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 ${\tilde{\tau }}_{st}^{\tilde{\mu }}(\tilde{1}-\mathcal{D})\le 1-r$. By Theorems (1) and (2), for each family, $\{{\tilde{\alpha }}_j\cap {\mathcal{C}}_j:\tilde{1}-\mathcal{D}={\bigcup }_{j\in \mathsf{\Gamma }}{\tilde{\alpha }}_j\cap {\mathcal{C}}_j\}$,

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

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

Let ${\tilde{\alpha }}_j=\tilde{0}$. Since $\mathcal{D}(x)=1$ for any $x\in$ supp$(\mathcal{D})$, we have

$$\begin{array}{l}(\tilde{1}-\mathcal{D})(x)={\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}({\tilde{\alpha }}_j\cap {\mathcal{C}}_j)(x)\Rightarrow 1=\mathcal{D}(x)={\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}(\tilde{1}-{\tilde{\alpha }}_j)(x)\cup (\tilde{1}-{\mathcal{C}}_j)(x).\end{array}$$

Hence, $(\tilde{1}-\mathcal{D})(x)={\bigcup }_{j\in \mathsf{\Gamma }}({\mathcal{C}}_j)(x)$ for $x\in supp(\mathcal{E})$. If $y\notin$ supp$(\mathcal{D})$, then

$$\begin{array}{l}1=(\tilde{1}-\mathcal{D})(y)=({\tilde{\alpha }}_j\cap {\mathcal{C}}_j)(y)\le {\displaystyle \bigcup _{j\in \mathsf{\Gamma }}}{\mathcal{C}}_j(y).\end{array}$$

Thus, for any family $\{{\tilde{\alpha }}_j\cap {\mathcal{C}}_j:\tilde{1}-\mathcal{D}={\bigcup }_{j\in \mathsf{\Gamma }}{\tilde{\alpha }}_j\cap {\mathcal{C}}_j\}$, we have $\tilde{1}-\mathcal{D}={\bigcup }_{j\in \mathsf{\Gamma }}{\mathcal{C}}_j$. It implies

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

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\eta }}(\tilde{1}-\mathcal{D})={\tilde{\tau }}^{\tilde{\eta }}(\tilde{1}-\mathcal{D})={\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\eta }}({\mathcal{C}}_j)\le 1-r,\end{array}$$

$$\begin{array}{l}{\tilde{\tau }}_{st}^{\tilde{\mu }}(\tilde{1}-\mathcal{D})={\tilde{\tau }}^{\tilde{\mu }}(\tilde{1}-\mathcal{D})={\displaystyle \bigcap _{j\in \mathsf{\Gamma }}}{\tilde{\tau }}^{\tilde{\mu }}({\mathcal{C}}_j)\le 1-r.\end{array}$$

Thus, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(\mathcal{D},r)=\mathcal{D}$. It implies

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

Similarly, $C_{{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }}}(C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{R},r),r)=C_{{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }}}(\mathcal{R},r)$. Thus, $\mathcal{E}$ and $\mathcal{R}$ are r-$\mathcal{SVNSEP}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ from

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

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

Thus, $\mathcal{S}$ is not r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$. It is a contradiction.

(3) Let $\mathcal{S}$ be a r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$. From (1), $\mathcal{S}$ is r-$\mathcal{SVNCON}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$. There exists a r-$\mathcal{SVNCOM}$$\mathcal{R}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ containing $\mathcal{S}$. From (2), $\mathcal{R}$ is r-$\mathcal{SVNCON}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$. Thus, $\mathcal{S}=\mathcal{R}$.

Let $\mathcal{R}$ be a r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$. Similarly, $\mathcal{R}$ is a r-$\mathcal{SVNCOM}$ in $(\tilde{X},{\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$. □

Example 6.

In Example 4, we proved that $({\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$ is a stratification for $\mathcal{SVNT}$$({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ on $\tilde{X}$. By Theorem 10, we can clearly see that each r-$\mathcal{SVNCOM}$ in $({\tilde{\tau }}^{\tilde{\gamma }},{\tilde{\tau }}^{\tilde{\eta }},{\tilde{\tau }}^{\tilde{\mu }})$ is r-$\mathcal{SVNCOM}$ in $({\tilde{\tau }}_{st}^{\tilde{\gamma }},{\tilde{\tau }}_{st}^{\tilde{\eta }},{\tilde{\tau }}_{st}^{\tilde{\mu }})$, The opposite is also true.

For the purpose of Symmetry, we can apply the idea in the following definition into Definition 3 and get the desired symmetry’s consequences.

Note A SVNS $\mathcal{R}$ in $\tilde{X}\times \tilde{X}$ is called a single valued neutrosophic relation ($\mathcal{SVNR}$, for short) in $\tilde{X}$, denoted by $\mathcal{R}=\{\langle (x,x_1){\tilde{\gamma }}_{\mathcal{R}}(x,x_1),{\tilde{\eta }}_{\mathcal{R}}(x,x_1),{\tilde{\mu }}_{\mathcal{R}}(x,x_1)\rangle :(x,x_1)\in \tilde{X}\times \tilde{X}\}$, where ${\tilde{\gamma }}_{\mathcal{R}}:\tilde{X}\times \tilde{X}\to [0,1]$, ${\tilde{\eta }}_{\mathcal{R}}:\tilde{X}\times \tilde{X}\to [0,1]$, ${\tilde{\mu }}_{\mathcal{R}}:\tilde{X}\times \tilde{X}\to [0,1]$ denote the truth-membership function, indeterminacy membership function and falsity-membership function of $\mathcal{R}$, respectively.

