£-Single Valued Extremally Disconnected Ideal Neutrosophic Topological Spaces

Abstract

This paper aims to mark out new concepts of r-single valued neutrosophic sets, called r-single valued neutrosophic £-closed and £-open sets. The definition of £-single valued neutrosophic irresolute mapping is provided and its characteristic properties are discussed. Moreover, the concepts of £-single valued neutrosophic extremally disconnected and £-single valued neutrosophic normal spaces are established. As a result, a useful implication diagram between the r-single valued neutrosophic ideal open sets is obtained. Finally, some kinds of separation axioms, namely r-single valued neutrosophic ideal-$R_i$ (r-$SVNI{R}_i$, for short), where $i=\{0,1,2,3\}$, and r-single valued neutrosophic ideal-$T_j$ (r-$SVNI{T}_j$, for short), where $j=\{1,2,2\frac{1}{2},3,4\}$, are introduced. Some of their characterizations, fundamental properties, and the relations between these notions have been studied.

1. Introduction

In 1999, Smarandache introduced the concept of a neutrosophy [1]. It has been used at various axes of mathematical theories and applications. In recent decades, the theory made an outstanding advancement in the field of topological spaces. Salama et al. and Hur et al. [2,3,4,5,6], for example, among many others, wrote their works in fuzzy neutrosophic topological spaces (FNTS), following Chang [7]’s discoveries in the way of fuzzy topological spaces (FTS).

Šostak, in 1985 [8], marked out a new definition of fuzzy topology as a crisp subfamily of family of fuzzy sets, which seems to be a drawback in the process of fuzzification of the concept of topological spaces. Yan, Wang, Nanjing, Liang, and Yan [9,10] developed a parallel theory in the context of intuitionistic I-fuzzy topological spaces.

The idea of “single-valued neutrosophic set” [11] was set out by Wang in 2010. Gayyar [12], in his 2016 paper, foregrounded the concept of a “smooth neutrosophic topological spaces”. The ordinary single-valued neutrosophic topology was presented by Kim [13]. Recently, Saber et al. [14,15] familiarized the concepts of single-valued neutrosophic ideal open local function, single-valued neutrosophic topological space, and the connectedness and stratification of single-valued neutrosophic topological spaces.

Neutrosophy, and especially neutrosophic sets, are powerful, general, and formal frameworks that generalize the concept of the ordinary sets, fuzzy sets, and intiuitionistic fuzzy sets from philosophical point of view. This paper sets out to introduce and examine a new class of sets called r-single valued £-closed in the single valued neutrosophic topological spaces in Šostak’s sense. More precisely, different attributes, like £-single valued neutrosophic irresolute mapping, £-single valued neutrosophic extremally disconnected, £-single valued neutrosophic normal spaces, and some kinds of separation axioms, were developed. It can be fairly claimed that we have achieved expressive definitions, distinguished theorems, important lemmas, and counterexamples to investigate, in-depth, our consequences and to find out the best results. It is notable to say that different crucial notions in single valued neutrosophic topology were generalized in this article. Different attributes, like extremally disconnected and some kinds of separation axioms, which have a significant impact on the overall topology’s notions, were also studied.

It is notable to say that the application aspects to this area of research can be further pointed to. There are many applications of neutrosophic theories in many branches of sciences. Possible applications are to control engineering and to Geographical Information Systems, and so forth, and could be secured, as mentioned by many authors, such as Reference [16,17,18,19,20].

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

2. Preliminaries

This section is devoted to provide a complete survey and trace previous studies related to the idea of this research article.

Definition 1 

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

$$\begin{array}{c}\hfill {\sigma }_n=\{\langle \nu ,{\tilde{\rho }}_{{\sigma }_n}\left(\nu \right),{\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right),{\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)\rangle :\nu \in \tilde{X}\},\end{array}$$

where

$$\tilde{\rho }:\tilde{X}\to {\rfloor }^{-}0,1^{+}{\lfloor ,\tilde{\varrho }:\tilde{X}\to \rfloor }^{-}0,1^{+}{\lfloor ,\tilde{\eta }:\tilde{X}\to \rfloor }^{-}0,1^{+}\lfloor$$

and

$${}^{-}0\le {\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)\le 3^{+}$$

represent the degree of membership (namely ${\tilde{\rho }}_{{\sigma }_n}\left(\nu \right))$, the degree of indeterminacy (namely ${\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right))$, and the degree of non-membership (namely ${\tilde{\eta }}_{{\sigma }_n}\left(\nu \right))$, respectively, of any $\nu \in \tilde{X}$ to the set ${\sigma }_n$.

Definition 2 

([11]). Let $\tilde{X}$ be a space of points (objects), with a generic element in $\tilde{X}$ denoted by ν. Then, ${\sigma }_n$ is called a single valued neutrosophic set (briefly, $SVNS$) in $\tilde{X}$, if ${\sigma }_n$ has the form ${\sigma }_n=\langle {\tilde{\rho }}_{{\sigma }_n},{\tilde{\varrho }}_{{\sigma }_n},{\tilde{\eta }}_{{\sigma }_n}\rangle$, where ${\tilde{\rho }}_{{\sigma }_n},{\tilde{\varrho }}_{{\sigma }_n},{\tilde{\eta }}_{{\sigma }_n}:\tilde{X}\to [0,1]$. In this case, ${\tilde{\rho }}_{{\sigma }_n},{\tilde{\varrho }}_{{\sigma }_n},{\tilde{\eta }}_{{\sigma }_n}$ are called truth membership function, indeterminancy membership function, and falsity membership function, respectively.

Let $\tilde{X}$ be a nonempty set and $\xi =[0,1]$ and ${\xi }_0=(0,1]$. A single-valued neutrosophic set ${\sigma }_n$ on $\tilde{X}$ is a mapping defined as ${\sigma }_n=\langle {\tilde{\rho }}_{{\sigma }_n},{\tilde{\varrho }}_{{\sigma }_n},{\tilde{\eta }}_{{\sigma }_n}\rangle :\tilde{X}\to \xi$ such that $0\le {\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)\le 3$.

We denote the single-valued neutrosophic sets $\langle 0,1,1\rangle$ and $\langle 1,0,0\rangle$ by $\tilde{0}$ and $\tilde{1}$, respectively.

Definition 3 

([11]). Let ${\sigma }_n=\langle {\tilde{\rho }}_{{\sigma }_n},{\tilde{\varrho }}_{{\sigma }_n},{\tilde{\eta }}_{{\sigma }_n}\rangle$ be an $SVNS$ on $\tilde{X}$. The complement of the set ${\sigma }_n$ (briefly ${\sigma }_n^c$) is defined as follows:

$$\begin{array}{c}\hfill {\tilde{\rho }}_{{\sigma }_n^c}\left(\nu \right)={\tilde{\eta }}_{{\sigma }_n}\left(\nu \right),{\tilde{\varrho }}_{{\sigma }_n^c}\left(\nu \right)={\left[{\tilde{\varrho }}_{{\sigma }_n}\right]}^c\left(\nu \right),{\tilde{\eta }}_{{\sigma }_n^c}\left(\nu \right)={\tilde{\rho }}_{{\sigma }_n}\left(\nu \right).\end{array}$$

Definition 4 

([22,23]). Let $\tilde{X}$ be a non-empty set and let ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ be given by ${\sigma }_n=\langle {\tilde{\rho }}_{{\sigma }_n},{\tilde{\varrho }}_{{\sigma }_n},{\tilde{\eta }}_{{\sigma }_n}\rangle$ and ${\gamma }_n=\langle {\tilde{\rho }}_{{\gamma }_n},{\tilde{\varrho }}_{{\gamma }_n},{\tilde{\eta }}_{{\gamma }_n}\rangle$. Then:

(1) 

We say that ${\sigma }_n\subseteq {\gamma }_n$ if ${\tilde{\rho }}_{{\sigma }_n}\le {\tilde{\rho }}_{{\gamma }_n},{\tilde{\varrho }}_{{\sigma }_n}\ge {\tilde{\varrho }}_{{\gamma }_n}$, ${\tilde{\eta }}_{{\sigma }_n}\ge {\tilde{\eta }}_{{\gamma }_n}$.

(2) 

The intersection of ${\sigma }_n$ and ${\gamma }_n$ denoted by ${\sigma }_n\cap {\gamma }_n$ is an SVNS and is given by

$${\sigma }_n\cap {\gamma }_n=\langle {\tilde{\rho }}_{{\sigma }_n}\cap {\tilde{\rho }}_{{\gamma }_n},{\tilde{\varrho }}_{{\sigma }_n}\cup {\tilde{\varrho }}_{{\gamma }_n},{\tilde{\eta }}_{{\sigma }_n}\cup {\tilde{\eta }}_{{\gamma }_n}\rangle .$$

(3) 

The union of ${\sigma }_n$ and ${\gamma }_n$ denoted by ${\sigma }_n\cup {\gamma }_n$ is an SVNS and is given by

$${\sigma }_n\cup {\gamma }_n=\langle {\tilde{\rho }}_{{\sigma }_n}\cup {\tilde{\rho }}_{{\gamma }_n},{\tilde{\varrho }}_{{\sigma }_n}\cap {\tilde{\varrho }}_{{\gamma }_n},{\tilde{\eta }}_{{\sigma }_n}\cap {\tilde{\eta }}_{{\gamma }_n}\rangle .$$

For any arbitrary family ${\left\{{\sigma }_n\right\}}_{i\in j}\subseteq {\xi }^{\tilde{X}}$ of SVNS, the union and intersection are given by

(4) 

${\bigcap }_{i\in j}{\left[{\sigma }_n\right]}_i=\langle {\cap }_{i\in j}{\tilde{\rho }}_{{\left[{\sigma }_n\right]}_i},{\cup }_{i\in j}{\tilde{\varrho }}_{{\left[{\sigma }_n\right]}_i},{\cup }_{i\in j}{\tilde{\eta }}_{{\left[{\sigma }_n\right]}_i}\rangle$,

(5) 

${\bigcup }_{i\in j}{\left[{\sigma }_n\right]}_i=\langle {\cup }_{i\in j}{\tilde{\rho }}_{{\left[{\sigma }_n\right]}_i},{\cap }_{i\in j}{\tilde{\varrho }}_{{\left[{\sigma }_n\right]}_i},{\cap }_{i\in j}{\tilde{\eta }}_{{\left[{\sigma }_n\right]}_i}\rangle$.

