$,

${\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S}\cap \mathcal{E})\le {\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})\cup {\tilde{\tau }}^{\tilde{\mu }}(\mathcal{E})$, for all $\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}$,

(SVNT3) 

${\tilde{\tau }}^{\tilde{\gamma }}({\cup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j)\ge {\cap }_{j\in \mathsf{\Gamma }}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{S}}_j)$,    ${\tilde{\tau }}^{\tilde{\eta }}({\cup }_{i\in \mathsf{\Gamma }}{\mathcal{S}}_j)\le {\cup }_{j\in \mathsf{\Gamma }}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{S}}_j)$,

${\tilde{\tau }}^{\tilde{\mu }}({\cup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j)\le {\cup }_{j\in \mathsf{\Gamma }}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{S}}_j)$ for all $\{{\mathcal{S}}_j,j\in \mathsf{\Gamma }\}\in {\zeta }^{\tilde{X}}$.

The quadruple $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is called $\mathcal{SVNTS}$. ${\tilde{\tau }}^{\tilde{\gamma }}$, ${\tilde{\tau }}^{\tilde{\eta }}$ and ${\tilde{\tau }}^{\tilde{\mu }}$ may be interpreted as the degree of openness, the degree of indeterminacy, and the degree of non-openness, respectively, and any single valued neutrosophic (briefly, $\mathcal{SVNS}$) set in $\tilde{X}$ is known as a single valued neutrosophic open set (briefly, r-$\mathbf{SVNO}$) set in $\tilde{X}$. The elements of ${\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}$ are called open single valued neutrosophic sets (such that, for any $\mathcal{SVNS}$$\mathcal{S}\in \tilde{X}$ and $r\in I_0$, we obtain ${\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})\le 1-r$, and ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})\le 1-r$]. Then, the complement of r-$\mathbf{SVNO}$ is a single valued neutrosophic closed set (briefly, r-$\mathbf{SVNC}$), and this will cause no ambiguity. Occasionally, we will write ${\tilde{\tau }}^{\tilde{\gamma }\tilde{\eta }\tilde{\mu }}$ for $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$, and it will be no ambiguity.

Definition 9

([17]). Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$. A mapping $C:{\zeta }^{\tilde{X}}\times {\zeta }_0\to {\zeta }^{\tilde{X}}$ is called a single-valued neutrosophic closure operator if, for every $\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}$ and $r,s\in {\zeta }_0$, it satisfies the following conditions:

(C₁)

$C(\tilde{0},r)=\tilde{0}$,

(C₂)

$\mathcal{S}\le C(\mathcal{S},r)$,

(C₃)

$C(\mathcal{S},r)\cup C(\mathcal{E},r)=C(\mathcal{S}\cup \mathcal{E},r)$,

(C₄)

$C(\mathcal{S},r)\le C(\mathcal{S},s)$ if $r\le s$.

(C₅)

$C(C(\mathcal{S},r),r)=C(\mathcal{S},r)$.

The pair $(\tilde{X},C)$ is a single-valued neutrosophic closure space (briefly, $\mathcal{SVNCS}$).

Theorem 1

([17]). Let $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}$ be an single-valued neutrosophic closure operator on $\tilde{X}$. Define the mappings ${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }}}}^{\tilde{\gamma }},{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\eta }}}}^{\tilde{\eta }},{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\mu }}:{\zeta }^{\tilde{X}}\to \zeta$ by

$$\begin{array}{l}{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }}}}^{\tilde{\gamma }}(\mathcal{S})=\bigcup \{r\in {\zeta }_0\mid C_{{\tilde{\tau }}^{\tilde{\gamma }}}({\mathcal{S}}^c,r)={\mathcal{S}}^c\},{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\eta }}}}^{\tilde{\eta }}(\mathcal{S})=\bigcap \{1-r\in {\zeta }_0\mid C_{{\tilde{\tau }}^{\tilde{\eta }}}({\mathcal{S}}^c,r)={\mathcal{S}}^c\},\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\mu }}(\mathcal{S})=\bigcap \{1-r\in {\zeta }_0\mid C_{{\tilde{\tau }}^{\tilde{\mu }}}({\mathcal{S}}^c,r)={\mathcal{S}}^c\}.\end{array}$$

Then, $({\tau }_{C_{{\tilde{\tau }}^{\tilde{\gamma }}}}^{\tilde{\gamma }},{\tau }_{C_{{\tilde{\tau }}^{\tilde{\eta }}}}^{\tilde{\eta }},{\tau }_{C_{{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\mu }})$ is an $\mathcal{SVNT}$ on $\tilde{X}$.

Definition 10

([25]). Let $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }})$ be a mapping and $r\in {\zeta }_0$. Then, f is said to be $\mathcal{SVN}$-continuous if ${\tilde{\tau }}_2^{\tilde{\gamma }}(\mathcal{S})\le {\tilde{\tau }}_1^{\tilde{\gamma }}(f^{-1}(\mathcal{S}))$, ${\tilde{\tau }}_2^{\tilde{\eta }}(\mathcal{S})\ge {\tilde{\tau }}_1^{\tilde{\eta }}(f^{-1}(\mathcal{S}))$, and ${\tilde{\tau }}_2^{\tilde{\mu }}(\mathcal{S})\ge {\tilde{\tau }}_1^{\tilde{\mu }}(f^{-1}(\mathcal{S}))$ for all $\mathcal{S}\in {\zeta }^{\tilde{Y}}$.

3. Connectedness in Single-Valued Neutrosophic Topological Spaces

The aim of this section is to introduce the r-single-valued neutrosophic separated (briefly, r-$\mathcal{SVNSEP}$), r-single-valued neutrosophic connected (briefly, r-$\mathcal{SVNCON}$), and r-single-valued neutrosophic component (briefly, r-$\mathcal{SVNCOM}$).

Definition 11.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$. For every $\mathcal{S},\mathcal{E},\mathcal{R}\in {\zeta }^{\tilde{X}}$, $\mathcal{S}$ and $\mathcal{E}$ are called r-single-valued neutrosophic separated (briefly, r-$\mathcal{SVNSEP}$) if for $r\in {\zeta }_0$,

$$\begin{array}{l}{C}_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)\cap \mathcal{E}=C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{E},r)\cap \mathcal{S}=\tilde{0}\end{array}$$

A $\mathcal{SVNS}$, $\mathcal{R}$ is called r-single-valued neutrosophic connected (briefly, r-$\mathcal{SVNCON}$) if r-$\mathcal{SVNSEP}$$\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}-\{\tilde{0}\}$ such that $\mathcal{R}=\mathcal{S}\cup \mathcal{E}$ does not exist. A $\mathcal{SVNS}$$\mathcal{R}$ is said to be $\mathcal{SVNCON}$ if it is r-$\mathcal{SVNCON}$ for any $r\in {\zeta }_0$. A quadruple $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is said to be r-$\mathcal{SVNCON}$ if $\tilde{1}$ is r-$\mathcal{SVNCON}$.

Remark 1.

Let $\mathcal{S}$ and $\mathcal{E}$ be r-$\mathcal{SVNSEP}$. Then for every $\mathcal{R}\in {\zeta }^{\tilde{X}}$ and $r_1\le r$. We have $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{R},r_1)\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{R},r)$, and $\mathcal{S}$ and $\mathcal{E}$ are said to be $r_1$-$\mathcal{SVNSEP}$. Conversely, from this fact, if $\mathcal{R}$ is $r_1$-$\mathcal{SVNCON}$ and $r\ge r_1$, then $\mathcal{R}$ is called r-$\mathcal{SVNCON}$.

Example 1.

Let $\tilde{X}=\{a,b,c\}$ be a set. Define ${\mathcal{E}}_1,{\mathcal{E}}_2\in {\zeta }^{\tilde{X}}$ as follows:

$$\begin{array}{l}{\mathcal{E}}_1=\langle (1,1,0),(1,1,0),(1,1,0)\rangle ;{\mathcal{E}}_2=\langle (0,0,1),(0,0,1),(0,0,1)\rangle .\end{array}$$

We define an $\mathcal{SVNT}$$({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ on $\tilde{X}$ as follows: for each $\mathcal{S}\in {\zeta }^{\tilde{X}}$,

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{0},\hfill \\ 1,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{3},if\mathcal{S}={\mathcal{E}}_1,\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2,\hfill \\ 0,otherwise,\hfill \end{array}\right.{\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{1},\hfill \\ \frac{2}{3},if\mathcal{S}={\mathcal{E}}_1,\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2,\hfill \\ 1,otherwise,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{0},\hfill \\ \frac{2}{3},if\mathcal{S}={\mathcal{E}}_1,\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2,\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

We thus obtain

$$\begin{array}{l}{C}_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)=\left\{\begin{array}{c}\tilde{0},if\mathcal{S}=\tilde{0},\mathrm{r}\in {\zeta }_0,\hfill \\ {\mathcal{E}}_2^c,if\mathcal{S}\le {\mathcal{E}}_1,r\le \frac{1}{2},1-r\ge \frac{1}{2},\hfill \\ {\mathcal{E}}_1^c,if\mathcal{S}\le {\mathcal{E}}_2,r\le \frac{1}{3},1-r\ge \frac{2}{3},\hfill \\ \tilde{0},otherwise.\hfill \end{array}\right.\end{array}$$

If $r\le \frac{1}{3}$ and $1-r\ge \frac{2}{3}$, then ${\mathcal{E}}_2^c=C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}({\mathcal{E}}_1,r)\cap {\mathcal{E}}_2=\tilde{0}$ and ${\mathcal{E}}_1^c=C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}({\mathcal{E}}_2,r)\cap {\mathcal{E}}_1=\tilde{0}$. Thus, ${\mathcal{E}}_1\cup {\mathcal{E}}_2=\tilde{1}$ is not r-$\mathcal{SVNCON}$ for $r\le \frac{1}{3}$ and $1-r\ge \frac{2}{3}$. If $r>\frac{1}{3}$ and $1-r<\frac{2}{3}$, $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is r-$\mathcal{SVNCON}$.

