On Single-Valued Neutrosophic Closure Spaces

Abstract

This paper aims to mark out new terms of single-valued neutrosophic notions in a Šostak sense called single-valued neutrosophic semi-closure spaces. To achieve this, notions such as $\beta \pounds$-closure operators and $\beta \pounds$-interior operators are first defined. More precisely, these proposed contributions involve different terms of single-valued neutrosophic continuous mappings called single-valued neutrosophic (almost $\beta \pounds$, faintly $\beta \pounds$, weakly $\beta \pounds$) and $\beta \pounds$-continuous. Finally, for the purpose of symmetry, we define the single-valued neutrosophic upper, single-valued neutrosophic lower and single-valued neutrosophic boundary sets of a rough single-valued neutrosophic set ${\alpha }_n$ in a single-valued neutrosophic approximation space $(\tilde{\mathcal{F}},\delta )$. Based on ${\alpha }_n$ and $\delta$, we also introduce the single-valued neutrosophic approximation interior operator ${\mathrm{int}}_{{\alpha }_n}^{\delta }$ and the single-valued neutrosophic approximation closure operator ${\mathrm{Cl}}_{{\alpha }_n}^{\delta }$.

1. Introduction

Neutrosophic set theory has a very powerful influence given that is a recent section of philosophy that is presented as the study of origin, nature and scope of neutralities. The idea of neutrosophy is initiated by Smarandache [1] in 1999 as a new mathematical approach that corresponds to degree of indeterminacy (uncertainty, etc.). Moreover, the soft set theory was successfully applied to several directions, such as smoothness of functions and architecture-based neuro-linguistic programming (NLP) in the papers of Bakbak et al. [2] and Mishra et al. [3]. The concept of continuous mappings plays a crucial role in many branches of mathematics, such as, fuzzy set theory, algebra and quantum gravity (see [4]). El-Naschie also has shown that both string theory and ${\epsilon }^{\infty }$ theory are kind of some applications in quantum particle physics especially in relation to heterotic strings and were influenced by the fuzzy topology in $\check{\mathrm{S}}$ostak sense. [5].

In current times, the theory of neutrosophy has been recycled at various junctions of mathematics. More precisely, this theory has made an exceptional advancement in the field of topological spaces. Salama et al. [6,7,8] dispatched their works of neutrosophic topological spaces, following the method of Chang [9] in the situation of fuzzy topological spaces $(\tilde{\mathcal{F}},\tau )$. Afterward, Hur et al. [10,11] presented NSet(H) and NCSet. Smarandache [12] defined the idea of neutrosophic topology on the non-standard interval. One can simply detect that the fuzzy topology familiarized by Chang is a crisp group of fuzzy subsets.

Šostak [13] determined that Chang’s style is crisp in nature and so he redefined the idea of fuzzy topology, frequently mentioned as smooth fuzzy topology, as a mapping from the group of all fuzzy subsets of $\tilde{\mathcal{F}}$ to $[0, 1]$. Fang Jin-ming et al. and Zahran et al. [14,15] discussed the notion of foundation as a function from an appropriate collection of fuzzy subsets of X to $[0, 1]$. Saber et al. [16] found a parallel theory in the context fuzzy ideal topological space.

Wang [17], in 2010, established the idea of a single-valued neutrosophic set. In 2016, Gayyar [18] presented the notion of fuzzy neutrosophic topological spaces in a Šostak sense. The concept of the foundation for an ordinary single-valued neutrosophic topology was explored by Kim [19]. Several authors [20,21,22,23,24,25] posted their efforts for the idea of single-valued neutrosophic topological spaces $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$. Others focusing their works on single valued neutrosophic relations, see [26,27]. Last but not least, in the sense that not only the objects are fuzzified, but also the axiomatics, the single-valued neutrsophic ideal theory was introduced in 1985 by Šostak [13] as a generalization of classical topological structures and as an extension of both crisp topology and Changs fuzzy topology.

In this article, preliminaries of single-value neutrosophic sets and single-valued neutrosophic topology are reviewed in Section 2. In Section 3, we define the notions of a single-valued neutrosophic semi-closure space. Some of their characteristic properties are considered. Further, we present and explore the properties and characterizations of the single-valued neutrosophic operators, namely $\beta \pounds$-closure $(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ and $\beta \pounds$-interior $(\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ in the single-valued neutrosophic ideal topological space $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$. The concepts of single-valued neutrosophic (almost, faintly, weakly) $\beta \pounds$-continuous mappings are introduced and studied in Section 4. In Section 5, we introduce a new improved single-valued neutrosophic lower and single-valued neutrosophic upper sets by which we obtain a more reliable single-valued neutrosophic boundary region set of a single-valued neutrosophic set ${\alpha }_n$. From these single-valued neutrosophic lower and fuzzy upper sets, we define new single-valued neutrosophic interior and single-valued neutrosophic closure operators associated with a specific single-valued neutrosophic set ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$.

2. Preliminaries

This section is devoted to bring a complete survey, some previous studies and important related notions to this work. Let us have a fixed universe $\tilde{\mathcal{F}}$ to be a finite set of objects and $\zeta$ a closed unit interval $[0, 1]$. We will also let ${\zeta }^{\mathcal{F}}$ to denote the set of all single-valued neutrosophic subsets of $\tilde{\mathcal{F}}$.

Definition 1

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

$$\begin{array}{c}\hfill {\alpha }_n=\{\langle \upsilon ,{\tilde{\varrho }}_{{\alpha }_n}(\upsilon ),{\tilde{\sigma }}_{{\sigma }_n}(\upsilon ),{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )\rangle :\upsilon \in \tilde{\mathcal{F}}\}\end{array}$$

where

$$\tilde{\varrho }:\tilde{\mathcal{F}}\to {\rfloor }^{-}0,1^{+}{\lfloor ,\tilde{\sigma }:\tilde{\mathcal{F}}\to \rfloor }^{-}0,1^{+}{\lfloor ,\tilde{\varsigma }:\tilde{\mathcal{F}}\to \rfloor }^{-}0,1^{+}\lfloor ,$$

and

$${}^{-}0\le {\tilde{\varrho }}_{{\alpha }_n}(\upsilon )+{\tilde{\sigma }}_{{\alpha }_n}(\upsilon )+{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )\le 3^{+}$$

represent the degree of membership (${\tilde{\varrho }}_{{\alpha }_n}$), the degree of indeterminacy (${\tilde{\sigma }}_{{\alpha }_n}$) and the degree of non-membership (${\tilde{\varsigma }}_{{\alpha }_n}$), respectively, of any $\upsilon \in \tilde{\mathcal{F}}$ to the set ${\alpha }_n$.

Definition 2

([17]). Suppose that $\tilde{\mathcal{F}}$ is a universal set a space of points (objects), with a generic element in $\tilde{\mathcal{F}}$ denoted by υ. Then, ${\alpha }_n$ is called a single-valued neutrosophic set (briefly, $\mathcal{SVNS}$) in $\tilde{\mathcal{F}}$, if ${\alpha }_n$ has the form

$$\begin{array}{c}\hfill {\alpha }_n=\{\langle \upsilon ,{\tilde{\varrho }}_{{\alpha }_n}(\upsilon ),{\tilde{\sigma }}_{{\alpha }_n}(\upsilon ),{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )\rangle :\upsilon \in \tilde{\mathcal{F}}\}.\end{array}$$

Now, ${\tilde{\varrho }}_{{\alpha }_n},{\tilde{\sigma }}_{{\sigma }_n},{\tilde{\varsigma }}_{{\alpha }_n}$ indicate the degree of non-membership, the degree of indeterminacy and the degree of membership, respectively, of any element $\upsilon \in \tilde{\mathcal{F}}$ to the set ${\alpha }_n$.

Definition 3

([17]). Let ${\alpha }_n=\{\langle \upsilon ,{\tilde{\varrho }}_{{\alpha }_n}(\upsilon ),{\tilde{\sigma }}_{{\sigma }_n}(\upsilon ),{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )\rangle :\upsilon \in \tilde{\mathcal{F}}\}$ be an SVNS on $\tilde{\mathcal{F}}$. The complement of the set ${\alpha }_n$ (briefly ${\alpha }_n^c$) defined as follows:

$${\tilde{\varrho }}_{{\alpha }_n^c}(\upsilon )={\tilde{\varsigma }}_{{\alpha }_n}(\upsilon ),{\tilde{\varrho }}_{{\alpha }_n^c}(\upsilon )={\left[{\tilde{\varrho }}_{{\alpha }_n}\right]}^c(\upsilon ),{\tilde{\varsigma }}_{{\alpha }_n^c}(\upsilon )={\tilde{\varrho }}_{{\alpha }_n}(\upsilon ).$$

Definition 4

([9]). Let $\tilde{\mathcal{F}}$ be a non-empty set, ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ be in the form: ${\alpha }_n=\{\langle \upsilon ,{\tilde{\varrho }}_{{\alpha }_n}(\upsilon ),$${\tilde{\sigma }}_{{\alpha }_n}(\upsilon ),{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )\rangle :\upsilon \in \tilde{\mathcal{F}}\}$ and ${\epsilon }_n=\{\langle \upsilon ,{\tilde{\varrho }}_{{\epsilon }_n}(\upsilon ),{\tilde{\sigma }}_{{\epsilon }_n}(\upsilon ),{\tilde{\varsigma }}_{{\epsilon }_n}(\upsilon )\rangle :\upsilon \in \tilde{\mathcal{F}}\}$ on $\tilde{\mathcal{F}}$ then,

(a) ${\alpha }_n\subseteq {\epsilon }_n$for every$\upsilon \in \tilde{\mathcal{F}}$;

$${\tilde{\varrho }}_{{\alpha }_n}(\upsilon )\le {\tilde{\varrho }}_{{\epsilon }_n}(\upsilon ),{\tilde{\sigma }}_{{\alpha }_n}(\upsilon )\ge {\tilde{\sigma }}_{{\epsilon }_n}(\upsilon ),{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )\ge {\tilde{\varsigma }}_{{\epsilon }_n}(\upsilon ).$$

(b) ${\alpha }_n={\epsilon }_n$iff${\sigma }_n\subseteq {\epsilon }_n$and${\sigma }_n\supseteq {\epsilon }_n$.

(c) $\tilde{0}=\langle 0, 1, 1\rangle$and$\tilde{1}=\langle 1, 0, 0\rangle$.

Definition 5

([26]). Let ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$. Then,

(a) ${\alpha }_n\cap {\epsilon }_n$ is an SVNS, if for every $\upsilon \in \tilde{\mathcal{F}}$,

$${\alpha }_n\cap {\epsilon }_n=\langle ({\tilde{\varrho }}_{{\alpha }_n}\cap {\tilde{\varrho }}_{{\epsilon }_n})(\upsilon ),({\tilde{\sigma }}_{{\alpha }_n}\cup {\tilde{\sigma }}_{{\epsilon }_n})(\upsilon ),({\tilde{\varsigma }}_{{\alpha }_n}\cup {\tilde{\varsigma }}_{{\epsilon }_n})(\upsilon )\rangle$$

where, $({\tilde{\varrho }}_{{\alpha }_n}\cap {\tilde{\varrho }}_{{\epsilon }_n})(\upsilon )={\tilde{\varrho }}_{{\alpha }_n}(\upsilon )\cap 5{\tilde{\varrho }}_{{\epsilon }_n}(\upsilon )$ and $({\tilde{\varsigma }}_{{\alpha }_n}\cup {\tilde{\varsigma }}_{{\epsilon }_n})(\upsilon )={\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )\cup {\tilde{\varsigma }}_{{\epsilon }_n}(\upsilon )$, for all $\upsilon \in \tilde{\mathcal{F}}$,

(b) ${\alpha }_n\cup {\epsilon }_n$ is an SVNS, if for every $\upsilon \in \tilde{\mathcal{F}}$,

$${\alpha }_n\cup {\epsilon }_n=\langle ({\tilde{\varrho }}_{{\alpha }_n}\cup {\tilde{\varrho }}_{{\epsilon }_n})(\upsilon ),({\tilde{\sigma }}_{{\alpha }_n}\cap {\tilde{\sigma }}_{{\epsilon }_n})(\upsilon ),({\tilde{\varsigma }}_{{\alpha }_n}\cap {\tilde{\varsigma }}_{{\epsilon }_n})(\upsilon )\rangle$$

Definition 6

([6]). For any arbitrary family ${\left\{{\alpha }_n\right\}}_{i\in j}\in {\zeta }^{\tilde{\mathcal{F}}}$ of SVNS the union and intersection are given by

(a) ${\bigcap }_{i\in j}{\left[{\alpha }_n\right]}_i=\langle {\cap }_{i\in j}{\tilde{\varrho }}_{{\left[{\alpha }_n\right]}_i}(\upsilon ),{\cup }_{i\in j}{\tilde{\sigma }}_{{\left[{\alpha }_n\right]}_i}(\upsilon ),{\cup }_{i\in j}{\tilde{\varsigma }}_{{\left[{\alpha }_n\right]}_i}(\upsilon )\rangle$,

(b) ${\bigcup }_{i\in j}{\left[{\alpha }_n\right]}_i=\langle {\cup }_{i\in j}{\tilde{\varrho }}_{{\left[{\alpha }_n\right]}_i}(\upsilon ),{\cap }_{i\in j}{\tilde{\sigma }}_{{\left[{\alpha }_n\right]}_i}(\upsilon ),{\cap }_{i\in j}{\tilde{\varsigma }}_{{\left[{\alpha }_n\right]}_i}(\upsilon )\rangle$.

Definition 7

([18]). A single-valued neutrosophic topological spaces is an ordered $(\tilde{\mathcal{F}},{\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\sigma }},{\tilde{\tau }}^{\tilde{\varsigma }})$ where ${\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\sigma }},{\tilde{\tau }}^{\tilde{\varsigma }}:{\zeta }^{\tilde{\mathcal{F}}}\to \zeta$ is a mapping satisfying the following axioms:

(SVNT1) ${\tilde{\tau }}^{\tilde{\varrho }}(\tilde{0})={\tilde{\tau }}^{\tilde{\varrho }}(\tilde{1})={\tilde{\tau }}^{\tilde{\sigma }}(\tilde{0})={\tilde{\tau }}^{\tilde{\sigma }}(\tilde{1})=0$ and ${\tilde{\tau }}^{\tilde{\varsigma }}(\tilde{0})={\tilde{\tau }}^{\tilde{\varsigma }}(\tilde{1})=1$.

