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DOI: 10.32604/cmes.2022.017782

ARTICLE

On Single Valued Neutrosophic Regularity Spaces

Yaser Saber1,2, Fahad Alsharari1,6,*, Florentin Smarandache3 and Mohammed Abdel-Sattar4,5

1Department of Mathematics, College of Science and Human Studies, Hotat Sudair, Majmaah University, Majmaah, 11952, Saudi Arabia
2Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt
3Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA
4Department of Mathematics, College of Science and Arts King Khaled University, Mhayal Asier, 61913, Saudi Arabia
5Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, 62511, Egypt
6Department of Mathematics, College of Science and Arts, Jouf University, Gurayat, 77455, Saudi Arabia
*Corresponding Author: Fahad Alsharari. Email: f.alsharari@mu.edu.sa
Received: 06 June 2021; Accepted: 03 August 2021

Abstract: This article aims to present new terms of single-valued neutrosophic notions in the Šostak sense, known as single-valued neutrosophic regularity spaces. Concepts such as r-single-valued neutrosophic semi £s-open, r-single-valued neutrosophic pre-£s-open, r-single valued neutrosophic regular-£s-open and r-single valued neutrosophic α£s-open are defined and their properties are studied as well as the relationship between them. Moreover, we introduce the concept of r-single valued neutrosophic θ£s-cluster point and r-single-valued neutrosophic γ£s-cluster point, r-θ£s-closed, and θ£s-closure operators and study some of their properties. Also, we present and investigate the notions of r-single-valued neutrosophic θ£s-connectedness and r-single valued neutrosophic δ£s-connectedness and investigate relationship with single-valued neutrosophic almost £s-regular. We compare all these forms of connectedness and investigate their properties in single-valued neutrosophic semiregular and single-valued neutrosophic almost regular in neutrosophic ideal topological spaces in Šostak sense. The usefulness of these concepts are incorporated to multiple attribute groups of comparison within the connectedness and separateness of X, and δ£s.

Keywords: Single valued neutrosophic θ£-closed; single valued neutrosophic θ£-separated; single valued neutrosophic δ£-separated; single-valued neutrosophic δ£-connected; single valued neutrosophic δ£-connected; single valued neutrosophic almost £-egular

1  Introduction

A neutrosophic set can be practical in addressing problems with indeterminate, imperfect, and inconsistent materials. The concept of neutrosophic set theory was introduced by Smarandache [1] as a new mathematical method that corresponds to the indeterminacy degree (uncertainty, etc.). Bakbak et al. [2] and Mishra et al. [3] applied the soft set theory successfully applied in several areas, such as the smoothness of functions, as well as architecture-based, neuro-linguistic programming. Wang et al. [4] proposed single-valued neutrosophic sets (SVNSs). Meanwhile, Kim et al. [5,6] inspected the single valued neutrosophic relations (SVNRs) and symmetric closure of SVNR, respectively. Recently, Saber et al. [79] introduced the concepts of single-valued neutrosophic ideal open local function and single-valued neutrosophic topological space. Many of their applications appear in the studies of Das et al. [10]. Alsharari et al. [1113]. Riaz et al. [14]. Salama et al. [1517]. Hur et al. [18,19]. Yang et al. [20]. El-Gayyar [21], AL-Nafee et al. [22]. Muhiuddin et al. [23,24] and Mukherjee et al. [25].

First, we define single-valued neutrosophic θ£s-closed and single-valued neutrosophic δ£s-closed sets as well as some of their core properties. We also present and explore the properties and characterizations of single valued neutrosophic operators namely θ£s-closure (CIτ~ϱ~σ~ς~θ£s) and δ£s-closure (CIτ~ϱ~σ~ς~δ£s) in the single valued neutrosophic ideal topological space (F~,τϱ~σ~ς~,£sϱ~σ~ς~). We then define the concept of single valued neutrosophic regularity spaces. Next, we study single-valued neutrosophic θ£s-separated and single-valued neutrosophic δ£s-separated with giving some definitions and theorems. Furthermore, we also introduce single-valued neutrosophic θ£s-connected and single valued neutrosophic δ£s-connected relying on the single valued neutrosophic θ£s-closure and δ£s-closure operators.

We define a fixed universe F~ to be a finite set of objects and ζ a closed unit interval [0, 1]. Additionally, we denote ζF as the set of all single-valued neutrosophic subsets of F~.

2  Preliminaries

This section provides a complete survey, some previous studies, and concepts associated with this study.

Definition 1. [1] Let F~ be a non-empty set. A neutrosophic set (briefly, NS) in F~ is an object having the form αn={υ,ϱ~αn(υ),σ~σn(ω),ς~αn(υ):υF~} where

ϱ~:F~0,1+,σ~:F~0,1+,ς~:F~0,1+ and 0ϱ~αn(υ)+σ~αn(υ)+ς~αn(υ)3+ (1)

Represent the degree of membership (ϱ~αn), the degree of indeterminacy (σ~αn), and the degree of non-membership (ς~αn) respectively of any υF~ to the set αn.

Definition 2. [4] Suppose that F~ is a universal set a space of points (objects), with a generic element in F~ denoted by υ. Then αn is called a single valued neutrosophic set (briefly, SVNS) in F~, if αn has the form αn={υ,ϱ~αn(υ),σ~αn(υ),ς~αn(υ):υF~}. Now, ϱ~αn,σ~σn,ς~αn indicate the degree of non-membership, the degree of indeterminacy, and the degree of membership, respectively of any υF~ to the set αn.

Definition 3. [4] Let αn={υ,ϱ~αn(υ),σ~σn(υ),ς~αn(υ):υF~} be an SVNS on F~. The complement of the set αn (briefly, αnc) defined as follows: ϱ~αnc(υ)=ς~αn(υ),σ~αn(υ)=[σ~αn]c(υ),ς~αnc(υ)=ϱ~αn(υ).

Definition 4. [26] Let F~ be a non-empty set and αn,εnζF~ be in the form: αn={υ,ϱ~αn(υ),σ~αn(υ),ς~αn(υ):υF~} and εn={υ,ϱ~εn(υ),σ~εn(υ),ς~εn(υ):υF~} on F~ then,

(a)   αnεn for every υF~; ϱ~αn(υ)ϱ~εn(υ),σ~αn(υ)σ~εn(υ),ς~αn(υ)ς~εn(υ).

(b)   αn=εn iff σnεn and σnεn.

(c)   0~=0,1,1 and 1~=1,0,0.

Definition 5. [20] Let αn,εnζF~. Then,

(a)   αnεn is an SVNS, if for every υF~,

αnεn=(ϱ~αnϱ~εn)(υ),(σ~αnσ~εn)(υ),(ς~αnς~εn)(υ), (2)

where, (ϱ~αnϱ~εn)(υ)=ϱ~αn(υ)ϱ~εn(υ) and (ς~αnς~εn)(υ)=ς~αn(υ)ς~εn(υ), for all υF~,

(b)   αnεn is an SVNS, if for every υF~,

αnεn=(ϱ~αnϱ~εn)(υ),(σ~αnσ~εn)(υ),(ς~αnς~εn)(υ). (3)

Definition 6. [15] For an any arbitrary family {αn}ijζF~ of SVNS the union and intersection are given by

(a)   ij[αn]i=ijϱ~[αn]i(υ),ijσ~[αn]i(υ),ijς~[αn]i(υ),

(b)   ij[αn]i=ijϱ~[αn]i(υ),ijσ~[αn]i(υ),ijς~[αn]i(υ).

Definition 7. [21] A single-valued neutrosophic topological spaces is an ordered (F~,τ~ϱ~,τ~σ~,τ~ς~) where τ~ϱ~,τ~σ~,τ~ς~:ζF~ζ is a mapping satisfying the following axioms:

(SVNT1) τ~ϱ~(0~)=τ~ϱ~(1~)=τ~σ~(0~)=τ~σ~(1~)=0 and τ~ς~(0~)=τ~ς~(1~)=1.

(SVNT2)τ~ϱ~(αnεn)τ~ϱ~(αn)τ~ϱ~(εn),τ~σ~(αnεn)τσ~(αn)τ~σ~(εn),τ~ς~(αnεn)τ~ς~(αn)τ~ς~(εn) for every, αn,εnζF~

(SVNT3)τ~ϱ~(jΓ[αn]j)jΓτ~ϱ~([αn]j), τ~σ~(iΓ[αn]j)jΓτ~σ~([αn]j),τ~ς~(jΓ[αn]j)jΓτ~ς~([αn]j), for every [αn]jζF~.

