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Computer Modeling in Engineering & Sciences
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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. [7–9] 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. [11–13]. Riaz et al. [14]. Salama et al. [15–17]. 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)⟩:υ∈F~} where

ϱ~:F~→⌋−0,1+⌊,σ~:F~→⌋−0,1+⌊,ς~:F~→⌋−0,1+⌊ and −0≤ϱ~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 υ∈F~ 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)⟩:υ∈F~}. 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 υ∈F~ to the set αn.

Definition 3. [4] Let αn={⟨υ,ϱ~αn(υ),σ~σn(υ),ς~αn(υ)⟩:υ∈F~} 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,εn∈ζF~ be in the form: αn={⟨υ,ϱ~αn(υ),σ~αn(υ),ς~αn(υ)⟩:υ∈F~} and εn={⟨υ,ϱ~εn(υ),σ~εn(υ),ς~εn(υ)⟩:υ∈F~} on F~ then,

(a)   αn⊆εn for every υ∈F~; ϱ~αn(υ)≤ϱ~εn(υ),σ~αn(υ)≥σ~εn(υ),ς~αn(υ)≥ς~εn(υ).

(b)   αn=εn iff σn⊆εn and σn⊇εn.

(c)   0~=⟨0,1,1⟩ and 1~=⟨1,0,0⟩.

Definition 5. [20] Let αn,εn∈ζF~. Then,

(a)   αn∩εn is an SVNS, if for every υ∈F~,

αn∩εn=⟨(ϱ~αn∩ϱ~εn)(υ),(σ~αn∪σ~εn)(υ),(ς~αn∪ς~εn)(υ)⟩, (2)

where, (ϱ~αn∩ϱ~εn)(υ)=ϱ~αn(υ)∩ϱ~εn(υ) and (ς~αn∪ς~εn)(υ)=ς~αn(υ)∪ς~εn(υ), for all υ∈F~,

(b)   αn∪εn is an SVNS, if for every υ∈F~,

αn∪εn=⟨(ϱ~αn∪ϱ~εn)(υ),(σ~αn∩σ~εn)(υ),(ς~αn∩ς~εn)(υ)⟩. (3)

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

(a)   ⋂i∈j[αn]i=⟨∩i∈jϱ~[αn]i(υ),∪i∈jσ~[αn]i(υ),∪i∈jς~[αn]i(υ)⟩,

(b)   ⋃i∈j[αn]i=⟨∪i∈jϱ~[αn]i(υ),∩i∈jσ~[αn]i(υ),∩i∈jς~[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~(αn∩εn)≥τ~ϱ~(αn)∩τ~ϱ~(εn),τ~σ~(αn∩εn)≤τσ~(αn)∪τ~σ~(εn),τ~ς~(αn∩εn)≤τ~ς~(αn)∪τ~ς~(εn) for every, αn,εn∈ζF~

(SVNT3)τ~ϱ~(∪j∈Γ[αn]j)≥∩j∈Γτ~ϱ~([αn]j), τ~σ~(∪i∈Γ[αn]j)≤∪j∈Γτ~σ~([αn]j),τ~ς~(∪j∈Γ[αn]j)≤∪j∈Γτ~ς~([αn]j), for every [αn]j∈ζF~.

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 αn∈ζF~ and r∈ζ0. Then the single valued neutrosophic closure and single valued neutrosophic interior of αn are define by:

Cτϱ~σ~ς~(αn,r)=⋂{εn∈ζF~:αn≤εn,τϱ~([εn]c)≥r,τσ~([εn]c)≤1−r,τς~([εn]c)≤1−r} (4)

intτϱ~σ~ς~(αn,r)=⋃{εn∈ζF~:αn≥εn,τϱ~(εn)≥r,τσ~(εn)≤1−r,τς~(εn)≤1−r} (5)

Definition 9. [7] Let (F~) be a nonempty set and υ∈F~, 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,p∈αn iff s<ϱ~αn(υ), t≥σ~αn(υ) and k≥ς~~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 υ∈F~ 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 σn≤γn, then £sϱ~(εn)≤£sϱ~(αn), £sσ~(εn)≥£sσ~(αn) and £sς~(εn)≥£sς~(αn), for εn,αn∈ζF~.