(SVNT2) ${\tilde{\tau }}^{\tilde{\varrho }}({\alpha }_n\cap {\epsilon }_n)\ge {\tilde{\tau }}^{\tilde{\varrho }}({\alpha }_n)\cap {\tilde{\tau }}^{\tilde{\varrho }}({\epsilon }_n)$,    ${\tilde{\tau }}^{\tilde{\sigma }}({\alpha }_n\cap {\epsilon }_n)\le {\tau }^{\tilde{\sigma }}({\alpha }_n)\cup {\tilde{\tau }}^{\tilde{\sigma }}({\epsilon }_n)$,

    ${\tilde{\tau }}^{\tilde{\varsigma }}({\alpha }_n\cap {\epsilon }_n)\le {\tilde{\tau }}^{\tilde{\varsigma }}({\alpha }_n)\cup {\tilde{\tau }}^{\tilde{\varsigma }}({\epsilon }_n)$, for every ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(SVNT3) ${\tilde{\tau }}^{\tilde{\varrho }}({\cup }_{j\in \Gamma }{\left[{\alpha }_n\right]}_j)\ge {\cap }_{j\in \Gamma }{\tilde{\tau }}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_j)$,    ${\tilde{\tau }}^{\tilde{\sigma }}({\cup }_{i\in \Gamma }{\left[{\alpha }_n\right]}_j)\le {\cup }_{j\in \Gamma }{\tilde{\tau }}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_j)$,

    ${\tilde{\tau }}^{\tilde{\varsigma }}({\cup }_{j\in \Gamma }{\left[{\alpha }_n\right]}_j)\le {\cup }_{j\in \Gamma }{\tilde{\tau }}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_j)$, for every ${\left[{\alpha }_n\right]}_j\in {\zeta }^{\tilde{\mathcal{F}}}$.

The quadruple $(\tilde{\mathcal{F}},{\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\sigma }},{\tilde{\tau }}^{\tilde{\varsigma }})$ is called a single-valued neutrosophic topological space (briefly, $SVNT$, for short). Occasionally we write ${\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}$ for $({\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\sigma }},{\tilde{\tau }}^{\tilde{\varsigma }})$ and it will cause no ambiguity.

Definition 8

([21]). Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNTS. Then, for every ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r\in {\zeta }_0$. Then the single-valued neutrosophic closure and single-valued neutrosophic interior of ${\alpha }_n$ are defined by:

$$\begin{array}{c}\hfill {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)=\bigcap \{{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}:{\alpha }_n\le {\epsilon }_n,{\tau }^{\tilde{\varrho }}({\left[{\epsilon }_n\right]}^c)\ge r,{\tau }^{\tilde{\sigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r,{\tau }^{\tilde{\varsigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r\},\end{array}$$

$$\begin{array}{c}\hfill {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)=\bigcup \{{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}:{\alpha }_n\ge {\epsilon }_n,{\tau }^{\tilde{\varrho }}({\epsilon }_n)\ge r,{\tau }^{\tilde{\sigma }}({\epsilon }_n)\le 1-r,{\tau }^{\tilde{\varsigma }}({\epsilon }_n)\le 1-r\}.\end{array}$$

Definition 9

([24,25]). Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an $\mathrm{SVNTS}$ and ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$, $r\in {\zeta }_0$. Then,

(1) 

${\alpha }_n$ is said to be r-single-valued neutrosophic semi-open (briefly, r-SVNSO) iff

$${\alpha }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}_r,r),r),$$

(2) 

${\alpha }_n$ is said to be r-single-valued neutrosophic β-open (briefly, r-SVNβO) iff

$${\alpha }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}_r,r),r),r).$$

(3) 

${\alpha }_n$ is said to be r-single-valued neutrosophic regular open (briefly, r-SVNRO) iff

$${\alpha }_n={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}_r,r),r),$$

The complement of r-SVNSO (resp, r-SVN$\beta$O) are said to be r-SVNSC(resp, r-SVN$\beta$C)), respectively.

Definition 10

([21]). Let $(\tilde{\mathcal{F}})$ be a non-empty set and $\upsilon \in \tilde{\mathcal{F}}$, let $s\in (0,1]$, $t\in [0,1)$ and $k\in [0,1)$, then the single-valued neutrosophic point $x_{s,t,k}$ in $\tilde{\mathcal{F}}$ given by

$$\begin{array}{c}\hfill x_{s,t,k}(\upsilon )=\left\{\begin{array}{c}(s,t,k),\mathit{if}\mathrm{x}=\upsilon ,\hfill \\ (0,1,1),\mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

We say that, $x_{s,t,p}\in {\alpha }_n$ iff $s<{\tilde{\varrho }}_{{\alpha }_n}(\upsilon )$, $t\ge {\tilde{\sigma }}_{{\alpha }_n}(\upsilon )$ and $k\ge {\tilde{\tilde{\varsigma }}}_{{\alpha }_n}(\upsilon )$. We indicate the set of all single-valued neutrosophic points in $\tilde{\mathcal{F}}$ as ${\mathrm{P}}_{x_{s,t,k}}(\tilde{\mathcal{F}})$. A single-valued neutrosophic set ${\alpha }_n$ is said to be quasi-coincident with another single-valued neutrosophic set ${\epsilon }_n$, denoted by ${\alpha }_{n}q{\epsilon }_n$, if there exists an element $\upsilon \in \tilde{\mathcal{F}}$ such that

$$\begin{array}{c}\hfill {\tilde{\varrho }}_{{\alpha }_n}(\upsilon )+{\tilde{\varrho }}_{{\epsilon }_n}(\upsilon )>1,{\tilde{\sigma }}_{{\alpha }_n}(\upsilon )+{\tilde{\sigma }}_{{\epsilon }_n}(\upsilon )\le 1,{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )+{\tilde{\varsigma }}_{{\epsilon }_n}(\upsilon )\le 1.\end{array}$$

Definition 11

([21]). A mapping ${\pounds }^{\tilde{\varrho }},{\pounds }^{\tilde{\sigma }},{\pounds }^{\tilde{\varsigma }}:{\zeta }^{\tilde{\mathcal{F}}}\to \zeta$ is called single-valued neutrosophic ideal ($\mathrm{SVNI}$) on $\tilde{\mathcal{F}}$ if it satisfies the following conditions:

(${\pounds }_1$) ${\pounds }^{\tilde{\varrho }}(\tilde{0})=1$ and ${\pounds }^{\tilde{\sigma }}(\tilde{0})={\pounds }^{\tilde{\varsigma }}(\tilde{0})=0$.

(${\pounds }_2$) If ${\sigma }_n\le {\gamma }_n,$ then ${\pounds }^{\tilde{\varrho }}({\epsilon }_n)\le {\pounds }^{\tilde{\varrho }}({\alpha }_n)$,   ${\pounds }^{\tilde{\sigma }}({\epsilon }_n)\ge {\pounds }^{\tilde{\sigma }}({\alpha }_n)$

  and ${\pounds }^{\tilde{\varsigma }}({\epsilon }_n)\ge {\pounds }^{\tilde{\varsigma }}({\alpha }_n)$, for every ${\epsilon }_n,{\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$.

(${\pounds }_3$) ${\pounds }^{\tilde{\varrho }}({\alpha }_n\cup {\epsilon }_n)\ge {\pounds }^{\tilde{\varrho }}({\alpha }_n)\cap {\pounds }^{\tilde{\varrho }}({\epsilon }_n)$,   ${\pounds }^{\tilde{\sigma }}({\alpha }_n\cup {\epsilon }_n)\le {\pounds }^{\tilde{\sigma }}({\alpha }_n)\cup {\pounds }^{\tilde{\sigma }}({\epsilon }_n)$

  and ${\pounds }^{\tilde{\varsigma }}({\alpha }_n\cup {\epsilon }_n)\le {\pounds }^{\tilde{\varsigma }}({\alpha }_n)\cup {\pounds }^{\tilde{\varsigma }}({\epsilon }_n)$, for every ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$.

The triable $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ is called a single-valued neutrosophic ideal topological space in the Šostak sense (briefly, $SVNITS$).

Definition 12

([21]). Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNITS for each ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$. Then, the single-valued neutrosophic ideal open local function ${\left[{\alpha }_n\right]}_r^{\odot }({\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ of ${\alpha }_n$ is the union of all single-valued neutrosophic points $x_{s,t,k}$ such that if ${\epsilon }_n\in Q_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ and ${\pounds }^{\tilde{\varrho }}({\omega }_n)\ge r$, ${\pounds }^{\tilde{\sigma }}({\omega }_n)\le 1-r$, ${\pounds }^{\tilde{\varsigma }}({\omega }_n)\le 1-r$, then there is at least one $\upsilon \in \tilde{\mathcal{F}}$ for which

$${\tilde{\varrho }}_{{\alpha }_n}(\upsilon )+{\tilde{\varrho }}_{{\epsilon }_n}(\nu )-1>{\tilde{\varrho }}_{{\omega }_n}(\upsilon ),{\tilde{\sigma }}_{{\alpha }_n}(\upsilon )+{\tilde{\sigma }}_{{\epsilon }_n}(\upsilon )-1\le {\tilde{\sigma }}_{{\omega }_n}(\upsilon ),{\tilde{\varsigma }}_{{\alpha }_n}(\upsilon )+{\tilde{\varsigma }}_{{\epsilon }_n}(\upsilon )-1\le {\tilde{\varsigma }}_{{\omega }_n}(\upsilon )$$

Occasionally, we will write ${\left[{\alpha }_n\right]}_r^{\odot }$ for ${\left[{\alpha }_n\right]}_r^{\odot }({\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ and it will have no ambiguity.

Remark 1

([21]). Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNITS and ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$, we can define

$$\begin{array}{c}\hfill {\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r)={\alpha }_n\cup {\left[{\alpha }_n\right]}_r^{\odot },{\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r)={\alpha }_n\cap {\left[{({\alpha }_n^c)}_r^{\odot }\right]}^c.\end{array}$$

Clearly, ${\mathrm{CI}}_{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}^{\odot }$ is a single-valued neutrosophic closure operator and $({\tau }^{\widetilde{\varrho \odot }}(\pounds ),{\tau }^{\widetilde{\sigma \odot }}(\pounds ),{\tau }^{\widetilde{\varsigma \odot }}(\pounds ))$ is the single-valued neutrosophic topology generated by ${\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }$, i.e.,

$$\begin{array}{c}\hfill {\tau }^{\odot }(\mathcal{I})({\alpha }_n)=\bigcup \{r|{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n^c,r)={\alpha }_n^c\}.\end{array}$$

Definition 13

(25). An SVNS δ in $\delta :\tilde{\mathcal{F}}\times \tilde{\mathcal{F}}$ is called a single-valued neutrosophic relation (SVNR) in $\tilde{\mathcal{F}}$, denoted by $\delta =\{\langle (\omega , \nu ),{\tilde{\varrho }}_{\delta }(\omega , \nu ),{\tilde{\sigma }}_{\delta }(\omega , \nu ),{\tilde{\varsigma }}_{\delta }(\omega , \nu )\rangle |(\omega , \nu )\in \tilde{\mathcal{F}}\times \tilde{\mathcal{F}}\}$, where ${\tilde{\varrho }}_{\delta }:\tilde{\mathcal{F}}\times \tilde{\mathcal{F}}\Rightarrow [0, 1]$, ${\tilde{\sigma }}_{\delta }:\tilde{\mathcal{F}}\times \tilde{\mathcal{F}}\Rightarrow [0, 1]$ and ${\tilde{\varsigma }}_{\delta }:\tilde{\mathcal{F}}\times \tilde{\mathcal{F}}\Rightarrow [0,1]$ denote the truth-membership function, indeterminacy membership function and falsity-membership function of δ, respectively. In what follows, SVNR($\tilde{\mathcal{F}}$) will denote the family of all single-valued neutrosophic relations in $\tilde{\mathcal{F}}$.

3. Single-Valued Neutrosophic Semi-Closure Spaces in Šostak Sense

We begin this section by defining the notion of single-valued neutrosophic semi-closure space. Some of its characteristic properties are considered. Further, we present and explore the properties and characterizations of the single-valued neutrosophic operators, namely $\beta \pounds$-closure $(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ and $\beta \pounds$-interior $(\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ in the single-valued neutrosophic ideal topological space $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$.

Definition 14.

A mapping $\mathbb{SC}:{\zeta }^{\tilde{\mathcal{F}}}\times {\zeta }_0\to {\zeta }^{\mathcal{F}}$ is called a single-valued neutrosophic semi-closure operator on $\mathcal{F}$ if, for every ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\mathcal{F}}$ and $r,s\in {\zeta }_0$, the following axioms are satisfied:

(${\mathbb{SC}}_1$) $\mathbb{SC}(\tilde{0}),r)=\tilde{0}$,

(${\mathbb{SC}}_2$) ${\alpha }_n\le \mathbb{SC}({\alpha }_n,r)$,

(${\mathbb{SC}}_3$) $\mathbb{SC}({\alpha }_n,r)\vee \mathbb{SC}({\epsilon }_n,r)=\mathbb{SC}({\alpha }_n\vee {\epsilon }_n,r)$,

(${\mathbb{SC}}_4$) $\mathbb{SC}({\alpha }_n,s)\le \mathbb{SC}({\alpha }_n,r)$ if $s\le r$,

(${\mathbb{SC}}_5$) $\mathbb{SC}(\mathbb{SC}({\alpha }_n,s),r)=\mathbb{SC}({\alpha }_n,r)$.

The pair $(X,\mathbb{SC})$ is a single-valued neutrosophic semi-closure space ($\mathcal{SVNSCS},forshort$).

If ${\mathbb{SC}}_1$ and ${\mathbb{SC}}_2$ are single-valued neutrosophic closure operators on $\mathcal{F}$. Then, ${\mathbb{SC}}_1$ is finer than ${\mathbb{C}}_2$, denoted by ${\mathbb{SC}}_2\le {\mathbb{SC}}_1$ iff ${\mathbb{SC}}_1({\alpha }_n,r)\le {\mathbb{SC}}_2({\alpha }_n,r)$, for every ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r\in {\zeta }_0$.