The quadruple (F~,τ~ϱ~,τ~σ~,τ~ς~) is called a single-valued neutrosophic topological spaces (briefly, SVNT, for short). Occasionally write τϱ~σ~ς~ for (τ~ϱ~,τ~σ~,τ~ς~) and it will cause no ambiguity.

Definition 8. [7] Let (F~,τϱ~σ~ς~) be an SVNTS. Then, for every αnζF~ and rζ0. Then the single valued neutrosophic closure and single valued neutrosophic interior of αn are define by:

Cτϱ~σ~ς~(αn,r)={εnζF~:αnεn,τϱ~([εn]c)r,τσ~([εn]c)1r,τς~([εn]c)1r} (4)

intτϱ~σ~ς~(αn,r)={εnζF~:αnεn,τϱ~(εn)r,τσ~(εn)1r,τς~(εn)1r} (5)

Definition 9. [7] Let (F~) be a nonempty set and υF~, let s(0,1], t[0,1) and k[0,1), then the single-valued neutrosophic point xs,t,k in F~ given by

xs,t,k(υ)={(s,t,k),if x=υ(0,1,1),otherwise.

We define that, xs,t,pαn iff s<ϱ~αn(υ), tσ~αn(υ) and kς~~αn(υ). We indicate the set of all single-valued neutrosophic points in F~ as Pxs,t,k(F~). A single-valued neutrosophic set αn is said to be quasi-coincident with another single-valued neutrosophic set εn, denoted by αnqεn, if there exists an element υF~ such that ϱ~αn(υ)+ϱ~εn(υ)>1,σ~αn(υ)+σ~εn(υ)1,ς~αn(υ)+ς~εn(υ)1.

Definition 10. [7] A mapping £sϱ~,£sσ~,£sς~:ζF~ζ is called single-valued neutrosophic ideal (SVNI) on F~ if, it satisfies the following conditions:

(£s1) £sϱ~(0~)=1 and  \pounds σ~(0~)=£sς~(0~)=0.

(£s2) If σnγn, then £sϱ~(εn)£sϱ~(αn), £sσ~(εn)£sσ~(αn) and £sς~(εn)£sς~(αn), for εn,αnζF~.

(£s3) £sϱ~(αnεn)£sϱ~(αn)£sϱ~(εn), £sσ~(αnεn)£sσ~(αn)£sσ~(εn) and £sς~(αnεn)£sς~(αn)£sς~(εn), for αn,εnζF~.

The tribal (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is called a single valued neutrosophic ideal topological space in Šostak sense (briefly, SVNITS).

Definition 11. [7] Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS for each αnζF~. Then, the single valued neutrosophic ideal open local function [αn]r(τϱ~σ~ς~,£sϱ~σ~ς~) of αn is the union of all single-valued neutrosophic points xs,t,k such that if εnQτϱ~σ~ς~(xs,t,k,r) and £sϱ~(ωn)r, £sσ~(ωn)1r, £sς~(ωn)1r, then there is at least one υF~ for which

ϱ~αn(υ)+ϱ~εn(ν)1>ϱ~ωn(υ),σ~αn(υ)+σ~εn(υ)1σ~ωn(υ),ς~αn(υ)+ς~εn(υ)1ς~ωn(υ) (6)

Occasionally, we will write [αn]r for [αn]r(τϱ~σ~ς~,£sϱ~σ~ς~) herein to avoid ambiguity.

Remark 1. [7] Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and αnζF~. Hence, we can write

CIτϱ~σ~ς~(αn,r)=αn[αn]r,intτϱ~σ~ς~(αn,r)=αn[(αnc)r]c (7)

Clearly, CIϱ~σ~ς~ is a single-valued neutrosophic closure operator and (τϱ~(£s),τσ~(£s),τς~(£s)) is the single-valued neutrosophic topology generated by CIτϱ~σ~ς~, i.e., τ(J)(αn)={r|CIτ~ϱ~σ~ς~(αnc,r)=αnc}.

Theorem 1. [7] Let {[αn]i}iJζF~ be a family of single-valued neutrosophic sets on F~ and (F~,τ~ϱ~σ~ς~,£sϱ~σ~ς~) be a SVNITS. Then,

(a)   (([αn]i)r:iJ)([αn]i:ij)r,

(b)   (([αn]i):ij)r(([αn]i)r:iJ).

Theorem 2. [7] Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and rζ, αn,εnζF~. Then,

(a)   intτ~ϱ~σ~ς~(αnεn,r)intτ~ϱ~σ~ς~(αn,r)intτ~ϱ~σ~ς~(εn,r),

(b)   intτ~ϱ~σ~ς~(αn,r)intτ~ϱ~σ~ς~(αn,r)αnCIτ~ϱ~σ~ς~(αn,r)Cτ~ϱ~σ~ς~(αn,r),

(c)   CIτ~ϱ~σ~ς~([αn]c,r)=[intτ~ϱ~σ~ς~(αn,r)]c,

(d)   [CIτ~ϱ~σ~ς~(αn,r)]c=intτ~ϱ~σ~ς~([αn]c,r),

(e)   intτ~ϱ~σ~ς~(αnεn,r)=intτ~ϱ~σ~ς~(αn,r)intτ~ϱ~σ~ς~(εn,r).

Definition 12. [8] Let (F~,τϱ~σ~ς~) be an SVNITS. For every αn,εn,ωnζF~, αn and εn are called r-single-valued neutrosophic separated if for rζ0,

CIτϱ~σ~ς~(αn,r)εn=CIτϱ~σ~ς~(εn,r)αn=0~ (8)

An SVNS, ωn is called r-single-valued neutrosophic connected if r-SVNSEP  αn,εnζF~{0~} such that ωn=αnεn does not exist. A SVNS  αn is said to be r-single-valued neutrosophic connected if it is r-single-valued neutrosophic connected for any rζ0. A (F~,τϱ~σ~ς~) is said to be r-single-valued neutrosophic connected if 1~ is r-single-valued neutrosophic connected.

3  Single Valued Neutrosophic δ£s-Cluster Point and Single Valued Neutrosophic θ£s-Cluster Point

In this section, we introduce the r-single-valued neutrosophic δ£s-cluster point (abbreviated SVNδ£s-cluster point) and r-single-valued neutrosophic £s-closed set (abbreviated SVN£sC). Furthermore, we analyze the single-valued neutrosophic δ£s-closure operator (δ£s-closure operator for brevity) and single-valued neutrosophic θ£s-closure operator (θ£s-closure operator for brevity).

Definition 13. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and αnζF~, rζ0. Then,

(a)   αn is said to be r-single valued neutrosophic £s-open (briefly, r-SVN£sO), if and only if αnintτ~τ~ϱ~σ~ς~([αn]r,r),

(b)   αn is said to be r-single valued neutrosophic semi-£s-open (briefly, r-SVNS£sO) if and only if αnCIτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~([αn]r,r),r),

(c)   αn is called r-single valued neutrosophic pre-£s-open (briefly, r-SVNP£sO) if and only if αnintτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~([αn]r,r),r),

(d)   αn is called r-single valued neutrosophic regular-£s-open (briefly, r-SVNR£sO) if and only if αn=intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~([αn]r,r),r),

(e)   αn is said to be r-single valued neutrosophic α£s-open (briefly, r-SVNα£sO) if and only if αnintτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(intτ~ϱ~([αn]r,r),r),

(f)   αn is said to be r-single valued neutrosophic -open set (briefly, r-SVN O) if and only if αn=CIτ~ϱ~σ~ς~(αn,r).

The complement of an rSVN\pounds O (resp, r-SVNS£sO, r-SVNP£sO, r-SVNR£sO, r-SVNα£sO, r-SVNO) is said to be an rSVN\pounds C (resp, r-SVNS£sC, r-SVNP£sC, r-SVNR£sC, r-SVNα£sC, r-SVNC) respectively.

Remark 2. r-single valued neutrosophic open set (rSVNO) and r-SVN£sO are independent notions as shown by the following example.

Example 1. Let F~={a,b,c} be a set. Define εn,πn,ωnζF~ as follows:

εn=(0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3);πn=(0.4,0.4,0.4),(0.4,0.4,0.4),(0.4,0.4,0.4), ωn=(0.5,0.5,0.5),(0.2,0.2,0.2),(0.1,0.1,0.1).