Definition 5 

([12]). A single-valued neutrosophic topological space is an ordered quadruple $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }},{\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\eta }})$ where ${\tilde{\tau }}^{\tilde{\rho }},{\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\eta }}:{\xi }^{\tilde{X}}\to \xi$ are mappings satisfying the following axioms:

(SVNT1) 

${\tilde{\tau }}^{\tilde{\rho }}\left(\tilde{0}\right)={\tilde{\tau }}^{\tilde{\rho }}\left(\tilde{1}\right)=1$ and ${\tilde{\tau }}^{\tilde{\rho }}\left(\tilde{0}\right)={\tilde{\tau }}^{\tilde{\rho }}\left(\tilde{1}\right)={\tilde{\tau }}^{\tilde{\eta }}\left(\tilde{0}\right)={\tilde{\tau }}^{\tilde{\eta }}\left(\tilde{1}\right)=0$,

(SVNT2) 

${\tilde{\tau }}^{\tilde{\rho }}({\sigma }_n\cap {\gamma }_n)\ge {\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\cap {\tilde{\tau }}^{\tilde{\rho }}\left({\gamma }_n\right)$, ${\tilde{\tau }}^{\tilde{\varrho }}({\sigma }_n\cap {\gamma }_n)\le {\tau }^{\tilde{\varrho }}\left({\sigma }_n\right)\cup {\tilde{\tau }}^{\tilde{\varrho }}\left({\gamma }_n\right)$,

${\tilde{\tau }}^{\tilde{\eta }}({\sigma }_n\cap {\gamma }_n)\le {\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\cup {\tilde{\tau }}^{\tilde{\eta }}\left({\gamma }_n\right)$, for all${\sigma }_n,{\gamma }_n\in {\zeta }^{\tilde{X}}$,

(SVNT3) 

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

${\tilde{\tau }}^{\tilde{\eta }}\left({\cup }_{j\in \Gamma }{\left[{\sigma }_n\right]}_j\right)\le {\cup }_{j\in \Gamma }{\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\sigma }_n\right]}_j\right)$ for all $\{{\left[{\sigma }_n\right]}_j,j\in \Gamma \}\in {\zeta }^{\tilde{X}}$.

The quadruple $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }},{\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\eta }})$ is called a single-valued neutrosophic topological space $(SVNTS$, for short). We will occasionally write ${\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}$ for $({\tau }^{\tilde{\rho }},{\tau }^{\tilde{\varrho }},{\tau }^{\tilde{\eta }})$ and it will cause no ambiguity

Definition 6 

([14]). Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }},{\tilde{\tau }}^{\tilde{\varrho }},{\tilde{\tau }}^{\tilde{\eta }})$ be an SVNTS. Then, for every ${\sigma }_n\in {\xi }^{\tilde{X}}$ and $r\in {\xi }_0$, the single valued neutrosophic closure and the single valued neutrosophic interior of ${\sigma }_n$ are defined by:

$$\begin{array}{c}\hfill {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,s)=\bigcap \{{\gamma }_n\in {\xi }^{\tilde{X}}:{\sigma }_n\le {\gamma }_n,{\tau }^{\tilde{\rho }}\left({\left[{\gamma }_n\right]}^c\right)\ge r,{\tau }^{\tilde{\varrho }}\left({\left[{\gamma }_n\right]}^c\right)\le 1-r,{\tau }^{\tilde{\eta }}\left({\left[{\gamma }_n\right]}^c\right)\le 1-r\},\end{array}$$

$$\begin{array}{c}\hfill {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,s)=\bigcup \{{\gamma }_n\in {\xi }^{\tilde{X}}:{\sigma }_n\ge {\gamma }_n,{\tau }^{\tilde{\rho }}\left({\gamma }_n\right)\ge r,{\tau }^{\tilde{\varrho }}\left({\gamma }_n\right)\le 1-r,{\tau }^{\tilde{\eta }}\left({\gamma }_n\right)\le 1-r\}.\end{array}$$

Definition 7 

([24]). Let $(\tilde{X},{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an $SVNTS$ and $r\in {\xi }_0,{\sigma }_n\in {\xi }^{\tilde{X}}$. Then,

(1) 

${\sigma }_n$ is r-single valued neutrosophic semiopen (r-SVNSO, for short) iff ${\sigma }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}$$({\sigma }_n,r),r)$,

(2) 

${\sigma }_n$ is r-single valued neutrosophic β-open (r-SVNβO, for short) iff ${\sigma }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}\right({\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}$$({\sigma }_n,r),r),r)$.

The complement of $r-SVNSO$ (resp. r-SVNβO) is said to be an $r-SVNSC$ (resp. r-SVNβC), respectively.

Definition 8 

([14]). Let $\tilde{X}$ be a nonempty set and $\nu \in \tilde{X}$. If $s\in (0,1]$, $t\in [0,1)$ and $p\in [0,1)$. Then, the single-valued neutrosophic point $x_{s,t,p}$ in $\tilde{X}$ is given by

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

We say $x_{s,t,p}\in {\sigma }_n$ iff $s<{\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)$, $t\ge {\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)$ and $p\ge {\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)$. To avoid the ambiguity, we denote the set of all neutrosophic points by $pt\left({\xi }^{\tilde{X}}\right)$.

A single-valued neutrosophic set ${\sigma }_n$ is said to be quasi-coincident with another single-valued neutrosophic set ${\gamma }_n$, denoted by ${\sigma }_{n}q{\gamma }_n$, if there exists an element $\nu \in \tilde{X}$ such that

$$\begin{array}{c}\hfill {\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\rho }}_{{\gamma }_n}\left(\nu \right)>1,{\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\varrho }}_{{\gamma }_n}\left(\nu \right)\le 1,{\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\eta }}_{{\gamma }_n}\left(\nu \right)\le 1.\end{array}$$

Definition 9 

([14]). A mapping ${\mathcal{I}}^{\widetilde{\rho \varrho \eta }}={\mathcal{I}}^{\tilde{\rho }},{\mathcal{I}}^{\tilde{\varrho }},{\mathcal{I}}^{\tilde{\eta }}:{\xi }^{\tilde{X}}\to \xi$ is called single-valued neutrosophic ideal (SVNI) on $\tilde{X}$ if it satisfies the following conditions:

($I_1$)

${\mathcal{I}}^{\tilde{\rho }}\left(\tilde{0}\right)=1$ and ${\mathcal{I}}^{\tilde{\varrho }}\left(\tilde{0}\right)={\mathcal{I}}^{\tilde{\eta }}\left(\tilde{0}\right)=0$.

($I_2$)

If ${\sigma }_n\le {\gamma }_n,$, then ${\mathcal{I}}^{\tilde{\rho }}\left({\gamma }_n\right)\le {\mathcal{I}}^{\tilde{\rho }}\left({\sigma }_n\right)$, ${\mathcal{I}}^{\tilde{\varrho }}\left({\gamma }_n\right)\ge {\mathcal{I}}^{\tilde{\varrho }}\left({\sigma }_n\right)$, and ${\mathcal{I}}^{\tilde{\eta }}\left({\gamma }_n\right)\ge {\mathcal{I}}^{\tilde{\eta }}\left({\sigma }_n\right)$, for ${\gamma }_n,{\sigma }_n\in {\xi }^{\tilde{X}}$.

($I_3$)

${\mathcal{I}}^{\tilde{\rho }}({\sigma }_n\cup {\gamma }_n)\ge {\mathcal{I}}^{\tilde{\rho }}\left({\sigma }_n\right)\cap {\mathcal{I}}^{\tilde{\rho }}\left({\gamma }_n\right)$, ${\mathcal{I}}^{\tilde{\varrho }}({\sigma }_n\cup {\gamma }_n)\le {\mathcal{I}}^{\tilde{\varrho }}\left({\sigma }_n\right)\cup {\mathcal{I}}^{\tilde{\varrho }}\left({\gamma }_n\right)$ and

${\mathcal{I}}^{\tilde{\eta }}({\sigma }_n\cup {\gamma }_n)\le {\mathcal{I}}^{\tilde{\eta }}\left({\sigma }_n\right)\cup {\mathcal{I}}^{\tilde{\eta }}\left({\gamma }_n\right)$, for each ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$.

The triple $(\tilde{X},{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is called a single valued neutrosophic ideal topological space in Šostak’s sense (SVNITS, for short).

Definition 10 

([14]). Let $(\tilde{X},{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS for each ${\sigma }_n\in {\xi }^{\tilde{X}}$. Then, the single valued neutrosophic ideal open local function ${\left[{\sigma }_n\right]}_r^{\pounds }({\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ of ${\sigma }_n$ is the union of all single-valued neutrosophic points $x_{s,t,k}$ such that, if ${\gamma }_n\in Q_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}(x_{s,t,k},r)$ and ${\mathcal{I}}^{\tilde{\rho }}\left({\varsigma }_n\right)\ge r$, ${\mathcal{I}}^{\tilde{\varrho }}\left({\varsigma }_n\right)\le 1-r$,${\mathcal{I}}^{\tilde{\eta }}\left({\varsigma }_n\right)\le 1-r$, then there is at least one $\nu \in \tilde{X}$ for which ${\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\rho }}_{{\gamma }_n}\left(\nu \right)-1>{\tilde{\rho }}_{{\varsigma }_n}\left(\nu \right)$,${\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\varrho }}_{{\gamma }_n}\left(\nu \right)-1\le {\tilde{\varrho }}_{{\varsigma }_n}\left(\nu \right)$, and ${\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\eta }}_{{\gamma }_n}\left(\nu \right)-1\le {\tilde{\eta }}_{{\varsigma }_n}\left(\nu \right)$.

Occasionally, we will write ${\left[{\sigma }_n\right]}_r^{\pounds }$ for ${\left[{\sigma }_n\right]}_r^{\pounds }({\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$, and it will cause no ambiguity.