Before we proceed further, we need to recall the following theorem given in [17] and prove its second part.

Theorem 2

([17]). Suppose that $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is an $\mathcal{SVNTS}$. For every $r\in {\zeta }_0$ and $\mathcal{S}\in {\zeta }^{\tilde{X}}$. Define an operator $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}:{\zeta }^{\tilde{X}}x{\zeta }_0\to \zeta$ as follows:

$$\begin{array}{l}{C}_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)=\bigcap \{\mathcal{E}\in {\zeta }^{\tilde{X}}:\mathcal{E}\le \mathcal{S},{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}^c)\ge r,{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}^c)\le 1-r,{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}^c)\le 1-r\}.\end{array}$$

Then,

(1) 

$(\tilde{X},C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}})$ is an $\mathcal{SVNCS}$,

(2) 

${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\gamma }}={\tilde{\tau }}^{\tilde{\gamma }}$, ${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\eta }}={\tilde{\tau }}^{\tilde{\eta }}$ and ${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\mu }}={\tilde{\tau }}^{\tilde{\mu }}$

Proof. 

(1) It has been proven in [17].

(2) Suppose that ${\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{E})=r$, ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{E})=1-r$ and ${\tilde{\tau }}^{\tilde{\mu }}(\mathcal{R})=1-r$. Then, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}({\mathcal{E}}^c,r)={\mathcal{E}}^c$. Therefore, ${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\gamma }}\ge {\tilde{\tau }}^{\tilde{\gamma }}$, ${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\eta }}\le {\tilde{\tau }}^{\tilde{\eta }}$ and ${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\mu }}\le {\tilde{\tau }}^{\tilde{\mu }}$. Suppose that

$${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\gamma }}\nleqq {\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\eta }}\ngeqq {\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\mu }}\ngeqq {\tilde{\tau }}^{\tilde{\mu }}.$$

Then, there exists $\mathcal{E}$ with $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}({\mathcal{E}}^c,r)={\mathcal{E}}^c$ such that

$${\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\gamma }}(\mathcal{E})\ge r>{\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{E}),{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\eta }}(\mathcal{E})\le 1-r<{\tilde{\tau }}^{\tilde{\eta }}(\mathcal{E}),{\tilde{\tau }}_{C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}}^{\tilde{\mu }}(\mathcal{E})\le 1-r<{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{E}).$$

(1)

By the definition of $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}$, we have ${\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{E})\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{E})\le 1-r$ and ${\tilde{\tau }}^{\tilde{\mu }}(\mathcal{E})\le 1-r$. It is a contradiction for Equation (1). □

Example 2.

Let $\tilde{X}=\{a,b\}$ be a set. Define ${\mathcal{E}}_1,{\mathcal{E}}_2\in {\zeta }^{\tilde{X}}$.

$$\begin{array}{l}{\mathcal{E}}_1=\langle (0.2,0.2),(0.3,0.3),(0.3,0.3)\rangle ;{\mathcal{E}}_2=\langle (0.5,0.5),(0.1,0.1),(0.1,0.1)\rangle .\end{array}$$

We define the mapping $C:{\zeta }^{\tilde{X}}\times {\zeta }_0\to {\zeta }^{\tilde{X}}$ as follows:

$$\begin{array}{l}C(\mathcal{S},r)=\left\{\begin{array}{c}\tilde{0},if\mathcal{S}=\tilde{0},r\in {\mathrm{I}}_0,\hfill \\ {\mathcal{E}}_1\cap {\mathcal{E}}_2,if0\ne \mathcal{S}\le {\mathcal{E}}_1\cap {\mathcal{E}}_2,0<r<\frac{1}{2},\hfill \\ {\mathcal{E}}_1,if\mathcal{S}\le {\mathcal{E}}_1,\mathcal{S}\nleqq {\mathcal{E}}_2,0<r<\frac{1}{2},\hfill \\ or0\ne \mathcal{S}\le {\mathcal{E}}_1\frac{1}{2}<r<\frac{2}{3},\hfill \\ {\mathcal{E}}_2,if\mathcal{S}\le {\mathcal{E}}_2,\mathcal{S}\nleqq {\mathcal{E}}_1,0<r<\frac{1}{2},\hfill \\ {\mathcal{E}}_1\cup {\mathcal{E}}_2,if0\ne \mathcal{S}\le {\mathcal{E}}_1\cup {\mathcal{E}}_2,0<r<\frac{1}{2},\hfill \\ \overline{1},otherwise.\hfill \end{array}\right.\end{array}$$

Then, C is a single-valued neutrosophic closure operator.

From Theorem 1, we have a single-valued neutrosophic topology $({\tau }_C^{\tilde{\gamma }},{\tau }_C^{\tilde{\eta }},{\tau }_C^{\tilde{\mu }})$ on $\tilde{X}$ as follows:

$$\begin{array}{l}{\tau }_C^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{1}or\tilde{0},\hfill \\ \frac{2}{3},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}}\cup {\mathcal{H}}^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}}\cap {\mathcal{E}}_2^{\mathrm{c}},\hfill \\ 0,otherwise.\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tau }_C^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{1}or\tilde{0},\hfill \\ \frac{1}{3},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}}\cup {\mathcal{E}}_2^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}}\cap {\mathcal{E}}_2^{\mathrm{c}},\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tau }_C^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{1}or\tilde{0},\hfill \\ \frac{1}{3},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}}\cup {\mathcal{E}}_2^{\mathrm{c}},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1^{\mathrm{c}}\cap {\mathcal{E}}_2^{\mathrm{c}},\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

Thus, the $({\tau }_C^{\tilde{\gamma }},{\tau }_C^{\tilde{\eta }},{\tau }_C^{\tilde{\mu }})$ is a single-valued neutrosophic topology on $\tilde{X}$.

Theorem 3.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$. Then, the following are equivalent.

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is r-$\mathcal{SVNCON}$.

(2) 

if $\mathcal{S}\cup \mathcal{E}=\tilde{1}$ and $\mathcal{S}\cap \mathcal{E}=\tilde{0}$ for $({\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{E})\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{E})\le 1-r,{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{E})\le 1-r)$ and $({\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})\le 1-r,{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})\le 1-r)$, then $\mathcal{E}=\tilde{0}$ or $\mathcal{S}=\tilde{0}$,

(3) 

if $\mathcal{S}\cup \mathcal{E}=\tilde{1}$, ${\mathcal{E}}_1\cap {\mathcal{E}}_2=\tilde{0}$ for $({\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}^c)\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}^c)\le 1-r,{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}^c)\le 1-r)$ and $({\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{S}}^c)\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}({\mathcal{S}}^c)\le 1-r,{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{S}}^c)\le 1-r)$, then $\mathcal{E}=\tilde{0}$ or $\mathcal{S}=\tilde{0}$.

Proof. 

(1)⇒(2): Let there exist $\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}-\{\tilde{0}\}$ such that for every $({\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{E})\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{E})\le 1-r,{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{E})\le 1-r)$ and $({\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})\le 1-r,{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})\le 1-r)$, $\mathcal{S}\cup \mathcal{E}=\tilde{1}$, $\mathcal{S}\cap \mathcal{E}=\tilde{0}$. It implies

$$\begin{array}{l}{\mathcal{S}}^c\cap {\mathcal{E}}^c=\tilde{0},{\mathcal{S}}^c\cup {\mathcal{E}}^c=\tilde{1}.\end{array}$$

Since $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}({\mathcal{S}}^c,r)={\mathcal{S}}^c$ and $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}({\mathcal{E}}^c,r)={\mathcal{E}}^c$ from Theorem 2, ${\mathcal{S}}^c$ and ${\mathcal{E}}^c$ are r-$\mathcal{SVNSEP}$. Suppose $\mathcal{S}=\tilde{1}$. Then, $\mathcal{E}=\mathcal{S}\cap \mathcal{E}=\tilde{0}$. It is a contradiction. Hence, ${\mathcal{S}}^c\in {\zeta }^{\tilde{X}}-\{\tilde{0}\}$. Similarly, ${\mathcal{E}}^c\in {\zeta }^{\tilde{X}}-\{\tilde{0}\}$. Furthermore, ${\mathcal{S}}^c\cup {\mathcal{E}}^c=\tilde{1}$. Thus, $\tilde{1}$ is not r-$\mathcal{SVNCON}$.

(2)⇒(3): It is trivial.

(3)⇒(1): Suppose that $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is not r-$\mathcal{SVNCON}$. Then, there exist r-$\mathcal{SVNSEP}$ $\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}-\{\tilde{0}\}$ such that $\mathcal{S}\cup \mathcal{E}=\tilde{1}$. Since $\mathcal{S}\cap \mathcal{E}\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)\cap \mathcal{E}=\tilde{0}$, we have $\mathcal{S}\cap \mathcal{E}=\tilde{0}$. Thus, ${\mathcal{S}}^c\cap {\mathcal{E}}^c=\tilde{0}$ implies ${\mathcal{E}}^c\le \mathcal{S}$. Hence, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)\cap \mathcal{E}=\tilde{0}$ implies, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)\le {\mathcal{E}}^c$. Thus, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)\le \mathcal{S}$. By Definition 9$(C_2)$, we have $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)=\mathcal{S}$. By Theorem 2, we obtain $({\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{S}}^c)\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}({\mathcal{S}}^c)\le 1-r$, ${\tilde{\tau }}^{\tilde{\mu }}({\mathcal{S}}^c)\le 1-r)$. Similarly, we have $({\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}^c)\ge r$, ${\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}^c)\le 1-r$, ${\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}^c)\le 1-r)$. It is a contradiction. □

Lemma 1.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$ and $\mathcal{S},\mathcal{E},\mathcal{R}\in {\zeta }^{\tilde{X}}$. If $\mathcal{E}$ and $\mathcal{R}$ are r-$\mathcal{SVNSEP}$, then $\mathcal{S}\cap \mathcal{E}$ and $\mathcal{S}\cap \mathcal{R}$ are r-$\mathcal{SVNSEP}$.