We define an SVNITS (τϱ~σ~ς~,£sϱ~σ~ς~) on F~ as follows: for each αnζF~,

τ~ϱ~(αn)={1,if αn={0~,1~},23,if αn={εn,πn},0,otherwise,£sϱ~(αn)={1,if αn=0~,23,if 0<αnωn0,otherwise, τ~σ~(αn)={0,if αn={0~,1~},13,if αn={εn,πn},1,otherwise,£sσ~(αn)={0,if αn=0~,13,if 0<αnωn,1,otherwise, τ~ς~(αn)={0,if αn={0~,1~},13,if αn={εn,πn},1,otherwise,£sς~(αn)={0,if αn=0~,13,if 0<αnωn,1,otherwise.

Based on εn=(0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3), it’s clear that, 23SVNO is set because τϱ~((0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3))23,τσ~((0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3))13,τς~((0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3))13.

However εn is not an r-SVN£sO set, and for that, we must prove that εnintτ~ϱ~σ~ς~([εn]23,23). So, we must first obtain [εn]23. Based on Eq. (11), 1~,εn,πnQτϱ~σ~ς~(xs,t,k,23) and £sϱ~((0.5,0.5,0.5),(0.2,0.2,0.2),(0.1,0.1,0.1))23, £sσ~((0.5,0.5,0.5),(0.2,0.2,0.2),(0.1,0.1,0.1))13, £sς~((0.5,0.5,0.5),(0.2,0.2,0.2),(0.1,0.1,0.1))13,

such that by using Eqs. (2), (3) and (6) we obtain,

ϱ~εn(υ)+ϱ~1~(ν)1>ϱ~ωn(υ),σ~εn(υ)+σ~1~(υ)1σ~ωn(υ),ς~εn(υ)+ς~1~(υ)1ς~ωn(υ). (0.3,0.3,0.3)(υ)+(1,1,1)(ν)1(0.5,0.5,0.5)(υ), (0.3,0.3,0.3)(υ)+(0,0,0)(υ)1(0.2,0.2,0.2)(υ),

(0.3,0.3,0.3)(υ)+(0,0,0)(υ)1(0.1,0.1,0.1)(υ),

ϱ~εn(υ)+ϱ~πn(ν)1>ϱ~ωn(υ),σ~εn(υ)+σ~πn(υ)1σ~ωn(υ),ς~εn(υ)+ς~πn(υ)1ς~ωn(υ). (0.3,0.3,0.3)(υ)+(0.4,0.4,0.4)(ν)1(0.5,0.5,0.5)(υ), (0.3,0.3,0.3)(υ)+(0.4,0.4,0.4)(υ)1(0.2,0.2,0.2)(υ), (0.3,0.3,0.3)(υ)+(0.4,0.4,0.4)(υ)1(0.1,0.1,0.1)(υ) ϱ~εn(υ)+ϱ~εn(ν)1>ϱ~ωn(υ),σ~εn(υ)+σ~εn(υ)1σ~ωn(υ),ς~εn(υ)+ς~εn(υ)1ς~ωn(υ). (0.3,0.3,0.3)(υ)+(0.3,0.3,0.3)(ν)1(0.5,0.5,0.5)(υ), (0.3,0.3,0.3)(υ)+(0.3,0.3,0.3)(υ)1(0.2,0.2,0.2)(υ), (0.3,0.3,0.3)(υ)+(0.3,0.3,0.3)(υ)1(0.1,0.1,0.1)(υ)

Therefore, [εn]23=0~. Subsequently, using Eq. (7) we obtain intτ~ϱ~σ~ς~([εn]23,23)=intτ~ϱ~σ~ς~(0~,23)=0~, which implies that

(0.3,0.3,0.3),(0.3,0.3,0.3),(0.3,0.3,0.3)=εnintτ~ϱ~σ~ς~([εn]23,23)=0~.

Hence, εn is not an r-SVN£sO set.

Definition 14. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, αnζF~, xs,t,kPs,t,k(F~) and rζ0. Then,

(a)   αn is an r-single valued neutrosophic Qτϱ~σ~ς~-neighborhood of xs,t,k if xs,t,kqαn with τϱ~(αn)r,τσ~(αn)1r,τς~(αn)1r;

(b)   xs,t,k is an r-single valued neutrosophic θ£s-cluster point (r-δ£s-cluster point) of αn if for every εnQτ~ϱ~σ~ς~(xs,t,k,r), we have αnqintτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(εn,r),r);

(c)   δ£s-closure operator is the mapping of CIτ~ϱ~σ~ς~δ£s:ζF~×ζ0ζF~ defined as

CIτ~ϱ~σ~ς~δ£s(αn,r)={xs,t,kPs,t,k(F~):xs,t,k is rδ£scluster point of αn}.

Definition 15. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, αnζF~, xs,t,kPs,t,k(F~) and rζ0. Then,

(a)   αn is called r-Single valued neutrosophic Rτϱ~σ~ς~£s-neighborhood of xs,t,k if xs,t,kqαn and αn is r-SVNRIO. We denote Rτϱ~σ~ς~£s={αnζF~|xs,t,kqαn,αn is rSVNRIO},

(b)   xs,t,k is called r-single valued neutrosophic θ£s-cluster point (r-θ£s-cluster point) of αn if for any εnQτ~ϱ~σ~ς~(xs,t,k,r), we have αnqCIτ~ϱ~σ~ς~(εn,r),

(c)   θ£s-closure operator is mapping CIτ~ϱ~σ~ς~θ£s:ζF~×ζ0ζF~ defined as

CIτ~ϱ~σ~ς~θ£s(αn,r)={xs,t,kPs,t,k(F~):xs,t,k is rθ£scluster point of αn} (9)

Example 2. Let F~={a,b,c} be a set. Define εn,πnζF~ as follows:

εn=(0.4,0.4,0.4),(0.4,0.4,0.4),(0.4,0.4,0.4);πn=(0.2,0.2,0.2),(0.2,0.2,0.2),(0.2,0.2,0.2).

We define an SVNITS (τϱ~σ~ς~,£sϱ~σ~ς~) on F~ as follows: for each αnζF~,

τ~ϱ~(αn)={1,if αn=0~,1,if αn=1~,23,if αn=εn,0,otherwise,£sϱ~(αn)={1,if αn=0~,13,if πn=εn23,if 0<αn<πn0,otherwise, τ~σ~(αn)={0,if αn=0~,0,if αn=1~,13,if αn=εn,1,otherwise,£sσ~(αn)={0,if αn=0~,23,if πn=εn13,if 0<αn<πn1,otherwise, τ~ς~(αn)={0,if αn=0~,0,if αn=1~,13,if αn=εn,1,otherwise,£sς~(αn)={0,if αn=0~,23,if πn=εn13,if 0<αn<πn1,otherwise,

From using (9) we get, we obtain

CIτ~ϱ~σ~ς~θ£s(αn,r)={0~,if αn=0~,εnc,if 0~αnεnc,r13,1r23,1,otherwise.

Theorem 3. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, rζ0 and αn,εnζF~. Then the following properties are holds:

(a)   αnCIτ~ϱ~σ~ς~δ£s(αn,r),

(b)   If αnεn, then CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(εn,r),

(c)   intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(αn,r),r) is r-SVNRIO,

(d)   CIτ~ϱ~σ~ς~δ£s(αn,r)={εnζF~|αnεn,εn is r-SVNRIC},

(e)   CIτ~ϱ~σ~ς~(αn,r)CIτ~ϱ~σ~ς~δ£s(αn,r).

Proof. (a) and (b) are easily proved from (9).

(c) Let εnζF~ and εn=intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(αn,r),r). Then, we have

intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(αn,r),r)=intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(αn,r),r),r),r) intτ˜ ϱ˜σ˜ς˜(CIτ˜ ϱ˜σ˜ς˜(CIτ˜ ϱ˜σ˜ς˜(αn,r),r),r) =intτ˜ ϱ˜σ˜ς˜(CIτ˜ ϱ˜σ˜ς˜(αn,r),r)=εn. 

Since εn=intτ~ϱ~σ~ς~(εn,r)intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(εn,r),r), we have intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(εn,r),r)=εn.