(£s3) £sϱ~(αn∪εn)≥£sϱ~(αn)∩£sϱ~(εn), £sσ~(αn∪εn)≤£sσ~(αn)∪£sσ~(εn) and £sς~(αn∪εn)≤£sς~(αn)∪£sς~(εn), for αn,εn∈ζF~.

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 αn∈ζF~. 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 εn∈Qτϱ~σ~ς~(xs,t,k,r) and £sϱ~(ωn)≥r, £sσ~(ωn)≤1−r, £sς~(ωn)≤1−r, then there is at least one υ∈F~ for which

ϱ~αn(υ)+ϱ~εn(ν)−1>ϱ~ωn(υ),σ~αn(υ)+σ~εn(υ)−1≤σ~ωn(υ),ς~αn(υ)+ς~εn(υ)−1≤ς~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 αn∈ζF~. 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}i∈J⊂ζF~ 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⊙:i∈J)≤(∪[αn]i:i∈j)r⊙,

(b)   (∩([αn]i):i∈j)r⊙≥(∩([αn]i)r⊙:i∈J).

Theorem 2. [7] Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS and r∈ζ, αn,εn∈ζF~. Then,

(a)   intτ~ϱ~σ~ς~⊙(αn∨εn,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)≤αn≤CIτ~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~⊙(αn∧εn,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,ωn∈ζF~, αn and εn are called r-single-valued neutrosophic separated if for r∈ζ0,

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,εn∈ζF~−{0~} such that ωn=αn∪εn 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 r∈ζ0. 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 αn∈ζF~, r∈ζ0. Then,

(a)   αn is said to be r-single valued neutrosophic £s-open (briefly, r-SVN£sO), if and only if αn≤intτ~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 αn≤CIτ~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 αn≤intτ~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 αn≤intτ~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 r−SVN\pounds O (resp, r-SVNS£sO, r-SVNP£sO, r-SVNR£sO, r-SVNα£sO, r-SVN⋆O) is said to be an r−SVN\pounds C (resp, r-SVNS£sC, r-SVNP£sC, r-SVNR£sC, r-SVNα£sC, r-SVN⋆C) respectively.

Remark 2. r-single valued neutrosophic open set (r−SVNO) 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,ωn∈ζF~ 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 αn∈ζF~,

τ~ϱ~(αn)={1,if αn={0~,1~},23,if αn={εn,πn},0,otherwise,£sϱ~(αn)={1,if αn=0~,23,if 0<αn≤ωn0,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<αn≤ωn,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<αn≤ωn,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, 23−SVNO 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 εn≰intτ~ϱ~σ~ς~([εn]23⊙,23). So, we must first obtain [εn]23⊙. Based on Eq. (11), 1~,εn,πn∈Qτϱ~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)−1≤σ~ωn(υ),ς~εn(υ)+ς~1~(υ)−1≤ς~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)−1≤σ~ωn(υ),ς~εn(υ)+ς~πn(υ)−1≤ς~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)−1≤σ~ωn(υ),ς~εn(υ)+ς~εn(υ)−1≤ς~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)⟩=εn≰intτ~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, αn∈ζF~, xs,t,k∈Ps,t,k(F~) and r∈ζ0. 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)≤1−r,τς~(αn)≤1−r;

(b)   xs,t,k is an r-single valued neutrosophic θ£s-cluster point (r-δ£s-cluster point) of αn if for every εn∈Qτ~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~×ζ0→ζF~ defined as

CIτ~ϱ~σ~ς~δ£s(αn,r)=∪{xs,t,k∈Ps,t,k(F~):xs,t,k is r−δ£s−cluster point of αn}.

Definition 15. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, αn∈ζF~, xs,t,k∈Ps,t,k(F~) and r∈ζ0. 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={αn∈ζF~|xs,t,kqαn,αn is r−SVNRIO},

(b)   xs,t,k is called r-single valued neutrosophic θ£s-cluster point (r-θ£s-cluster point) of αn if for any εn∈Qτ~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~×ζ0→ζF~ defined as

CIτ~ϱ~σ~ς~θ£s(αn,r)=∪{xs,t,k∈Ps,t,k(F~):xs,t,k is r−θ£s−cluster point of αn} (9)

Example 2. Let F~={a,b,c} be a set. Define εn,πn∈ζF~ 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 αn∈ζF~,

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~≠αn≤εnc,r≤13,1−r≥23,1,otherwise.