Proof. 

Let $\mathcal{E}$ and $\mathcal{R}$ be r-$\mathcal{SVNSEP}$. Thus,

$$\begin{array}{l}{C}_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S}\cap \mathcal{E},r)\cap (\mathcal{S}\cap \mathcal{R})\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{E},r)\cap \mathcal{R}=\tilde{0}\end{array}$$

Similarly, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S}\cap \mathcal{R},r)\cap (\mathcal{S}\cap \mathcal{E})=\tilde{0}$. Thus, $\mathcal{S}\cap \mathcal{E}$ and $\mathcal{S}\cap \mathcal{R}$ are r-$\mathcal{SVNSEP}$. □

Theorem 4.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$ and $\mathcal{S}\in {\zeta }^{\tilde{X}}$. Then, the following are equivalent.

(1) 

$\mathcal{S}$ is r-$\mathcal{SVNCON}$,

(2) 

If $\mathcal{E}$ and $\mathcal{R}$ are r-$\mathcal{SVNSEP}$ such that $\mathcal{S}\le \mathcal{E}\cup \mathcal{R}$, then $\mathcal{S}\cap \mathcal{E}=\tilde{0}$ or $\mathcal{S}\cap \mathcal{R}=\tilde{0}$,

(3) 

If $\mathcal{E}$ and $\mathcal{R}$ are r-$\mathcal{SVNSEP}$ such that $\mathcal{S}\le \mathcal{E}\cup \mathcal{R}$, then $\mathcal{S}\le \mathcal{E}$ or $\mathcal{S}\le \mathcal{R}$.

Proof. 

$(1)\Rightarrow (2)$: Let $\mathcal{E},\mathcal{R}\in {\zeta }^{\tilde{X}}$ be r-$\mathcal{SVNSEP}$ such that $\mathcal{S}\le \mathcal{E}\cup \mathcal{R}$. By Lemma 1, $\mathcal{S}\cap \mathcal{E}$ and $\mathcal{S}\cap \mathcal{R}$ are r-$\mathcal{SVNSEP}$. Since $\mathcal{S}$ is r-$\mathcal{SVNCON}$ and $\mathcal{S}=\mathcal{S}\cap (\mathcal{E}\cup \mathcal{R})=(\mathcal{S}\cap \mathcal{E})\cup (\mathcal{S}\cap \mathcal{R})$, then $\mathcal{S}\cap \mathcal{E}=\tilde{0}$ or $\mathcal{S}\cap \mathcal{R}=\tilde{0}$.

$(2)\Rightarrow (3)$: It is easily proved.

$(3)\Rightarrow (1)$: Let $\mathcal{E}$ and $\mathcal{R}$ be r-$\mathcal{SVNSEP}$ such that $\mathcal{S}=\mathcal{E}\cup \mathcal{R}$. By (3), $\mathcal{S}\le \mathcal{E}$ or $\mathcal{S}\le \mathcal{R}$. If $\mathcal{S}\le \mathcal{E}$ and $\mathcal{E},\mathcal{R}$ are r-$\mathcal{SVNSEP}$, then

$$\begin{array}{l}\mathcal{R}=\mathcal{R}\cap \mathcal{S}\le \mathcal{R}\cap \mathcal{E}\le \mathcal{R}\cap C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{E},r)=\tilde{0}.\end{array}$$

Hence, $\mathcal{R}=\tilde{0}$. If $\mathcal{S}\le \mathcal{R}$, similarly $\mathcal{E}=\tilde{0}$. □

Theorem 5.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$ and $\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}$.

(1) 

If $\mathcal{S}$ is r-$\mathcal{SVNCON}$, $\mathcal{S}\le \mathcal{E}\le C_{{\tilde{\tau }}^{\tilde{\gamma }\tilde{\eta }\tilde{\mu }}}(\mathcal{S},r)$, then $\mathcal{E}$ is r-$\mathcal{SVNCON}$.

(2) 

If $\mathcal{S}$ and $\mathcal{E}$ are r-$\mathcal{SVNCON}$ single-valued neutrosophic sets which are not r-$\mathcal{SVNSEP}$, then $\mathcal{S}\cup \mathcal{E}$ is r-$\mathcal{SVNCON}$.

Proof. 

(1) Let $\mathcal{R},\mathcal{D}\in {\zeta }^{\tilde{X}}$ be r-$\mathcal{SVNSEP}$ such that $\mathcal{E}=\mathcal{R}\cup \mathcal{D}$. Put, ${\mathcal{R}}_1=\mathcal{S}\cap \mathcal{R}$ and ${\mathcal{D}}_1=\mathcal{S}\cap \mathcal{D}$, then ${\mathcal{R}}_1$ and ${\mathcal{D}}_1$ are r-$\mathcal{SVNSEP}$ such that $\mathcal{S}={\mathcal{R}}_1\cup {\mathcal{D}}_1$. Since $\mathcal{S}$ is r-$\mathcal{SVNCON}$, ${\mathcal{R}}_1=\tilde{0}$ or ${\mathcal{D}}_1=\tilde{0}$. If ${\mathcal{R}}_1=\tilde{0}$, then $\mathcal{S}={\mathcal{D}}_1=\mathcal{S}\cap \mathcal{D}$⇒$\mathcal{S}\le \mathcal{D}$. It implies

$$\begin{array}{l}\mathcal{E}\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{D},r).\end{array}$$

Hence, $\mathcal{R}=\mathcal{R}\cap \mathcal{E}\le \mathcal{R}\cap C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{D},r)=\tilde{0}$.

If ${\mathcal{D}}_1=\tilde{0}$, similarly $\mathcal{D}=\tilde{0}$. Therefore, $\mathcal{E}$ is r-$\mathcal{SVNCON}$.

(2) Let $\mathcal{R}$ and $\mathcal{D}$ be r-$\mathcal{SVNSEP}$ such that $\mathcal{S}\cup \mathcal{E}=\mathcal{R}\cup \mathcal{D}$. Since $\mathcal{S}$ is r-$\mathcal{SVNCON}$, by Theorem 4 (3), $\mathcal{S}\le \mathcal{R}$ or $\mathcal{S}\le \mathcal{D}$. Say, $\mathcal{S}\le \mathcal{R}$. Suppose that $\mathcal{E}\le \mathcal{D}$. Since $(\mathcal{S}\cup \mathcal{E})\cap \mathcal{R}=\mathcal{S}$ and $(\mathcal{S}\cup \mathcal{E})\cap \mathcal{D}=\mathcal{E}$, by Lemma 1, $\mathcal{S}$ and $\mathcal{E}$ are r-$\mathcal{SVNSEP}$. It is a contradiction. Thus, $\mathcal{E}\le \mathcal{R}$ Hence, $\mathcal{S}\cup \mathcal{E}\le \mathcal{R}$, by Theorem 4 (3), $\mathcal{S}\cup \mathcal{E}$ is r-$\mathcal{SVNCON}$. □

Theorem 6.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$. Let $\mathcal{B}=\{{\mathcal{S}}_j\in {\zeta }^{\tilde{X}}\mid {\mathcal{S}}_{j}isr-\mathcal{SVNCON}sets,j\in \mathsf{\Gamma }\}$ be a family in $\tilde{X}$ such that no two members of $\mathcal{B}$ are r-$\mathcal{SVNSEP}$, then ${\bigcup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j$ is r-$\mathcal{SVNCON}$.

Proof. 

Put $\mathcal{S}={\bigcup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j$. Let $\mathcal{E},\mathcal{R}\in {\zeta }^{\tilde{X}}$ be r-$\mathcal{SVNSEP}$ such that $\mathcal{S}=\mathcal{E}\cup \mathcal{R}$. Since any two members ${\mathcal{S}}_j,{\mathcal{S}}_i\in \mathcal{B}$ are not r-$\mathcal{SVNSEP}$, by Theorem 5 (2), ${\mathcal{S}}_j\cup {\mathcal{S}}_i$ is r-$\mathcal{SVNCON}$. From Theorem 4 (3), ${\mathcal{S}}_j\cup {\mathcal{S}}_i\le \mathcal{E}$ or ${\mathcal{S}}_j\cup {\mathcal{S}}_i\le \mathcal{R}$. Say, ${\mathcal{S}}_j\cup {\mathcal{S}}_i\le \mathcal{E}$. It implies that $\mathcal{S}\le \mathcal{E}$. Thus, $\mathcal{S}$ is r-$\mathcal{SVNCON}$. □

Corollary 1.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$. Let $\mathcal{B}=\{{\mathcal{S}}_j\in {\zeta }^{\tilde{X}}\mid {\mathcal{S}}_{j}isr-\mathcal{SVNCON}sets,j\in \mathsf{\Gamma }\}$ be a family in $\tilde{X}$. If ${\bigcap }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j\ne \tilde{0}$, then ${\bigcup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j$ is r-$\mathcal{SVNCON}$.

Lemma 2.

Let $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }})$ be a mapping from an $\mathcal{SVNTS}$$(\tilde{X},{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }})$ to another $\mathcal{SVNTS}$$(\tilde{Y},{\tilde{\tau }}_2^{\tilde{\gamma }}.{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }})$. Then, the following are equivalent, ∀$\mathcal{S}\in {\zeta }^{\tilde{X}}$, $\mathcal{E}\in {\zeta }^{\tilde{Y}}$ and $r\in {\zeta }_0$

(1) 

f is $\mathcal{SVN}-continuous$.

(2) 

$f(C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(\mathcal{S},r))\le C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(f(\mathcal{S}),r)$.

(3) 

$C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(f^{-1}(\mathcal{E}),r)\le f^{-1}(C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(\mathcal{E},r))$.