(d) Based on P={εnζF~|αnεn,εn is r-SVNRIC}, let CIτ~ϱ~σ~ς~δ£s(αn,r)P; therefore, υF~ and s(0,1], t[0,1),k[0,1)] exist such that

ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s<ϱ~P(υ)σ~CIτ~σ~δ£s(αn,r)(υ)tσ~P(υ)ς~CIτ~ς~δ£s(αn,r)(υ)kς~P(υ)} (10)

Therefore, xs,t,k is not an r-δ£s-cluster point of αn. As such, εnQτ~ϱ~σ~ς~(xs,t,k,r) and αn[intτ~ϱ~σ~ς~(εn,r)]c. Consequently, αn[intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(εn,r),r)]c=Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~([εn]c,r),r).

Since Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~([εn]c,r),r) is r-SVNRIC, we have ϱ~P(υ)ϱ~Clτ~ϱ~(intτ~ϱ~([εn]c,r),r)(υ)<s,σ~P(υ)σ~Clτ~σ~(intτ~σ~~([εn]c,r),r)(υ)>t and ς~P(υ)ς~Clτ~ς~(intτ~ς~~([εn]c,r),r)(υ)>k. This is a contradiction to Eq. (10). Therefore, CIτ~ϱ~σ~ς~δ£s(αn,r)P.

Meanwhile, by setting CIτ~ϱ~σ~ς~δ£s(αn,r)P, then an r-δ£s-cluster point of ys1,t1,k1Ps,t,k(F~) of αn exists such that

ϱ~CIτ~ϱ~δ£s(αn,r)(y)>s1>ϱ~P(y)σ~CIτ~σ~δ£s(αn,r)(y)t1σ~P(y)ς~CIτ~ς~δ£s(αn,r)(y)k1ς~P(y)} (11)

Owing to P, there exists r-SVNRIC  εnζF~  with  αnεn such that ϱ~CIτ~ϱ~δ£s(αn,r)(y)>s1>ϱ~εnϱ~P(y), σ~CIτ~σ~δ£s(αn,r)(y)t1ϱ~εnσ~P(y) and ς~CIτ~ς~δ£s(αn,r)(y)k1ϱ~εnς~P(y). Therefore, [εn]cQτ~ϱ~σ~ς~(ys1,t1,k1). So, αnεn=[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([εn]c,r),r)]c. Hence, αnq¯intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([εn]c,r),r).

Additionally, ys1,t1,k1 is not an r-δ£s-cluster point of αn, that is, ϱ~CIτ~ϱ~δ£s(αn,r)(y)<s1,σ~CIτ~σ~δ£s(αn,r)(y)t1, ς~CIτ~ς~δ£s(αn,r)(y)k1. This is a contradiction to Eq. (11). Therefore, CIτ~ϱ~σ~ς~δ£s(αn,r)P,

(e) Suppose that Clτ~ϱ~σ~ς~(αn,r)CIτ~ϱ~σ~ς~δ£s(αn,r); therefore, υF~ and [s(0,1],t[0,1), k[0,1)] exist such that

ϱ~CIτ~ϱ~(αn,r)(υ)>s>ϱ~CIτ~ϱ~δ£s(αn,r)(υ)σ~CIτ~σ~(αn,r)(υ)tσ~CIτ~σ~δ£s(αn,r)(υ)ς~CIτ~σ~(αn,r)(υ)kς~CIτ~ς~δ£s(αn,r)(υ),} (12)

Since, ϱ~CIτ~ϱ~(αn,r)(υ)<s,σ~CIτ~σ~(αn,r)(υ)t,ς~CIτ~σ~(αn,r)(υ)k, we have xs,t,k not r-δ£s-cluster point of αn. Therefore, there exists εnQτ~ϱ~σ~ς~(xs,t,k,r) and αn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)]c. Hence, ϱ~Clτ~ϱ~(αn,r)(υ)ϱ~[intτ~ϱ~(Clτ~ϱ~(εn,r),r)]c(υ)<s,  σ~Clτ~σ~(αn,r)(υ)ϱ~[intτ~σ~(Clτ~σ~(εn,r),r)]c(υ)t and ς~Clτ~ς~(αn,r)(υ)ϱ~[intτ~ς~(Clτ~ς~(εn,r),r)]c(υ)k. It is a contradiction for Eq. (12). Thus Clτ~ϱ~σ~ς~(αn,r)CIτ~ϱ~σ~ς~δ£s(αn,r).

Theorem 4. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, for each rζ0 and αn,εnζF~. Then the following properties hold:

(a) αnCIτ~ϱ~σ~ς~θ£s(αn,r),

(b) If αnεn, then CIτ~ϱ~σ~ς~θ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(εn,r),

(c) CIτ~ϱ~σ~ς~(αn,r){xs,t,kPs,t,k(F~)|xs,t,k is r-δ£s-cluster point of αn},

(d) CIτ~ϱ~σ~ς~θ£s(αn,r)={εnζF~|αnintτ~ϱ~σ~ς~(εn,r), τϱ~([εn]c)r, τσ~([εn]c)1r, τς~([εn]c)1r},

(e) CIτ~ϱ~σ~ς~δ£s(αn,r)={εnζF~|αnεn,εn is r-δ£s-cluster point of αn}

(f) xs,t,k is r-θ£s-cluster point of αn iff xs,t,kCIτ~ϱ~σ~ς~θ£s(αn,r),

(g) xs,t,k is r-δ£s-cluster point of αn iff xs,t,kCIτ~ϱ~σ~ς~δ£s(αn,r),

(h) If αn=Clτ~ϱ~σ~ς~(intlτ~ϱ~σ~ς~(αn,r),r), then CIτ~ϱ~σ~ς~δ£s(αn,r)= αn,

(i) αnCIτ~ϱ~σ~ς~(αn,r)CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(αn,r),

(j) W(αnεn,r)=W(αn,r)W(εn,r)for each W={CIτ~ϱ~σ~ς~θ£s,CIτ~ϱ~σ~ς~δ£s},

(k) CIτ~ϱ~σ~ς~δ£s(CIτ~ϱ~σ~ς~δ£s(αn,r),r)=CIτ~ϱ~σ~ς~δ£s(αn,r).

Proof. (a) and (b) are easily proved from Definition 14.

(c) Set P={xs,t,kPs,t,k(F~)|xs,t,k as an r-δ£s-cluster point of αn}. Suppose that CIτ~ϱ~σ~ς~(αn,r)P. Then there exists υF~, and [s(0,1], t[0,1), k[0,1)] such that

ϱ~CIτ~ϱ~(αn,r)(υ)>s>ϱ~P(υ)σ~CIτ~σ~(αn,r)(υ)tσ~P(υ)ς~CIτ~ς~(αn,r)(υ)kς~P(υ)} (13)

Consequently, xs,t,k is not r-δ£s-cluster point of αn. So, there exists εnQτϱ~σ~ς~(xs,t,k,r) and

αn[intτ~ϱ~σ~ς~(CIτ~ϱ~σ~ς~(εn,r),r)]c[εn]c

Based on Eq. (4), ϱ~CIτ~ϱ~(αn,r)(υ)ϱ~[εn]c(υ)<s,σ~CIτ~σ~(αn,r)(υ)σ~[εn]c(υ)t and ς~CIτ~ς~(αn,r)(υ)ς~[εn]c(υ)k.

It is a contradiction for Eq. (13). Thus CIτ~ϱ~σ~ς~(αn,r)P.

(d) γ={εnζF~|αnintτ~ϱ~σ~ς~(εn,r), τϱ~([εn]c)r, τσ~([εn]c)1r, τς~([εn]c)1r}.

Suppose that CIτ~ϱ~σ~ς~θ£s(αn,r)γ, then there exists υF~ and [s(0,1], t[0,1), k[0,1)] such that

ϱ~CIτ~ϱ~σ~ς~θ£s(αn,r)(υ)<sϱ~γ(υ)σ~CIτ~σ~θ£s(αn,r)(υ)>tσ~γ(υ)ς~CIτ~ς~θ£s(αn,r)(υ)>kς~γ(υ)} (14)

Consequently, xs,t,k is not r-θ£s-cluster point of αn. So, there exists εnQτϱ~σ~ς~(xs,t,k,r) , αn[(Clτ~ϱ~σ~ς~(εn,r),r)]c. Thus, αn[(Clτ~ϱ~σ~ς~(εn,r),r)]c=(intτ~ϱ~σ~ς~([εn]c,r),r) , τϱ~(εn)r, τσ~(εn)1r, τς~(εn)1r}. Hence, ϱ~γ(υ)ϱ~[εn]c(υ)<s,σ~γ(υ)σ~[εn]c(υ)<t,ς~γ(υ)ς~[εn]c(υ)<k.