Theorem 1.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNTS. Then, for any ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r\in {\zeta }_0$, we define an operator ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}:{\zeta }^{\tilde{\mathcal{F}}}\times {\zeta }_0\to {\zeta }^{\tilde{\mathcal{F}}}$ as follows:

$$\begin{array}{c}\hfill {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,s)=\bigwedge \{{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}:{\alpha }_n\le {\epsilon }_n,{\epsilon }_{n}isr-\mathrm{SVNSC}\}.\end{array}$$

Then, $(\mathcal{F},{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ is an $\mathcal{SVNSCS}$.

Proof. 

Suppose that $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ is an $SVNTS$. Then, $({\mathbb{SC}}_1)$, $({\mathbb{SC}}_2)$ and (${\mathbb{SC}}_4$) follows directly from the definition of ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$.

(${\mathbb{SC}}_3$) Since ${\alpha }_n,{\epsilon }_n\le {\alpha }_n\cup {\epsilon }_n$ we obtain ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)\le {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n\cup {\epsilon }_n,r)$ and ${\mathbb{SC}}_{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}$$({\alpha }_n,r)\le {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n\cup {\epsilon }_n,r)$, therefore,

$$\begin{array}{c}\hfill {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,s)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)\le {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n\cup {\epsilon }_n,r).\end{array}$$

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an $SVNTS$. From (${\mathbb{SC}}_2$), we have

$$\begin{array}{ccc}& {\alpha }_n\le {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r),& {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)\right]}^c\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)\right]}^c,r),r)\hfill \\ & & {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)\right]}^c\ge {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\sigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)\right]}^c,r),r),\hfill \\ & & {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c\ge {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varsigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c,r),r)\hfill \end{array}$$

$$\begin{array}{ccc}& {\epsilon }_n\le {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r),& {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r)\right]}^c\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \\ & & {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)\right]}^c\ge {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\sigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)\right]}^c,r),r),\hfill \\ & & {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\epsilon }_n,r)\right]}^c\ge {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varsigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \end{array}$$

It implies that ${\alpha }_n\cup {\epsilon }_n\le {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)$ and

$$\begin{array}{ccc}& {[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r)]}^c& =[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}{({\alpha }_n]}^c\cap {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r)\right]}^c\hfill \\ & & \le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)\right]}^c,r),r)\cap {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)\right]}^c\cap {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }}}({[{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\epsilon }_n,r)]}^c,r),r),\hfill \end{array}$$

$$\begin{array}{ccc}& {[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)]}^c& =[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}{({\alpha }_n]}^c\cap {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)\right]}^c\hfill \\ & & \ge {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\sigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)\right]}^c,r),r)\cap {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\sigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\sigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)\right]}^c\cap {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\sigma }}}({[{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)]}^c,r),r)\hfill \end{array}$$

$$\begin{array}{ccc}& {[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\epsilon }_n,r)]}^c& =[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}{({\alpha }_n]}^c\cap {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\epsilon }_n,r)\right]}^c\hfill \\ & & \ge {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varsigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c,r),r)\cap {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varsigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\sigma }}}({\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c\cap {\left[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\epsilon }_n,r)\right]}^c,r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varsigma }}}({[{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\epsilon }_n,r)]}^c,r),r)\hfill \end{array}$$

Hence, ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,s)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)\ge {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n\cup {\epsilon }_n,r)$; therefore,

$$\begin{array}{c}\hfill {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,s)\cup {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)={\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n\cup {\epsilon }_n,r).\end{array}$$

(${\mathbb{SC}}_5$) Suppose that there exists $r\in {\zeta }_0$, ${\alpha }_n\in {\zeta }^{\mathcal{F}}$ and $\kappa \in \mathcal{F}$ such that

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r),r)}^{\tilde{\varrho }}(\upsilon )>{\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)}^{\tilde{\varrho }}(\upsilon ).\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r),r)}^{\tilde{\sigma }}(\upsilon )\le {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)}^{\tilde{\sigma }}(\upsilon ).\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r),r)}^{\tilde{\varsigma }}(\upsilon )\le {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)}^{\tilde{\varsigma }}(\upsilon ).\end{array}$$

By the definition of ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$, there exists an ${\alpha }_n\in {\zeta }^{\mathcal{F}}$ with ${\pi }_n\ge {\alpha }_n$ and ${\pi }_n$ that is r-SVNSC such that

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r),r)}^{\tilde{\varrho }}(\kappa )>{\tau }_{{\pi }_n}^{\tilde{\varrho }}(\kappa )\ge {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)}^{\tilde{\varrho }}(\kappa ).\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r),r)}^{\tilde{\sigma }}(\kappa )\le {\tau }_{{\pi }_n}^{\tilde{\sigma }}(\kappa )<{\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)}^{\tilde{\sigma }}(\kappa ).\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r),r)}^{\tilde{\varsigma }}(\kappa )\le {\tau }_{{\pi }_n}^{\tilde{\varsigma }}(\kappa )<{\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)}^{\tilde{\varsigma }}(\kappa ).\end{array}$$

Since ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,s)\le {\pi }_n$ and ${\pi }_n$ is r-SVNSC, by the definition of ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$, we have

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r),r)}^{\tilde{\varrho }}(\kappa )\le {\tau }_{{\pi }_n}^{\tilde{\varrho }}(\kappa ),{\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r),r)}^{\tilde{\sigma }}(\kappa )>{\tau }_{{\pi }_n}^{\tilde{\sigma }}(\kappa ),\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathbb{SC}}_{{\tau }^{\tilde{\sigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r),r)}^{\tilde{\varsigma }}(\kappa )>{\tau }_{{\pi }_n}^{\tilde{\varsigma }}(\kappa ).\end{array}$$

It is a contradiction. Thus, ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)={\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$. Hence, ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ is a single-valued neutrosophic semi-closure operator on $\mathcal{F}$. □

Theorem 2.

Let $(\mathcal{F},{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ be an SVNSCS and ${\alpha }_n\in {\zeta }^{\mathcal{F}}$. Define the mapping ${\tau }_{\mathbb{SC}}^{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}:{\zeta }^{\mathcal{F}}\to \zeta$ on $\mathcal{F}$ by

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\alpha }_n)=\bigcup \{r\in {\zeta }_0\mid {\mathbb{SC}}_{{\tau }^{\varrho }}({\left[{\alpha }_n\right]}^c,r)={\left[{\alpha }_n\right]}^c\},\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\alpha }_n)=\bigcap \{1-r\in {\zeta }_0\mid {\mathbb{SC}}_{{\tau }^{\sigma }}({\left[{\alpha }_n\right]}^c,r)={\left[{\alpha }_n\right]}^c\},\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\alpha }_n)=\bigcap \{1-r\in {\zeta }_0\mid {\mathbb{SC}}_{{\tau }^{\varsigma }}({\left[{\alpha }_n\right]}^c,r)={\left[{\alpha }_n\right]}^c\},\end{array}$$

Then,

(1) 

${\tau }_{\mathbb{SC}}^{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ is an SVNTS on $\mathcal{F}$;

(2) 

${\mathbb{SC}}_{\mathbb{SC}}^{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ is finer than $\mathbb{SC}$.

Proof. 

(SVNT1) Let $(\mathcal{F},{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ be an $SVNSCS$. Since $\mathbb{SC}(\tilde{0},r)=\tilde{0}$ and $\mathbb{SC}(\tilde{1},r)=\tilde{1}$ for every $r\in {\zeta }_0$,

(SVNT2) Let $(\mathcal{F},{\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}})$ be an $SVNSCS$. Suppose that there exists ${\left[{\alpha }_n\right]}_1,{\left[{\alpha }_n\right]}_2\in {\zeta }^{\mathcal{F}}$ such that

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)<{\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_1)\cap {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_2),{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)>{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_1)\cup {\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_2),\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)>{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_1)\cup {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_2)\end{array}$$

There exists $r\in {\zeta }_0$ such that

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)<r<{\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_1)\cap {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_2),{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)>1-r>{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_1)\cup {\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_2),\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)>1-r>{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_1)\cup {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_2)\end{array}$$

For each $i\in \{1,2\}$, there exists $r\in {\zeta }_0$ with $\mathbb{SC}({\left[{\alpha }_n\right]}_i,r_i)={\left[{\alpha }_n\right]}_i$ such that

$$\begin{array}{c}\hfill r<r_i\le {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_i),{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_i)\le 1-r_i<1-r,{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_i)\le 1-r_i<1-r.\end{array}$$

In addition, since ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}_i,r)={\left[{\alpha }_n\right]}_i$ by ${\mathbb{SC}}_2$ and ${\mathbb{SC}}_4$ of Definition 13, for any $i\in \{1,2\}$,

$$\begin{array}{c}\hfill {\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}_1\cup {\left[{\alpha }_n\right]}_2,r)={\left[{\alpha }_n\right]}_1\cup {\left[{\alpha }_n\right]}_2\end{array}$$

It follows that ${\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)\ge r$, ${\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)\le 1-r$ and ${\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_1\cap {\left[{\alpha }_n\right]}_2)\le 1-r$. It is a contradiction. Thus, for every ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\mathcal{F}}$, ${\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\alpha }_n\cap {\epsilon }_n)\ge {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\alpha }_n)\cap {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\epsilon }_n)$, ${\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\alpha }_n\cap {\epsilon }_n)\le {\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\alpha }_n)\cup {\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\epsilon }_n)$ and ${\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\alpha }_n\cap {\epsilon }_n)\le {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\alpha }_n)\cup {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\epsilon }_n)$.

(SVNT3) Suppose that there exists ${\alpha }_n={\bigcup }_{i\in \zeta },{\left[{\alpha }_n\right]}_i\in {\zeta }^{\mathcal{F}}$ such that

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\alpha }_n)<\bigcup _{i\in I}{\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_i),{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\alpha }_n)>\bigcup _{i\in I}{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_i),{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\alpha }_n)>\bigcup _{i\in I}{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_i).\end{array}$$

There exists $r_0\in {\zeta }_0$ such that

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\alpha }_n)<r_0<\bigcup _{i\in I}{\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_i),{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\alpha }_n)>1-r_0>\bigcup _{i\in I}{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_i),{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\alpha }_n))>1-r_0>\bigcup _{i\in I}{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_i).\end{array}$$

For every $i\in I$, there exists $\mathbb{SC}({\left[{\alpha }_n\right]}_i^c,r_i)={\left[{\alpha }_n\right]}_i^c$ and $r_i\in {\zeta }_0$ such that

$$\begin{array}{c}\hfill r_0<r_i\le {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[{\alpha }_n\right]}_i),1-r_0>1-r_i\ge {\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[{\alpha }_n\right]}_i),1-r_i>1-r_0\ge {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}_i).\end{array}$$

In addition, since $\mathbb{SC}({\left[{\alpha }_n\right]}^c,r_0)\le \mathbb{SC}({\left[{\alpha }_n\right]}_i^c,r_i)={\left[{\alpha }_n\right]}_i^c$, by ${\mathbb{SC}}_2$ of Definition 13,

$$\begin{array}{c}\hfill \mathbb{SC}({\left[{\alpha }_n\right]}_i^c,r_0)={\left[{\alpha }_n\right]}_i^c.\end{array}$$

It implies, for all $i\in I$,

$$\begin{array}{c}\hfill \mathbb{SC}{({\alpha }_n]}^c,s_0)\le \mathbb{SC}({\left[{\alpha }_n\right]}_i^c,s_0)={\left[{\alpha }_n\right]}_i^c.\end{array}$$

It follows that

$$\begin{array}{c}\hfill \mathbb{SC}({\left[{\alpha }_n\right]}^c,r_0)\le \bigcap _{i\in J}({\left[{\alpha }_n\right]}_i^c)={\left[{\alpha }_n\right]}^c.\end{array}$$

Thus, $\mathbb{SC}({\left[{\alpha }_n\right]}^c,r_0)={\left[{\alpha }_n\right]}^c$, that is, ${\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\alpha }_n)\ge r_0$, ${\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\alpha }_n)\le 1-r_0$ and ${\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\alpha }_n)\le 1-r_0$. It is a contradiction. Hence, ${\tau }_{\mathbb{SC}}^{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ is an $SVNTS$ on $\mathcal{F}$.

Since ${\alpha }_n\le \mathbb{SC}({\alpha }_n,r)$,

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\left[\mathbb{SC}({\alpha }_n,r)\right]}^c)\ge r,{\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\left[\mathbb{SC}({\alpha }_n,r)\right]}^c))\le 1-r,{\tau }_{\mathbb{SC}}^F({\left[\mathbb{SC}({\alpha }_n,r)\right]}^c)\le 1-r.\end{array}$$

From ${\mathbb{SC}}_5$ of Definition 9, we have ${\mathbb{SC}}_{{\tau }_{\mathbb{SC}}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\le \mathbb{SC}({\alpha }_n,r)$. Thus, ${\mathbb{SC}}_{{\tau }_{\mathbb{SC}}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ is finer than $\mathbb{SC}$. □

Example 1.

Let $\mathcal{F}=\{a,b\}$. Define ${\epsilon }_n,{\pi }_n\in {\zeta }^{\mathcal{F}}$ as follows:

$$\begin{array}{c}\hfill {\epsilon }_n=\langle (0.1, 0.1),(0.3, 0.3),(0.3, 0.3)\rangle ;{\pi }_n=\langle (0.4, 0.4),(0.1, 0.1),(0.1, 0.1)\rangle .\end{array}$$

We define the mapping $\mathbb{SC}:{\zeta }^{\mathcal{F}}\times {\zeta }_0\to {\zeta }^{\mathcal{F}}$ as follows:

$$\begin{array}{c}\hfill \mathbb{SC}({\alpha }_n,s)=\left\{\begin{array}{c}\tilde{0},\mathit{if}{\alpha }_n=\tilde{0},s\in {\zeta }_0,\hfill \\ {\epsilon }_n\cap {\pi }_n,\mathit{if}0\ne {\alpha }_n\le {\epsilon }_n\cap {\pi }_n,0<r<\frac{1}{2},\hfill \\ {\epsilon }_n,\mathit{if}{\alpha }_n\le {\epsilon }_n,{\alpha }_n\nleqq {\pi }_n,0<r<\frac{1}{2},\hfill \\ \mathit{or}0\ne {\alpha }_n\le {\epsilon }_n\frac{1}{2}<r<\frac{2}{3},\hfill \\ {\pi }_n,\mathit{if}{\alpha }_n\le {\pi }_n,{\alpha }_n\nleqq {\epsilon }_n,0<r<\frac{1}{2},\hfill \\ {\epsilon }_n\cup {\pi }_n,\mathit{if}0\ne {\alpha }_n\le {\epsilon }_n\cup {\pi }_n,0<r<\frac{1}{2},\hfill \\ \tilde{1},\mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

Then, $\mathbb{SC}$ is a single-valued neutrosophic closure operator.