Proof. 

(1)⇒(2): Suppose that f is $\mathcal{SVN}-continuous$. Then, ${\tilde{\tau }}_1^{\tilde{\gamma }}({(f^{-1}(\mathcal{S}))}^c)\ge {\tilde{\tau }}_2^{\tilde{\gamma }}({\mathcal{S}}^c)$, ${\tilde{\tau }}_1^{\tilde{\eta }}({(f^{-1}(\mathcal{S}))}^c)\le {\tilde{\tau }}_2^{\tilde{\eta }}({\mathcal{S}}^c)$ and ${\tilde{\tau }}_1^{\tilde{\mu }}({(f^{-1}(\mathcal{S}))}^c)\le {\tilde{\tau }}_2^{\tilde{\mu }}({\mathcal{S}}^c)$. Hence,

$$\begin{array}{ll}{C}_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(f(\mathcal{S}),r)& =\bigcap \{\mathcal{E}\in {\zeta }^{\tilde{Y}}\mid f(\mathcal{S})\le \mathcal{E},{\tilde{\tau }}_2^{\tilde{\gamma }}({\mathcal{E}}^c)\ge r,{\tilde{\tau }}_2^{\tilde{\eta }}({\mathcal{E}}^c)\le 1-r,{\tilde{\tau }}_2^{\tilde{\mu }}({\mathcal{E}}^c)\le 1-r\}\\ & \ge \bigcap \{\mathcal{E}\in {\zeta }^{\tilde{Y}}\mid \mathcal{S}\le f^{-1}(\mathcal{E}),{\tilde{\tau }}_1^{\tilde{\gamma }}({(f^{-1}(\mathcal{E}))}^c)\ge r,{\tilde{\tau }}_1^{\tilde{\eta }}({(f^{-1}(\mathcal{E}))}^c)\le 1-r,\\ & {\tilde{\tau }}_1^{\tilde{\mu }}({(f^{-1}(\mathcal{E}))}^c)\le 1-r\}\\ & \ge \bigcap \{f(f^{-1}(\mathcal{E}))\in {\zeta }^{\tilde{Y}}\mid \mathcal{S}\le f^{-1}(\mathcal{E}),{\tilde{\tau }}_1^{\tilde{\gamma }}({(f^{-1}(\mathcal{E}))}^c)\ge r,{\tilde{\tau }}_1^{\tilde{\eta }}({(f^{-1}(\mathcal{E}))}^c)\le 1-r,\\ & {\tilde{\tau }}_1^{\tilde{\mu }}({(f^{-1}(\mathcal{E}))}^c)\le 1-r\}\\ & \ge f[\bigcap \{f^{-1}(\mathcal{E}))\in {\zeta }^{\tilde{Y}}\mid \mathcal{S}\le f^{-1}(\mathcal{E}),{\tilde{\tau }}_1^{\tilde{\gamma }}({(f^{-1}(\mathcal{E}))}^c)\ge r,{\tilde{\tau }}_1^{\tilde{\eta }}({(f^{-1}(\mathcal{E}))}^c)\le 1-r,\\ & {\tilde{\tau }}_1^{\tilde{\mu }}({(f^{-1}(\mathcal{E}))}^c)\le 1-r\}]\\ & \ge f(C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(f(\mathcal{S}),r)).\end{array}$$

(2)⇒(3). For all $\mathcal{E}\in {\zeta }^{\tilde{Y}}$. By (2),

$$f(C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(\mathcal{S},r))\le C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(f(\mathcal{S}),r).$$

Putting $\mathcal{S}=f^{-1}(\mathcal{E})$, we obtain

$$f(C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(f^{-1}(\mathcal{E}),r))\le C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(f(f^{-1}(\mathcal{E})),r)\le C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(\mathcal{E},r)$$

Hence, $C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(f^{-1}(\mathcal{E}),r)\le f^{-1}(C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(\mathcal{E},r))$.

(3)⇒(1). It follows that $C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(\mathcal{E},r)=\mathcal{E}$ implies $C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(f^{-1}(\mathcal{E}),r)=f^{-1}(\mathcal{E})$. □

Theorem 7.

Let $(\tilde{X},{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }})$, $(\tilde{Y},{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }})$ be two ${\mathcal{SVNTS}}^{\prime }s$ and $f:\tilde{X}\to \tilde{Y}$ is $\mathcal{SVN}$-continuous mapping. If $\mathcal{S}$ is r-$\mathcal{SVNCON}$, then $f(\mathcal{S})$ is r-$\mathcal{SVNCON}$.

Proof. 

Let $\mathcal{E},\mathcal{R}\in {\zeta }^{\tilde{Y}}$ be two r-${\mathcal{SVNSEP}}^{\prime }s$ such that $f(\mathcal{S})=\mathcal{E}\cup \mathcal{R}$. We obtain

$$\begin{array}{l}\mathcal{S}\le f(f^{-1})(\mathcal{S})=f^{-1}(\mathcal{E}\cup \mathcal{R})=f^{-1}(\mathcal{E})\cup f^{-1}(\mathcal{R}).\end{array}$$

Since f is $\mathcal{SVN}$-continuous, by Lemma 2,

$$\begin{array}{l}{C}_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(f^{-1}(\mathcal{E},r))\le f^{-1}(C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(\mathcal{E},r)).\end{array}$$

Thus,

$$\begin{array}{lll}& C_{{\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }}}(f^{-1}(\mathcal{E},r)\cap f^{-1}(\mathcal{R}))& \le f^{-1}(C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(\mathcal{E},r))\cap f^{-1}(\mathcal{R})\hfill \\ & & =f^{-1}(C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(\mathcal{E},r)\cap \mathcal{R})\hfill \\ & & =f^{-1}(\tilde{0})=\tilde{0}.\hfill \end{array}$$

Likewise, we obtain $f^{-1}(\mathcal{E})\cap C_{{\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }}}(\mathcal{R},r)=\tilde{0}$. It implies that $f^{-1}(\mathcal{E}),f^{-1}(\mathcal{R})\in {\zeta }^{\tilde{X}}$ are r-${\mathcal{SVNSEP}}^{\prime }s$. Since $\mathcal{S}$ is r-$\mathcal{SVNCON}$, then we have by Theorem 4 (3), $\mathcal{S}\le f^{-1}(\mathcal{E})$ or $\mathcal{S}\le f^{-1}(\mathcal{R})$, so $\mathcal{S}\le f^{-1}(\mathcal{E})$. Thus, $f(\mathcal{S})\le f(f^{-1}(\mathcal{E}))\le \mathcal{E}$. Hence, $f(\mathcal{S})$ is r-$\mathcal{SVNCON}$. □

Example 3.

Let $\tilde{X}=\tilde{Y}=\{a,b\}$ be a set. Define ${\mathcal{E}}_1,{\mathcal{E}}_2,{\mathcal{E}}_3,{\mathcal{B}}_1,{\mathcal{B}}_2,{\mathcal{B}}_3\in {\zeta }^{\tilde{X}}$:

$$\begin{array}{l}{\mathcal{E}}_1=\langle (0.5,0.4),(0.5,0.5),(0.9,0.6)\rangle ,{\mathcal{E}}_2=\langle (0.4,0.4),(0.1,0.1),(0.1,0.1)\rangle ,\end{array}$$

$$\begin{array}{l}{\mathcal{E}}_3=\langle (0.3,0.1),(0.1,0.1),(0.1,0.1),{\mathcal{B}}_1=\langle (0.4,0.5),(0.5,0.5),(0.6,0.9)\rangle ,\end{array}$$

$$\begin{array}{l}{\mathcal{B}}_2=\langle (0.2,0.2),(0.2,0.2),(0.1,0.1),{\mathcal{B}}_3=\langle (0.1,0.1),(0.1,0.1),(0.1,0.1)\rangle .\end{array}$$

Define ${\tilde{\tau }}^{\tilde{\gamma }\tilde{\eta }\tilde{\mu }},{\tilde{\sigma }}^{\tilde{\gamma }\tilde{\eta }\tilde{\mu }}:{\zeta }^{\tilde{X}}\to {\zeta }^{\tilde{X}}$ as follows:

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{0},\hfill \\ 1,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1,\hfill \\ 0,otherwise.\hfill \end{array}\right.{\tilde{\sigma }}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{0}\rangle ,\hfill \\ 1,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{B}}_1,\hfill \\ 0,otherwise.\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2,\hfill \\ 1,otherwise.\hfill \end{array}\right.{\tilde{\sigma }}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{B}}_2,\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_3,\hfill \\ 1,otherwise.\hfill \end{array}\right.{\tilde{\sigma }}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{B}}_3,\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

Define $f:(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }\tilde{\eta }\tilde{\mu }}))\to (\tilde{Y},{\tilde{\sigma }}^{\tilde{\gamma }\tilde{\eta }\tilde{\mu }})$ be a map as follows $f(a)=b$ and $f(b)=a$. If ${\tilde{\mathcal{J}}}^{\tilde{\gamma }}({\mathcal{B}}_1)\ge \frac{1}{2}$, ${\tilde{\mathcal{J}}}^{\tilde{\eta }}({\mathcal{B}}_1)\le 1-\frac{1}{2}$ and ${\tilde{\mathcal{J}}}^{\tilde{\mu }}({\mathcal{B}}_1)\le 1-\frac{1}{2}$. Then, $f^{-1}({\mathcal{B}}_1)=\langle (0.5,0.4),(0.5,0.5),(0.9,0.6)\rangle$ is $\frac{1}{2}$-single-valued neutrosophic open set in $\tilde{X}$. Thus, f is $\mathcal{SVN}$-continuous. However, by Theorem 7, for every $\mathcal{S}\in {\zeta }^{\tilde{X}}$ is r-$\mathcal{SVNCON}$, then $f(\mathcal{S})$ is r-$\mathcal{SVNCON}$ in $\tilde{Y}$.