It is a contradiction to Eq. (14). Thus CIτ~ϱ~σ~ς~θ£s(αn,r)γ.

Suppose that CIτ~ϱ~σ~ς~θ£s(αn,r)γ, then there exists r-θ£s-cluster point of αn. ys1,t1,k1Ps,t,k(F~) of αn, such that

ϱ~CIτ~ϱ~θ£s(αn,r)(y)>s1>ϱ~γ(y)σ~CIτ~σ~θ£s(αn,r)(y)<t1σ~γ(y)ς~CIτ~ς~θ£s(αn,r)(y)<k1ς~γ(y)} (15)

By the definition of γ, there exists εnζF~ with τϱ~(εn)r, τσ~(εn)1r,τς~(εn)1r and αnintτ~ϱ~σ~ς~(εn,r), s.t ϱ~CIτ~ϱ~θ£s(αn,r)(y)>s1>ϱ~εn(y)ϱ~γ(y), σ~CIτ~σ~θ£s(αn,r)(y)<t1σ~εn(y)σ~γ(y) and ς~CIτ~ς~θ£s(αn,r)(y)<k1ς~εn(y)ς~γ(y). Additionally, [εn]cQτϱ~σ~ς~(ys1,t1,k1,r). αnintτ~ϱ~σ~ς~(εn,r)=[Clτ~ϱ~σ~ς~([εn]c,r)]c, implies αnq¯Clτ~ϱ~σ~ς~([εn]c,r). Hence ys1,t1,k1 is not an r-θ£s-cluster point of αn. It is a contradiction for Eq. (15). Thus CIτ~ϱ~σ~ς~θ£s(αn,r)γ.

(e) Similar results are shown in (c) and (d).

(f) (), clear.

() Suppose that xs,t,k is not an r-θ£s-cluster point of αn. There exists εnQτϱ~σ~ς~(xs,t,k,r) such that Clτ~ϱ~σ~ς~(εn,r)αn. Thus αn[Clτ~ϱ~σ~ς~(εn,r)]c=Clτ~ϱ~σ~ς~([εn]c,r). By (d), ϱ~CIτ~ϱ~θ£s(αn,r)(υ)ϱ~[εn]c(υ)<s, σ~CIτ~σ~θ£s(αn,r)(υ)σ~[εn]c(υ)>t and ς~CIτ~ς~θ£s(αn,r)(υ)ς~[εn]c(υ)>t. Hence xs,t,kCIτ~ϱ~σ~ς~θ£s(αn,r).

(g) is similarly proved as in (f).

(h) The validity of this axiom is obvious from Theorem 3 (4).

(i) Based on Theorem 3(e), we show that CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(αn,r). Suppose that CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(αn,r), then there exists υζ and [s(0,1], t[0,1), k[0,1)] such that

ϱ~CIτ~ϱ~δ£s(αn,r)(υ)>s>ϱ~CIτ~ϱ~θ£s(αn,r)(υ)σ~CIτ~σ~δ£s(αn,r)(υ)>tσ~CIτ~σ~θ£s(αn,r)(υ)ς~CIς~δ£s(αn,r)(υ)>kς~CIτ~ς~θ£s(αn,r)(υ)} (16)

Since ϱ~CIτ~ϱ~θ£s(αn,r)(υ)<s,σ~CIτ~σ~θ£s(αn,r)(υ)t and ς~CIτ~ς~θ£s(αn,r)(υ)k, then we have xs,t.k is not r-θ£s-cluster point of αn So, there exists εnQτϱ~σ~ς~(ys1,t1,k1,r), αn[Clτ~ϱ~σ~ς~(εn,r)]c, implies Aq¯intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r). Hence, xs,t,k is not r-δ£s-cluster point of αn, by (7), we can get than, ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s,σCIτ~σ~δ£s(αn,r)(υ)t,ς~CIτ~ς~δ£s(αn,r)(υ)k. It is a contradiction for Eq. (16). Thus, CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(αn,r).

(j) Let CIτ~ϱ~σ~ς~δ£s(εn,r)CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(αnεn,r). Then there exists υF~ such that

ϱ~CIτ~ϱ~δ£s(εn,r)(υ)ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s<ϱ~CIτ~ϱ~δ£s(αnεn,r)(υ)σ~CIτ~σ~δ£s(εn,r)(υ)σ~CIτ~σ~δ£s(αn,r)(υ)>t>σ~CIτ~ϱ~δ£s(αnεn,r)(υ)ς~CIτ~ς~δ£s(εn,r)(υ)ς~CIτ~σ~δ£s(αn,r)(υ)>t>ς~CIτ~ς~δ£s(αnεn,r)(υ)} (17)

Since ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s, σ~CIτ~σ~δ£s(αn,r)(υ)>t, ς~CIτ~ς~δ£s(αn,r)(υ)>k and ϱ~CIτ~ϱ~δ£s(εn,r)(υ)<s, σ~CIτ~σ~δ£s(εn,r)(υ)>t, ς~CIτ~ς~δ£s(εn,r)(υ)>k. We obtain, xs,t,k is not r-δ£s-cluster point of αn and εn So, there exists [αn]1,[εn]1Qτϱ~σ~ς~(xs,t,k,r), and αn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([αn]1,r),r)]c,εn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([εn]1,r),r)]c. Thus, [αn]1[εn]1Qτϱ~σ~ς~(xs,t,k,r).

Using Eqs. (4) and (5) we obtain,

αnεn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([αn]1,r),r)intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([εn]1,r),r)]c [intτ˜ ϱ˜σ˜ς˜(Clτ˜ ϱ˜σ˜ς˜([αn]1,r)Clτ˜ ϱ˜σ˜ς˜([εn]1,r),r)]c [intτ˜ ϱ˜σ˜ς˜(Clτ˜ ϱ˜σ˜ς˜([αn]1[εn]1,r),r)]c. 

Therefore, αnεnq¯ intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([αn]1[εn]1,r),r). Hence, xs,t,k is not r-δ£s-cluster point of αnεn, by (g), ϱ~Clτ~ϱ~(αnεn,r)(υ)<s,σ~Clτ~σ~αnεn(,r)(υ)>t and ς~Clτ~ς~(αnεn,r)(υ)>k. It is a contradiction for Eq. (17), and hence, CIτ~ϱ~σ~ς~δ£s(αnεn,r)CIτ~ϱ~σ~ς~δ£s(εn,r)CIτ~ϱ~σ~ς~δ£s(αn,r).

Meanwhile, αnεnαn and αnεnεn. Hence CIτ~ϱ~σ~ς~δ£s(αnεn,r)CIτ~ϱ~σ~ς~δ£s(εn,r)CIτ~ϱ~σ~ς~δ£s(αn,r). Therefore, CIτ~ϱ~σ~ς~δ£s(εn,r)CIτ~ϱ~σ~ς~δ£s(αn,r)=CIτ~ϱ~σ~ς~δ£s(αnεn,r).

(k) Since αnCIτ~ϱ~σ~ς~δ£s(αn,r), we have CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(CIτ~ϱ~σ~ς~δ£s(αn,r),r). On the other hand, suppose that CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(CIτ~ϱ~σ~ς~δ£s(αn,r),r). Then there exists υF~ and [s(0,1], t[0,1), k[0,1)] such that

ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s<ϱ~CIτ~ϱ~δ£s(CIτ~ϱ~δ£s(αn,r),r)(υ)σ~CIτ~σ~δ£s(αn,r)(υ)>tσ~CIτ~ϱ~δ£s(CIτ~σ~δ£s(αn,r),r)(υ)ς~CIτ~ς~δ£s(αn,r)(υ)>kς~CIτ~ϱ~δ£s(CIτ~ς~δ£s(αn,r),r)(υ)} (18)

Since ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s,σ~CIτ~σ~δ£s(αn,r)(υ)>t,ς~CIτ~ς~δ£s(αn,r)(υ)>k, we have xs,t,k is not an r-δ£s-cluster point of αn. So, there exists εnQτϱ~σ~ς~(xs,t,k,r) such that αn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)]c=Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~(εn,r,r), since, Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~(εn,r,r) is rSVNRIC. Then by Theorem 3(d), CIτ~ϱ~σ~ς~δ£s(αn,r)Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~(εn,r,r).