From Theorem 2, we have a single-valued neutrosophic topology $({\tau }_{\mathbb{SC}}^{\tilde{\varrho }},{\tau }_{\mathbb{SC}}^{\tilde{\sigma }},{\tau }_{\mathbb{SC}}^{\tilde{\varsigma }})$ on $\mathcal{F}$ as follows:

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varrho }}({\alpha }_n)=\left\{\begin{array}{c}1,\mathit{if}{\alpha }_n=\tilde{1}or\tilde{0},\hfill \\ \frac{2}{3},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}}\cup {\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}}\cap {\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ 0,\mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\sigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{1}or\tilde{0},\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}}\cup {\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}}\cap {\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ 1,\mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\mathbb{SC}}^{\tilde{\varsigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{1}or\tilde{0},\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}}\cup {\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\left[{\epsilon }_n\right]}^{\mathrm{c}}\cap {\left[{\pi }_n\right]}^{\mathrm{c}},\hfill \\ 1,\mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

Thus, the ${\tau }_{\mathbb{SC}}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}$ is a single-valued neutrosophic topology on $\mathcal{F}$.

Theorem 3.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNTS. Then, for any ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r\in {\zeta }_0$, we define an operator ${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}:{\zeta }^{\tilde{\mathcal{F}}}\times {\zeta }_0\to {\zeta }^{\tilde{\mathcal{F}}}$ as follows:

$$\begin{array}{c}\hfill {\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,s)=\bigwedge \{{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}:{\alpha }_n\ge {\epsilon }_n,{\epsilon }_{n}isr-SVNSO\}.\end{array}$$

For each ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r,s\in {\zeta }_0$ the operator ${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ satisfies the following conditions:

(1) 

${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(\tilde{1},r)=\tilde{1}$,

(2) 

${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\le {\alpha }_n$,

(3) 

${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\wedge {\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n\wedge {\epsilon }_n,r)$,

(4) 

${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\ge {\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,s)$ if $r\le s$,

(5) 

${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)={\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$.

Proof. 

From the Definition of ${\mathbb{SINT}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$, the proof can be performed □

Definition 15.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNTS, ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$, $x_{s,t,k}\in {\mathrm{P}}_{s,t,k}(\tilde{\mathcal{F}})$ and $r\in {\zeta }_0$. Then,

(1) 

${\alpha }_n$ is called r-single-valued neutrosophic $Q_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$-neighborhood of $x_{s,t,k}$ if $x_{s,t,k}q{\alpha }_n$ with

$${\tau }^{\tilde{\varrho }}({\alpha }_n)\ge r,{\tau }^{\tilde{\sigma }}({\alpha }_n)\le 1-r,{\tau }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r,$$

(2) 

$x_{s,t,k}$${\alpha }_n$ is called r-single-valued neutrosophic θ-cluster point (r-$\theta$-cluster point) of ${\alpha }_n$ if for any ${\epsilon }_n\in Q_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$, we have ${\alpha }_{n}q{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)$,

(3) 

r-θ-closure operator is a mapping ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }:{\zeta }^{\tilde{\mathcal{F}}}\times {\zeta }_0\to {\zeta }^{\tilde{\mathcal{F}}}$ defined as:

$${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)=\bigvee \{x_{s,t,k}\in P_{s,t,k}(\tilde{\mathcal{F}}):x_{s,t,k}isanr-\theta -clusterpointof{\alpha }_n\},$$

(4) 

${\alpha }_n$ is said to be r-$\theta$-closed iff ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta \pounds }({\alpha }_n,r)={\alpha }_n$. We define

$${\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)=\bigwedge \left\{{\epsilon }_n\right|{\alpha }_n\le {\epsilon }_n,{\epsilon }_n={\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\epsilon }_n,r)\}.$$

Theorem 4.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNTS. For $r\in {\zeta }_0$ and ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$. The following properties hold:

(1) 

If ${\alpha }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)$,

(2) 

If ${\alpha }_n\le {\epsilon }_n$, then ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\epsilon }_n,r)$

(3) 

${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)=\bigwedge \left\{{\epsilon }_n\right|{\alpha }_n\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\epsilon }_n,r)$, ${\tau }^{\tilde{\varrho }}({\left[{\epsilon }_n\right]}^c)\ge r$, ${\tau }^{\tilde{\sigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r$, ${\tau }^{\tilde{\varsigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r$}.

(4) 

${\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)={\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\delta }({\alpha }_n,r),r)$,

(5) 

${\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)$ is r-$\theta$-closed,

(6) 

${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)\le {\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)$.

Proof. 

(1) and (2) are easily proved from Definition 14.

(3) $\gamma =\bigwedge \left\{{\epsilon }_n\right|{\alpha }_n\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\epsilon }_n,r)$, ${\tau }^{\tilde{\varrho }}({\left[{\epsilon }_n\right]}^c)\ge r$, ${\tau }^{\tilde{\sigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r$, ${\tau }^{\tilde{\varsigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r$}. Suppose that ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)\nleqq \gamma$, then there exists $\upsilon \in \mathcal{F}$ and $s,t,k\in {\zeta }_0$ such that

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\theta }({\alpha }_n,r)}^{\tilde{\varrho }}(\upsilon )<s\le {\tau }_{\gamma }^{\tilde{\varrho }}(\upsilon ).\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\sigma }}}^{\theta }({\alpha }_n,r)}^{\tilde{\sigma }}(\upsilon )>t\ge {\tau }_{\gamma }^{\tilde{\sigma }}(\upsilon )\end{array}$$

(1)

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)}^{\tilde{\varsigma }}(\upsilon )>k\ge {\tau }_{\gamma }^{\tilde{\varsigma }}(\upsilon ).\end{array}$$

Then $x_{S,t,K}$ is not r-θ-cluster point of ${\alpha }_n$. So, there exists ${\epsilon }_n\in Q_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$, and ${\alpha }_n\le [{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{([{\epsilon }_n,r)]}^c$. Thus, ${\alpha }_n\le {\left[{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)\right]}^c={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\epsilon }_n\right]}^c,r)$ and ${\tau }^{\tilde{\varrho }}({\epsilon }_n)\ge r$, ${\tau }^{\tilde{\sigma }}({\epsilon }_n)\le 1-r$, ${\tau }^{\tilde{\varsigma }}({\epsilon }_n)\le 1-r$. Hence,

$$\begin{array}{c}\hfill {\tau }_{\gamma }^{\tilde{\varrho }}(\upsilon )\le {\tau }_{[{\epsilon }_n){]}^c}^{\tilde{\varrho }}(\upsilon )<s,{\tau }_{\gamma }^{\tilde{\sigma }}(\upsilon )\ge {\tau }_{[{\epsilon }_n){]}^c}^{\tilde{\varrho }}(\upsilon )>t,{\tau }_{\gamma }^{\tilde{\varsigma }}(\upsilon )\ge {\tau }_{[{\epsilon }_n){]}^c}^{\tilde{\varsigma }}(\upsilon )>k.\end{array}$$

It is a contradiction for Equation (1). Thus ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)\ge \gamma$.

Suppose that ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)\nleqq \gamma$, then there exists r-θ-cluster point of $y_{s,t,k}\in P_{x_{s,t,k}}(\tilde{\mathcal{F}})$ of ${\alpha }_n$ such that

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\theta }({\alpha }_n,r)}^{\tilde{\varrho }}(y)>s\ge {\tau }_{\gamma }^{\tilde{\varrho }}(y).\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\sigma }}}^{\theta }({\alpha }_n,r)}^{\tilde{\sigma }}(y)<t\le {\tau }_{\gamma }^{\tilde{\sigma }}(y)\end{array}$$

(2)

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)}^{\tilde{\varsigma }}(y)<k\le {\tau }_{\gamma }^{\tilde{\varsigma }}(\upsilon ).\end{array}$$

By definition of $\gamma$, there exists ${\epsilon }_n\in {\zeta }^{tilde\mathcal{F}}$ with ${\alpha }_n\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)$ and ${\tau }^{\tilde{\varrho }}({\left[{\epsilon }_n\right]}^c)\ge r$, ${\tau }^{\tilde{\sigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r$, ${\tau }^{\tilde{\varsigma }}({\left[{\epsilon }_n\right]}^c)\le 1-r$ such that

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\theta }({\alpha }_n,r)}^{\tilde{\varrho }}(y)>s>{\tau }_{{\epsilon }_n}^{\tilde{\varrho }}(y)\ge {\tau }_{\gamma }^{\tilde{\varrho }}(y),\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\sigma }}}^{\theta }({\alpha }_n,r)}^{\tilde{\sigma }}(y)<t<{\tau }_{{\epsilon }_n}^{\tilde{\sigma }}(y)\le {\tau }_{\gamma }^{\tilde{\sigma }}(y),\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)}^{\tilde{\varsigma }}(y)<k<{\tau }_{{\epsilon }_n}^{\tilde{\varsigma }}(y)\le {\tau }_{\gamma }^{\tilde{\varsigma }}(\upsilon ).\end{array}$$

Then ${\left[{\epsilon }_n\right]}^c\in Q_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(y_{s,t,k},r)$. Furthermore, ${\alpha }_n\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)={\left[{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\epsilon }_n\right]}^c,r)\right]}^c$ which implies ${\alpha }_n\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\epsilon }_n\right]}^c,r)$. Hence $y_{s,t,k}$ is not an r-θ-cluster point of ${\alpha }_n$. It is a contradiction for Equation (2). Thus ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)\le \gamma$.

(4) Let ${\alpha }_n\le {\left[{\epsilon }_n\right]}_i={\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\left[{\epsilon }_n\right]}_i,r)$ for each $i\in \Gamma$. Then

$$\underset{i\in \Gamma }{\bigwedge }{\left[{\epsilon }_n\right]}_i\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }(\underset{i\in \Gamma }{\bigwedge }{\left[{\epsilon }_n\right]}_i,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\left[{\epsilon }_n\right]}_i,r)={\left[{\epsilon }_n\right]}_i.$$

So, ${\bigwedge }_{i\in \Gamma }{\left[{\epsilon }_n\right]}_i\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\bigwedge }_{i\in \Gamma }{\left[{\epsilon }_n\right]}_i,r)$. Hence, ${\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)={\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta \pounds }({\alpha }_n,r),r)$.

(5) It is directly obtained from (4).

(6) Since ${\alpha }_n\le {\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)$, by (5), we have ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r),r)={\Theta }_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\theta }({\alpha }_n,r)$. □

Definition 16.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNITS and ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$, $r\in {\zeta }_0$. Then, ${\alpha }_n$ is said to be r-single-valued neutrosophic $\pounds \beta$-open (briefly, r-SVN$\pounds \beta$O) iff ${\alpha }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}_r,r),r),r)$. The complement of r-SVNβO is said to be r-SVNβC.

Remark 2.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNITS. For ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$, $x_{s,t,k}\in P_{x_{s,t,k}}(\tilde{\mathcal{F}})$ and $r\in {\zeta }_0$. Then, ${\alpha }_n$ is called r-open $Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$-neighborhood of $x_{s,t,k}$ if $x_{s,t,k}q{\alpha }_n$ with ${\alpha }_n$ is r-SVN$\pounds \beta$O set, denoted as:

$$Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)=\{{\alpha }_n\in {\zeta }^{\mathcal{F}}|x_{s,t,k}q{\alpha }_n\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)\}.$$

Definition 17.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNITS. For each ${\alpha }_n,{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$and $r\in {\zeta }_0$, we define the operators $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}},\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}:{\zeta }^{\mathcal{F}}\times {\zeta }_0\to {\zeta }^{\mathcal{F}}$ as follows:

$$\begin{array}{c}\hfill \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)=\bigwedge \{{\epsilon }_n\in {\zeta }^{\mathcal{F}}|{\alpha }_n\le {\epsilon }_n,{\epsilon }_{n}isr-SVN\pounds \beta C\},\\ \hfill \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)=\bigvee \{{\epsilon }_n\in {\zeta }^{\mathcal{F}}|{\epsilon }_n\le {\alpha }_n,{\epsilon }_{n}isr-SVN\pounds \beta O\}.\end{array}$$

Theorem 5.

Let $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be an SVNITS and $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}},\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}:{\zeta }^{\mathcal{F}}\times {\zeta }_0\to {\zeta }^{\mathcal{F}}$. Then,

(1) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(\tilde{0},r)=\tilde{0}$,

(2) 

${\alpha }_n\le \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$,

(3) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\alpha }_n^c,r)={\left[\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c$,

(4) 

${\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\le {\alpha }_n\le \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$,

(5) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\vee \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)\le \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n\vee {\epsilon }_n,r)$,

(6) 

${\alpha }_n$ isr-SVN$\pounds \beta$Ciff ${\alpha }_n=\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$,

(7) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)=\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$,

(8) 

$\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\alpha }_n\wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)$.

Proof. 

(1), (2), (3), (4), (5) and (6) are easily proved from the definitions of $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ and $\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$.