Remark 1 

([14]). Let $(\tilde{X},{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and ${\sigma }_n\in {\xi }^{\tilde{X}}$. Then,

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

It is clear that ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$ is a single-valued neutrosophic closure operator and $({\tau }^{\tilde{\rho }\pounds }({\mathcal{I}}^{\rho },{\tau }^{\tilde{\varrho }\pounds }({\mathcal{I}}^{\varrho },{\tau }^{\tilde{\eta }\pounds }$ $\left({\mathcal{I}}^{\eta }\right)$ is the single-valued neutrosophic topology generated by ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$, i.e.,

$$\begin{array}{c}\hfill {\tau }^{\pounds }\left(\mathcal{I}\right)\left({\sigma }_n\right)=\bigcup \left\{r\right|{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n^c,r)={\sigma }^c\}.\end{array}$$

Theorem 1 

([14]). Let ${\left\{{\left[{\sigma }_n\right]}_i\right\}}_{i\in J}\subset {\xi }^{\tilde{X}}$ be a family of single-valued neutrosophic sets on $\tilde{X}$ and $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an r-SVNITS. Then,

(1) 

$(\bigcup {\left({\left[{\sigma }_n\right]}_i\right)}_r^{\pounds }:i\in J)\le {(\bigcup {\left[{\sigma }_n\right]}_i:i\in j)}_r^{\pounds }$,

(2) 

${(\bigcap \left({\left[{\sigma }_n\right]}_i\right):i\in j)}_r^{\pounds }\ge (\bigcap {\left({\left[{\sigma }_n\right]}_i\right)}_r^{\pounds }:i\in J)$.

Theorem 2 

([14]). Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$, $r\in {\xi }_0$. Then,

(1) 

${\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n\vee {\gamma }_n,r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\vee {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$,

(2) 

${\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\sigma }_n\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)$,

(3) 

${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}^c,r)={\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$, and ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}^c,r)$,

(4) 

${\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n\wedge {\gamma }_n,r)={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$.

3. £-Single Valued Neutrosophic Ideal Irresolute Mapping

This section provides the definitions of the r-single-valued neutrosophic £-open set (SVN£O, for short), the r-single-valued neutrosophic £-closed set (SVN£C, for short) and the £-single valued neutrosophic ideal irresolute mapping (£-SVNI-irresolute, for short), in the sense of Šostak. To understand the aim of this section, it is essential to clarify its content and elucidate the context in which the definitions, theorems, and examples are performed. Some results follow.

Definition 11. 

Let$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$be an r-SVNITS for every${\sigma }_n\in {\xi }^{\tilde{X}}$and$r\in {\xi }_0$. Then, ${\sigma }_n$is called r-SVN£C iff${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)={\sigma }_n$. The complement of the r-SVN£C is called r-SVN£O.

Proposition 1. 

Let$(\tilde{X},{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$be an r-SVNITS and${\sigma }_n\in {\xi }^{\tilde{X}}$. Then,

(1) 

${\sigma }_n$ is r-SVN£C iff ${\left[{\sigma }_n\right]}_r^{\pounds }\le {\sigma }_n$,

(2) 

${\sigma }_n$ is r-SVN£O iff ${\left({\left[{\sigma }_n\right]}_r^{\pounds }\right)}^c\ge {\left[{\sigma }_n\right]}^c$,

(3) 

If ${\tau }^{\tilde{\rho }}\left({\left[{\sigma }_n\right]}^c\right)\ge r$, ${\tau }^{\varrho }\left({\left[{\sigma }_n\right]}^c\right)\le 1-r$, ${\tau }^{\eta }\left({\left[{\sigma }_n\right]}^c\right)\le 1-r$, then ${\sigma }_n$ is r-SVN£C,

(4) 

If ${\tau }^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tau }^{\varrho }\left({\sigma }_n\right)\le 1-r$, ${\tau }^{\eta }\left({\sigma }_n\right)\le 1-r$, then ${\sigma }_n$ is r-SVN£O,

(5) 

If ${\sigma }_n$ is r-SVNSC (resp. r-SVNβC), then ${\mathrm{int}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\left[{\sigma }_n\right]}_r^{\pounds },r)\le {\sigma }_n$$(resp.{\mathrm{int}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}(\left[{\mathrm{int}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)\right]$${}_r^{\pounds },r)\le {\sigma }_n)$.

Proof. 

The proof of (1) and (2) are straightforward from Definition 11.

(3) Let ${\tau }^{\tilde{\rho }}\left({\left[{\sigma }_n\right]}^c\right)\ge r$, ${\tau }^{\varrho }\left({\left[{\sigma }_n\right]}^c\right)\le 1-r$, ${\tau }^{\eta }\left({\left[{\sigma }_n\right]}^c\right)\le 1-r$. Then,

$$\begin{array}{c}\hfill {\sigma }_n={\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)\ge {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)={\sigma }_n\cup {\left[{\sigma }_n\right]}_r^{\pounds }\ge {\left[{\sigma }_n\right]}_r^{\pounds }.\end{array}$$

Hence, ${\sigma }_n$ is an r-SVN£C.

(4) The proof is direct consequence of (1).

(5) Let ${\sigma }_n$ be an r-SVNSC. Then,

$$\begin{array}{ccc}& {\sigma }_n\ge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r),r)& \ge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}([{\sigma }_n\cup {\left[{\sigma }_n\right]}_r^{\pounds }],r)\hfill \\ & & \ge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\left[{\sigma }_n\right]}_r^{\pounds },r).\hfill \end{array}$$

The another case is similarly proved. □

Example 1. 

Suppose that $\tilde{X}=\{a,b\}$. Define ${\epsilon }_n,{\gamma }_n,{\varsigma }_n\in {\xi }^{\tilde{X}}$ as follows:

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

$$\begin{array}{c}\hfill {\varsigma }_n=\langle (0.2,0.2),(0.2,0.2),(0.2,0.2)\rangle .\end{array}$$

Define ${\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}:{\xi }^{\tilde{X}}\to \xi$ as follows:

$$\begin{array}{c}\hfill {\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)=\left\{\begin{array}{c}1,\mathit{if}{\sigma }_n=\tilde{0},\hfill \\ 1,\mathit{if}{\sigma }_n=\tilde{1},\hfill \\ \frac{1}{3},\mathit{if}{\sigma }_n={\gamma }_n;\hfill \\ \frac{1}{3},\mathit{if}{\sigma }_n={\epsilon }_n;\hfill \\ 0,\mathit{if}\mathit{otherwise};\hfill \end{array}\right.{\mathcal{I}}^{\tilde{\rho }}\left({\sigma }_n\right)=\left\{\begin{array}{c}1,\mathit{if}{\sigma }_n=(0,1,1),\hfill \\ \frac{1}{3},\mathit{if}{\sigma }_n={\varsigma }_n,\hfill \\ \frac{2}{3},\mathit{if}\tilde{0}<{\sigma }_n<{\varsigma }_n;\hfill \\ 0,\mathit{if}\mathit{otherwise};\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)=\left\{\begin{array}{c}0,\mathit{if}{\sigma }_n=\tilde{0},\hfill \\ 0,\mathit{if}{\sigma }_n=\tilde{1},\hfill \\ \frac{2}{3},\mathit{if}{\sigma }_n={\gamma }_n;\hfill \\ \frac{2}{3},\mathit{if}{\sigma }_n={\epsilon }_n;\hfill \\ 1,\mathit{if}\mathit{otherwise};\hfill \end{array}\right.{\mathcal{I}}^{\tilde{\varrho }}\left({\sigma }_n\right)=\left\{\begin{array}{c}0,\mathit{if}{\sigma }_n=(0,1,1),\hfill \\ \frac{2}{3},\mathit{if}{\sigma }_n={\varsigma }_n,\hfill \\ \frac{1}{3},\mathit{if}\tilde{0}<{\sigma }_n<{\varsigma }_n;\hfill \\ 1,\mathit{if}\mathit{otherwise};\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)=\left\{\begin{array}{c}0,\mathit{if}{\sigma }_n=\tilde{0},\hfill \\ 0,\mathit{if}{\sigma }_n=\tilde{1},\hfill \\ \frac{2}{3},\mathit{if}{\sigma }_n={\gamma }_n;\hfill \\ \frac{2}{3},\mathit{if}{\sigma }_n={\epsilon }_n;\hfill \\ 1,\mathit{if}\mathit{otherwise};\hfill \end{array}\right.{\mathcal{I}}^{\tilde{\eta }}\left({\sigma }_n\right)=\left\{\begin{array}{c}0,\mathit{if}{\sigma }_n=(0,1,1),\hfill \\ \frac{2}{3},\mathit{if}{\sigma }_n={\varsigma }_n,\hfill \\ \frac{1}{3},\mathit{if}\tilde{0}<{\sigma }_n<{\varsigma }_n;\hfill \\ 1,\mathit{if}\mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

(1) 

$G_n=\langle (0.6,0.6),(0.6,0.6),(0.6,0.6)\rangle$ is $\frac{1}{3}$-SVN£C but ${\tilde{\tau }}^{\tilde{\rho }}\left({\left[G_n\right]}^c\right)\ngeqq \frac{1}{3}$, ${\tilde{\tau }}^{\varrho }\left({\left[G_n\right]}^c\right)\nleqq \frac{2}{3},$ and ${\tilde{\tau }}^{\eta }\left({\left[G_n\right]}^c\right)\nleqq \frac{2}{3},$

(2) 

$G_n=\langle (0.6,0.6),(0.6,0.6),(0.6,0.6)\rangle \ge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\left[G_n\right]}_{\frac{1}{3}}^{\pounds },\frac{1}{3})=\tilde{0}$ but $G_n$ is not is $\frac{1}{3}$-SVNSC.

Lemma 1. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS. Then, we have the following.

(1) 

Every intersection of r-SVN£C’s is r-SVN£C.

(2) 

Every union of r-SVN£O’s is r-SVN£O.

Proof.