Similarly, CIτ~ϱ~σ~ς~δ£s(CIτ~ϱ~σ~ς~δ£s(αn,r),r)CIτ~ϱ~σ~ς~δ£s(Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~(εn,r),r),r)=Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~(εn,r),r). Hence,

CIτ~ϱ~σ~ς~δ£s(CIτ~ϱ~σ~ς~δ£s(αn,r),r)Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~(εn,r),r)<xs,t,k. It is a contradiction for Eq. (18).

Theorem 5. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, for rζ0 and αn,εnζF~. Then the following properties hold:

(a)   αn is r-SVNPIC iff CIτ~ϱ~σ~ς~(αn,r)=CIτ~ϱ~σ~ς~δ£s(αn,r),

(b)   αn is r-SVNSIC iff CIτ~ϱ~σ~ς~(αn,r)=CIτ~ϱ~σ~ς~δ£s(αn,r),

(c)   αn is r-SVNαIO iff CIτ~ϱ~σ~ς~(αn,r)=CIτ~ϱ~σ~ς~δ£s(αn,r)=CIτ~ϱ~σ~ς~θ£s(αn,r).

Proof. (a) Let αn be an r-SVNPIC. Then αnCIτ~ϱ~σ~ς~(αn,r), and by Theorem 3 (3) and (4), we have

CIτ˜ ϱ˜σ˜ς˜δ£(αn,r)CIτ˜ ϱ˜σ˜ς˜δ£(CIτ˜ ϱ˜σ˜ς˜(intτ˜ ϱ˜σ˜ς˜(αn,r),r),r)=Clτ˜ ϱ˜σ˜ς˜(intτ˜ ϱ˜σ˜ς˜(αn,r),r)Clτ˜ ϱ˜σ˜ς˜(αn,r)CIτ˜ ϱ˜σ˜ς˜δ£(αn,r).

Conversely, suppose that there exist υF~ and [s(0,1], t[0,1), k[0,1)] such that ϱ~CIτ~ϱ~δ£s(αn,r)(υ)>s>ϱ~Clτ~ϱ~(αn,r)(υ),σ~CIτ~σ~δ£s(αn,r)(υ)<tσ~Clτ~σ~(αn,r)(υ) and ς~CIτ~ς~δ£s(αn,r)(υ)<kς~Clτ~ς~(αn,r)(υ). Then xs,t,k is not r-δ-cluster point of αn. So, there exists εnQτϱ~σ~ς~(xs,t,k,r), with αn[εn]c Since xs,t,k is r-δ£s-cluster point of αn, for εnQτϱ~σ~ς~(xs,t,k,r), we have intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)qαn. Since,

intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([αn]c,r),r),

we obtain, αn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)]c[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([αn]c,r),r)]c=Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~([αn],r),r).

Hence, αn is not r-SVNIC set.

(b) Let αn is an r-SVNSIC set. Then, αnintτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([αn]c,r),r) andτϱ~([Clτ~ϱ~([αn,r)]cr,τσ~([Clτ~σ~([αn,r)]cr,τς~([Clτ~ς~([αn,r)]cr. By Theorem 4(d), we have CIτ~ϱ~σ~ς~θ£s(αn,r)CIτ~ϱ~σ~ς~(αn,r),

Conversely, suppose that there exist αnζF~,rζ0, υF~ and [s(0,1], t[0,1), k[0,1)] such that ϱ~CIτ~ϱ~θ£s(αn,r)(υ)>t>ϱ~Clτ~ϱ~(αn,r)(υ),σ~CIτ~σ~θ£s(αn,r)(υ)<tσ~Clτ~σ~(αn,r)(υ) and ς~CIτ~ς~θ£s(αn,r)(υ)<tς~Clτ~ς~(αn,r)(υ). Then [Clτ~ϱ~σ~ς~(αn,r]c)=intτ~ϱ~σ~ς~([αn]c,r)Qτϱ~σ~ς~(xs,t,k,r) Since xs,t,k is r-θ£s-cluster point of αn, we have Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~([αn]c,r),r)qαn. It implies αn[Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~([αn]c,r),r)]c=intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(αn,r),r). Thus, αn is not an r-SVNSIC.

(c) Similar results are shown in (a) and (b).

4  r-δ£s-Closed and r-θ£s-Closed

In this section, we firstly introduce and analyze the r-δ£s-closed and r-θ£s-closed of an SVNITS (F~,τϱ~σ~ς~,£sϱ~σ~ς~). Subsequently, we define and analyze the single-valued neutrosophic £s-regular and the single-valued neutrosophic almost £s-regular of F~. The findings have resulted in many theorems.

Definition 16. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS. For rζ0 and αn,εnζF~. Therefore,

(a) αn is said to be r-δ£s-closed ([αn]δ£s) [resp. r-θ£s-closed [αn]θ£s] iff CIτ~ϱ~σ~ς~δ£s(αn,r)=αn (resp. CIτ~ϱ~σ~ς~θ£s(αn,r)=αn). We define

Δτ~ϱ~σ~ς~δ£s(αn,r)={εn|αnεn,εn=CIτ~ϱ~σ~ς~δ£s(εn,r)} (19)

Θτ~ϱ~σ~ς~θ£s(αn,r)={εn|αnεn,εn=CIτ~ϱ~σ~ς~θ£s(εn,r)} (20)

(b) The complement of r-δ£s-closed (resp. r-θ£s-closed) set is called r-δ£s-open (resp. r-θ£s-open).

Theorem 6. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS. For rζ0 and αnζF~. Then the following properties are holds:

(c). Δτ~ϱ~σ~ς~δ£s(αn,r)=CIτ~ϱ~σ~ς~δ£s(αn,r),

(d). Δτ~ϱ~σ~ς~δ£s(αn,r) is r-δ£s-closed,

(e). Θτ~ϱ~σ~ς~θ£s(αn,r)=CIτ~ϱ~σ~ς~θ£s(Θτ~ϱ~σ~ς~δ£s(αn,r),r),

(f). Θτ~ϱ~σ~ς~θ£s(αn,r) is r-θ£s-closed,

(g). CIτ~ϱ~σ~ς~θ£s(αn,r)Θτ~ϱ~σ~ς~θ£s(αn,r).

Proof. (1) Based on Theorem 4(i,j), αnCIτ~ϱ~σ~ς~δ£s(αn,r)=CIτ~ϱ~σ~ς~δ£s(CIτ~ϱ~σ~ς~δ£s(αn,r),r), which implies Δτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(αn,r). Suppose that Δτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(αn,r). Then there exist υF~ and [s(0,1], t[0,1), k[0,1)] such that ϱ~Δτ~ϱ~δ£s(αn,r)(υ)<s<ϱ~CIτ~ϱ~δ£s(αn,r)(υ),σ~Δτ~σ~δ£s(αn,r)(υ)>t>σ~CIτ~σ~δ£s(αn,r)(υ) and ς~Δτ~ς~δ£s(αn,r)(υ)>k>ς~CIτ~ς~δ£s(αn,r)(υ). Based on Eq. (19), there exist εnζF~ and αnεn=CIτ~ϱ~σ~ς~δ£s(εn,r) such that ϱ~Δτ~ϱ~δ£s(αn,r)(υ)ϱ~εn(υ)<s<ϱ~CIτ~ϱ~δ£s(αn,r)(υ),σ~Δτ~σ~δ£s(αn,r)(υ)ϱ~εn(υ)>t>σ~CIτ~σ~δ£s(αn,r)(υ) and ς~Δτ~ς~δ£s(αn,r)(υ)ϱ~εn(υ)>k>ς~CIτ~ς~δ£s(αn,r)(υ).

Meanwhile, CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(εn,r)=εn, which is a contradiction. Hence, Δτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(αn,r).

(b) is similar to Theorem 4 (k).

(c) Let αn[εn]i=CIτ~ϱ~σ~ς~θ£s([εn]i,r). Therefore, iΓ[εn]iCIτ~ϱ~σ~ς~θ£s(iΓ[εn]i,r)CIτ~ϱ~σ~ς~θ£s([εn]i,r)=[εn]i. Consequently, iΓ[εn]iCθJτ(iΓ[εn]i,r). Hence, Θτ~ϱ~σ~ς~θ£s(αn,r)=CIτ~ϱ~σ~ς~θ£s(Θτ~ϱ~σ~ς~θ£s(αn,r),r).

(d) It is directly obtained from (c).