Theorem 3. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, r∈ζ0 and αn,εn∈ζF~. Then the following properties are holds:

(a)   αn≤CIτ~ϱ~σ~ς~δ£s(αn,r),

(b)   If αn≤εn, 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)=∩{εn∈ζF~|αn≤εn,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 εn∈ζF~ 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=∩{εn∈ζF~|αn≤εn,εn is r-SVNRIC}, let CIτ~ϱ~σ~ς~δ£s(αn,r)≱P; therefore, υ∈F~ 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)≥t≥σ~P(υ)ς~CIτ~ς~δ£s(αn,r)(υ)≥k≥ς~P(υ)} (10)

Therefore, xs,t,k is not an r-δ£s-cluster point of αn. As such, εn∈Qτ~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,k1∈Ps,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)≤t1≤σ~P(y)ς~CIτ~ς~δ£s(αn,r)(y)≤k1≤ς~P(y)} (11)

Owing to P, there exists r-SVNRIC  εn∈ζF~  with  αn≤εn such that ϱ~CIτ~ϱ~δ£s(αn,r)(y)>s1>ϱ~εn≥ϱ~P(y), σ~CIτ~σ~δ£s(αn,r)(y)≤t1≤ϱ~εn≤σ~P(y) and ς~CIτ~ς~δ£s(αn,r)(y)≤k1≤ϱ~εn≤ς~P(y). Therefore, [εn]c∈Qτ~ϱ~σ~ς~(ys1,t1,k1). So, αn≤εn=[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, υ∈F~ 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)≤t≤σ~CIτ~σ~δ£s(αn,r)(υ)ς~CIτ~σ~(αn,r)(υ)≤k≤ς~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 εn∈Qτ~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 r∈ζ0 and αn,εn∈ζF~. Then the following properties hold:

(a) αn≤CIτ~ϱ~σ~ς~θ£s(αn,r),

(b) If αn≤εn, 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,k∈Ps,t,k(F~)|xs,t,k is r-δ£s-cluster point of αn},

(d) CIτ~ϱ~σ~ς~θ£s(αn,r)=∩{εn∈ζF~|αn≤intτ~ϱ~σ~ς~⊙(εn,r), τϱ~([εn]c)≥r, τσ~([εn]c)≤1−r, τς~([εn]c)≤1−r},

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

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

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

(j) W(αn∨εn,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,k∈Ps,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 υ∈F~, 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)≤t≤σ~P(υ)ς~CIτ~ς~(αn,r)(υ)≤k≤ς~P(υ)} (13)

Consequently, xs,t,k is not r-δ£s-cluster point of αn. So, there exists εn∈Qτϱ~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=∩{εn∈ζF~|αn≤intτ~ϱ~σ~ς~⊙(εn,r), τϱ~([εn]c)≥r, τσ~([εn]c)≤1−r, τς~([εn]c)≤1−r}.

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

ϱ~CIτ~ϱ~σ~ς~θ£s(αn,r)(υ)<s≤ϱ~γ(υ)σ~CIτ~σ~θ£s(αn,r)(υ)>t≥σ~γ(υ)ς~CIτ~ς~θ£s(αn,r)(υ)>k≥ς~γ(υ)} (14)

Consequently, xs,t,k is not r-θ£s-cluster point of αn. So, there exists εn∈Qτϱ~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)≤1−r, τς~(εn)≤1−r}. 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,k1∈Ps,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)<t1≤σ~γ(y)ς~CIτ~ς~θ£s(αn,r)(y)<k1≤ς~γ(y)} (15)

By the definition of γ, there exists εn∈ζF~ with τϱ~(εn)≥r, τσ~(εn)≤1−r,τς~(εn)≤1−r and αn≤intτ~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)<t1≤σ~εn(y)≤σ~γ(y) and ς~CIτ~ς~θ£s(αn,r)(y)<k1≤ς~εn(y)≤ς~γ(y). Additionally, [εn]c∈Qτϱ~σ~ς~(ys1,t1,k1,r). αn≤intτ~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 εn∈Qτϱ~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,k∉CIτ~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 υ∈ζ 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~CIς~δ£s(αn,r)(υ)>k≥ς~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 εn∈Qτϱ~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(αn∨εn,r). Then there exists υ∈F~ such that