Definition 12.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$. A $\mathcal{SVNs}$$\mathcal{S}$ is called r-single-valued neutrosophic component (r-$\mathcal{SVNCOM}$, for short) in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ if $\mathcal{S}$ is a maximal r-$\mathcal{SVNCON}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$, i.e., if $\mathcal{E}\ge \mathcal{S}$ and $\mathcal{E}$ is r-$\mathcal{SVNCON}$, then $\mathcal{E}=\mathcal{S}$.

Corollary 2.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$.

(1) 

If $\mathcal{S}$ is a r-$\mathcal{SVNCOM}$, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)=\mathcal{S}$.

(2) 

If ${\mathcal{S}}_1,{\mathcal{S}}_2\in {\zeta }^{\tilde{X}}$ are r-$\mathcal{SVNCOM}$ in $\tilde{X}$ such that ${\mathcal{S}}_1\cap {\mathcal{S}}_2=\tilde{0}$, then ${\mathcal{S}}_1,{\mathcal{S}}_2\in {\zeta }^{\tilde{X}}$ are r-$\mathcal{SVNSEP}$.

(3) 

Each single-valued neutrosophic point $x_{t,s,k}$ is $\mathcal{SVNCON}$.

(4) 

Every r-$\mathcal{SVNCOM}$ is a crisp set.

Proof. 

Straightforward. □

4. Stratification of Single-Valued Neutrosophic Topological Spaces

In this section, we obtain crucial results in the stratification of the single-valued neutrosophic topology as follows.

Definition 13.

The stratification of the single-valued neutrosophic topology $(\mathcal{SVNT})$ on $\tilde{X}$ is a mapping from ${\zeta }^{\tilde{X}}$ to ζ such that

(SVNT1) 

${\tilde{\tau }}^{\tilde{\gamma }}(\tilde{\alpha })=1$ and ${\tilde{\tau }}^{\tilde{\eta }}(\tilde{\alpha })={\tilde{\tau }}^{\tilde{\mu }}(\tilde{\alpha })=0$, ∀$\alpha \in \zeta$,

(SVNT2) 

${\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S}\cap \mathcal{E})\ge {\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})\cap {\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{E})$,    ${\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S}\cap \mathcal{E})\le {\tau }^{\tilde{\eta }}(\mathcal{S})\cup {\tilde{\tau }}^{\tilde{\eta }}(\mathcal{E})$,

${\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S}\cap \mathcal{E})\le {\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})\cup {\tilde{\tau }}^{\tilde{\mu }}(\mathcal{E})$, for all $\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}$,

(SVNT3) 

${\tilde{\tau }}^{\tilde{\gamma }}({\cup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j)\ge {\cap }_{j\in \mathsf{\Gamma }}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{S}}_j)$,    ${\tilde{\tau }}^{\tilde{\eta }}({\cup }_{i\in \mathsf{\Gamma }}{\mathcal{S}}_j)\le {\cup }_{j\in \mathsf{\Gamma }}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{S}}_j)$,

${\tilde{\tau }}^{\tilde{\mu }}({\cup }_{j\in \mathsf{\Gamma }}{\mathcal{S}}_j)\le {\cup }_{j\in \mathsf{\Gamma }}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{S}}_j)$, for all $\{{\mathcal{S}}_j,j\in \mathsf{\Gamma }\}\in {\zeta }^{\tilde{X}}$.

The ordered pair $\mathcal{SVNTS}$$(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is called stratified. Let $({\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }})$ and $({\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }})$ be ${\mathcal{SVNGO}}^{\prime }s$ on $\tilde{X}$. We say that $({\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }})$ is finer then $({\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }})$ [$({\tilde{\tau }}_2^{\tilde{\gamma }},{\tilde{\tau }}_2^{\tilde{\eta }},{\tilde{\tau }}_2^{\tilde{\mu }})$ is coarser then $({\tilde{\tau }}_1^{\tilde{\gamma }},{\tilde{\tau }}_1^{\tilde{\eta }},{\tilde{\tau }}_1^{\tilde{\mu }})$] if ${\tilde{\tau }}_1^{\tilde{\gamma }}(\mathcal{S})\le {\tilde{\tau }}_2^{\tilde{\gamma }}(\mathcal{S})$, ${\tilde{\tau }}_1^{\tilde{\eta }}(\mathcal{S})\ge {\tilde{\tau }}_2^{\tilde{\eta }}(\mathcal{S})$ and ${\tilde{\tau }}_1^{\tilde{\mu }}(\mathcal{S})\ge {\tilde{\tau }}_2^{\tilde{\mu }}(\mathcal{S})$ for all $\mathcal{S}\in {\zeta }^{\tilde{X}}$.

Theorem 8.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ be an $\mathcal{SVNTS}$. Define the mappings ${\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}:{\zeta }^{\tilde{X}}\to \zeta$ as follows: for all $\mathcal{S}\in {\zeta }^{\tilde{X}}$,

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})=\bigcup \{{\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{S}}_j)\mid \mathcal{S}={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{S}}_j\cap {\tilde{\alpha }}_j)\}\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})=\bigcap \{{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{S}}_j)\mid \mathcal{S}={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{S}}_j\cap {\tilde{\alpha }}_j)\}\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})=\bigcap \{{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{S}}_j)\mid \mathcal{S}={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{S}}_j\cap {\tilde{\alpha }}_j)\}.\end{array}$$

Then, $({\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ is the coarsest stratified $\mathcal{SVNT}$ on $\tilde{X}$ which is finer than $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$.

Proof. 

Firstly, we will show that $({\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ is a stratified $\mathcal{SVNT}$ on $\tilde{X}$.

(SVNT1) For every $\alpha \in \zeta$, there exists a collection $\{\tilde{1}\}$ with $\tilde{\alpha }=\tilde{\alpha }\cap \tilde{1}$, we obtain ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\tilde{\alpha })\ge {\tilde{\tau }}^{\tilde{\gamma }}(\tilde{1})=1$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\tilde{\alpha })\le {\tilde{\tau }}^{\tilde{\eta }}(\tilde{1})=0$ and ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\tilde{\alpha })\le {\tilde{\tau }}^{\tilde{\mu }}(\tilde{1})=0$. Thus, ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\tilde{\alpha })=1$ and ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\tilde{\alpha })={\tilde{\tau }}_{st}^{\tilde{\mu }}(\tilde{\alpha })=0$.

(SVNT2) Suppose there exists $\mathcal{E},\mathcal{R}\in {\zeta }^{\tilde{X}}$ and $r\in {\zeta }_0$ with

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E}\cap \mathcal{R})<r<{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})\cap {\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{R}),\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E}\cap \mathcal{R})>1-r>{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E})\cup {\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{R}),\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E}\cap \mathcal{R})>1-r>{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E})\cup {\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{R}).\end{array}$$

Since $[{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})>r$, ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{R})>r]$, $[{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E})<1-r$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{R})<1-r]$ and $[{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})<1-r$, ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{R})<1-r]$, by the definition of $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$, there exist $\{{\mathcal{E}}_j\mid j\in \mathsf{\Gamma }\}$ with $\mathcal{E}={\bigcup }_{j\in \mathsf{\Gamma }}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j)$ and $\{{\mathcal{R}}_k\mid k\in K\}$ with $\mathcal{R}={\bigcup }_{k\in K}({\mathcal{R}}_k\cap {\tilde{\alpha }}_k)$ such that

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j)>r,{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E})\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j)<1-rand{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E})\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j)<1-r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{R})\ge {\displaystyle \bigcap _{k\in K}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{R}}_k)>r,{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{R})\le {\displaystyle \bigcup _{k\in K}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{R}}_k)<1-rand{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{R})\le {\displaystyle \bigcup _{k\in K}}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{R}}_k)<1-r.\end{array}$$

Since $\zeta$ is completely distributive lattice, we have

$$\begin{array}{lll}& \mathcal{E}\cap \mathcal{R}& =[{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j)]\cap [{\displaystyle \bigcup _{k\in K}}({\mathcal{R}}_k\cap {\tilde{\alpha }}_k)]={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\cap ({\tilde{\alpha }}_j\cap {\tilde{\alpha }}_k)\hfill \\ & & ={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\cap {\tilde{\alpha }}_{jk}.({\tilde{\alpha }}_{jk}={\tilde{\alpha }}_j\cap {\tilde{\alpha }}_k).\hfill \end{array}$$

Moreover, since ${\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\ge {\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j)\cap {\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{R}}_k)$, ${\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\le {\tau }^{\tilde{\eta }}({\mathcal{E}}_j)\cup {\tilde{\tau }}^{\tilde{\eta }}({\mathcal{R}}_k)$ and ${\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\le {\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j)\cup {\tilde{\tau }}^{\tilde{\mu }}({\mathcal{R}}_k)$, we obtain

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E}\cap \mathcal{R})\ge {\displaystyle \bigcap _{j,k}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\ge {\displaystyle \bigcap _{j,k}}({\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j)\cap {\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{R}}_k))=[{\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}({\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j)]\cap [{\displaystyle \bigcap _{k\in K}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{R}}_k)]>r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E}\cap \mathcal{R})\le {\displaystyle \bigcup _{j,k}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\le {\displaystyle \bigcup _{j,k}}({\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j)\cup {\tilde{\tau }}^{\tilde{\eta }}({\mathcal{R}}_k))=[{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j)]\cup [{\displaystyle \bigcup _{k\in K}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{R}}_k)]<1-r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E}\cap \mathcal{R})\le {\displaystyle \bigcup _{j,k}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j\cap {\mathcal{R}}_k)\le {\displaystyle \bigcup _{j,k}}({\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j)\cup {\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{R}}_k))=[{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j)]\cup [{\displaystyle \bigcup _{k\in K}}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{R}}_k)]<1-r\end{array}$$

It is a contradiction. Hence, for each $\mathcal{E},\mathcal{R}\in {\zeta }^{\tilde{X}}$,