(7) From (4) we only show $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)\ge \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$. Suppose that

$$\begin{array}{c}\hfill \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)\nleqq \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),\end{array}$$

there exist $\upsilon \in \mathcal{F}$ and $s,t,k\in {\zeta }_0$ such that

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)}^{\tilde{\varrho }}(\upsilon )<s<{\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r),r)}^{\tilde{\varrho }}(\upsilon ),\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)}^{\tilde{\sigma }}(\upsilon )\ge t\ge {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r),r)}^{\tilde{\sigma }}(\upsilon ),\end{array}$$

(3)

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)}^{\tilde{\varsigma }}(\upsilon )\ge k\ge {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r),r)}^{\tilde{\varsigma }}(\upsilon ),\end{array}$$

Since ${\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)}^{\tilde{\varrho }}(\upsilon )<s,{\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)}^{\tilde{\sigma }}(\upsilon )\ge t$ and ${\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)}^{\tilde{\varsigma }}(\upsilon )\ge k$ by definition $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ there exists r-SVN$\pounds \beta$C set ${\epsilon }_n\in {\zeta }^{\mathcal{F}}$ with ${\alpha }_n\le {\epsilon }_n$ such that

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)}^{\tilde{\varrho }}(\upsilon )\le {\tau }_{{\epsilon }_n}^{\tilde{\varrho }}(\upsilon )<s,\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)}^{\tilde{\sigma }}(\upsilon )\ge {\tau }_{{\epsilon }_n}^{\tilde{\sigma }}(\upsilon )\ge t,\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)}^{\tilde{\varsigma }}(\upsilon )\ge {\tau }_{{\epsilon }_n}^{\tilde{\varsigma }}(\upsilon )\ge k.\end{array}$$

Since ${\alpha }_n\le {\epsilon }_n$, $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\le {\epsilon }_n.$ Again, by the definition of $\beta \mathcal{I}{C}_{\tau },$ we have $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)\le {\epsilon }_n$. Hence

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r),r)}^{\tilde{\varrho }}(\upsilon )\le {\tau }_{{\epsilon }_n}^{\tilde{\varrho }}(\upsilon )<s,\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r),r)}^{\tilde{\sigma }}(\upsilon )\ge {\tau }_{{\epsilon }_n}^{\tilde{\sigma }}(\upsilon )\ge t,\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r),r)}^{\tilde{\varsigma }}(\upsilon )\ge {\tau }_{{\epsilon }_n}^{\tilde{\varsigma }}(\upsilon )\ge k.\end{array}$$

It is a contradiction for Equation (1).

(8) Since

$$\begin{array}{ccc}& {\alpha }_n& \wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r)\wedge {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r)],r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r)\wedge {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r)],r),r)\hfill \\ & & \le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r)\wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)],r),r)\hfill \\ & & ={\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }([{\alpha }_n\wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)],r),r),r).\hfill \end{array}$$

Hence, ${\alpha }_n\wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)$ is an r-SVN$\pounds \beta$O set contained in ${\alpha }_n$ and so ${\alpha }_n\wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)\le \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$. Since $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$ is an r-SVN$\pounds \beta$O set, we have

$$\begin{array}{c}\hfill \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r),r),r)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r).\end{array}$$

So, ${\alpha }_n\wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r)\ge \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$. Hence

$$\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\alpha }_n\wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\alpha }_n,r),r),r).$$

□

4. New Terms of Single-Valued Neutrosophic Continuity

In this section, we introduce and characterize new classes of mappings called single-valued neutrosophic almost $\beta \pounds$, faintly $\beta \pounds$, weakly $\beta \pounds$ and $\beta \pounds$-continuous mappings. These findings lead to many theorems and consequences. Using these different attributes, we provide an example at the end of this section to show the difference between these kinds of mappings.

Definition 18.

Let $f:(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be a mapping. Then, f is called single-valued neutrosophic almost $\beta \pounds$-continuous mapping (SVNA$\beta \pounds$C, for short) iff for each ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$ there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\epsilon }_n)\le {\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$.

Lemma 1.

For any single-valued neutrosophic set ${\alpha }_n\in {\zeta }^{\mathcal{F}}$ in an SVNTS $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ and $r\in {\zeta }_0$, if ${\tau }^{\tilde{\varrho }}({\alpha }_n)\ge r,{\tau }^{\tilde{\sigma }}({\alpha }_n)\le 1-r,{\tau }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$, then ${\mathbb{SC}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$.

Lemma 2.

For any single-valued neutrosophic set ${\alpha }_n\in {\zeta }^{\mathcal{F}}$ in an SVNTS $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ and $r\in {\zeta }_0$, if ${\alpha }_n$ is r-SVNβO, then ${\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_N,r),r),r)$.

Theorem 6.

Let $f:(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be a mapping for each ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$ and $r\in {\zeta }_0$. Then the following statements are equivalent:

(1) 

f is SVNA$\beta \pounds$C,

(2) 

For every $x_{s,t,k}q{\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r)$,

(3) 

$f^{-1}({\alpha }_n)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$ for every ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)$$\le 1-r$.

Proof. 

(1)⇒(2): Let ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$. Then by (1), there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\epsilon }_n)\le {\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$. Since, ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$. by Lemma 1, ${\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$. Hence, $f({\epsilon }_n)\le S{\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$. Since ${\epsilon }_n$ is r-SVN$\pounds \beta$O set,

$${\epsilon }_n\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\epsilon }_n,r),r),r)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r).$$

Since, $x_{s,t,k}q{\epsilon }_n$ we obtain $x_{s,t,k}q{\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r)$.

(2)⇒(3): Let $x_{s,t,k}q{f}^{-1}({\alpha }_n)$ and ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$, by (2), we have

$$\begin{array}{c}\hfill x_{s,t,k}q{\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r).\end{array}$$

Since $x_{s,t,k}q{f}^{-1}({\alpha }_n)$ we obtain $x_{s,t,k}q{f}^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)).$ Hence, by Theorem 5(8),

$$\begin{array}{ccc}& x_{S,t,K}q{f}^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))& \wedge {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r)\hfill \\ & & =\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r).\hfill \end{array}$$

Thus, $x_{S,t,K}q{f}^{-1}({\alpha }_n)$ implies $x_{S,t,K}q\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$. Hence

$$\begin{array}{c}\hfill f^{-1}({\alpha }_n)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r).\end{array}$$

(3)⇒(1): Let ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$. and since $f^{-1}({\alpha }_n)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),$$r)$. Then by (3), $x_{s,t,k}q\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$. Since, $\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$ is r-SVN$\pounds \beta$O set, $\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$. Moreover,

$$\begin{array}{c}\hfill f(\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r))\le f(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)))\le {\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),\end{array}$$

by Lemma 1, $f(\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r))\le {\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$. □

Theorem 7.

Let $f:(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be a mapping for each ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$, ${\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r\in {\zeta }_0$. Then the following statements are equivalent:

(1) 

f is SVNA$\beta \pounds$C,

(2) 

$f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))$ is an r-SVN$\pounds \beta$O set in $\tilde{\mathcal{F}}$, for every ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$,

(3) 

$f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))$ is an r-SVN$\pounds \beta$C set in $\tilde{\mathcal{F}}$, for each ${\phi }^{\tilde{\varrho }}({\left[{\alpha }_n\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$,

(4) 

$f^{-1}({\alpha }_n)$ is an r-SVN$\pounds \beta$O set in $\tilde{\mathcal{F}}$, for each r-SVNRO set ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$,

(5) 

$f^{-1}({\alpha }_n)$ is an r-SVN$\pounds \beta$C set in $\tilde{\mathcal{F}}$, for each r-SVNRC set ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$,

(6) 

For each ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$ there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\epsilon }_n)\le {\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$$({\alpha }_n,r)$,

(7) 

$f^{-1}({\alpha }_n)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$ for each ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)$$\le 1-r$,

(8) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SINT}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)\le f^{-1}({\alpha }_n),$ for each ${\phi }^{\tilde{\varrho }}({\left[{\alpha }_n\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$,

(9) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r)\le f^{-1}({\alpha }_n)$ for each ${\phi }^{\tilde{\varrho }}({\left[{\alpha }_n\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$,

(10) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\alpha }_n),r)\le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$ for each r-SVNβO set ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$,

(11) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\alpha }_n),r)\le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$ for each r-SVNSO set ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$.

Proof. 

(1)⇒(2): Let $x_{S,t,K}q{f}^{-1}({\alpha }_n)$ and ${\alpha }_n$ is r-SVNRO set in $\tilde{\mathcal{G}}$. Then ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$. Since f is SVNA$\beta \pounds$C, then there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\epsilon }_n)\le {\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$$({\alpha }_n,r),r)$. So,

$$\begin{array}{c}\hfill x_{s,t,k}q{\epsilon }_n\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }({\epsilon }_n,r),r),r)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r),r),r).\end{array}$$

Thus, $x_{s,t.,k}q{f}^{-1}({\alpha }_n)$ implies $x_{s,t,k}q{\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r),$$r),r)$, Hence,

$$\begin{array}{c}\hfill f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r),r),r).\end{array}$$

Therefore $f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))$ is an r-SVN$\pounds \beta$O set in $\tilde{\mathcal{F}}$.

(2)⇒(3): Let ${\phi }^{\tilde{\varrho }}({\left[{\alpha }_n\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$. Then, by (2), $f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}^C,r),r))={\left[f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))\right]}^c$ is r-SVN$\pounds \beta$O set. This yields $f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))$ to be an r-SVN$\pounds \beta$C set in $\tilde{\mathcal{F}}$.

(3)⇒(4): Let ${\alpha }_n$ be an r-SVNRO set in $\tilde{\mathcal{G}}$. Then, ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$. From (3), we have $f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}^C,r),r))={\left[f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))\right]}^c$ is r-SVN$\pounds \beta$C set. Hence, $f^{-1}({\alpha }_n)$ is r-SVN$\pounds \beta$O set in $\tilde{\mathcal{F}}$.

(4)⇒(5): It is easily proved from (4) and the fact that $f^{-1}({\left[{\alpha }_n\right]}^c)={\left[f^{-1}({\alpha }_n)\right]}^c$.

(5)⇒(6): Let ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$. Then $x_{s,t,k}q{f}^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))$ and ${\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$$({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$ is r-SVNRO, which implies that ${\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}^c,r),r)$ is r-SVNRC. By (5), $f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}^c,r),r))$ is an r-SVN$\pounds \beta$C set. Then, $f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))$ is r-SVN$\pounds \beta$O. Put ${\epsilon }_n=f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))$. Then ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ and

$$\begin{array}{c}\hfill f({\epsilon }_n)\le f(f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)))={\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r).\end{array}$$

Since ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$ and by Lemma 1, we have

$$\begin{array}{c}\hfill f({\epsilon }_n)\le {\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)={\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r).\end{array}$$

(6)⇒(7): Let $x_{S,t,K}q{f}^{-1}({\alpha }_n)$ and ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$. Then, ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$ by (6), there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\epsilon }_n)\le {\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$. Thus, $\mathcal{B}\le f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)).$ Since ${\epsilon }_n$ is r-SVN$\pounds \beta$O set,

$$\begin{array}{c}\hfill x_{s,t,k}q{\epsilon }_n=\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r).\end{array}$$

Thus, $x_{t}q{f}^{-1}({\alpha }_n)$ implies $x_{s,t,k}q\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$. Hence,

$$\begin{array}{c}\hfill f^{-1}({\alpha }_n)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r).\end{array}$$

(7)⇒(8): Let ${\phi }^{\tilde{\varrho }}({\left[{\alpha }_n\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$. Then by (7),

$$\begin{array}{c}\hfill f^{-1}({\left[{\alpha }_n\right]}^c)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\alpha }_n\right]}^c,r)),r)={\left[\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SINT}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)\right]}^c.\end{array}$$

Then, $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SINT}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)\le f^{-1}({\alpha }_n)$.

(8)⇒(9): Since ${\phi }^{\tilde{\varrho }}({\left[{\alpha }_n\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, by (8) and Lemma 1, we have

$$\begin{array}{c}\hfill \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SINT}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)=\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r)\le f^{-1}({\alpha }_n).\end{array}$$

(9)⇒(10): Let ${\alpha }_n$ be an r-SVNβO set in $\mathcal{G}$ Then by Lemma 2,

$${\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)$$

and hence ${\phi }^{\tilde{\varrho }}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{({\alpha }_n]}^c,r))\ge r$, ${\phi }^{\tilde{\sigma }}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{({\alpha }_n]}^c,r))\le 1-r$, ${\phi }^{\tilde{\varsigma }}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{({\alpha }_n]}^c,r))\le 1-r$ by (9), we have

$$\begin{array}{c}\hfill \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)),r)\le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)).\end{array}$$

Since ${\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)$ we obtain

$$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\alpha }_n),r)\le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r).$$

(10)⇒(11): It is easily proved from Definition 9.

(11)⇒(1): Obvious. □

Theorem 8.

Let $f:(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be a mapping for each ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$, ${\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r\in {\zeta }_0$. Then the following statements are equivalent:

(1) 

f is SVNA$\beta \pounds$C,

(2) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)),r)\le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))$,

(3) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r)\le f^{-1}({\alpha }_n)$, for every ${\phi }^{\tilde{\varrho }}({\left[{\alpha }_n\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\alpha }_n\right]}^c)\le 1-r$,

(4) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)\le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))$, for every ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n])\le 1-r$,

(5) 

$f^{-1}({\alpha }_n)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$, for every ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n])\le 1-r$,

(6) 

$f^{-1}({\alpha }_n)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r)$, for every ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)$$\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$,

(7) 

$f^{-1}({\alpha }_n)\le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r),r),r)$, for every ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$.

Proof. 

(1)⇒(2): Let ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}}$ and $x_{s,t,k}q{\left[f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))\right]}^c.$ Then, $f(x_{s,t,k})q[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)){]}^c$ and since ${\phi }^{\tilde{\varrho }}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c)\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c)\le 1-r$. Then, $[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)){]}^c\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$, By SVNA$\beta \pounds$C of $\tilde{\mathcal{F}}$, there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that

$$\begin{array}{c}\hfill f({\epsilon }_n)\le {\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c,r),r)=[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)]}^c.\end{array}$$

It implies that ${\epsilon }_n\le [f^{-1}{({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r))]}^c$. Since ${\epsilon }_n$ is SVNA$\beta \pounds$O set in $\tilde{\mathcal{F}}$

$$\begin{array}{ccc}& x_{s,t,k}q{\epsilon }_n& \le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)]}^c),r)\hfill \\ & & =[\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}{(f^{-1}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)),r)]}^c.\hfill \end{array}$$

Thus, $x_{s,t,k}q{\left[f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))\right]}^c$ implies $x_{s,t,k}q[\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),$${r),r)),r)]}^c$. Hence,

$$\begin{array}{c}\hfill {\left[f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))\right]}^c\le {\left[\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)),r)\right]}^c.\end{array}$$

Thus,

$$\begin{array}{c}\hfill \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}([{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)),r)\le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)).\end{array}$$

(2)⇒(3): It is trivial.