(1) Let ${\left\{{\left[{\sigma }_n\right]}_i\right\}}_{i\in j}$ be a family of r-SVN£C’s. Then, for every $i\in j$, we obtain ${\left[{\sigma }_n\right]}_i={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}_i,r)$, and, by Theorem 1(2), we have

$$\begin{array}{ccc}& \bigcap _{i\in j}{\left[{\sigma }_n\right]}_i& =\bigcap _{i\in j}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}_i,r),r)=\bigcap _{i\in j}({\left[{\sigma }_n\right]}_i\cup {\left({\left[{\sigma }_n\right]}_i\right)}_r^{\pounds })\ge \bigcap _{i\in j}{\left[{\sigma }_n\right]}_i\cup \bigcap _{i\in j}{\left({\left[{\sigma }_n\right]}_i\right)}_r^{\pounds }\hfill \\ & & \ge \bigcap _{i\in j}{\left[{\sigma }_n\right]}_i\cup {\left(\bigcap _{i\in j}{\left[{\sigma }_n\right]}_i\right)}_r^{\pounds }={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(\bigcap _{i\in j}{\left[{\sigma }_n\right]}_i,r).\hfill \end{array}$$

Therefore, ${\bigcap }_{i\in \Gamma }{\left[{\sigma }_n\right]}_i$ is r-SVN£C.

(2) From Theorem 1(1). □

Lemma 2. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS for each $r\in {\xi }_0$. Then,

(1) 

For each r-SVN£O ${\sigma }_n\in {\xi }^{\tilde{X}}$, ${\sigma }_{n}q{\gamma }_n$ iff ${\sigma }_{n}q{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$,

(2) 

$x_{s,t,k}q{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$ iff ${\sigma }_{n}q{\gamma }_n$ for every r-SVN£O ${\sigma }_n\in {\xi }^{\tilde{X}}$ with $x_{s,t,k}\in {\sigma }_n$.

Proof.

(1) Let ${\sigma }_n$ be an r-SVN£O and ${\sigma }_n\overline{q}{\gamma }_n$. Then, for any $\nu \in \tilde{X}$, we obtain

$$\begin{array}{c}\hfill {\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\rho }}_{{\gamma }_n}\left(\nu \right)>1,{\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\varrho }}_{{\gamma }_n}\left(\nu \right)\le 1,{\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)+{\tilde{\eta }}_{{\gamma }_n}\left(\nu \right)\le 1.\end{array}$$

This implies that ${\tilde{\rho }}_{{\gamma }_n}\le {\tilde{\rho }}_{{\left[{\sigma }_n\right]}^c},{\tilde{\varrho }}_{{\gamma }_n}\ge {\tilde{\varrho }}_{{\left[{\sigma }_n\right]}^c}$ and ${\tilde{\eta }}_{{\gamma }_n}\ge {\tilde{\eta }}_{{\left[{\sigma }_n\right]}^c}$; hence, ${\gamma }_n\le {\left[{\sigma }_n\right]}^c$. Since ${\sigma }_n$ is r-SVN£O, ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}^c,r)={\left[{\sigma }_n\right]}^c$, it follows that ${\sigma }_n\overline{q}{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$.

(2) Let $x_{s,t,k}q{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$. Then, ${\sigma }_{n}q{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$ with $x_{s,t,k}\in {\sigma }_n$. By (1), we have ${\gamma }_{n}q{\sigma }_n$ for each r-SVN£O ${\sigma }_n\in {\xi }^{\tilde{X}}$. On the other hand, let ${\sigma }_n\overline{q}{\gamma }_n$. Then, ${\gamma }_n\le {\left[{\sigma }_n\right]}^c$. Since ${\sigma }_n$ is r-SVN£O,

$$\begin{array}{c}\hfill {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}^c,r)={\left[{\sigma }_n\right]}^{c}and{\sigma }_n\overline{q}{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r).\end{array}$$

Since $x_{s,t,k}\in {\sigma }_n$, we obtain $x_{s,t,k}\overline{q}{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$ □

Definition 12. 

Suppose that $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is a mapping. Then,

(1) 

f is called £-SVNI-irresolute iff $f^{-1}\left({\sigma }_n\right)$ is r-SVN£O in $\tilde{X}$ for any r-SVN£O ${\sigma }_n$ in $\tilde{Y}$,

(2) 

f is called £-SVNI-irresolute open iff $f\left({\sigma }_n\right)$ is r-SVN£O in $\tilde{Y}$ for any r-SVN£O ${\sigma }_n$ in $\tilde{X}$,

(3) 

f is called £-SVNI-irresolute closed iff $f\left({\sigma }_n\right)$ is r-SVN£C in $\tilde{Y}$ for any r-SVN£C ${\sigma }_n$ in $\tilde{X}$.

Theorem 3. 

Let $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be a mapping. Then, the following conditions are equivalent:

(1) 

f is £-SVNI-irresolute,

(2) 

$f^{-1}\left({\sigma }_n\right)$ is r-SVN£C, for each r-SVN£C ${\sigma }_n\in \tilde{Y}$,

(3) 

$f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le {\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)$ for each ${\sigma }_n\in {\xi }^{\tilde{X}},r\in {\xi }_0$,

(4) 

${\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\gamma }_n\right),r)\le f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right)$ for each ${\gamma }_n\in {\xi }^{\tilde{Y}},r\in {\xi }_0$.

Proof.

(1)⇒(2): Let ${\sigma }_n$ be an r-SVN£C in $\tilde{Y}$. Then, ${\left[{\sigma }_n\right]}^c$ is r-SVN£O in $\tilde{Y}$ by (1), we obtain $f^{-1}\left({\left[{\sigma }_n\right]}^c\right)$ is r-SVN£O. But, $f^{-1}\left({\left[{\sigma }_n\right]}^c\right)={\left[f^{-1}\left({\sigma }_n\right)\right]}^c$. Then, $f^{-1}\left({\sigma }_n\right)$ is r-SVN£C in $\tilde{X}$.

(2)⇒(3): For each ${\sigma }_n\in {\xi }^{\tilde{X}}$ and $r\in {\xi }_0$, since ${\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)={\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)$. From Definition 11, ${\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)$ is r-SVN£C in $\tilde{Y}$. By (2), $f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)\right)$ is r-SVN£C in $\tilde{X}$. Since

$${\sigma }_n\le f^{-1}\left(f\left({\sigma }_n\right)\right)\le f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)\right),$$

by Definition 11, we get,

$${\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)\right),r)=f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)\right).$$

Hence,

$$f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le f\left(f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)\right)\right)\le {\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r).$$

(3)⇒(4): For each ${\gamma }_n\in {\xi }^{\tilde{Y}}$ and $r\in {\xi }_0$, put ${\sigma }_n=f^{-1}\left({\gamma }_n\right)$. By (3),

$$f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\gamma }_n\right),r)\right)\le {\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left(f^{-1}\left({\gamma }_n\right)\right),r)\le {\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r).$$

It implies that ${\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\gamma }_n\right),r)\le f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right)$.

(4)⇒(1): Let ${\gamma }_n$ be an r-SVN£O in $\tilde{Y}$. Then, ${\left[{\gamma }_n\right]}^c$ is an r-SVN£C in $\tilde{Y}$. Hence, ${\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}^c,r)={\left[{\gamma }_n\right]}^c$, and, by (4), we have,

$$f^{-1}\left({\left[{\gamma }_n\right]}^c\right)=f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}^c,r)\right)\ge {\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\left[{\gamma }_n\right]}^c\right),r).$$

On the other hand, $f^{-1}\left({\left[{\gamma }_n\right]}^c\right)\le {\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\left[{\gamma }_n\right]}^c\right),r)$. Thus, $f^{-1}\left({\left[{\gamma }_n\right]}^c\right)={\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}$$\left({\left[{\gamma }_n\right]}^c\right),r)$, that is $f^{-1}\left({\left[{\gamma }_n\right]}^c\right)$ is an r-SVN£C set in $\tilde{X}$. Hence, $f^{-1}\left({\gamma }_n\right)$ is an r-SVN£O set in $\tilde{X}$. □

Theorem 4. 

Let $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be a mapping. Then, the following conditions are equivalent:

(1) 

f is £-SVNI-irresolute open,

(2) 

$f\left({\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le {\mathrm{int}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r)$ for each ${\sigma }_n\in {\xi }^{\tilde{X}},r\in {\xi }_0$,

(3) 

${\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }((f^{-1}\left({\gamma }_n\right),r)\le f^{-1}\left({\mathrm{int}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right)$ for each ${\gamma }_n\in {\xi }^{\tilde{Y}},r\in {\xi }_0$,

(4) 

For any ${\gamma }_n\in {\xi }^{\tilde{Y}}$ and any r-SVN£C ${\sigma }_n\in {\xi }^{\tilde{X}}$ with $f^{-1}\left({\gamma }_n\right)\le {\sigma }_n$, there exists an r-SVN£C ${\varsigma }_n\in {\xi }^{\tilde{Y}}$ with ${\gamma }_n\le {\varsigma }_n$ such that $f^{-1}\left({\varsigma }_n\right)\le {\sigma }_n$.

Proof.

(1)⇒(2): For every ${\sigma }_n\in {\xi }^{\tilde{X}},r\in {\xi }_0$ and ${\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\sigma }_n$ from Theorem 2(2), we have $f\left({\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le f\left({\sigma }_n\right).$ By (1), $f\left({\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)$ is r-SVN£O in $\tilde{Y}$. Hence,

$$f\left({\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)={\mathrm{int}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }\left(f\left({\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\right)\le {\mathrm{int}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\sigma }_n\right),r).$$

(2)⇒(3): For each ${\gamma }_n\in {\xi }^{\tilde{Y}}$ and $r\in {\xi }_0$, put ${\sigma }_n=f^{-1}\left({\gamma }_n\right)$ from (2),

$$f\left({\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\gamma }_n\right),r)\right)\le {\mathrm{int}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left(f^{-1}\left({\gamma }_n\right)\right),r)\le {\mathrm{int}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r).$$

It implies that

$${\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\gamma }_n\right),r)\le f^{-1}\left(f\left({\mathrm{int}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\gamma }_n\right),r)\right)\right)\le f^{-1}\left({\mathrm{int}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right).$$

(3)⇒(4): Obvious.