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E}\cap \mathcal{R})\ge {\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})\cap {\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{R}),{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E}\cap \mathcal{R})\le {\tau }_{st}^{\tilde{\eta }}(\mathcal{E})\cup {\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{R}),{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E}\cap \mathcal{R})\le {\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E})\cup {\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{R}).\end{array}$$

(SVNT3) Suppose there exists a family $\{{\mathcal{E}}_j\in {\zeta }^{\tilde{X}}\mid j\in \mathsf{\Gamma }\}$ and $r\in {\zeta }_0$ with

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{E}}_j)<r<{\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\mathcal{E}}_j),\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}({\displaystyle \bigcup _{i\in \mathsf{\Gamma }}}{\mathcal{E}}_j)>1-r>{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}_{st}^{\tilde{\eta }}({\mathcal{E}}_j),\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{E}}_j)>1-r>{\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{E}}_j)\end{array}$$

Since $[{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\mathcal{E}}_j)\ge r$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}({\mathcal{E}}_j)\le 1-r$ and ${\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{E}}_j)\le 1-r]$ for all $j\in \mathsf{\Gamma }$, there exists a family $\{{\mathcal{E}}_{jk}\mid k\in K_j\}$ with ${\mathcal{E}}_j={\bigcup }_{k\in K_j}{\mathcal{E}}_{jk}\cap {\tilde{\alpha }}_k$ such that

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\mathcal{E}}_j)\ge {\displaystyle \bigcap _{k\in K_j}}{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\mathcal{E}}_{jk})>r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}({\mathcal{E}}_j)\le {\displaystyle \bigcup _{k\in K_j}}{\tilde{\tau }}_{st}^{\tilde{\eta }}({\mathcal{E}}_{jk})<1-r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{E}}_j)\le {\displaystyle \bigcup _{k\in K_j}}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{E}}_{jk})<1-r.\end{array}$$

Since ${\bigcup }_{j\in \mathsf{\Gamma }}{\mathcal{E}}_j={\bigcup }_{j\in \mathsf{\Gamma }}({\bigcup }_{k\in K_j}({\mathcal{E}}_{jk}\cap {\tilde{\alpha }}_k))={\bigcup }_{j,k}({\mathcal{E}}_{jk}\cap {\tilde{\alpha }}_k)$, we obtain

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{E}}_j)\ge {\displaystyle \bigcap _{j,k}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_{jk})={\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}({\displaystyle \bigcap _{k\in K_j}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_{jk})\ge r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{E}}_j)\le {\displaystyle \bigcup _{j,k}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_{jk})={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\displaystyle \bigcup _{k\in K_j}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_{jk})\le 1-r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{E}}_j)\le {\displaystyle \bigcup _{j,k}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_{jk})={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\displaystyle \bigcup _{k\in K_j}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_{jk})\le 1-r\end{array}$$

It is a contradiction. Hence, for each ${\{{\mathcal{E}}_j\}}_{j\in \mathsf{\Gamma }}\in {\zeta }^{\tilde{X}}$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{E}}_j)\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}_{st}^{\tilde{\gamma }}({\mathcal{E}}_j),{\tilde{\tau }}_{st}^{\tilde{\eta }}({\displaystyle \bigcup _{i\in \mathsf{\Gamma }}}{\mathcal{S}}_j)\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}_{st}^{\tilde{\eta }}({\mathcal{E}}_j),{\tilde{\tau }}_{st}^{\tilde{\mu }}({\displaystyle \bigcup _{i\in \mathsf{\Gamma }}}{\mathcal{S}}_j)\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}_{st}^{\tilde{\mu }}({\mathcal{E}}_j).\end{array}$$

Secondly, for each $\mathcal{S}\in {\zeta }^{\tilde{X}}$, there exists a family $\{\tilde{1}\}$ with $\mathcal{S}=\mathcal{S}\cap \tilde{1}$, such that $[{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})\ge {\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})\le {\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})$ and ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})\le {\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})]$. Hence, $({\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ is finer than $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$. Finally, if a stratified $\mathcal{SVNT}$$({\tilde{U}}^{\tilde{\gamma }},{\tilde{U}}^{\tilde{\eta }},{\tilde{U}}^{\tilde{\mu }})$ is finer than $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$, we show that $[{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})\le {\tilde{U}}^{\tilde{\gamma }}(\mathcal{S})$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})\ge {\tilde{U}}^{\tilde{\eta }}(\mathcal{S})$ and ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})\ge {\tilde{U}}^{\tilde{\mu }}(\mathcal{E})]$ for each $\mathcal{S}\in {\zeta }^{\tilde{X}}$.

Suppose that there exist $\mathcal{E}\in {\zeta }^{\tilde{X}}$ and $r\in {\zeta }_0$ such that

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})>r>{\tilde{U}}^{\tilde{\gamma }}(\mathcal{E}),{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E})<1-r<{\tilde{U}}^{\tilde{\eta }}(\mathcal{E}),{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E})<1-r<{\tilde{U}}^{\tilde{\mu }}(\mathcal{E}).\end{array}$$

Since $[{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})>r,{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E})<1-r$ and ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E})<1-r]$, there exists $\{{\mathcal{E}}_j\mid j\in \mathsf{\Gamma }\}$ with ${\mathcal{E}}_j={\bigcup }_{j\in \mathsf{\Gamma }}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j)$ such that

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{E})\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j)>r,{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{E})\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j)<1-r,{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{E})\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j)<1-r.\end{array}$$

On the other hand, since $[{\tilde{U}}^{\tilde{\gamma }}({\mathcal{E}}_j)\ge {\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j)$, ${\tilde{U}}^{\tilde{\eta }}({\mathcal{E}}_j)\le {\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j)$ and ${\tilde{U}}^{\tilde{\mu }}({\mathcal{E}}_j)\le {\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j)]$ for all $j\in \mathsf{\Gamma }$, we have

$$\begin{array}{l}{\tilde{U}}^{\tilde{\gamma }}(\mathcal{E})={\tilde{U}}^{\tilde{\gamma }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j))\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\gamma }}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j)\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}[{\tilde{U}}^{\tilde{\gamma }}({\mathcal{E}}_j)\cap {\tilde{U}}^{\tilde{\gamma }}({\tilde{\alpha }}_j)]={\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\gamma }}({\mathcal{E}}_j)\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_j)>r,\end{array}$$

$$\begin{array}{l}{\tilde{U}}^{\tilde{\eta }}(\mathcal{E})={\tilde{U}}^{\tilde{\eta }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\eta }}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j)\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}[{\tilde{U}}^{\tilde{\eta }}({\mathcal{E}}_j)\cup {\tilde{U}}^{\tilde{\eta }}({\tilde{\alpha }}_j)]={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\eta }}({\mathcal{E}}_j)\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_j)<1-r\end{array}$$

$$\begin{array}{l}{\tilde{U}}^{\tilde{\mu }}(\mathcal{E})={\tilde{U}}^{\tilde{\eta }}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\mu }}({\mathcal{E}}_j\cap {\tilde{\alpha }}_j)\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}[{\tilde{U}}^{\tilde{\mu }}({\mathcal{E}}_j)\cup {\tilde{U}}^{\tilde{\mu }}({\tilde{\alpha }}_j)]={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\mu }}({\mathcal{E}}_j)\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_j)<1-r\end{array}$$

It is a contradiction. □

Remark 2.

From Defintion 13 and Theorem 8, we have $({\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ as a stratification for $\mathcal{SVNT}$$({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ on $\tilde{X}$.

Example 4.

Let $\tilde{X}=\{a,b,c\}$ be a set. Define ${\mathcal{E}}_1,{\mathcal{E}}_2\in {\zeta }^{\tilde{X}}$ as follows:

$$\begin{array}{l}{\mathcal{E}}_1=\langle (0.5,0.5,0.5),(0.5,0.5,0.5),(0.5,0.5,0.5)\rangle ;{\mathcal{E}}_2=\langle (0.4,0.4,0.4),(0.4,0.4,0.4),(0.6,0.6,0.6)\rangle .\end{array}$$

We define ${\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}:{\zeta }^{\tilde{X}}x{\zeta }_0\to \zeta$ as follows: for every $\mathcal{S}\in {\zeta }^{\tilde{X}}$,

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{0},\hfill \\ 1,if\mathcal{S}=\tilde{1},\hfill \\ \frac{1}{3},if\mathcal{S}={\mathcal{E}}_1,\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2,\hfill \\ \frac{3}{4},if\mathcal{S}={\mathcal{E}}_1\cup {\mathcal{E}}_2,\hfill \\ \frac{2}{3},if\mathcal{S}={\mathcal{E}}_1\cap {\mathcal{E}}_2,\hfill \\ 0,otherwise,\hfill \end{array}\right.{\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{1},\hfill \\ \frac{2}{3},if\mathcal{S}={\mathcal{E}}_1,\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2,\hfill \\ \frac{1}{4},if\mathcal{S}={\mathcal{E}}_1\cup {\mathcal{E}}_2,\hfill \\ \frac{1}{3},if\mathcal{S}={\mathcal{E}}_1\cap {\mathcal{E}}_2,\hfill \\ 1,otherwise,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\hfill \\ 0,if\mathcal{S}=\tilde{1},\hfill \\ \frac{2}{3},if\mathcal{S}={\mathcal{E}}_1,\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_2,\hfill \\ \frac{1}{2},if\mathcal{S}={\mathcal{E}}_1\cup {\mathcal{E}}_2,\hfill \\ \frac{1}{3},if\mathcal{S}={\mathcal{E}}_1\cap {\mathcal{E}}_2,\hfill \\ 1,otherwise,\hfill \end{array}\right.\end{array}$$

If ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha$ for any $0.5<\alpha <0.6$, ${\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha$ for every $0.5<\alpha <0.6$ and ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=0.6$ for all $\beta \ge 0.6$, since

$$\begin{array}{l}\mathcal{S}=(\tilde{\alpha }\cap \tilde{1})\cup (\tilde{\beta }\cap ({\mathcal{E}}_1\cup {\mathcal{E}}_2))=(\tilde{\alpha }\cap \tilde{1})\cup (\tilde{\beta }\cap {\mathcal{E}}_2).\end{array}$$

Then,

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})=[{\tilde{\tau }}^{\tilde{\gamma }}(\tilde{1})\cap {\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_1\cup {\mathcal{E}}_2)]\cup [{\tilde{\tau }}^{\tilde{\gamma }}(\tilde{1})\cap {\tilde{\tau }}^{\tilde{\gamma }}({\mathcal{E}}_2)]=\frac{3}{4},\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})=[{\tilde{\tau }}^{\tilde{\eta }}(\tilde{1})\cup {\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_1\cup {\mathcal{E}}_2)]\cap [{\tilde{\tau }}^{\tilde{\eta }}(\tilde{1})\cup {\tilde{\tau }}^{\tilde{\eta }}({\mathcal{E}}_2)]=\frac{1}{4},\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})=[{\tilde{\tau }}^{\tilde{\mu }}(\tilde{1})\cup {\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_1\cup {\mathcal{E}}_2)]\cap [{\tilde{\tau }}^{\tilde{\mu }}(\tilde{1})\cup {\tilde{\tau }}^{\tilde{\mu }}({\mathcal{E}}_2)]=\frac{1}{2}.\end{array}$$

If ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha$∀$0.5<\alpha <0.6$, ${\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha$∀$0.5<\alpha <0.6$ and ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=\beta$∀$0.5<\alpha$, $\beta \ge 0.6$, we have ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})=\frac{3}{4}$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})=\frac{1}{4}$, ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})=\frac{1}{2}$.

If ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=0.5$, ${\tilde{\eta }}_{\mathcal{S}}(\omega )=0.5$ and ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=0.6$, since ∀$\beta \ge 0.6$, $\alpha \ge 0.5$,

$$\begin{array}{l}\mathcal{S}=(\tilde{\beta }\cap ({\mathcal{E}}_1\cup {\mathcal{E}}_2))=(\tilde{\alpha }\cap {\mathcal{E}}_1)\cup (\tilde{\beta }\cap {\mathcal{E}}_2),\end{array}$$

we obtain ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})=\frac{3}{4}$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})=\frac{1}{4}$, ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})=\frac{1}{2}$.

If ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=0.5$, ${\tilde{\eta }}_{\mathcal{S}}(\omega )=0.5$, and ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=\beta$, ∀$0.5<\beta <0.6$, since

$$\begin{array}{l}\mathcal{S}=(\tilde{\beta }\cap ({\mathcal{E}}_1\cup {\mathcal{E}}_2))=(\tilde{\beta }\cap {\mathcal{E}}_1)\cup (\tilde{\beta }\cap {\mathcal{E}}_2),\end{array}$$

we obtain ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})=\frac{3}{4}$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})=\frac{1}{4}$, ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})=\frac{1}{2}$.

If ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha$, ${\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha$, and ${\tilde{\gamma }}_{\mathcal{S}}(\omega )=\beta$, ∀$0.4<\alpha$$\beta <0.5$ and $\alpha <\beta$, since for every ${\mathcal{S}}_1=\{\tilde{1},{\mathcal{E}}_1,{\mathcal{E}}_1\cup {\mathcal{E}}_2\}$ and ${\mathcal{S}}_2=\{{\mathcal{E}}_2,{\mathcal{E}}_1\cap {\mathcal{E}}_2\}$

$$\begin{array}{l}\mathcal{S}=(\tilde{\alpha }\cap {\mathcal{E}}_1)\cup (\tilde{\beta }\cap {\mathcal{E}}_2),\end{array}$$

we have ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})=\frac{2}{3}$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})=\frac{1}{3}$, ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})=\frac{1}{3}$. We can obtain the following:

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{ll}1,& if\mathcal{S}=\tilde{\alpha },\\ \frac{3}{4},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta for0.5\le \alpha ,\beta \le 0.6,\alpha <\beta ,\\ \frac{1}{2},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha for0.4\le \alpha <0.5and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta ,for0.5<\beta \le 0.6,\\ \frac{2}{3},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta for0.4\le \alpha ,\beta \le 0.5,\alpha <\beta ,\\ 0,& otherwise,\end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{ll}0,& 0,if\mathcal{S}=\tilde{\alpha },\\ \frac{1}{4},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta for0.5\le \alpha ,\beta \le 0.6,\alpha <\beta ,\\ \frac{1}{2},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha for0.4\le \alpha <0.5and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta ,for0.5<\beta \le 0.6,\\ \frac{1}{3},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta for0.4\le \alpha ,\beta \le 0.5,\alpha <\beta ,\\ 1,& otherwise,\end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{ll}0,& 0,if\mathcal{S}=\tilde{\alpha },\\ \frac{1}{4},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta for0.5\le \alpha ,\beta \le 0.6,\alpha <\beta ,\\ \frac{1}{2},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha for0.4\le \alpha <0.5and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta ,for0.5<\beta \le 0.6,\\ \frac{1}{3},& if{\tilde{\gamma }}_{\mathcal{S}}(\omega )=\alpha ,{\tilde{\eta }}_{\mathcal{S}}(\omega )=\alpha and{\tilde{\mu }}_{\mathcal{S}}(\omega )=\beta for0.4\le \alpha ,\beta \le 0.5,\alpha <\beta ,\\ 1,& otherwise.\end{array}\right.\end{array}$$

Theorem 9.

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$, $(\tilde{Y},{\tilde{U}}^{\tilde{\gamma }},{\tilde{U}}^{\tilde{\eta }},{\tilde{U}}^{\tilde{\mu }})$ be two ${\mathcal{SVNTS}}^{\prime }s$ and let $({\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ and $({\tilde{U}}_{st}^{\tilde{\gamma }},{\tilde{U}}_{st}^{\tilde{\eta }},{\tilde{U}}_{st}^{\tilde{\mu }})$ be stratification for $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ and $({\tilde{U}}^{\tilde{\gamma }},{\tilde{U}}^{\tilde{\eta }},{\tilde{U}}^{\tilde{\mu }})$, respectively. If $f:(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})\to (\tilde{Y},{\tilde{U}}^{\tilde{\gamma }},{\tilde{U}}^{\tilde{\eta }},{\tilde{U}}^{\tilde{\mu }})$ is r-$\mathcal{SVN}$-continuous, then $f:(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})\to (\tilde{Y},{\tilde{U}}_{st}^{\tilde{\gamma }},{\tilde{U}}_{st}^{\tilde{\eta }},{\tilde{U}}_{st}^{\tilde{\mu }})$ is r-$\mathcal{SVN}$-continuous.

Proof. 

Suppose there exist $r\in {\zeta }_0$, $\mathcal{R}\in {\zeta }^{\tilde{X}}$, such that

$$\begin{array}{l}{\tilde{U}}_{st}^{\tilde{\gamma }}(\mathcal{R})>r>{\tilde{\tau }}_{st}^{\tilde{\gamma }}(f^{-1}(\mathcal{R})),\end{array}$$

$$\begin{array}{l}{\tilde{U}}_{st}^{\tilde{\eta }}(\mathcal{R})<1-r<{\tilde{\tau }}_{st}^{\tilde{\eta }}(f^{-1}(\mathcal{R}))\end{array}$$

$$\begin{array}{l}{\tilde{U}}_{st}^{\tilde{\mu }}(\mathcal{R})<1-r<{\tilde{\tau }}_{st}^{\tilde{\mu }}(f^{-1}(\mathcal{R}))\end{array}$$

Since ${\tilde{U}}_{st}^{\tilde{\gamma }}(\mathcal{R})>r$, ${\tilde{U}}_{st}^{\tilde{\eta }}(\mathcal{R})<1-r$ and ${\tilde{U}}_{st}^{\tilde{\mu }}(\mathcal{R})<1-r$, by the definition of $({\tilde{U}}_{st}^{\tilde{\gamma }},{\tilde{U}}_{st}^{\tilde{\eta }},{\tilde{U}}_{st}^{\tilde{\mu }})$, there exists a family ${\{{\mathcal{R}}_j\}}_{j\in \mathsf{\Gamma }}$ with $\mathcal{R}={\bigcup }_{j\in \mathsf{\Gamma }}({\mathcal{R}}_j\cap {\tilde{\alpha }}_j)$ such that

$$\begin{array}{l}{\tilde{U}}_{st}^{\tilde{\gamma }}(\mathcal{R})\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\gamma }}({\mathcal{R}}_j)>r,{\tilde{U}}_{st}^{\tilde{\eta }}(\mathcal{R})\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\eta }}({\mathcal{R}}_j)<1-r,\end{array}$$

$$\begin{array}{l}{\tilde{U}}_{st}^{\tilde{\mu }}(\mathcal{R})\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\mu }}({\mathcal{R}}_j)<1-r.\end{array}$$

Since

$$\begin{array}{l}{f}^{-1}(\mathcal{R})=f^{-1}({\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\mathcal{R}}_j\cap {\tilde{\alpha }}_j))={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{f}^{-1}(\mathcal{R})\cap {\tilde{\alpha }}_j,\end{array}$$

and by Remark 2 and Theorem 8, we obtain

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(f^{-1}(\mathcal{R}))\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\gamma }}(f^{-1}({\mathcal{R}}_j)),\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(f^{-1}(\mathcal{R}))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\eta }}(f^{-1}({\mathcal{R}}_j)),\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(f^{-1}(\mathcal{R}))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\mu }}(f^{-1}({\mathcal{R}}_j)).\end{array}$$