Definition 14

([17]). Let $\mathcal{S},\mathcal{E}\in {\zeta }^{\tilde{X}}$. If $\mathcal{S}$ is a single-valued neutrosophic relation on a set $\tilde{X}$, then $\mathcal{S}$ is called a single-valued neutrosophic relation on $\mathcal{E}$ if, for every $x,x_1\in \tilde{X}$, ${\tilde{\gamma }}_{\mathcal{S}}(x,x_1)\le min({\tilde{\gamma }}_{\mathcal{E}}(x),{\tilde{\gamma }}_{\mathcal{E}}(x_1))$, ${\tilde{\eta }}_{\mathcal{S}}(x,x_1)\ge max({\tilde{\eta }}_{\mathcal{E}}(x),{\tilde{\eta }}_{\mathcal{E}}(x_1))$ and ${\tilde{\mu }}_{\mathcal{S}}(x,x_1)\ge max({\tilde{\mu }}_{\mathcal{E}}(x),{\tilde{\mu }}_{\mathcal{E}}(x_1))$.

Moreover, a single-valued neutrosophic relation $\mathcal{S}$ on $\tilde{X}$ is called symmetric if, for any $k,k_1\in \tilde{X}$,

${\tilde{\gamma }}_{\mathcal{S}}(k,k_1)={\tilde{\gamma }}_{\mathcal{S}}(k_1,k)$, ${\tilde{\eta }}_{\mathcal{S}}(k,k_1)={\tilde{\eta }}_{\mathcal{S}}(k_1,k)$, ${\tilde{\mu }}_{\mathcal{S}}(k,k_1)={\tilde{\mu }}_{\mathcal{S}}(k_1,k)$; and ${\tilde{\gamma }}_{\mathcal{E}}(k,k_1)={\tilde{\gamma }}_{\mathcal{E}}(k_1,k)$${\tilde{\eta }}_{\mathcal{E}}(k,{\kappa }_1)={\tilde{\eta }}_{\mathcal{E}}(k_1,k)$, ${\tilde{\mu }}_{\mathcal{E}}(k,k_1)={\tilde{\mu }}_{\mathcal{E}}(k_1,k)$.

Example 7.

Let $\tilde{X}=\{x_1,x_2,x_3,x_4,x_5\}$. A a single valued neutrosophic relation $\mathcal{S}$ on $\tilde{X}$ is given in the following table.

$\mathcal{S}$	${\mathit{x}}_{\mathbf{1}}$	${\mathit{x}}_{\mathbf{2}}$	${\mathit{x}}_{\mathbf{3}}$	${\mathit{x}}_{\mathbf{4}}$	${\mathit{x}}_{\mathbf{5}}$
$x_1$	(0.2, 0.6, 0.4)	(0, 0.3, 0.7)	(0.9, 0.2, 0.4)	(0.3, 0.9, 1)	(0.3, 0.9, 1)
$x_2$	(0.4, 0.5, 0.1)	(0.1, 0.7, 0)	(1, 1, 1)	(1, 0.3, 0)	(0.5, 0.6, 1)
$x_3$	(0, 1, 1)	(1, 0.5, 0)	(0, 0, 0)	(0.2, 0.8, 0.1)	(1, 0.8, 1)
$x_4$	(1, 0, 0)	(0, 0, 1)	(0.5, 0.7, 0.1)	(0.1, 0.4, 1)	(1, 0.8, 0.8)
$x_5$	(0, 1, 0)	(0.9, 0, 0)	(0, 0.1, 0.7)	(0.8, 0.9, 1)	(0.6, 1, 0)

5. Conclusions

In this paper, authors have made a study of the connectedness, the idea of component, and the stratification of single-valued neutrosophic topological spaces which are different from the study taken so far and obtained some of their basic properties. Next, the concepts of an r-$\mathcal{SVNSEP}$ and r-$\mathcal{SVNCOM}$ were introduced and studied. It has been proven that every r-$\mathcal{SVNCOM}$ in an single-valued neutrosophic topological spaces is r-$\mathcal{SVNCOM}$ in the stratification of it. We will now go into detail on some of the conclusions of the research. Firstly, a single-valued neutrosophic connected (r-$\mathcal{SVNCON}$) has the same properties in a single-valued neutrosophic topological spaces (see Theorem 3). Secondly, a single-valued neutrosophic separated (r-$\mathcal{SVNSEP}$) has the same properties in a single-valued neutrosophic topological spaces (see Theorems 4 and 5). Finally, it has been proven that every single-valued neutrosophic component (r-$\mathcal{SVNCOM}$) in single-valued neutrosophic topological spaces is r-$\mathcal{SVNCOM}$ in the stratification of it.

Discussion for Further Works

It is known that the notion of boundedness in topological spaces (see [26]) plays a significant role in topological aspects. It is also well known that the collection of bounded sets is an ideal. This concept is generalized to the concept of bornology (which is essentially an interesting ideal). There is also the corresponding generalized notion in fuzzy topics (the concept of fuzzy bornology (see [27]).

Therefore, the following ideas could be applied to the notion of single-valued neutrosophic topological spaces.

(a)

The collection of bounded single-valued sets;

(b)

The concept of fuzzy bornology;

(c)

The notion of boundedness in topological spaces.