(4)⇒(1): Let ${\epsilon }_n$ be an r-SVN£O in $\tilde{X}$. Put ${\gamma }_n={\left[f\left({\epsilon }_n\right)\right]}^c$ and ${\sigma }_n={\left[{\epsilon }_n\right]}^c$ such that ${\sigma }_n$ is r-SVN£C in $\tilde{X}$. We obtain

$$f^{-1}\left({\gamma }_n\right)=f^{-1}\left({\left[f\left({\epsilon }_n\right)\right]}^c\right)={\left[f^{-1}\left(f\left({\epsilon }_n\right)\right)\right]}^c\le {\left[{\epsilon }_n\right]}^c={\sigma }_n.$$

From (4), there exists r-SVN£O ${\varsigma }_n\in {\xi }^{\tilde{Y}}$ with ${\gamma }_n\le {\varsigma }_n$ such that $f^{-1}\left({\varsigma }_n\right)\le {\sigma }_n={\left[{\epsilon }_n\right]}^c$. It implies ${\epsilon }_n\le {\left[f^{-1}\left(\varsigma \right)\right]}^c=f^{-1}\left({\left[{\varsigma }_n\right]}^c\right)$. Thus, $f\left({\epsilon }_n\right)\le f\left(f^{-1}\left({\left[\varsigma \right]}^c\right)\right)\le {\left[{\varsigma }_n\right]}^c$. On the other hand, since ${\gamma }_n\le {\varsigma }_n$, we have

$$f\left({\epsilon }_n\right)={\left[\gamma \right]}^c\ge {\left[{\varsigma }_n\right]}^c.$$

Hence, $f\left({\epsilon }_n\right)={\left[{\varsigma }_n\right]}^c$, that is, $f\left({\epsilon }_n\right)$ is r-SVN£O in $\tilde{Y}$. □

Theorem 5. 

Let $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be a mapping. Then, the following conditions are equivalent:

(1) 

f is £-SVNI-irresolute closed.

(2) 

$f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right)\le {\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\gamma }_n\right),r)$ for each ${\gamma }_n\in {\xi }^{\tilde{X}},r\in {\xi }_0$.

Proof. 

Obvious. □

Theorem 6. 

Let $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be a bijective mapping. Then, the following conditions are equivalent:

(1) 

f is £-SVNI-irresolute closed,

(2) 

${\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\sigma }_n\right),r)\le f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)$ for each ${\sigma }_n\in {\xi }^{\tilde{Y}},r\in {\xi }_0$.

Proof.

$\left(1\right)\Rightarrow \left(2\right):$ Suppose that f is an £-SVNI-irresolute closed. From Theorem 5(2), we claim that, for each ${\gamma }_n\in {\xi }^{\tilde{X}}$ and $r\in {\xi }_0$,

$$f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right)\le {\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\gamma }_n\right),r).$$

Now, for all ${\sigma }_n\in {\xi }^{\tilde{Y}},r\in {\xi }_0$, put ${\gamma }_n=f^{-1}\left({\sigma }_n\right)$, since f is onto, it implies that $f\left(f^{-1}\left({\sigma }_n\right)\right)={\sigma }_n$. Thus,

$$f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\sigma }_n\right),r)\right)\le {\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left(f^{-1}\left({\sigma }_n\right)\right),r)={\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r).$$

Again, since f is onto, it follows:

$${\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\sigma }_n\right),r)=f^{-1}\left(f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left({\sigma }_n\right),r)\right)\right)\le f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right).$$

$\left(2\right)\Rightarrow \left(1\right):$ Put ${\sigma }_n=f\left({\gamma }_n\right)$. By the injection of f, we get

$${\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)={\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f^{-1}\left(f\left({\gamma }_n\right)\right),r)\le f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\gamma }_n\right),r)\right),$$

for the reason that f is onto, which implies that

$$f\left({\mathrm{CI}}_{{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right)\le f\left(f^{-1}\left({\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\gamma }_n\right),r)\right)\right)={\mathrm{CI}}_{{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(f\left({\gamma }_n\right),r).$$

□

4. £-Single Valued Neutrosophic Extremally Disconnected and £-Single Valued Neutrosophic Normal

This section is devoted to introducing £-single valued neutrosophic extremally disconnected (£-SVNE-disconnected, for short) and £-single valued neutrosophic normal (£-SVN-normal, for short), in the sense of Šostak. These definitions and their components, together with a set of criteria for identifying the spaces, are provided to illustrate how the ideas are applied.

Definition 13. 

An SVNITS $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is called £-SVNE-disconnected if ${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$ for each ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$.

Definition 14. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, ${\sigma }_n\in {\xi }^{\tilde{X}}$ is said to be:

(1) 

r-single valued neutrosophic semi-ideal open set (r-SVNSIO) iff ${\sigma }_n\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r),r)$,

(2) 

r-single valued neutrosophic pre-ideal open set (r-SVNPIO) iff ${\sigma }_n\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)$,

(3) 

r-single valued neutrosophic α-ideal open set (r-SVNαIO) iff ${\sigma }_n\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}{\sigma }_n,r),$$r),r)$,

(4) 

r-single valued neutrosophic β-ideal open set (r-SVNβIO) iff ${\sigma }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}\right({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$$({\sigma }_n,$$r),r),r)$,

(5) 

r-single valued neutrosophic β-ideal open (r-SVNSβIO) iff ${\sigma }_n\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}\right({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$$({\sigma }_n,r),r),r)$,

(6) 

r-single valued neutrosophic regular ideal open set (r-SVNRIO) iff ${\sigma }_n={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$$({\sigma }_n,r),r)$.

The complement of r-SVNSIO (resp. r-SVNPIO, r-SVNαIO, r-SVNβIO, r-SVNSβIO, r-SVNRIO) are called r-SVNSIC (resp. r-SVNPIC, r-SVNαIC, r-SVNβIC, r-SVNSβIC, r-SVNRIC).

Remark 2. 

The following diagram can be easily obtained from the above definition:

$$\begin{array}{c}\hfill r-SVN\alpha IO\Rightarrow r-SVNSIO\Rightarrow r-SVNSO\\ \hfill \Downarrow \Downarrow \Downarrow \\ \hfill r-SVNRIO\Rightarrow r-SVNPIO\Rightarrow r-SVN\beta IO\Rightarrow r-SVN\beta O\\ \hfill \Downarrow \\ \hfill r-SVNSIO\Rightarrow r-SVNS\beta IO\Rightarrow r-SVN\beta IO.\end{array}$$

Theorem 7. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, the following properties are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected,

(2) 

${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right]}^c\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right]}^c\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c\right)\le 1-r$ for each ${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\sigma }_n\right]}^c\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\sigma }_n\right]}^c\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\sigma }_n\right]}^c\right)\le 1-r$,

(3) 

${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n.r),r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),$ for each ${\sigma }_n\in {\xi }^{\tilde{X}}$,

(4) 

Every r-SVNSIO set is r-SVNPIO,

(5) 

${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$ for each r-SVNSβIO ${\sigma }_n\in {\xi }^{\tilde{X}}$,

(6) 

Every r-SVNSβIO set is r-SVNPIO,

(7) 

For each ${\sigma }_n\in {\xi }^{\tilde{X}}$, ${\sigma }_n$ is r-SVNαIO set iff it is r-SVNSIO.

Proof.

(1) ⇒ (2):The proof is direct consequence of Definition 14.

(2)⇒(3): For each ${\sigma }_n\in {\xi }^{\tilde{X}}$, ${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r)\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\sigma }_n,r)\right)$$\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\sigma }_n,r)\right)\le 1-r$, and, by (2), we have

$${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r)\right]}^c,r)\right]}^c\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\sigma }_n,r)\right]}^c,r)\right]}^c\right)\le 1-r,$$

$${\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\sigma }_n,r)\right]}^c,r)\right]}^c\right)\le 1-r.$$

Thus,

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r),r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}(\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\sigma }_n,r),r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}(\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\sigma }_n,r),r)\right)\le 1-r;$$

hence,

$${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n.r),r)={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r),r),r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r).$$

(3)⇒(4): Let ${\sigma }_n$ be an r-SVNSIO set. Then, by (4), we have

$${\sigma }_n\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r),r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r).$$

Thus, ${\sigma }_n$ is an r-SVNPIO set.

(4)⇒(5): Since ${\sigma }_n$ is an r-SVNS$\beta$IO set, ${\sigma }_n\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({CI}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)$. Then, ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$ is r-SVNSIO, and, by (4), ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({CI}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)$; hence, ${\tilde{\tau }}^{\tilde{\rho }}{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r))\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r))\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r))\le 1-r$.

(5)⇒(6): Let ${\sigma }_n$ be an r-SVN$\beta$IO set, then, by (5), ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}(C{l}^{🟉}({\sigma }_n,r),r)$. Thus,

$${\sigma }_n\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r).$$

Therefore, ${\sigma }_n$ is an r-SVNPIO set.

(6)⇒(7): Let ${\sigma }_n$ be an r-SVNSIO. Then, ${\sigma }_n$ is r-SVNS$\beta$IO, by (6), ${\sigma }_n$ is an r-SVNPIO set. Since ${\sigma }_n$ is r-SVNSIO and r-SVNPIO, ${\sigma }_n$ is r-SVN$\alpha$IO.

(7) ⇒ (1): Suppose that ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$, then ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$ is r-SVNSIO, and, by (7), ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$ is r-SVN$\alpha$IO. Hence,

$${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}(C{l}^{🟉}({\sigma }_n,r),r),r),r)={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r).$$

Hence,

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r.$$

Thus, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected. □

Theorem 8. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS $r\in {\xi }_0$ and ${\sigma }_n\in {\xi }^{\tilde{X}}$. Then, the following are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected,

(2) 

${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)$, for every ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$ and every r-SVN£O ${\gamma }_n\in {\xi }^{\tilde{X}}$ with ${\sigma }_n\overline{q}{\gamma }_n$,

(3) 

${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)$, for every ${\sigma }_n\in {\xi }^{\tilde{X}}$ and r-SVN£O ${\gamma }_n\in {\xi }^{\tilde{X}}$ with ${\sigma }_n\overline{q}{\gamma }_n$.

Proof.