Since $f:(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})\to (\tilde{Y},{\tilde{U}}^{\tilde{\gamma }},{\tilde{U}}^{\tilde{\eta }},{\tilde{U}}^{\tilde{\mu }})$ is r-$\mathcal{SVN}$-continuous, that is, ${\tilde{\tau }}^{\tilde{\gamma }}(f^{-1}({\mathcal{R}}_j))\ge {\tilde{U}}^{\tilde{\gamma }}({\mathcal{R}}_j)$, ${\tilde{\tau }}^{\tilde{\eta }}(f^{-1}({\mathcal{R}}_j))\le {\tilde{U}}^{\tilde{\eta }}({\mathcal{R}}_j)$, ${\tilde{\tau }}^{\tilde{\mu }}(f^{-1}({\mathcal{R}}_j))\le {\tilde{U}}^{\tilde{\mu }}({\mathcal{R}}_j)$ for every $j\in \mathsf{\Gamma }$,

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(f^{-1}(\mathcal{R}))\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\gamma }}(f^{-1}({\mathcal{R}}_j))\ge {\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\gamma }}({\mathcal{R}}_j)>r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(f^{-1}(\mathcal{R}))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\eta }}(f^{-1}({\mathcal{R}}_j))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\eta }}({\mathcal{R}}_j)<1-r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(f^{-1}(\mathcal{R}))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\mu }}(f^{-1}({\mathcal{R}}_j))\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\tilde{U}}^{\tilde{\mu }}({\mathcal{R}}_j)<1-r.\end{array}$$

It is contradiction. Hence, $f:(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})\to (\tilde{Y},{\tilde{U}}_{st}^{\tilde{\gamma }},{\tilde{U}}_{st}^{\tilde{\eta }},{\tilde{U}}_{st}^{\tilde{\mu }})$ is r-$\mathcal{SVN}$-continuous. □

The converse of the previous theorem is not true in general as it will be shown by the following example.

Example 5.

Let $\tilde{X}$ be a nonempty set. Define ${\mathcal{SVNT}}^{\prime }s$$({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ and $({\tilde{U}}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{U}},{\tilde{U}}^{\tilde{\mu }})$, for each $\mathcal{S}\in {\zeta }^{\tilde{X}}$ and define $\mathcal{R}\in {\zeta }^{\tilde{X}}$ as follows: $\mathcal{R}=\langle (0.5,0.5,0.5),(0.5,0.5,0.5),(0.5,0.5,0.5)\rangle$,

$$\begin{array}{l}{\tilde{\tau }}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{0},\tilde{1},\hfill \\ 0,otherwise,\hfill \end{array}\right.{\tilde{\tau }}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\tilde{1},\hfill \\ 1,otherwise,\hfill \end{array}\right.{\tilde{\tau }}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\tilde{1},\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{U}}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{0},\tilde{1},\hfill \\ \frac{1}{3},if\mathcal{S}=\mathcal{R},\hfill \\ 0,otherwise,\hfill \end{array}\right.{\tilde{U}}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\tilde{1},\hfill \\ \frac{2}{3},if\mathcal{S}=\mathcal{R},\hfill \\ 1,otherwise,\hfill \end{array}\right.{\tilde{U}}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{0},\tilde{1},\hfill \\ \frac{2}{3},if\mathcal{S}=\mathcal{R},\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

Since $0={\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{R})<{\tilde{U}}^{\tilde{\gamma }}(\mathcal{R})=\frac{1}{3}$, $0={\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{R})>{\tilde{U}}^{\tilde{\eta }}(\mathcal{R})=\frac{2}{3}$ and $0={\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{R})>{\tilde{U}}^{\tilde{\mu }}(\mathcal{R})=\frac{2}{3}$, then the identity mapping $i{d}_x:(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})\to (\tilde{X},{\tilde{U}}^{\tilde{\gamma }},{\tilde{U}}^{\tilde{U}},{\tilde{U}}^{\tilde{\mu }})$ is not r-$\mathcal{SVN}$-continuous. Since for every a family $\{\tilde{1}\}$ and $\mathcal{R}=\mathcal{R}\cap \tilde{1}$, we have ${\tilde{U}}_{st}^{\tilde{\gamma }}(\mathcal{R})\ge {\tilde{U}}^{\tilde{\gamma }}(\tilde{1})=1$, ${\tilde{U}}_{st}^{\tilde{\eta }}(\mathcal{R})\le {\tilde{U}}^{\tilde{\eta }}(\tilde{1})=0$ and ${\tilde{U}}_{st}^{\tilde{\mu }}(\mathcal{R})\le {\tilde{U}}^{\tilde{\mu }}(\tilde{1})=0$. Thus, ${\tilde{U}}_{st}^{\tilde{\gamma }}(\mathcal{R})=1$, ${\tilde{U}}_{st}^{\tilde{\eta }}(\mathcal{R})=0$ and ${\tilde{U}}_{st}^{\tilde{\mu }}(\mathcal{R})=0$. Hence,

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\gamma }}(\mathcal{S})={\tilde{U}}_{st}^{\tilde{\gamma }}(\mathcal{S})=\left\{\begin{array}{c}1,if\mathcal{S}=\tilde{\alpha },\forall \alpha \in {\zeta }_0,\hfill \\ 0,otherwise,\hfill \end{array}\right.{\tilde{\tau }}_{st}^{\tilde{\eta }}(\mathcal{S})={\tilde{U}}_{st}^{\tilde{\eta }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{\alpha },\forall \alpha \in {\zeta }_0\hfill \\ 1,otherwise,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(\mathcal{S})={\tilde{U}}_{st}^{\tilde{\mu }}(\mathcal{S})=\left\{\begin{array}{c}0,if\mathcal{S}=\tilde{\alpha },\forall \alpha \in {\zeta }_0\hfill \\ 1,otherwise.\hfill \end{array}\right.\end{array}$$

Therefore, $i{d}_x:(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})\to (\tilde{X},{\tilde{U}}_{st}^{\tilde{\gamma }},{\tilde{U}}_{st}^{\tilde{\eta }},{\tilde{U}}_{st}^{\tilde{\mu }})$ is r-$\mathcal{SVN}$-continuous.

In the following, we will show that every r-$\mathcal{SVNCOM}$ in the single-valued neutrosophic is r-$\mathcal{SVNCOM}$ in the stratification of it.

Theorem 10.

Let $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ be a stratification of an $\mathcal{SVNTS}$ $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$. A $\mathcal{SVNS}$$\mathcal{S}$ is a r-$\mathcal{SVNCOM}$ in $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ iff $\mathcal{S}$ is a r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$

Proof. 

(1) Let $\mathcal{S}$ be r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$. Suppose that $\mathcal{S}$ is not r-$\mathcal{SVNCON}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$. Then, $\mathcal{E}\ne \tilde{0}$ and $\mathcal{R}\ne \tilde{0}$ are r-$\mathcal{SVNSEP}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ such that $\mathcal{S}=\mathcal{E}\cup \mathcal{D}$. Since ${\tilde{\tau }}^{\tilde{\gamma }}\le {\tilde{\tau }}_{st}^{\tilde{\gamma }}$, ${\tilde{\tau }}^{\tilde{\eta }}\ge {\tilde{\tau }}_{st}^{\tilde{\eta }}$ and ${\tilde{\tau }}^{\tilde{\mu }}\ge {\tilde{\tau }}_{st}^{\tilde{\mu }}$, then, from Theorem 8, we get

$$\begin{array}{l}{C}_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{E},r)\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{E},r),C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{R},r)\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{R},r).\end{array}$$

Hence, $\mathcal{E},\mathcal{R}$ are r-$\mathcal{SVNSEP}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$. Thus, $\mathcal{S}$ is not r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$. We reach a contradiction.

(2) Now, we show that, if $\mathcal{S}$ is r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$, then $\mathcal{S}$ is r-$\mathcal{SVNCON}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$. Let $\mathcal{S}$ be a r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$. Then, by Corollary 2 (1), we have $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)=\mathcal{S}$.

Supposing that $\mathcal{S}$ is not r-$\mathcal{SVNCON}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$, then $\mathcal{E}\ne \tilde{0},\mathcal{R}\ne \tilde{0}$ are r-$\mathcal{SVNSEP}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ such that $\mathcal{S}=\mathcal{E}\cup \mathcal{R}$. Since ${\tilde{\tau }}^{\tilde{\gamma }}\le {\tilde{\tau }}_{st}^{\tilde{\gamma }}$, ${\tilde{\tau }}^{\tilde{\eta }}\ge {\tilde{\tau }}_{st}^{\tilde{\eta }}$, ${\tilde{\tau }}^{\tilde{\mu }}\ge {\tilde{\tau }}_{st}^{\tilde{\mu }}$, then $C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{S},r)\le C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{S},r)=\mathcal{S}$. Thus, $C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{S},r)=\mathcal{S}$. Since $\mathcal{E}\le \mathcal{S}$, we have $C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{E},r)\le \mathcal{S}$. It implies that $\mathcal{S}=C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{E},r)\cup \mathcal{R}$. Put $C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{E},r)=\mathcal{D}$. If $x\in$ supp$(\mathcal{E})$, then $x\in supp(\mathcal{S})$. Since $\mathcal{S}$ is a r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$, by Corollary 2 (4), $x_1\in \mathcal{S}=\mathcal{D}\cup \mathcal{R}$, that is, $\mathcal{D}(x)\cup \mathcal{R}(x)=1$. Since $\mathcal{D}\cap \mathcal{R}=\tilde{0}$, thus, $\mathcal{R}(x)=0$. It implies that $\mathcal{D}(x)=1$. Therefore, $\mathcal{D}$ is a crisp set. Since ${\tilde{\tau }}_{st}^{\tilde{\gamma }}(\tilde{1}-\mathcal{D})\ge r$, ${\tilde{\tau }}_{st}^{\tilde{\eta }}(\tilde{1}-\mathcal{D})\le 1-r$, and ${\stackrel{}{\tau }}_{}^{}$