ϱ~CIτ~ϱ~δ£s(εn,r)(υ)∨ϱ~CIτ~ϱ~δ£s(αn,r)(υ)<s<ϱ~CIτ~ϱ~δ£s(αn∨εn,r)(υ)σ~CIτ~σ~δ£s(εn,r)(υ)∨σ~CIτ~σ~δ£s(αn,r)(υ)>t>σ~CIτ~ϱ~δ£s(αn∨εn,r)(υ)ς~CIτ~ς~δ£s(εn,r)(υ)∨ς~CIτ~σ~δ£s(αn,r)(υ)>t>ς~CIτ~ς~δ£s(αn∨εn,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]1∈Qτϱ~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]1∈Qτϱ~σ~ς~(xs,t,k,r).

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

αn∨εn≤[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, αn∨εnq¯ 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 αn∨εn, by (g), ϱ~Clτ~ϱ~⊙(αn∨εn,r)(υ)<s,σ~Clτ~σ~⊙αn∨εn(,r)(υ)>t and ς~Clτ~ς~⊙(αn∨εn,r)(υ)>k. It is a contradiction for Eq. (17), and hence, CIτ~ϱ~σ~ς~δ£s(αn∨εn,r)≤CIτ~ϱ~σ~ς~δ£s(εn,r)∨CIτ~ϱ~σ~ς~δ£s(αn,r).

Meanwhile, αn∨εn≥αn and αn∨εn≥εn. Hence CIτ~ϱ~σ~ς~δ£s(αn∨εn,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(αn∨εn,r).

(k) Since αn≤CIτ~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 υ∈F~ 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)>t≥σ~CIτ~ϱ~δ£s(CIτ~σ~δ£s(αn,r),r)(υ)ς~CIτ~ς~δ£s(αn,r)(υ)>k≥ς~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 εn∈Qτϱ~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 r−SVNRIC. 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 r∈ζ0 and αn,εn∈ζF~. 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 αn≤CIτ~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 υ∈F~ 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)<t≤σ~Clτ~σ~(αn,r)(υ) and ς~CIτ~ς~δ£s(αn,r)(υ)<k≤ς~Clτ~ς~(αn,r)(υ). Then xs,t,k is not r-δ-cluster point of αn. So, there exists εn∈Qτϱ~σ~ς~(xs,t,k,r), with αn≤[εn]c Since xs,t,k is r-δ£s-cluster point of αn, for εn∈Qτϱ~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, αn≤intτ~ϱ~σ~ς~⊙(Clτ~ϱ~σ~ς~([αn]c,r),r) andτϱ~([Clτ~ϱ~([αn,r)]c≥r,τσ~([Clτ~σ~([αn,r)]c≤r,τς~([Clτ~ς~([αn,r)]c≤r. 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 αn∈ζF~,r∈ζ0, υ∈F~ 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)<t≤σ~Clτ~σ~(αn,r)(υ) and ς~CIτ~ς~θ£s(αn,r)(υ)<t≤ς~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 r∈ζ0 and αn,εn∈ζF~. 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|αn≤εn,εn=CIτ~ϱ~σ~ς~δ£s(εn,r)} (19)

Θτ~ϱ~σ~ς~θ£s(αn,r)=∩{εn|αn≤εn,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 r∈ζ0 and αn∈ζF~. 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), αn≤CIτ~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 υ∈F~ 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 εn∈ζF~ and αn≤εn=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, ∧i∈Γ[εn]i≤CIτ~ϱ~σ~ς~θ£s(∧i∈Γ[εn]i,r)≤CIτ~ϱ~σ~ς~θ£s([εn]i,r)=[εn]i. Consequently, ∧i∈Γ[εn]i≤CθJτ(∧i∈Γ[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 αn≤Θτ~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,εn∈ζF~, and r∈ζ0. Then F~~ is called,

(a)   single valued neutrosophic £s-regular (SVN£s-regular) if for any αn∈Qτ~ϱ~σ~ς~(xs,t,k,r), there exists εn∈Qτ~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 αn∈Rτϱ~σ~ς~£s(xs,t,k,r), then there exists εn∈Rτϱ~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,εn∈ζF~ and r∈ζ0. Then the following statements are equivalent:

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

(b)   For each xs,t,k∈Ps,t,k(F~) and αn∈Qτϱ~σ~ς~(xs,t,k,r), there exists εn∈Rτϱ~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,k∈Ps,t,k(F~) and each αn∈Qτϱ~σ~ς~(xs,t,k,r), there exists εn∈Qτϱ~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,k∈Ps,t,k(F~) and r-SVNRIC set ωn∈ζF~ with xs,t,k∉ωn, there exists εn∈Qτϱ~σ~ς~(xs,t,k,r) and αn is r-SVN⋆-open set such that ωn≤αn and Clτ~ϱ~σ~ς~⊙(αn,r)q¯Clτ~ϱ~σ~ς~⊙(εn,r),

(d)   For each xs,t,k∈Ps,t,k(F~) and r-SVNRIC set ωn∈ζF~ with xs,t,k∉ωn, there exists εn∈Qτϱ~σ~ς~(xs,t,k,r) and αn is r-SVN ⋆-open set such that ωn≤αn and Clτ~ϱ~σ~ς~⊙(εn,r)q¯αn,

(e)   For each r-SVNRIO set αn∈ζF~ with ωnqαn, there exists r-SVNRIO set εn∈ζF~ such that ωnqεn≤Clτ~ϱ~σ~ς~⊙(εn,r)≤αn.

(f)   For each r-SVNRIC set αn∈ζF~ with ωn≰αn, there exists r-SVNRIO set εn∈ζF~ and is r-SVN ⋆-open set πn∈ζF~ such that ωnqεn, αn≤πn and εnq¯πn.

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

(c)⇒(a): xs,t,k∈Ps,t,k(F~) and αn∈Rτϱ~σ~ς~£s(xs,t,k,r). Then, by (c), there exists εn∈Qτϱ~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, εn∈Qτϱ~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,kqωn≤Clτ~ϱ~σ~ς~⊙(ωn,r)≤Clτ~ϱ~σ~ς~⊙(εn,r)≤αn where ωn∈Rτϱ~σ~ς~£s(xs,t,k,r).

(c)⇒(d): Let ωn be an r-SVNRIC set in F~ and xt∈Ps,t,k(F~) with xs,t,k∉ωn. Then xs,t,kq[ωn]c and [ωn]c∈Rτϱ~σ~ς~£s(xs,t,k,r)⊂Qτϱ~σ~ς~(xs,t,k,r). By (c), there exists πn∈Qτϱ~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 εn∈Qτϱ~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,k∈Ps,t,k(F~) such that xs,t,k∈ωn and ωn≰[αn]c where [αn]c is r-SVNRIC set. By (e), there exists εn∈Qτϱ~σ~ς~(xs,t,k,r) and πn∈ζF~ is r-SVN ⋆ O set such that [αn]c≤πn and Clτ~ϱ~σ~ς~⊙(εn,r)q¯πn. From εn∈Qτϱ~σ~ς~(xs,t,k,r) we have xs,t,kqεn≤intτ~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]1≤Clτ~ϱ~σ~ς~⊙([εn]1,r)≤Clτ~ϱ~σ~ς~⊙(εn,r)≤1_−πn≤αn

(f)⇒(g): Let αn be an r-SVNRIC set αn∈ζF~ with ωn≰αn. Therefore, ωnq[αn]c and hence by, then there exists an r-SVNRIO set εn∈ζF~ such that ωnqεn≤Clτ~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 αn∈Rτϱ~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 εn∈ζF~ and it is r-SVN ⋆ O set πn∈ζF~ such that xs,t,kqεn, [αn]c≤πn and εnq¯πn. Then,εn∈Rτϱ~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,kqεn≤Clτ~ϱ~σ~ς~⊙(εn,r)≤[πn]c≤αn. 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, αn∈ζF~ and r∈ζ0. Then the following statements are equivalent:

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

(b)   For each xs,t,k∈Ps,t,k(F~), αn∈ζF~ with τϱ~([αn]c)≥r,τσ~([αn]c)≤1−r,τς~([αn]c)≤1−r, and xs,t,k∉αn, there exists εn∈ζF~ with εn is r-SVN ⋆ O such that xs,t,k∉Clτ~ϱ~σ~ς~(εn,r) and αn≤εn,