(3)⇒(4): Since ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$, we have ${\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r),r)$ By (3), we have

$$\begin{array}{ccc}& \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))& =\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r)\hfill \\ & & \le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)).\hfill \end{array}$$

(4)⇒(5): Since ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$, we have ${\phi }^{\tilde{\varrho }}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c)$$\ge r$, ${\phi }^{\tilde{\sigma }}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c)\le 1-r$. and by Lemma 1, we have ${\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)={\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$. From (4), we have

$$\begin{array}{ccc}& {\left[\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r)\right]}^c& =\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^n,r)),r)\hfill \\ & & \le f^{-1}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\left[{\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)\right]}^c,r))\hfill \\ & & \le {\left[f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r))\right]}^c.\hfill \end{array}$$

It implies that $f^{-1}({\alpha }_n)\le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)$.

(5)⇒(6): Let ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$. Then by (5), we have

$$\begin{array}{ccc}& f^{-1}({\alpha }_n)& \le \beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r)\hfill \\ & & \le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r),r)\hfill \\ & & \le {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r).\hfill \end{array}$$

(6)⇒(7): It is easily proved from Lemma 1.

(7)⇒(1): Let ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$, then $x_{s,t,k}q{f}^{-1}({\alpha }_n)$ and ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$. From (7), $x_{s,t,k}q{\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)),r),r),r)$. Since ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$, ${\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)=i{\mathrm{int}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r),r)$ and $x_{s,t,k}q{\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{CI}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}^{\odot }(f^{-1}({\mathbb{SC}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)),r),r),r).$ By Theorem 6(2), we have f is SVNA$\beta \pounds$C. □

Definition 19.

Let $f:(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be a mapping. Then,

(1) 

f is called single-valued neutrosophic faintly $\beta \pounds$-continuous (SVNF$\beta \pounds$C, for short) iff for every ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$, there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\epsilon }_n)\le {\alpha }_n$,

(2) 

f is called single-valued neutrosophic weakly $\beta \pounds$-continuous (SVNW$\beta \pounds$C, for short) iff for every ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$, there exists ${\epsilon }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\epsilon }_n)\le {\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$,

(3) 

f is called single-valued neutrosophic $\beta \pounds$-continuous (SVN$\beta \pounds$C, for short) iff $f^{-1}({\alpha }_n)$ is r-SVN$\pounds \beta$O, for every ${\phi }^{\tilde{\varrho }}({\alpha }_n)\ge r$, ${\phi }^{\tilde{\sigma }}({\alpha }_n)\le 1-r$, ${\phi }^{\tilde{\varsigma }}({\alpha }_n)\le 1-r$.

Remark 3.

From the above definition we obtain the following diagram:

$$\begin{array}{c}\hfill SVN\beta \pounds C\Rightarrow SVNA\beta \pounds C\\ \hfill \Downarrow \\ \hfill SVNF\beta \pounds C\Leftarrow SVNW\beta \pounds C\end{array}$$

Some supporting examples will be shown after the following two theorems.

Theorem 9.

Let $f:(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ be a mapping. Then the following statements are equivalent:

(1) 

f is SVNW$\beta \pounds$C,

(2) 

$f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))\le {\Theta }_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f({\alpha }_n),r)$, for each ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}},r\in {\zeta }_0$,

(3) 

$\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f^{-1}({\epsilon }_n),r)\le f^{-1}({\Theta }_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)),$ for each ${\epsilon }_n\in {\zeta }^{\tilde{\mathcal{G}}},r\in {\zeta }_0$,

(4) 

$f^{-1}({\epsilon }_n)$ is r-SVN$\beta \pounds$C set in $\tilde{\mathcal{F}}$ for each r-θ-closed set,

(5) 

$f^{-1}({\epsilon }_n)$ isr-SVN$\beta \pounds$O set in $\tilde{\mathcal{F}}$ for each r-θ-open set.

Proof. 

(1)⇒(2) Suppose there exists ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and $r\in {\zeta }_0$ such that $f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))\nleqq {\Theta }_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f({\alpha }_n),r)$. Then there exists $\upsilon \in \tilde{\mathcal{G}}$ and $s,t,k\in {\zeta }_0$ such that

$$\begin{array}{c}\hfill {\phi }_{f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r))}^{\tilde{\varrho }}(\upsilon )>s>{\phi }_{{\Theta }_{{\phi }^{\tilde{\varrho }}}(f({\alpha }_n),r)}^{\tilde{\varrho }}(\upsilon )\end{array}$$

$$\begin{array}{c}\hfill {\phi }_{f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r))}^{\tilde{\sigma }}(\upsilon )\le t\le {\phi }_{{\Theta }_{{\phi }^{\tilde{\sigma }}}(f({\alpha }_n),r)}^{\tilde{\sigma }}(\upsilon )\end{array}$$

$$\begin{array}{c}\hfill {\phi }_{f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r))}^{\tilde{\varsigma }}(\upsilon )\le k\le {\phi }_{{\Theta }_{{\phi }^{\tilde{\varsigma }}}(f({\alpha }_n),r)}^{\tilde{\varsigma }}(\upsilon )\end{array}$$

If $f^{-1}(\left\{\upsilon \right\})=\varnothing$, provides a contradiction that $f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r))=0$. If $f^{-1}(\left\{\upsilon \right\})\ne \varnothing$, there exists $\omega \in f^{-1}(\left\{\upsilon \right\})$ such that

$$\begin{array}{c}\hfill {\phi }_{f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r))}^{\tilde{\varrho }}(\upsilon )\ge {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)}^{\tilde{\varrho }}(\omega )>s>{\phi }_{{\Theta }_{{\phi }^{\tilde{\varrho }}}(f({\alpha }_n),r)}^{\tilde{\varrho }}(f(\omega ))\end{array}$$

$$\begin{array}{c}\hfill {\phi }_{f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r))}^{\tilde{\sigma }}(\upsilon )\le {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)}^{\tilde{\sigma }}(\omega )\le t\le {\phi }_{{\Theta }_{{\phi }^{\tilde{\sigma }}}(f({\alpha }_n),r)}^{\tilde{\sigma }}(f(\omega ))\end{array}$$

(4)

$$\begin{array}{c}\hfill {\phi }_{f(\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r))}^{\tilde{\varsigma }}(\upsilon )\le {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)}^{\tilde{\varsigma }}(\omega )\le k\le {\phi }_{{\Theta }_{{\phi }^{\tilde{\varsigma }}}(f({\alpha }_n),r)}^{\tilde{\varsigma }}(f(\omega )).\end{array}$$

Since ${\phi }_{{\Theta }_{{\phi }^{\tilde{\varrho }}}(f({\alpha }_n),r)}^{\tilde{\varrho }}(f(\omega ))<s$, ${\phi }_{{\Theta }_{{\phi }^{\tilde{\sigma }}}(f({\alpha }_n),r)}^{\tilde{\sigma }}(f(\omega ))\ge t$ and ${\phi }_{{\Theta }_{{\phi }^{\tilde{\varsigma }}}(f({\alpha }_n),r)}^{\tilde{\varsigma }}(f(\omega ))\ge k$. Then, $f{(x)}_{s,.t,k}$ is not r-θ-cluster point of $f({\alpha }_n)$, there exists ${\epsilon }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$ such that $f({\alpha }_n)\le {\left[{\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)\right]}^n$. By SVNW$\beta \pounds$C of f, there exists ${\pi }_n\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ such that $f({\pi }_n)\le {\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\epsilon }_n,r)$. Thus, $f({\alpha }_n)\le {\left[f({\pi }_n)\right]}^c$ implies ${\alpha }_n\le {\left[{\pi }_n\right]}^c$. Hence

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }}}({\alpha }_n,r)}^{\tilde{\varrho }}(\omega )\le {\tau }_{{\left[{\pi }_n\right]}^c}^{\tilde{\varrho }}(\omega )<s,\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\sigma }}}({\alpha }_n,r)}^{\tilde{\sigma }}(\omega )\ge {\tau }_{{\left[{\pi }_n\right]}^c}^{\tilde{\sigma }}(\omega )\ge t,\end{array}$$

$$\begin{array}{c}\hfill {\tau }_{\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varsigma }}}({\alpha }_n,r)}^{\tilde{\varsigma }}(\omega )\ge {\tau }_{{\left[{\pi }_n\right]}^c}^{\tilde{\varsigma }}(\omega )\ge t.\end{array}$$

It is a contradiction for Equation (2).

(2)⇒(3), (3)⇒(4) and (4)⇒(5): are obvious.

(5)⇒(1): Let ${\alpha }_n\in Q_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(f(x_{s,t,k}),r)$. Then $x_{s,t,k}q{f}^{-1}({\alpha }_n)$ and ${\alpha }_n$ is r- $\theta$-open set. By (5), we have $f^{-1}({\alpha }_n)$ is r-SVN$\beta \pounds$O set in $\tilde{\mathcal{F}}$. Since $x_{s,t,k}q{f}^{-1}({\alpha }_n)$ we obtain $f^{-1}({\alpha }_n)\in Q_{{\tau }_{\pounds \beta }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}(x_{s,t,k},r)$ and hence from $f(f^{-1}({\alpha }_n))\le {\alpha }_n\le {\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)$. Then, f is SVNW$\beta \pounds$C. □

Theorem 10.

A mapping $f:(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ is SVNF$\beta \pounds$C iff for eachr-$\theta$-closed set ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{G}}},$$f^{-1}({\alpha }_n)$ is r-SVN$\beta \pounds$C.

Proof. 

Obvious. □

Example 2.

Let $\mathcal{F}=\{a,b\}$. Define ${\epsilon }_n,{\pi }_n\in {\zeta }^{\mathcal{F}}$ as follows:

$$\begin{array}{c}\hfill {\epsilon }_n=\langle (0.3, 0.3),(0.7, 0.7),(0.3, 0.3)\rangle ;{\pi }_n=\langle (0.3, 0.3),(0.3, 0.3),(0.3, 0.3)\rangle .\end{array}$$

We define the mapping ${\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}:{\zeta }^{\mathcal{F}}\times {\zeta }_0\to {\zeta }^{\mathcal{F}}$ as follows:

$$\begin{array}{c}\hfill {\tau }^{\tilde{\varrho }}({\alpha }_n)=\left\{\begin{array}{c}1,\mathit{if}{\alpha }_n=\tilde{1}\mathit{or}\tilde{0},\hfill \\ \frac{2}{3},\mathit{if}{\alpha }_n={\epsilon }_n\mathit{or}{\pi }_n\hfill \\ 0,\mathit{otherwise},\hfill \end{array}\right.{\phi }^{\tilde{\varrho }}({\alpha }_n)=\left\{\begin{array}{c}1,\mathit{if}{\alpha }_n=\tilde{1}\mathrm{or}\tilde{0},\hfill \\ \frac{2}{3},\mathit{if}{\alpha }_n={\pi }_n\hfill \\ 0,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tau }^{\tilde{\sigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{1}\mathrm{or}\tilde{0},\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n={\epsilon }_n\mathrm{or}{\pi }_n\hfill \\ 1,\mathit{otherwise},\hfill \end{array}\right.{\phi }^{\tilde{\sigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{1}\mathrm{or}\tilde{0},\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n={\pi }_n\hfill \\ 0,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tau }^{\tilde{\varsigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{1}\mathrm{or}\tilde{0},\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n={\epsilon }_n\mathrm{or}{\pi }_n\hfill \\ 1,\mathit{otherwise},\hfill \end{array}\right.{\phi }^{\tilde{\varsigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{1}\mathrm{or}\tilde{0},\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n={\pi }_n\hfill \\ 1,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\pounds }^{\tilde{\varrho }}({\alpha }_n)=\left\{\begin{array}{c}1,\mathit{if}{\alpha }_n=\tilde{0},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\pi }_n,\hfill \\ \frac{2}{3},\mathit{if}{\alpha }_n=0<{\alpha }_n<{\pi }_n,\hfill \\ 0,\mathit{otherwise},\hfill \end{array}\right.{\pounds }^{\tilde{\sigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{0},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\pi }_n,\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n=0<{\alpha }_n<{\pi }_n,\hfill \\ 1,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\pounds }^{\tilde{\varsigma }}({\alpha }_n)=\left\{\begin{array}{c}0,\mathit{if}{\alpha }_n=\tilde{0},\hfill \\ \frac{1}{2},\mathit{if}{\alpha }_n={\pi }_n,\hfill \\ \frac{1}{3},\mathit{if}{\alpha }_n=0<{\alpha }_n<{\pi }_n,\hfill \\ 1,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

From Theorems 4 and 5, we obtain $\beta \mathcal{I}{C}_{\tau },T_{\tau }:I^X\times I_0\to I^X$ as follows:

$$\begin{array}{c}\hfill {\Theta }^{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)=\left\{\begin{array}{c}\tilde{0},\mathit{if}{\alpha }_n=\tilde{0},\hfill \\ {\epsilon }_n,\mathit{if}\tilde{0}\ne {\alpha }_n\le {\pi }_n,0<r\le \frac{2}{3},\hfill \\ \tilde{1},\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill \beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\alpha }_n,r)=\left\{\begin{array}{c}\tilde{0},\mathit{if}\mathcal{B}=\underline{0},\hfill \\ {\pi }_n,\mathit{if}\tilde{0}\ne {\alpha }_n\le {\pi }_n,0<r\le \frac{2}{3},\hfill \\ \tilde{1},\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

By Theorem 9(2), the identity mapping $i{d}_X::(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }},{\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})\to (\tilde{\mathcal{G}},{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$ is SVNW$\beta \pounds$C but is not SVNA$\beta \pounds$C, because by Theorem 8(5), for each ${\phi }^{\tilde{\varrho }}({\pi }_n)\ge \frac{2}{3}$, ${\phi }^{\tilde{\sigma }}({\pi }_n)\le \frac{1}{3}$, ${\phi }^{\tilde{\varsigma }}({\pi }_n)\le \frac{1}{3}$ and ${\phi }^{\tilde{\varsigma }}({\pi }_n)\le \frac{1}{3}$,

$$\begin{array}{c}\hfill \tilde{1}=\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\pi }_n,\frac{2}{3}))\nleqq {\mathrm{C}}_{{\phi }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}({\pi }_n,\frac{2}{3})={\epsilon }_n.\end{array}$$

5. Single-Valued Neutrosophic Approximation Space

In this section, and for symmetrical purposes, we establish the definition of the single-valued neutrosophic upper, single-valued neutrosophic lower and single-valued neutrosophic boundary sets of a rough single-valued neutrosophic set ${\alpha }_n$ in a single-valued neutrosophic approximation space $(\tilde{\mathcal{F}},\delta )$. Based on ${\alpha }_n$ and $\delta$, we introduce the single-valued neutrosophic approximation interior operator ${\mathrm{int}}_{{\alpha }_n}^{\delta }$ and the single-valued neutrosophic approximation closure operator ${\mathrm{Cl}}_{{\alpha }_n}^{\delta }$.