(1)⇒(2): Let ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$. Then, by (1),

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r.$$

Since ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$ is an r-SVN£O and ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$, it implies that

$${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c,r).$$

(2)⇒(1): Let ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$. Since ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$ is an r-SVN£O, then, by (2),

$${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c,r).$$

This implies that ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$, so

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r.$$

(2)⇒(3): Suppose that ${\sigma }_n\in {\xi }^{\tilde{X}}$ and ${\gamma }_n$ is an r-SVN£O with ${\sigma }_n\overline{q}{\gamma }_n$. Since

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r),r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r),r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\right)\le 1-r.$$

By (2), we have ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)$.

(3)⇒(2): Let ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$ and ${\gamma }_n$ be an r-SVN£O with ${\sigma }_n\overline{q}{\gamma }_n$. Then, by (3), we obtain ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)$. Since

$${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{iny}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r),$$

then, we have ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)$. □

Definition 15. 

An SVNITS $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is called £-SVN-normal if, for every ${\left[{\sigma }_n\right]}_1\overline{q}{\left[{\sigma }_n\right]}_2$ with ${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\sigma }_n\right]}_1\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\sigma }_n\right]}_1\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\sigma }_n\right]}_1\right)\le 1-r$ and ${\left[{\sigma }_n\right]}_2$ is r-SVN£O, there exists ${\left[{\gamma }_n\right]}_j\in {\xi }^{\tilde{X}}$, for $j=\{1,2\}$ with ${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\gamma }_n\right]}_1^c\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\gamma }_n\right]}_1^c\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\gamma }_n\right]}_1^c\right)\le 1-r$, ${\left[{\gamma }_n\right]}_2$ is r-SVN£C such that ${\left[{\sigma }_n\right]}_2\le {\left[{\gamma }_n\right]}_1,$ ${\left[{\sigma }_n\right]}_1\le {\left[{\gamma }_n\right]}_2$ and ${\left[{\gamma }_n\right]}_1\overline{q}{\left[{\gamma }_n\right]}_2$.

Theorem 9. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS; then, the following are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is an £-SVN-normal.

(2) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is an £-SVNE-disconnected.

Proof.

(1)⇒(2): Let ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$ and ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$ be an r-SVN£O. Then, ${\sigma }_n\overline{q}{\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$. By the £-SVN-normality of $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$, there exist ${\left[{\gamma }_n\right]}_i\in {\xi }^{\tilde{X}}$, for $i=\{1,2\}$ with

$${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\gamma }_n\right]}_1^c\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\gamma }_n\right]}_1^c\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\gamma }_n\right]}_1^c\right)\le 1-r,$$

and ${\left[{\gamma }_n\right]}_2^c$r-SVN£C such that ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c\le {\left[{\gamma }_n\right]}_1$, ${\sigma }_n\le {\left[{\gamma }_n\right]}_2$ and ${\left[{\gamma }_n\right]}_1\overline{q}{\left[{\gamma }_n\right]}_2$. Since

$${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}_2,r)={\left[{\gamma }_n\right]}_2\le {\left[{\gamma }_n\right]}_1^c\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),$$

we have ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)={\left[{\gamma }_n\right]}_2$. Since ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c\le {\left[{\gamma }_n\right]}_1\le {\left[{\gamma }_n\right]}_2^c={\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$, so ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c={\left[{\gamma }_n\right]}_1$. Hence, ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)={\left[{\gamma }_n\right]}_1^c$ and

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r.$$

Thus, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is an £-SVNE-disconnected.

(2)⇒(1): Suppose that ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$ and ${\gamma }_n$ is an r-SVN£O with ${\sigma }_n\overline{q}{\gamma }_n.$ By the £-SVNE-disconnected of $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$, we have

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,$$

and ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$ is r-SVN£O. Since ${\sigma }_n\overline{q}{\gamma }_n$, ${\sigma }_n\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$ and ${\gamma }_n\le {\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$. Thus, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is an £-SVN-normal. □

Theorem 10. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS, ${\sigma }_n,{\sigma }_n{\xi }^{\tilde{X}}$ and $r\in {\xi }_0$. Then, the following properties are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected.

(2) 

If ${\sigma }_n$ is r-SVNRIO, then ${\sigma }_n$ is r-SVN£C.

(3) 

If ${\sigma }_n$ is r-SVNRIC, then ${\sigma }_n$ is r-SVN£O.

Proof.

(1)⇒(2): Let ${\sigma }_n$ be an r-SVNRIO. Then, ${\sigma }_n={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)$ and ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$. By (1),

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r.$$

Hence ${\sigma }_n={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)={\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$.

(2)⇒(1): Suppose that ${\sigma }_n={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)$, then ${\tilde{\tau }}^{\tilde{\rho }}\left({\sigma }_n\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\sigma }_n\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\sigma }_n\right)\le 1-r$, by (2), ${\sigma }_n$ is r-SVN£C. This implies that

$$\begin{array}{c}\hfill {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r).\end{array}$$

Thus,

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,$$

then $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is an £-SVNE-disconnected.

(2) ⇔ (3): Obvious. □

Remark 3. 

The union of two r-SVNRIO sets need not to be an r-SVNRIO.

Theorem 11. 

If $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected and ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$, $r\in {\xi }_0.$ Then, the following properties hold:

(1) 

If ${\sigma }_n$ and ${\gamma }_n$ are r-SVNRIC, then ${\sigma }_n\wedge {\gamma }_n$ is r-SVNRIC.

(2) 

If ${\sigma }_n$ and ${\gamma }_n$ are r-SVNRIO, then ${\sigma }_n\vee {\gamma }_n$ is r-SVNRIO.

Proof. 

Let ${\sigma }_n$ and ${\gamma }_n$ be r-SVNRIC. Then, ${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\sigma }_n\right]}^c\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\sigma }_n\right]}^c\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\sigma }_n\right]}^c\right)\le 1-r$ and ${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\gamma }_n\right]}^c\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\gamma }_n\right]}^c\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\gamma }_n\right]}^c\right)\le 1-r$, by Theorem 7, we have

$${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right]}^c\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right]}^c\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c\right)\le 1-r,$$

and

$${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\gamma }_n,r)\right]}^c\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\gamma }_n,r)\right]}^c\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\right]}^c\right)\le 1-r.$$

This implies that

$$\begin{array}{ccc}& {\sigma }_n\wedge {\gamma }_n& ={\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\wedge {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r)\hfill \\ & & ={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n\wedge {\gamma }_n,r)\hfill \\ & & \le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n\wedge {\gamma }_n,r),r).\hfill \end{array}$$

On the other hand,

$$\begin{array}{ccc}& {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n\wedge {\gamma }_n,r),r)& ={\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r)\hfill \\ & & \le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\wedge {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r)\hfill \\ & & ={\sigma }_n\wedge {\gamma }_n.\hfill \end{array}$$

Thus, ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n\wedge {\gamma }_n,r),r)={\sigma }_n\wedge {\gamma }_n$. Therefore, ${\sigma }_n\wedge {\gamma }_n$ is an r-SVNRIC.

(2) The proof is similar to that of (1). □

Theorem 12. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, the following properties are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected,

(2) 

${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$, for every r-SVNSIO ${\sigma }_n\in {\xi }^{\tilde{X}}$,

(3) 

${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$, for every r-SVNPIO ${\sigma }_n\in {\xi }^{\tilde{X}}$,

(4) 

${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$, for every r-SVNRIO ${\sigma }_n\in {\xi }^{\tilde{X}}$.

Proof.

$\left(1\right)\Rightarrow \left(2\right)$ and $\left(1\right)\Rightarrow \left(3\right)$. Let ${\sigma }_n$ be an r-SVNSIO (r-SVNPIO). Then, ${\sigma }_n$ is r-SVNS$\beta$IO, and, by Theorem 7, we have,

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r.$$

(2)⇒(4) and (3)⇒(4). Let ${\sigma }_n$ be an r-SVNRIO. Then, ${\sigma }_n$ is r-SVNPIO and r-SVNSIO. Thus,

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r.$$

(4)⇒(1). Suppose that

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r),r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r),r)\right)\ge r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\right)\ge r.$$

Then, by (4), we have

$${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r),r),r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r),r),r)\right)\ge r,$$

$${\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)\right)\ge r.$$

Hence,

$$\begin{array}{ccc}& {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)& \le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)\hfill \\ & & ={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r),r)\hfill \\ & & ={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)\le {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r).\hfill \end{array}$$

Thus, ${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r,{\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r,{\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$; hence, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is an £-SVNE-disconnected. □

Definition 16. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS. Then, ${\sigma }_n$ is said to be an r-SVN£SO if ${\sigma }_n\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)$.

Definition 17. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS for each $r\in {\xi }_0,$${\sigma }_n\in {\xi }^{\tilde{X}}$ and $x_{s,t,p}\in Pt\left({\xi }^{\tilde{X}}\right)$. Then, $x_{s,t,p}$ is called an r-SVN$\delta \mathcal{I}$-cluster point of ${\sigma }_n$ if, for every ${\gamma }_n\in Q_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}(x_{s,t,p},r)$, we have ${\sigma }_{n}q{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r)$.

Definition 18. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS for each $r\in {\xi }_0,$${\sigma }_n\in {\xi }^{\tilde{X}}$ and $x_{s,t,p}\in Pt\left({\xi }^{\tilde{X}}\right)$. Then, the single-valued neutrosophic $\delta \mathcal{I}$-closure operator is a mapping ${\mathrm{C}}_{\delta \mathit{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}:{\xi }^{\tilde{X}}\times {\xi }_0\to {\xi }^{\tilde{X}}$ that is defined as: ${\mathrm{C}}_{\delta \mathit{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)=\bigvee \{x_{s,t,p}\in Pt\left({\xi }^{\tilde{X}}\right)$ is r-SVN$\delta \mathcal{I}$-cluster point of ${\sigma }_n\}$.

Lemma 3. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS. Then, ${\sigma }_n$ is r-SVN£SO iff ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)={\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}$$({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)$.

Proof. 

Obvious. □

Lemma 4. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS for each ${\sigma }_n\in {\xi }^{\tilde{x}}$ and $r\in {\xi }_0$. Then, ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)\le {\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)$.

Proof. 

Obvious. □

Lemma 5. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and ${\sigma }_n$ be an r-SVN£SO. Then, ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)={\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)$.

Proof. 