(e) Since αnΘτ~ϱ~σ~ς~θ£s(αn,r), by (c) and Eq. (19), CIτ~ϱ~σ~ς~θ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(Θτ~ϱ~σ~ς~θ£s(αn,r),r)=Θτ~ϱ~σ~ς~θ£s(αn,r).

Definition 17. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, αn,εnζF~, and rζ0. Then F~~ is called,

(a)   single valued neutrosophic £s-regular (SVN£s-regular) if for any αnQτ~ϱ~σ~ς~(xs,t,k,r), there exists εnQτ~ϱ~σ~ς~(xs,t,k,r) such that Clτ~ϱ~σ~ς~(εn,r)αn,

(b)   single valued neutrosophic almost £s-regular (SVNA£s-regular), if for any αnRτϱ~σ~ς~£s(xs,t,k,r), then there exists εnRτϱ~σ~ς~£s(xs,t,k,r) such that Clτ~ϱ~σ~ς~(εn,r)αn.

Theorem 7. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, αn,εnζF~ and rζ0. Then the following statements are equivalent:

(a)   (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is called SVN£s-regular,

(b)   For each xs,t,kPs,t,k(F~) and αnQτϱ~σ~ς~(xs,t,k,r), there exists εnRτϱ~σ~ς~£s(xs,t,k,r) such that Clτ~ϱ~σ~ς~(εn,r)intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(αn,r),r),

(c)   For each xs,t,kPs,t,k(F~) and each αnQτϱ~σ~ς~(xs,t,k,r), there exists εnQτϱ~σ~ς~(xs,t,k,r) such that Clτ~ϱ~σ~ς~(εn,r)intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(αn,r),r),

(c)   For each xs,t,kPs,t,k(F~) and r-SVNRIC set ωnζF~ with xs,t,kωn, there exists εnQτϱ~σ~ς~(xs,t,k,r) and αn is r-SVN-open set such that ωnαn and Clτ~ϱ~σ~ς~(αn,r)q¯Clτ~ϱ~σ~ς~(εn,r),

(d)   For each xs,t,kPs,t,k(F~) and r-SVNRIC set ωnζF~ with xs,t,kωn, there exists εnQτϱ~σ~ς~(xs,t,k,r) and αn is r-SVN -open set such that ωnαn and Clτ~ϱ~σ~ς~(εn,r)q¯αn,

(e)   For each r-SVNRIO set αnζF~ with ωnqαn, there exists r-SVNRIO set εnζF~ such that ωnqεnClτ~ϱ~σ~ς~(εn,r)αn.

(f)   For each r-SVNRIC set αnζF~ with ωnαn, there exists r-SVNRIO set εnζF~ and is r-SVN -open set πnζF~ such that ωnqεn, αnπn and εnq¯πn.

Proof. The proof of (a)(b) and (b)(c) are clear.

(c)(a): xs,t,kPs,t,k(F~) and αnRτϱ~σ~ς~£s(xs,t,k,r). Then, by (c), there exists εnQτϱ~σ~ς~(xs,t,k,r) such that Clτ~ϱ~σ~ς~(εn,r)intτ~ϱ~σ~ς~((Clτ~ϱ~σ~ς~(αn,r),r)=αn. since, εnQτϱ~σ~ς~(xs,t,k,r) we have  intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)Rτϱ~σ~ς~£s(xs,t,k,r).

Moreover, since,ωn=intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)Clτ~ϱ~σ~ς~(εn,r), we have Clτ~ϱ~σ~ς~(ωn,r)Clτ~ϱ~σ~ς~(εn,r), and hence xs,t,kqωnClτ~ϱ~σ~ς~(ωn,r)Clτ~ϱ~σ~ς~(εn,r)αn where ωnRτϱ~σ~ς~£s(xs,t,k,r).

(c)(d): Let ωn be an r-SVNRIC set in F~ and xtPs,t,k(F~) with xs,t,kωn. Then xs,t,kq[ωn]c and [ωn]cRτϱ~σ~ς~£s(xs,t,k,r)Qτϱ~σ~ς~(xs,t,k,r). By (c), there exists πnQτϱ~σ~ς~(xs,t,k,r) such that

Clτ~ϱ~σ~ς~(πn,r)intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~([ωn]c,r),r)=[ωn]c.

Next, xs,t,kqintτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(πn,r),r), then intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(πn,r),r)Qτϱ~σ~ς~(xs,t,k,r), and hence by hypothesis, there exists εnQτϱ~σ~ς~(xs,t,k,r) such that Clτ~ϱ~σ~ς~(εn,r)intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(πn,r),r). Then, ωn[Clτ~ϱ~σ~ς~(πn,r),r)]c. Put αn=[Clτ~ϱ~σ~ς~(πn,r),r)]c then αn is r-SVN O set. Hence

Clτ~ϱ~σ~ς~(αn,r)[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(πn,r),r)]c[Clτ~ϱ~σ~ς~(εn,r).

Therefore, Clτ~ϱ~σ~ς~(εn,r)q¯Clτ~ϱ~σ~ς~(αn,r).

(d)(e): It is trivial.

(e)(f): Suppose that αn is an r-SVNRIO set with ωnqαn, then ωn[αn]c. Hence there exists xs,t,kPs,t,k(F~) such that xs,t,kωn and ωn[αn]c where [αn]c is r-SVNRIC set. By (e), there exists εnQτϱ~σ~ς~(xs,t,k,r) and πnζF~ is r-SVN O set such that [αn]cπn and Clτ~ϱ~σ~ς~(εn,r)q¯πn. From εnQτϱ~σ~ς~(xs,t,k,r) we have xs,t,kqεnintτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r).

By setting [εn]1=intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r), we have ωnq[εn]1 and [εn]1 is r-SVNRIO set such that

ωnq[εn]1Clτ~ϱ~σ~ς~([εn]1,r)Clτ~ϱ~σ~ς~(εn,r)1_πnαn

(f)(g): Let αn be an r-SVNRIC set αnζF~ with ωnαn. Therefore, ωnq[αn]c and hence by, then there exists an r-SVNRIO set εnζF~ such that ωnqεnClτ~ϱ~σ~ς~(εn,r)[αn]c. Then, εn is an r-SVNRIO set and [Clτ~ϱ~σ~ς~(εn,r)]c is an r-SVN O set such that ωnqεn, αn[Clτ~ϱ~σ~ς~(εn,r)]c and εnq¯[Clτ~ϱ~σ~ς~(εn,r)]c.

(g)(a): Let αnRτϱ~σ~ς~£s(xs,t,k,r) Then xs,t,k[αn]c and [αn]c is an r-SVNRIC set. By (g), there exist r-SVNRIO set εnζF~ and it is r-SVN O set πnζF~ such that xs,t,kqεn, [αn]cπn and εnq¯πn. Then,εnRτϱ~σ~ς~£s(xs,t,k,r). Since, πn is r-SVN O set, Clτ~ϱ~σ~ς~(εn,r)q¯πn. Therefore, xs,t,kqεnClτ~ϱ~σ~ς~(εn,r)[πn]cαn. Hence (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVN£s-regular.

Theorem 8. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, αnζF~ and rζ0. Then the following statements are equivalent:

(a)   (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is called SVN£s-regular,

(b)   For each xs,t,kPs,t,k(F~), αnζF~ with τϱ~([αn]c)r,τσ~([αn]c)1r,τς~([αn]c)1r, and xs,t,kαn, there exists εnζF~ with εn is r-SVN O such that xs,t,kClτ~ϱ~σ~ς~(εn,r) and αnεn,

(c)   For each xs,t,kPs,t,k(F~), αnζF~ with τϱ~([αn]c)r, τσ~([αn]c)1r, τς~([αn]c)1r, and xs,t,kαn, there exists, εnQτϱ~σ~ς~(xs,t,k,r) and πnζF~ with πn is r-SVN O such that αnεn and εnq¯πn,

(d)   For each ωn,αnζF~ with τϱ~([αn]c)r, τσ~([αn]c)1r, τς~([αn]c)1r, and ωnαn, then there exists εnQτϱ~σ~ς~(xs,t,k,r) and εn,πnζF~ with τϱ~(εn)r,τσ~(εn)1r,τς~(εn)1r and πn is r-SVN O sets such that ωnqεn, αnπn and εnq¯πn.

Proof. Similar to the proof of Theorem 7.

Theorem 9. An SVNITS (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVNA£s-regular iff for each αnζF~ and rζ0, CIτ~ϱ~σ~ς~δ£s(αn,r)=CIτ~ϱ~σ~ς~θ£s(αn,r).