Definition 20.

Assume that an SVNR δ is defined so that ${\tilde{\varrho }}_{\delta }(\omega ,\omega )=1,{\tilde{\sigma }}_{\delta }(\omega ,\omega )$$=0,{\tilde{\varsigma }}_{\delta }(\omega ,\omega )=0$ for every $\omega \in \tilde{\mathcal{F}}$, ${\tilde{\varrho }}_{\delta }(\omega ,\nu )={\tilde{\varrho }}_{{\alpha }_n}(\nu ,\omega )$, ${\tilde{\sigma }}_{\delta }(\omega ,\nu )={\tilde{\sigma }}_{\delta }(\nu ,\omega )$, ${\tilde{\varsigma }}_{\delta }(\omega ,\nu )={\tilde{\varsigma }}_{{\alpha }_n}(\nu ,\omega )$ for every $\omega ,\nu \in \tilde{\mathcal{F}}$ and ${\tilde{\varrho }}_{\delta }(\omega ,\nu )\ge ({\tilde{\varrho }}_{\delta }(\omega ,\mu )\wedge {\tilde{\varrho }}_{\delta }(\mu ,\nu ))$, ${\tilde{\sigma }}_{\delta }(\omega ,\nu )\le ({\tilde{\sigma }}_{\delta }(\omega ,\mu )\vee {\tilde{\sigma }}_{\delta }(\mu ,\nu ))$ and ${\tilde{\varsigma }}_{\delta }(\omega ,\nu )\le ({\tilde{\varsigma }}_{\delta }(\omega ,\mu )\vee {\tilde{\varsigma }}_{\delta }(\mu ,\nu ))$ for every $\omega ,\nu ,\mu \in \tilde{\mathcal{F}}$. That is, δ is a single-valued neutrosophic equivalence relation on $\tilde{\mathcal{F}}$. Then $(\tilde{\mathcal{F}},\delta )$ is called a single-valued neutrosophic approximation space based on the single-valued neutrosophic equivalence relation (briefly, SVN-equivalence relation) δ on $\tilde{\mathcal{F}}$.

Definition 21.

For each $\omega \in \tilde{\mathcal{F}}$, define a single-valued neutrosophic coset $\left[\omega \right]:\tilde{\mathcal{F}}\Rightarrow [0,1]$ by:

$${\tilde{\varrho }}_{\left[\omega \right]}(\nu )={\tilde{\sigma }}_{\delta }(\omega ,\nu ),{\tilde{\sigma }}_{\left[\omega \right]}(\nu )={\tilde{\sigma }}_{\delta }(\omega ,\nu ),{\tilde{\varsigma }}_{\left[\omega \right]}(\nu )={\tilde{\varsigma }}_{\delta }(\omega ,\nu )\forall \nu \in .\tilde{\mathcal{F}}.$$

All elements $\nu \in \tilde{\mathcal{F}}$ with SVNR value ${\tilde{\varrho }}_{\delta }(\omega ,\nu )>0, {\tilde{\sigma }}_{\delta }(\omega ,\nu )\le 1, {\tilde{\varsigma }}_{\delta }(\omega ,\nu )\le 1$ are elements having a membership value in the single-valued neutrosophic coset $\left[\omega \right]$, and any element $\nu \in \tilde{\mathcal{F}}$ with ${\tilde{\varrho }}_{\delta }(\omega ,\omega )=1, {\tilde{\sigma }}_{\delta }(\omega ,\omega )=0, {\tilde{\varsigma }}_{\delta }(\omega ,\omega )=0$ is not included in the single-valued neutrosophic coset $\left[\omega \right]$. Any single-valued neutrosophic coset $\left[\omega \right]$ surely include the element $\omega \in \tilde{\mathcal{F}}$, and consequently

$${\tilde{\varrho }}_{{\bigvee }_{\mu \in \tilde{\mathcal{F}}}}(\left[\omega \right](\mu ))=1,{\tilde{\sigma }}_{{\bigwedge }_{\mu \in \tilde{\mathcal{F}}}}(\left[\omega \right](\mu ))=0,{\tilde{\varsigma }}_{{\bigwedge }_{\mu \in \tilde{\mathcal{F}}}}(\left[\omega \right](\mu ))=0,\forall \omega \in \tilde{\mathcal{F}}$$

Further,

$${\tilde{\varrho }}_{{\bigvee }_{\mu \in \tilde{\mathcal{F}}}}(\left[\omega \right](\nu ))=1,{\tilde{\sigma }}_{{\bigwedge }_{\mu \in \tilde{\mathcal{F}}}}(\left[\omega \right](\nu ))=0,{\tilde{\varsigma }}_{{\bigwedge }_{\mu \in \tilde{\mathcal{F}}}}(\left[\omega \right](\nu ))=0,\forall \nu \in \tilde{\mathcal{F}},$$

such that ${\bigvee }_{\nu \in \tilde{\mathcal{F}}}(\left[\nu \right])=\langle 0, 1, 1\rangle$. Clearly, if ${\tilde{\varrho }}_{\delta }(\omega ,\nu )>0, {\tilde{\sigma }}_{\delta }(\omega ,\nu )\le 1, {\tilde{\varsigma }}_{\delta }(\omega ,\nu )\le 1$, then the single-valued neutrosophic cosets $\left[\omega \right],\left[\nu \right]$ (as SVNSs) are containing the same elements of $\tilde{\mathcal{F}}$ with some non-zero membership values, and moreover, if ${\tilde{\varrho }}_{\left[\nu \right]}(\mu )=0,{\tilde{\sigma }}_{\left[\nu \right]}(\mu )=1$ and ${\tilde{\varsigma }}_{\left[\omega \right]}(\mu )=1$, then it must be that ${\tilde{\varrho }}_{\left[\omega \right]}(\mu )=0,{\tilde{\sigma }}_{\left[\omega \right]}(\mu )=1$ and ${\tilde{\varsigma }}_{\left[\nu \right]}(\mu )=1$ whenever ${\tilde{\varrho }}_{\delta }(\omega ,\nu )>0, {\tilde{\sigma }}_{\delta }(\omega ,\nu )\le 1, {\tilde{\varsigma }}_{\delta }(\omega ,\nu )\le 1$. That is, any two single-valued neutrosophic cosets are either two single-valued neutrosophic sets containing the same elements of $\tilde{\mathcal{F}}$ with some non-zero membership values or containing completely different elements of $\tilde{\mathcal{F}}$ with some non-zero membership values.

Definition 22.

Let ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and δ be SVN-equivalence relation on $\tilde{\mathcal{F}}$ and the single-valued neutrosophic cosets. Then, the single-valued neutrosophic lower set (briefly, SVN-lower) ${({\alpha }_n)}^{\delta }$, the single-valued neutrosophic upper set (briefly, SVN-upper) ${({\alpha }_n)}_{\delta }$ and the single-valued neutrosophic boundary region set (briefly, SVN-boundary region) ${({\alpha }_n)}^B$ are defined as follows: for $\omega \in \tilde{\mathcal{F}}$,

$${\tilde{\varrho }}_{{({\alpha }_n)}_{\delta }}(\omega )={\tilde{\varrho }}_{{\alpha }_n}(\omega )\vee \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne \omega }{\bigwedge }{\tilde{\varsigma }}_{\left[\omega \right]}(\mu ),{\tilde{\sigma }}_{{({\alpha }_n)}_{\delta }}(\omega )={\tilde{\sigma }}_{{\alpha }_n}(\omega )\wedge \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne \omega }{\bigvee }(1-{\tilde{\sigma }}_{\left[\omega \right]})(\mu )$$

(5)

$${\tilde{\varsigma }}_{{({\alpha }_n)}_{\delta }}(\omega )={\tilde{\varsigma }}_{{\alpha }_n}(\omega )\wedge \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne \omega }{\bigvee }{\tilde{\varrho }}_{\left[\omega \right]}(\mu )$$

$${\tilde{\varrho }}_{{({\alpha }_n)}^{\delta }}(\omega )={\tilde{\varrho }}_{{\alpha }_n}(\omega )\wedge \underset{{\alpha }_n(\mu )>0,\mu \ne \omega }{\bigvee }{\tilde{\varrho }}_{\left[\omega \right]}(\mu ),{\tilde{\sigma }}_{{({\alpha }_n)}^{\delta }}(\omega )={\tilde{\sigma }}_{{\alpha }_n}(\omega )\vee \underset{{\alpha }_n(\mu )>0,\mu \ne \omega }{\bigwedge }({\tilde{\sigma }}_{\left[\omega \right]}(\mu )$$

(6)

$${\tilde{\varsigma }}_{{({\alpha }_n)}^{\delta }}(\omega )={\tilde{\varsigma }}_{{\alpha }_n}(\omega )\vee \underset{{\alpha }_n(\mu )>0,\mu \ne \omega }{\bigwedge }{\tilde{\varsigma }}_{\left[\omega \right]}(\mu )$$

$$\begin{array}{c}\hfill {\tilde{\varrho }}_{{({\alpha }_n)}^B}(\omega )={\tilde{\varrho }}_{{({\alpha }_n)}^{\delta }}\overline{\wedge }{\tilde{\varrho }}_{{({\alpha }_n)}_{\delta }}=\left\{\begin{array}{c}\tilde{0},\mathit{if}{({\alpha }_n)}^{\delta }\le {({\alpha }_n)}_{\delta },\hfill \\ {({\alpha }_n)}^{\delta }\wedge {\left[{({\alpha }_n)}_{\delta }\right]}^c,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\sigma }}_{{({\alpha }_n)}^B}(\omega )={\tilde{\sigma }}_{{({\alpha }_n)}^{\delta }}\overline{\vee }{\tilde{\sigma }}_{{({\alpha }_n)}_{\delta }}=\left\{\begin{array}{c}\tilde{1},\mathit{if}{({\alpha }_n)}^{\delta }\ge {({\alpha }_n)}_{\delta },\hfill \\ {({\alpha }_n)}^{\delta }\vee {\left[{({\alpha }_n)}_{\delta }\right]}^c,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

(7)

$$\begin{array}{c}\hfill {\tilde{\varsigma }}_{{({\alpha }_n)}^B}(\omega )={\tilde{\varsigma }}_{{({\alpha }_n)}^{\delta }}\overline{\vee }{\tilde{\varsigma }}_{{({\alpha }_n)}_{\delta }}=\left\{\begin{array}{c}\tilde{1},\mathit{if}{({\alpha }_n)}^{\delta }\ge {({\alpha }_n)}_{\delta },\hfill \\ {({\alpha }_n)}^{\delta }\vee {\left[{({\alpha }_n)}_{\delta }\right]}^c,\mathit{otherwise},\hfill \end{array}\right.\end{array}$$

${({\alpha }_n)}^{\delta }$, ${({\alpha }_n)}_{\delta }$ and ${({\alpha }_n)}^B$ are then called SVN-lower, SVN-upper and SVN-boundary region sets associated with the SVNS ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ and based on the SVN-equivalence relation δ in a single-valued neutrosophic approximation space $(\tilde{\mathcal{F}},\delta )$.

From (5) and (6), we obtain that ${\tilde{\varrho }}_{{({\alpha }_n)}_{\delta }}(\omega )\le {\tilde{\varrho }}_{{\alpha }_n}(\omega )\le {\tilde{\varrho }}_{{({\alpha }_n)}^{\delta }}(\omega )$, ${\tilde{\sigma }}_{{({\alpha }_n)}_{\delta }}(\omega )\ge {\tilde{\sigma }}_{{\alpha }_n}(\omega )\ge {\tilde{\sigma }}_{{({\alpha }_n)}^{\delta }}(\omega )$ and ${\tilde{\varsigma }}_{{({\alpha }_n)}_{\delta }}(\omega )\ge {\tilde{\varsigma }}_{{\alpha }_n}(\omega )\ge {\tilde{\varsigma }}_{{({\alpha }_n)}^{\delta }}(\omega )$ for each ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$. Whenever ${({\alpha }_n)}^{\delta }$ so that ${\tilde{\varrho }}_{{({\alpha }_n)}^{\delta }}(\omega )\le {\tilde{\varrho }}_{{({\alpha }_n)}_{\delta }(\omega )}$, ${\tilde{\sigma }}_{{({\alpha }_n)}^{\delta }}(\omega )\ge {\tilde{\sigma }}_{{({\alpha }_n)}_{\delta }(\omega )}$ and ${\tilde{\varsigma }}_{{({\alpha }_n)}^{\delta }}(\omega )\ge {\tilde{\varsigma }}_{{({\alpha }_n)}_{\delta }(\omega )}$ we obtain that ${\tilde{\varrho }}_{{\alpha }_n}(\omega )={\tilde{\varrho }}_{{({\alpha }_n)}_{\delta }}(\omega )={\tilde{\varrho }}_{{({\alpha }_n)}^{\delta }}(\omega )$, ${\tilde{\sigma }}_{{\alpha }_n}(\omega )={\tilde{\sigma }}_{{({\alpha }_n)}_{\delta }}(\omega )={\tilde{\sigma }}_{{({\alpha }_n)}^{\delta }}(\omega )$ and ${\tilde{\varsigma }}_{{\alpha }_n}(\omega )={\tilde{\varsigma }}_{{({\alpha }_n)}_{\delta }}(\omega )={\tilde{\varsigma }}_{{({\alpha }_n)}^{\delta }}(\omega )$, and then from (7), we obtain ${\tilde{\varrho }}_{{({\alpha }_n)}^B}(\omega )=0$, ${\tilde{\sigma }}_{{({\alpha }_n)}^B}(\omega )=1$ and ${\tilde{\varsigma }}_{{({\alpha }_n)}^B}(\omega )=1$. Otherwise, ${\tilde{\varrho }}_{{({\alpha }_n)}^B}(\omega )={({\alpha }_n)}^{\delta }\wedge {\left[{({\alpha }_n)}_{\delta }\right]}^c$, ${\tilde{\sigma }}_{{({\alpha }_n)}^B}(\omega )={({\alpha }_n)}^{\delta }\vee {\left[{({\alpha }_n)}_{\delta }\right]}^c$ and ${\tilde{\varsigma }}_{{({\alpha }_n)}^B}(\omega )={({\alpha }_n)}^{\delta }\vee {\left[{({\alpha }_n)}_{\delta }\right]}^c$.