We show that ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)\le {\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)$. Suppose that ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)\ngeqq {\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)$,; then, there exist $\nu \in \tilde{X}$ and $s,t,p\in {\xi }_0$ such that

$$\begin{array}{c}\hfill {\tilde{\rho }}_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r)}\left(\nu \right)<s\le {\tilde{\rho }}_{{\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r)}\left(\nu \right),{\tilde{\varrho }}_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\sigma }_n,r)}\left(\nu \right)\ge t>{\tilde{\varrho }}_{{\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\varrho }}}({\sigma }_n,r)}\left(\nu \right),\end{array}$$

(1) 

$${\tilde{\eta }}_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\sigma }_n,r)}\left(\nu \right)\ge p>{\tilde{\eta }}_{{\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\eta }}}({\sigma }_n,r)}\left(\nu \right).$$

By the definition of ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}$, there exists ${\tilde{\tau }}^{\tilde{\rho }}\left({\left[{\gamma }_n\right]}^c\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\left[{\gamma }_n\right]}^c\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\left[{\gamma }_n\right]}^c\right)\le 1-r$ with ${\sigma }_n\le {\gamma }_n$ such that

$${\tilde{\rho }}_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r)}\left(\nu \right)\le {\tilde{\rho }}_{{\gamma }_n}\left(\nu \right)<s<{\tilde{\rho }}_{{\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r)}\left(\nu \right),{\tilde{\varrho }}_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\varrho }}}({\sigma }_n,r)}\left(\nu \right)\ge {\tilde{\varrho }}_{{\gamma }_n}\left(\nu \right)>t>{\tilde{\varrho }}_{{\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }}}({\sigma }_n,r)}\left(\nu \right),$$

$${\tilde{\eta }}_{{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\eta }}}({\sigma }_n,r)}\left(\nu \right)\ge {\tilde{\rho }}_{{\gamma }_n}\left(\nu \right)>p>{\tilde{\eta }}_{{\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\eta }}}({\sigma }_n,r)}\left(\nu \right).$$

Then, ${\left[{\gamma }_n\right]}^c\in Q_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}(x_{s,t,p},r)$ and

$$\begin{array}{ccc}& {\left[{\sigma }_n\right]}^c\ge {\left[{\gamma }_n\right]}^c& \Rightarrow {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}^c,r)\ge {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}^c,r)\hfill \\ & & \Rightarrow {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\sigma }_n\right]}^c,r)\ge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\left[{\gamma }_n\right]}^c,r)\hfill \\ & & \Rightarrow {\left[{\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c\ge {\left[{\gamma }_n\right]}^c.\hfill \end{array}$$

Thus, ${\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\left[{\gamma }_n\right]}^c$. Hence, ${\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}^c,r),r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r),r)$. Since ${\sigma }_n$ is an r-SVN£SO, we have ${\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r)\overline{q}{\sigma }_n$. So, $x_{s,t,p}$ is not an r-SVN$\delta \mathcal{I}$-cluster point of ${\sigma }_n$. It is a contradiction for equation 3. Thus, ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)\ge {\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)$. By Lemma 4, we have ${\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)={\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n,r)$. □

Theorem 13. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS. Then, the following properties are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected,

(2) 

If ${\sigma }_n$ is r-SVNSβIO and ${\gamma }_n$ is r-SVN£SO, then ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}$$({\sigma }_n\wedge {\gamma }_n)$,

(3) 

If ${\sigma }_n$ is r-SVNSIO and ${\gamma }_n$ is r-SVN£SO, then ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}$$({\sigma }_n\wedge {\gamma }_n)$,

(4) 

${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)$, for every r-SVNSIO set ${\sigma }_n\in {\xi }^{\tilde{X}}$ and every r-SVN£SO ${\gamma }_n\in {\xi }^{\tilde{X}}$ with ${\sigma }_n\overline{q}{\gamma }_n$,

(5) 

If ${\sigma }_n$ is an r-SVNPIO and ${\gamma }_n$ is an r-SVN£SO, then ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}$$({\sigma }_n\wedge {\gamma }_n)$.

Proof.

(1)⇒(2): Let ${\sigma }_n$ be an r-SVNS$\beta$IO and ${\gamma }_n$ be an r-SVN£SO, by Theorem 7, ${\tilde{\tau }}^{\tilde{\rho }}$$\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)$$\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$. Then,

$$\begin{array}{ccc}& {\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge & {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\wedge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r]\hfill \\ & & \le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }[{\gamma }_n\wedge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r],r]\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}[{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}[{\gamma }_n\wedge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r],r]\hfill \\ & & \le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}[{\gamma }_n\wedge {\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r),r]\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}[{\gamma }_n\wedge {\gamma }_n,r].\hfill \end{array}$$

Hence, ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n\wedge {\gamma }_n)$.

(2)⇒(3): It follows from the fact that every r-SVNSIO set is an r-SVNS$\beta$IO.

(3)⇒(4): Clear.

(4)⇒(1): Let ${\sigma }_n$ be an r-SVNSIO. Since ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }\right({\left[{\sigma }_n\right]}^c,r),r),r)$ we have, ${\left[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c$ is an r-SVN£SO. Then, by (4), ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}\left(\right[{\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$$({\sigma }_n,r){]}^c,r)$. Thus, ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le [{\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}{\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right]}^c,r){]}^c={\mathrm{int}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r),r)$. Therefore, ${\tilde{\tau }}^{\tilde{\rho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }}}^{\pounds }({\sigma }_n,r)\right)\ge r$, ${\tilde{\tau }}^{\tilde{\varrho }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\varrho }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$, ${\tilde{\tau }}^{\tilde{\eta }}\left({\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\right)\le 1-r$. Thus, by Theorem 12, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected.

(2)⇒(5): It follows from the fact that every r-SVNPIO is an r-SVNS$\beta$IO. □

Corollary 1. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS. Then, the following properties are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is £-SVNE-disconnected.

(2) 

If ${\sigma }_n$ is an r-SVNSβIO and ${\gamma }_n$ is an r-SVN£SO, then ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n\wedge {\gamma }_n)$.

(3) 

If ${\sigma }_n$ is an r-SVNSIO and ${\gamma }_n$ is an r-SVN£SO, then ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n\wedge {\gamma }_n).$

(4) 

If ${\sigma }_n$ is an r-SVNPIO and ${\gamma }_n$ is an r-SVN£SO, then ${\mathrm{CI}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\wedge {\mathrm{C}}_{\delta \mathbf{I}{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\gamma }_n,r)\le {\mathrm{C}}_{{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}({\sigma }_n\wedge {\gamma }_n)$.

Proof. 

It follows directly from Lemma 3 and 5. □

5. Some Types of Separation Axioms

In this section, some kinds of separation axioms, namely r-single valued neutrosophic ideal-$R_i$ (r-$SVNI{R}_i$, for short), where $i=\{0,1,2,3\}$, and r-single valued neutrosophic ideal-$T_j$ (r-$SVNI{T}_j$, for short), where $j=\{1,2,2\frac{1}{2},3,4\}$, in the sense of Šostak are defined. Some of their characterizations, fundamental properties, and the relations between these notions have been studied.

Definition 19. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, $\tilde{X}$ is called:

(1) 

r-$SVNI{R}_0$ iff $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$ implies $y_{s_1,t_1,p_1}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)$ for any $x_{s,t,p}\ne y_{s_1,}$$t_1,p_1$.

(2) 

r-$SVNI{R}_1$ iff $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$ implies that there exist r-SVN£O sets $\sigma n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n$, $y_{s_1,t_1,p_1}\in {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$.

(3) 

r-$SVNI{R}_2$ iff $x_{s,t.p}\overline{q}{\varsigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\varsigma }_n,r)$ implies there exist r-SVN£O sets $\sigma n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n$, ${\varsigma }_n\le {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$.

(4) 

r-$SVNI{R}_3$ iff ${\left[{\varsigma }_n\right]}_1={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\varsigma }_n\right]}_1,r)\overline{q}{\left[{\varsigma }_n\right]}_2={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\varsigma }_n\right]}_2,r)$ implies that there exist r-SVN£O sets ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that ${\left[{\varsigma }_n\right]}_1\le {\sigma }_n$, ${\left[{\varsigma }_n\right]}_2\le {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$.

(5) 

r-$SVNI{T}_1$ iff $x_{s,t,p}\overline{q}{y}_{s_1,t_1,p_1}$ implies that there exists r-SVN£O $\sigma n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n$ and $y_{s_1,t_1,p_1}\overline{q}{\sigma }_n$.

(6) 

r-$SVNI{T}_2$ iff $x_{s,t,p}\overline{q}{y}_{s_1,t_1,p_1}$ implies that there exist r-SVN£O sets $\sigma n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n$, $y_{s_1,t_1,p_1}\in {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$.

(7) 

r-$SVNI{T}_{2\frac{1}{2}}$ iff $x_{s,t,p}\overline{q}{y}_{s_1,t_1,p_1}$ implies that there exist r-SVN£O sets $\sigma n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n$, $y_{s_1,t_1,p_1}\in {\gamma }_n$ and ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$.

(8) 

r-$SVNI{T}_3$ iff it is r-$SVNIT{R}_2$ and r-$SVNI{T}_1$.

(9) 

r-$SVNI{T}_4$ iff it is r-$SVNIT{R}_3$ and r-$SVNI{T}_1$.

Theorem 14. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, the following statements are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_0$.

(2) 

If $x_{s,t,p}\overline{q}{\sigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$, then there exists r-SVN£O ${\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\overline{q}{\gamma }_n$ and ${\sigma }_n\le {\gamma }_n$.

(3) 

If $x_{s,t,p}\overline{q}{\sigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$, then ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)\overline{q}{\sigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$.

(4) 

If $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$, then ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$.

Proof.