Proof. From Theorem 4(i), we only show that CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(αn,r).

Suppose that CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~θ£s(αn,r). Then there exist υF~ and [s(0,1], t[0,1), k[0,1)] such that

ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s<ϱ~CIτ~ϱ~θ£s(αn,r)(υ)σ~CIτ~σ~δ£s(αn,r)(υ)>t>σ~CIτ~σ~θ£s(αn,r)(υ)ς~CIτ~ς~δ£s(αn,r)(υ)>k>ς~CIτ~ς~θ£s(αn,r)(υ)} (21)

Because ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s,σ~CIτ~σ~δ£s(αn,r)(υ)>t,ς~CIτ~ς~δ£s(αn,r)(υ)>k, and xs,t,k is not an r-δ£s-cluster point of αn. So, there exists εnQτ~ϱ~σ~ς~(xs,t,k,r) with αn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)]c Since εnQτ~ϱ~σ~ς~(xs,t,k,r) we have intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)Rτϱ~σ~ς~£s(xs,t,k,r). By SVNA£s-regularity of F~, there exists ωnRτϱ~σ~ς~£s(xs,t,k,r) such that Clτ~ϱ~σ~ς~(ωn,r),r)intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r). Thus,

αn[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(εn,r),r)]c[Clτ~ϱ~σ~ς~(ωn,r)]c=intτ~ϱ~σ~ς~([ωn]c,r),

and τϱ~(ωn)r,τσ~(ωn)1r,τς~(ωn)1r. By Theorem 4(d), ϱ~CIτ~ϱ~θ£s(αn)(υ)ϱ~[ωn]c(υ)<s, σ~CIτ~σ~θ£s(αn,r)(υ)σ~[ωn]c(υ)>t and ς~CIτ~ς~θ£s(αn,r)(υ)ς~[ωn]c(υ)>k. It is a contradiction for Eq. (21).

Conversely, let αnRτϱ~σ~ς~£s(xs,t,k,r)Qτ~ϱ~σ~ς~(xs,t,k). Then by Theorem 4(h), s>ϱ~[αn]n(υ)=ϱ~CIτ~ϱ~δ£s([αn]c,r)(υ), s>ϱ~[αn]n(υ)=ϱ~CIτ~ϱ~δ£s([αn]c,r)(υ) and k<σ~[αn]n(υ)=σ~CIτ~σ~δ£s([αn]c,r)(υ). Since, CIτ~ϱ~σ~ς~δ£s([αn]c,r)=CIτ~ϱ~σ~ς~θ£s([αn]c,r), xs,t,k is not an r-θJ-cluster point of [αn]c. Then there exists εnQτ~ϱ~σ~ς~(xs,t,k,r) such that [αn]cq¯Clτ~ϱ~σ~ς~(εn,r) implies Clτ~ϱ~σ~ς~(εn,r)αn=intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(αn,r),r) and by Theorem 7(c), (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVNA£s-regular.

Theorem 10. An SVNITS (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVNA£s-regular iff for each r-SVNRIC set αnζF~ and ζ0, CIτ~ϱ~σ~ς~θ£s(αn,r)=αn.

Proof. The proof is similar to Theorem 9; additionally, r-SVNRIC set is r-δ£s-closed.

Conversely, let αn be any r-FRIC set with xtαn. Then, xtCIτ~ϱ~σ~ς~θ£s(αn,r) and hence, xt is not r-θ£s-cluster point of αn so, there there exists εnQτϱ~σ~ς~(xs,t,k,r) such that αnq¯Clτ~ϱ~σ~ς~(εn,r). Thus, αn[Clτ~ϱ~σ~ς~(εn,r)]c=ωn and ωn is r-SVNO implies ωnq¯Clτ~ϱ~σ~ς~(εn,r). Hence, by Theorem 4(e), (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVNA£s-regular.

Lemma 1. If αn,εnζF~, rζ0 such that αnq¯εn where εn is r-δ£s-open, then CIτ~ϱ~σ~ς~δ£s(αn,r)q¯εn.

Proof. Let αnq¯εn where εn is r-δ£s-open. Then, αn[εn]c=CIτ~ϱ~σ~ς~δ£s([εn]c, by Theorem 4(k), CIτ~ϱ~σ~ς~δ£s(αn,r)CIτ~ϱ~σ~ς~δ£s(CIτ~ϱ~σ~ς~δ£s([εn]c,r),r)=CIτ~ϱ~σ~ς~δ£s([εn]c,r)=[εn]c. Hence, CIτ~ϱ~σ~ς~δ£s(A,r)q¯εn.

Lemma 2. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and αnζ(F~ is δ£s-open iff for each xx,t,kQτϱ~σ~ς~(xs,t,k,r) with xs,t,kqαn, there exists r-SVNRIO set εnζF~ such that xx,t,kqεnαn.

Proof. Let xx,t,kPs,t,k((F~) with xx,t,kqαn Then xx,t,kαn]c. Since αn is an r-δ£s-open set, xx,t,k[αn]c=CIτ~ϱ~σ~ς~δ£s([αn]c,r). Thus, xx,t,k is not r-δ£s-cluster point of [αn]c. So, there exists ωnQτϱ~σ~ς~(xs,t,k,r) such that [αn]cq¯CIτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(ωn,r),r). Put εn=intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~(ωn,r),r), so, εn is an r-SVNRIO set with xx,t,kqεnαn.

Conversely, let [αn]cCIτ~ϱ~σ~ς~δ£s([αn]c,r), then there exist υF~ and s,t,kζ0 such that

ϱ~[αn]c(υ)<s<ϱ~CIτ~ϱ~δ£s([αn]c,r)(υ)σ~[αn]c(υ)>t>σ~CIτ~σ~δ£s([αn]c,r)(υ)ς~[αn]c(υ)>k>ς~CIτ~ς~δ£s([αn]c,r)(υ).} (22)

Because of xx,t,kqαn, then there exists an r-SVNRIO set εn such that xx,t,kqεnαn. This implies [αn]c[εn]n=Clτ~ϱ~σ~ς~(intτ~ϱ~σ~ς~([εn]n,r),r). By Theorem 3(d), we have ϱ~CIτ~ϱ~δ£s([αn]c,r)(υ)ϱ~([εn]n)(υ)<s,σ~CIτ~σ~δ£s([αn]c,r)(υ)σ~([εn]n)(υ)>t and ς~CIτ~ς~δ£s([αn]c,r)(υ)ς~([εn]n)(υ)>k. It is a contradiction for Eq. (22). Hence, [αn]c=CIτ~ϱ~σ~ς~δ£s([αn]c,r), i.e., αn is an r-δ£s-open set.

Lemma 3. If τϱ~(αn)r,τσ~(αn)1r,τς~(αn)1r, then CIτ~ϱ~σ~ς~(αn,r)=CIτ~ϱ~σ~ς~δ£s(αn,r).

Proof. Follows easily by virtue of Theorem 4.

Theorem 11. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS. Then the following statements are equivalent:

(a)   (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVNA£s-regular,

(b)   For each rδ£s-open set αnζF~ and each xx,t,kPs,t,k(F~) with xs,t,kqA, there exists r-δ£s-open set εnζF~ such that xx,t,kqεnClτ~ϱ~σ~ς~(εn,r)αn.

Proof. (a)(b): Let αn be r-fuzzy δJ-open set such each xs,t,kqαn. Then by Lemma 3, there exists an r-SVNRIO set πnζF~ such that xs,t,kqπnαn. By SVNA £s-regularity of θ£s there exists an r-FRIO set εn (which is also r-δ£s-open such that xs,t,kqεnClτ~ϱ~σ~ς~(εn,r)πnαn.

Therefore, (b) (a) is clear.

5  Single Valued Neutrosophic θ£s-Connected

The aim of this section is to introduce the r-single-valued neutrosophic θ£s-separated and r-single-valued neutrosophic δ£s-separated. Moreover, we introduce r-single-valued neutrosophic θ£s-connected and r-single valued neutrosophic δ£s-connected related to the r-single valued neutrosophic operator θ and δ defined on the set F~.

Definition 18. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS. For rζ0 and αn,εnζF~. Then,

(a)   Two non-null SVNSs  αn,εnζF~ are said to be r-single-valued neutrosophic θ£s-separated if αnq¯[εn]θ£s and εnq¯[αn]θ£s,

(b)   Two non-null SVNSs  αn,ε