Theorem 11.

For any SVNS ${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ we find that

(1) 

${\tilde{0}}_{\delta }={\tilde{0}}^{\delta }=\tilde{0}$ and ${\tilde{1}}_{\delta }={\tilde{1}}^{\delta }=\tilde{1}$,

(2) 

${({\alpha }_n\vee {\epsilon }_n)}_{\delta }\ge {({\alpha }_n)}_{\delta }\vee {({\epsilon }_n)}_{\delta }$,

(3) 

${({\alpha }_n\wedge {\epsilon }_n)}^{\delta }\le {({\alpha }_n)}^{\delta }\wedge {({\epsilon }_n)}^{\delta }$,

(4) 

${\alpha }_n\le {\epsilon }_n$, implies that ${({\alpha }_n)}^{\delta }\le {({\epsilon }_n)}^{\delta }$ and ${({\alpha }_n)}_{\delta }\le {({\epsilon }_n)}_{\delta }$,

(5) 

${({\alpha }_n\vee {\epsilon }_n)}^{\delta }={({\alpha }_n)}^{\delta }\vee {({\epsilon }_n)}^{\delta }$,

(6) 

${({\alpha }_n\wedge {\epsilon }_n)}_{\delta }={({\alpha }_n)}_{\delta }\wedge {({\epsilon }_n)}_{\delta }$,

(7) 

${\left[{({\alpha }_n)}^{\delta }\right]}^c={({\left[{\alpha }_n\right]}^c)}_{\delta }$ and ${\left[{({\alpha }_n)}_{\delta }\right]}^c={({\left[{\alpha }_n\right]}^c)}^{\delta }$,

(8) 

${\left[{({\alpha }_n)}_{\delta }\right]}^{\delta }\ge {({\alpha }_n)}_{\delta }={\left[{({\alpha }_n)}_{\delta }\right]}_{\delta }$,

(9) 

${\left[{({\alpha }_n)}^{\delta }\right]}_{\delta }\le {({\alpha }_n)}^{\delta }={\left[{({\alpha }_n)}^{\delta }\right]}^{\delta }$.

Proof. 

Obvious. □

Example 3.

Let δ be an SVNR on a set $\tilde{\mathcal{F}}=\{{\omega }_1,{\omega }_2,{\omega }_3\}$ as shown below.

${\mathbf{\alpha }}_{\mathbf{n}}$	${\mathbf{\omega }}_{\mathbf{1}}$	${\mathbf{\omega }}_{\mathbf{2}}$	${\mathbf{\omega }}_{\mathbf{3}}$
${\omega }_1$	$\langle 0, 0, 1\rangle$	$\langle 0.3, 0.1, 0.6\rangle$	$\langle 0.1, 0, 0.4\rangle$
${\omega }_2$	$\langle 0, 0.2, 0.4\rangle$	$\langle 0.6, 0.5, 1\rangle$	$\langle 0.6, 0, 1\rangle$
${\omega }_3$	$\langle 1, 0, 1\rangle$	$\langle 1, 0.5, 1\rangle$	$\langle 1, 1, 0\rangle$

Assume that ${\alpha }_n=\langle (0.3, 0.4, 1),(0.5, 0.2, 0.2),(0.3, 0.5, 0.7)\rangle$. Then, the single-valued neutrosophic cosets are as follows:

$$\begin{array}{ccc}& & {\tilde{\varrho }}_{{({\alpha }_n)}_{\delta }}({\omega }_1)={\tilde{\varrho }}_{{\alpha }_n}({\omega }_1)\vee \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne {\omega }_1}{\bigwedge }{\tilde{\varsigma }}_{\left[\omega \right]}(\mu )=0.4\hfill \\ & & {\tilde{\sigma }}_{{({\alpha }_n)}_{\delta }}({\omega }_1)={\tilde{\sigma }}_{{\alpha }_n}({\omega }_1)\wedge \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne {\omega }_1}{\bigvee }(1-{\tilde{\sigma }}_{\left[{\omega }_1\right]})(\mu )=0.4\hfill \\ & & {\tilde{\varsigma }}_{{({\alpha }_n)}_{\delta }}({\omega }_1)={\tilde{\varsigma }}_{{\alpha }_n}({\omega }_1)\wedge \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne {\omega }_1}{\bigvee }{\tilde{\varrho }}_{\left[{\omega }_1\right]}(\mu )=0\hfill \end{array}$$

Hence, ${({\alpha }_n)}_{\delta }({\omega }_1)=(0.4, 0.4, 0)$ and

$$\begin{array}{ccc}& & {\tilde{\varrho }}_{{({\alpha }_n)}_{\delta }}({\omega }_2)={\tilde{\varrho }}_{{\alpha }_n}({\omega }_2)\vee \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne {\omega }_2}{\bigwedge }{\tilde{\varsigma }}_{\left[\omega \right]}(\mu )=0.6\hfill \\ & & {\tilde{\sigma }}_{{({\alpha }_n)}_{\delta }}({\omega }_2)={\tilde{\sigma }}_{{\alpha }_n}({\omega }_2)\wedge \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne {\omega }_2}{\bigvee }(1-{\tilde{\sigma }}_{\left[{\omega }_2\right]})(\mu )=0.2\hfill \\ & & {\tilde{\varsigma }}_{{({\alpha }_n)}_{\delta }}({\omega }_2)={\tilde{\varsigma }}_{{\alpha }_n}({\omega }_2)\wedge \underset{{({\alpha }_n)}^c(\mu )>0,\mu \ne {\omega }_2}{\bigvee }{\tilde{\varrho }}_{\left[{\omega }_2\right]}(\mu )=0.2\hfill \end{array}$$

Hence, ${({\alpha }_n)}_{\delta }({\omega }_2)=(0.6, 0.2, 0.2)$. Similarly, we can obtain ${({\alpha }_n)}_{\delta }({\omega }_2)=(0.4, 0.5, 0.7)$; therefore, ${({\alpha }_n)}_{\delta }=\langle (0.4, 0.4, 0),(0.6, 0.2, 0.2),(0.4, 0.5, 0.7)\rangle$, ${({\alpha }_n)}^{\delta }=\langle (0.3, 0.4, 0.4),(0.5, 0.5,$$0.6),(0.3, 0.5, 0.7)\rangle$ and then ${({\alpha }_n)}^B=\langle (0, 0.6, 0.4),(0.2, 0.8, 0.6),(0.3, 0.5, 0.7)\rangle$.

Definition 23.

The single-valued neutrosophic approximation interior operator ${\mathrm{int}}_{\delta }^{{\alpha }_n}:{\zeta }^{\tilde{\mathcal{F}}}\to {\zeta }^{\tilde{\mathcal{F}}}$ is defined as follows:

$$\begin{array}{ccc}& & {\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)={({\alpha }_n)}_{\delta }\wedge {({\epsilon }_n)}_{\delta },\forall {\epsilon }_n\ne \tilde{1}\mathit{and}{\mathrm{int}}_{\delta }^{{\alpha }_n}(\tilde{1})=\tilde{1}.\hfill \end{array}$$

That is associated with an SVNS${\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$ in a single-valued neutrosophic approximation space $(\tilde{\mathcal{F}},\delta )$.

Theorem 12.

The following conditions are satisfied

(1) 

${\mathrm{int}}_{\delta }^{{\alpha }_n}(\tilde{0})=\tilde{0}$,

(2) 

${\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\le {\epsilon }_n$$\forall {\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(3) 

${\epsilon }_n\le {\pi }_n$ ⇒ ${\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\le {\mathrm{int}}_{\delta }^{{\alpha }_n}({\pi }_n)$, $\forall {\epsilon }_n,{\pi }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(4) 

${\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n\wedge {\pi }_n)={\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\wedge {\mathrm{int}}_{\delta }^{{\alpha }_n}({\pi }_n)$$\forall {\epsilon }_n,{\pi }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(5) 

${\mathrm{int}}_{\delta }^{{\alpha }_n}({\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n))={\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)$$\forall {\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$.

Proof. 

For (1): ${\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)={({\alpha }_n)}_{\delta }\wedge {(\tilde{0})}_{\delta }=\tilde{0}$.

For (2): ${\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)={({\alpha }_n)}_{\delta }\wedge {({\epsilon }_n)}_{\delta }\le {({\alpha }_n)}_{\delta }\wedge {\epsilon }_n\le {\epsilon }_n$.

For (3): ${\epsilon }_n\le {\pi }_n$ then ${({\epsilon }_n)}_{\delta }\le {({\pi }_n)}_{\delta }$⇒${\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\le {\mathrm{int}}_{\delta }^{{\alpha }_n}({\pi }_n)$.

For (4): ${\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n\wedge {\pi }_n)={({\alpha }_n)}_{\delta }\wedge {({\epsilon }_n\wedge {\pi }_n)}_{\delta }={({\alpha }_n)}_{\delta }\wedge {({\epsilon }_n)}_{\delta }\wedge {({\alpha }_n)}_{\delta }\wedge {({\pi }_n)}_{\delta }={\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\wedge {\mathrm{int}}_{\delta }^{{\alpha }_n}({\pi }_n)$.

For (5): Similarly to (4). □

Thus, this is called a single-valued neutrosophic interior associated with ${\alpha }_n$ in the single-valued neutrosophic approximation space $(\tilde{\mathcal{F}},\delta )$ generating a single-valued neutrosophic topology defined by:

$${\varpi }_{\delta }^{{\alpha }_n}=\{{\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}:{\epsilon }_n={\mathrm{int}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\}.$$

(8)

Definition 24.

The single-valued neutrosophic approximation closure operator ${\mathrm{Cl}}_{\delta }^{{\alpha }_n}:{\zeta }^{\tilde{\mathcal{F}}}\to {\zeta }^{\tilde{\mathcal{F}}}$ is defined as follows:

$$\begin{array}{ccc}& & {\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n)={\left[{({\alpha }_n)}_{\delta }\right]}^c\vee {({\epsilon }_n)}^{\delta },\forall {\epsilon }_n\ne \tilde{0} \mathit{and} {\mathrm{Cl}}_{\delta }^{{\alpha }_n}(\tilde{0})=\tilde{0}.\hfill \end{array}$$

Theorem 13.

The single-valued neutrosophic approximation closure operator satisfies the following conditions:

(1) 

${\mathrm{Cl}}_{\delta }^{{\alpha }_n}(\tilde{1})=\tilde{1}$$\forall {\alpha }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(2) 

${\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\ge {\epsilon }_n$$\forall {\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(3) 

${\epsilon }_n\le {\pi }_n$⇒${\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\le {\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\pi }_n)$, $\forall {\epsilon }_n,{\pi }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(4) 

${\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n\wedge {\pi }_n)={\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\wedge {\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\pi }_n)$ and ${\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n\vee {\pi }_n)={\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n)\vee {\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\pi }_n)$$\forall {\epsilon }_n,{\pi }_n\in {\zeta }^{\tilde{\mathcal{F}}}$,

(5) 

${\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n))={\mathrm{Cl}}_{\delta }^{{\alpha }_n}({\epsilon }_n)$$\forall {\epsilon }_n\in {\zeta }^{\tilde{\mathcal{F}}}$.

Proof. 

Similar to the proof of Theorem 12. □

6. Conclusions

In this paper, we defined the single-valued neutrosophic semi-closure space (SVNSCS). It has been proven that every ${\tau }_{\mathbb{SC}}^{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ is a single-valued neutrosophic ideal topological space on $\mathcal{F}$. It has also been proven that every ${\mathbb{SC}}_{\mathbb{SC}}^{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ is finer than $\mathbb{SC}$ (see Theorem 2). In addition, single-valued neutrosophic operators, namely $\beta \pounds {\mathrm{C}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$ and $\beta \pounds {\mathrm{int}}_{{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}}$, are constructed from a single-valued neutrosophic topological space $(\tilde{\mathcal{F}},{\tau }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }})$) (see Theorem 5). Next, the concepts of a single-valued neutrosophic almost $\beta \pounds$-continuous, single-valued neutrosophic faintly $\beta \pounds$-continuous and single-valued neutrosophic weakly $\beta \pounds$-continuous based on a single-valued neutrosophic ideal ${\pounds }^{\tilde{\varrho }\tilde{\sigma }\tilde{\varsigma }}$ were introduced and studied (see Theorems 6–8). Finally, we introduced the single-valued neutrosophic sets ${({\alpha }_n)}_{\delta }$, ${({\alpha }_n)}^{\delta }$, ${({\alpha }_n)}^B$ for a single-valued neutrosophic set ${\alpha }_n$ that explains the single-valued neutrosophic roughness of the single-valued neutrosophic set ${\alpha }_n$. We introduced the notion of single-valued neutrosophic approximation space and the related single-valued neutrosophic topology.

7. Discussion for Further Works

The theories that were used in this article could be extended to study some similar notions in the neutrosophic metric topological spaces.