(1)⇒(2): Let $x_{s,t,p}\overline{q}{\sigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$. Then,

$$\begin{array}{c}\hfill s+{\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)<1,t+{\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)\ge 1,p+{\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)\ge 1,\end{array}$$

for every $y_{s_1,t_1,p_1}\in {\sigma }_n$, we have $s_1<{\tilde{\rho }}_{{\sigma }_n}\left(\nu \right)$, $t_1\ge {\tilde{\varrho }}_{{\sigma }_n}\left(\nu \right)$ and $p_1\ge {\tilde{\eta }}_{{\sigma }_n}\left(\nu \right)$. Thus, $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$. Since $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is an r-$SVNI{R}_0$, we obtain $y_{s_1,t_1,p_1}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$$(x_{s,t,p},r)$. By Lemma 2(2), there exists an r-SVN£O ${\varsigma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\overline{q}{\varsigma }_n$ and $y_{s_1,t_1,p_1}\le {\varsigma }_n$. Let

$${\gamma }_n=\underset{y_{s_1,t_1,p_1}\in {\sigma }_n}{\bigvee }\{{\varsigma }_n:x_{s,t,p}\overline{q}{\varsigma }_n,y_{s_1,t_1,p_1}\in {\varsigma }_n\}.$$

From Lemma 1(1), ${\gamma }_n$ is an r-SVN£O. Then, $x_{s,t,p}\overline{q}{\gamma }_n$, ${\sigma }_n\le {\gamma }_n$.

(2)⇒(3): Let $x_{s,t,p}\overline{q}{\sigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$. Then, there exists an r-SVN£O ${\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\overline{q}{\gamma }_n$ and ${\sigma }_n\le {\gamma }_n$. Since for every $\nu \in \tilde{X}$,

$$\begin{array}{c}\hfill s<1-{\tilde{\rho }}_{{\gamma }_n}\left(\nu \right),t\ge 1-{\tilde{\varrho }}_{{\gamma }_n}\left(\nu \right),p\ge 1-{\tilde{\eta }}_{{\gamma }_n}\left(\nu \right),\end{array}$$

we obtain

$$\begin{array}{c}\hfill {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)\le {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}^c,r)={\left[{\gamma }_n\right]}^c\le {\left[{\sigma }_n\right]}^c.\end{array}$$

Therefore, ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)\overline{q}{\sigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$.

(3)⇒(4): Let $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$. Then, $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$$(y_{s_1,t_1,p_1},r),r)$. By (3), $s_1,t_1,p_1(x_{s,t,p},r)\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$.

(4)⇒(1): Clear. □

Theorem 15. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, if $\tilde{X}$ is

(1) 

[r-$SVNI{R}_3$ and r-$SVNI{R}_0$] ${\Rightarrow }^{\left(a\right)}$r-$SVNI{R}_2$ ${\Rightarrow }^{\left(b\right)}$r-$SVNI{R}_1$${\Rightarrow }^{\left(c\right)}$r-$SVNI{R}_0$.

(2) 

r-$SVNI{T}_2\Rightarrow$ r-$SVNI{R}_1$.

(3) 

r-$SVNI{T}_3\Rightarrow$ r-$SVNI{R}_2$.

(4) 

r-$SVNI{T}_4\Rightarrow$ r-$SVNI{R}_3$.

(5) 

r-$SVNI{T}_4$${\Rightarrow }^{\left(a\right)}$r-$SVNI{T}_3$${\Rightarrow }^{\left(b\right)}$r-$SVNI{T}_{2\frac{1}{2}}$${\Rightarrow }^{\left(c\right)}$r-$SVNI{T}_2$${\Rightarrow }^{\left(d\right)}$r-$SVNI{T}_1$.

Proof.

$\left(1_a\right)$. Let $x_{s,t.p}\overline{q}{\varsigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\varsigma }_n,r)$, by Theorem 14(3), ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)\overline{q}{\varsigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }$$({\varsigma }_n,r)$. Since $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_3$ and ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r),r)$, there exist r-SVN£O sets ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)\le {\sigma }_n,$ ${\varsigma }_n\le {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$. Hence, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_2$.

$\left(1_b\right)$. For each $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$, by r-$SVNI{R}_2$ of $\tilde{X}$, there exist r-SVN£O sets ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n,$ $y_{s_1,t_1,p_1},r\in {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r,r)\le {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$. Thus, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_1$.

$\left(1_c\right)$. Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be r-$SVNI{R}_1$. Then, for every $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r,r)$ and $x_{s,t,p}\ne y_{s_1,t_1,p_1}$, there exist r-SVN£O sets ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n,$$y_{s_1,t_1,p_1}\in {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$. Hence, $x_{s,t,p}\in {\sigma }_n\le {\left[{\gamma }_n\right]}^c$. Since ${\gamma }_n$ is an r-SVN£O set, we obtain ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)\le {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}^c,r)={\left[{\gamma }_n\right]}^c\le {\left[y_{s_1,t_1,p_1}\right]}^c$. Thus, $y_{s_1,t_1,p_1}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(x_{s,t,p},r)$ and $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_0$.

(2). Let $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_2,p_1},r)$. Then, $x_{s,t,p}\overline{q}{y}_{s_1,t_1,p_1}$. By r-$SVNI{T}_2$ of $\tilde{X}$, there exist r-SVN£O sets ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{st,p}\in {\sigma }_n,$$y_{s_1,t_1,p_1}\in {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$. Hence, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_1$.

(3) and (4) The proofs are direct consequence of (2) .

$\left(5_a\right)$. The proof is direct consequence of (1).

$\left(5_b\right)$. For each $x_{s,t,p}\overline{q}{y}_{s_1,t_1,p_1}$, since $\tilde{X}$ is both r-$SVNI{R}_2$ and r-$SVNI{T}_1$, then, there exists an r-SVN£O set ${\varsigma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\varsigma }_n$ and $y_{s_1,t_1,p_1}\overline{q}{\varsigma }_n$. Then,

$$x_t\in {\varsigma }_n={\mathrm{int}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\varsigma }_n,r)\le {\mathrm{int}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[y_{s_1,t_1,p_1}\right]}^c,r)={\left[{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)\right]}^c.$$

Hence, $x_{s,t,p}\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$. By r-$SVNI{R}_2$ of $\tilde{X}$, there exist r-SVN£O sets ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n$, ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)\le {\gamma }_n$ and ${\sigma }_n\overline{q}{\gamma }_n$. Thus, ${\sigma }_n\le {\left[{\gamma }_n\right]}^c$, so

$${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\le {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}^c,r)={\left[{\gamma }_n\right]}^c\le {\left[{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)\right]}^c.$$

It implies ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$ with $x_{s,t,p}\in {\sigma }_n$ and $y_{s_1,t_1,p_1}\in {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }(y_{s_1,t_1,p_1},r)$. Thus, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{T}_{2\frac{1}{2}}$.

$\left(5_c\right)$. Let $x_{s,t,p}\overline{q}{y}_{s_1,t_1,p_1}$. Then, by r-$SVNI{T}_{2\frac{1}{2}}$ of $\tilde{X}$, there exist r-SVN£O sets ${\sigma }_n,{\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\sigma }_n$, $y_{s_1,t_1,p_1}\in {\gamma }_n$ and ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)$, which implies that ${\sigma }_n\overline{q}{\gamma }_n$. Thus, $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{T}_2$.

$\left(5_d\right)$. Similar to the proof of $\left(5_c\right)$. □

Theorem 16. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, the following statements are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_2$.

(2) 

If $x_{s,t,p}\in {\sigma }_n$ and ${\sigma }_n$ is r-SVN£O set, then there exists r-SVN£O set ${\gamma }_n\in {\xi }^{\tilde{X}}$ such that $x_{s,t,p}\in {\gamma }_n\le {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\le {\sigma }_n$.

(3) 

If $x_{s,t,p}\overline{q}{\sigma }_n={\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\sigma }_n,r)$, then there exists r-SVN£O set ${\left[{\gamma }_n\right]}_j\in {\xi }^{\tilde{X}},$$j=\{1,2\}$ such that $x_{s,t,p}\in {\left[{\gamma }_n\right]}_1,$ ${\sigma }_n\le {\left[{\gamma }_n\right]}_2$ and ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}_1,r)\overline{q}{\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\left[{\gamma }_n\right]}_2,r).$

Proof. 

Similar to the proof of Theorem 14. □

Theorem 17. 

Let $(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be an SVNITS and $r\in {\xi }_0$. Then, the following statements are equivalent:

(1) 

$(\tilde{X},{\tilde{\tau }}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_3$.

(2) 

If ${\left[{\sigma }_n\right]}_1\overline{q}{\left[{\sigma }_n\right]}_2$ and ${\left[{\sigma }_n\right]}_1,{\left[{\sigma }_n\right]}_2$ are r-SVN£C sets, then there exists r-SVN£O set ${\gamma }_n\in {\xi }^{\tilde{X}}$ such that ${\left[{\sigma }_n\right]}_1\le {\gamma }_n$ and ${\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\le {\left[{\sigma }_n\right]}_2$.

(3) 

For any ${\left[{\sigma }_n\right]}_1\le {\left[{\sigma }_n\right]}_2$, where ${\left[{\sigma }_n\right]}_1$ is an r-SVN£O set, and ${\left[{\sigma }_n\right]}_2$ is an r-SVN£C set, then, there exists an r-SVN£O set ${\gamma }_n\in {\xi }^{\tilde{X}}$ such that ${\left[{\sigma }_n\right]}_1\le {\gamma }_n\le {\mathrm{CI}}_{{\tau }^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }}}^{\pounds }({\gamma }_n,r)\le {\left[{\sigma }_n\right]}_2$.

Proof. 

Similar to the proof of Theorem 15. □

Theorem 18. 

Let $f:(\tilde{X},{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})\to (\tilde{Y},{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ be a £-SVNI-irresolute, bijective, £-SVNI-irresolute open mapping and $(\tilde{X},{\tilde{\tau }}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_1^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_2$. Then, $(\tilde{Y},{\tilde{\tau }}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }},{\mathcal{I}}_2^{\tilde{\rho }\tilde{\varrho }\tilde{\eta }})$ is r-$SVNI{R}_2$.

Proof. 

Let $y_{s,t,p}\overline{q}{\varsigma }_n=C{l}^{🟉}({\varsigma }_n,r)$. Then, by Definition 11, ${\varsigma }_n$ is an r-SVN£C set in $\tilde{Y}$. By Theorem 3(2), $f^{-1}\left({\varsigma }_n\right)$ is an r-SVN£C set in $\tilde{X}$. Put $y_{s,t,p}=f\left(x_{s,t,p}\right)$. Then, $x_{s,t,p}$