(c)   For each xs,t,k∈Ps,t,k(F~), αn∈ζF~ with τϱ~([αn]c)≥r, τσ~([αn]c)≤1−r, τς~([αn]c)≤1−r, and xs,t,k∉αn, there exists, εn∈Qτϱ~σ~ς~(xs,t,k,r) and πn∈ζF~ with πn is r-SVN ⋆ O such that αn≤εn and εnq¯πn,

(d)   For each ωn,αn∈ζF~ with τϱ~([αn]c)≥r, τσ~([αn]c)≤1−r, τς~([αn]c)≤1−r, and ωn≰αn, then there exists εn∈Qτϱ~σ~ς~(xs,t,k,r) and εn,πn∈ζF~ with τϱ~(εn)≥r,τσ~(εn)≤1−r,τς~(εn)≤1−r and πn is r-SVN ⋆ O sets such that ωnqεn, αn≤πn 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 αn∈ζF~ and r∈ζ0, 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 υ∈F~ 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 εn∈Qτ~ϱ~σ~ς~(xs,t,k,r) with αn≤[intτ~ϱ~σ~ς~(Clτ~ϱ~σ~ς~⊙(εn,r),r)]c Since εn∈Qτ~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 ωn∈Rτϱ~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)≤1−r,τς~(ωn)≤1−r. 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 αn∈Rτϱ~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 εn∈Qτ~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 αn∈ζF~ 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 xt∉αn. Then, xt∉CIτ~ϱ~σ~ς~θ£s(αn,r) and hence, xt is not r-θ£s-cluster point of αn so, there there exists εn∈Qτϱ~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-SVN⋆O 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,εn∈ζF~, r∈ζ0 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 αn∈ζ(F~ is δ£s-open iff for each xx,t,k∈Qτϱ~σ~ς~(xs,t,k,r) with xs,t,kqαn, there exists r-SVNRIO set εn∈ζF~ such that xx,t,kqεn≤αn.

Proof. Let xx,t,k∈Ps,t,k((F~) with xx,t,kqαn Then xx,t,k∉αn]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 ωn∈Qτϱ~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,kqεn≤αn.

Conversely, let [αn]c≠CIτ~ϱ~σ~ς~δ£s([αn]c,r), then there exist υ∈F~ and s,t,k∈ζ0 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,kqεn≤αn. 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)≤1−r,τς~(αn)≤1−r, 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 r−δ£s-open set αn∈ζF~ and each xx,t,k∈Ps,t,k(F~) with xs,t,kqA, there exists r-δ£s-open set εn∈ζF~ such that xx,t,kqεn≤Clτ~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 πn∈ζF~ such that xs,t,kqπn≤αn. 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,kqεn≤Clτ~ϱ~σ~ς~⊙(εn,r)≤πn≤αn.

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 r∈ζ0 and αn,εn∈ζF~. Then,

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

Remark 2. For any two non-null SVNSs αn,εn∈ζF~, and by Eq. (8). The following implications hold: r-single-valued neutrosophic θ£s-separated ⇒ r-single-valued neutrosophic δ£s-separated ⇒ r-single-valued neutrosophic separated.

The following example shows that the concept of r-single-valued neutrosophic δ£s-separated is weaker than that of r-single-valued neutrosophic θ£s-separated.

Example 3. Let F~={a,b,c} be a set. Define [εn]1,[εn]2∈ζF~ as follows:

[εn]1=⟨(1,1,0),(1,1,0),(1,1,0)⟩;[εn]2=⟨(0,0,1),(0,0,1),(0,0,1)⟩.

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

τ~ϱ~(αn)={1,if αn=0~,1,if αn=1~,13,if αn=[εn]1,12,if αn=[εn]2,0,otherwise,£sϱ~(αn)={1,if αn=0~,0,otherwise, τ~σ~(αn)={0,if αn=0~,0,if αn=1~,23,if αn=[εn]1,12,if αn=[εn]2,1,otherwise£sσ~(αn)={0,if αn=1~,1,otherwise, τ~ς~(αn)={0,ifαn=0~,0,if αn=0~,23,if αn=[εn]1,12,if αn=[εn]2,1,otherwise,£sς~(αn)={0,if αn=1~,1,otherwise.

Therefore, we obtain

CIτ~ς~θ£s(αn,r)={0~,if αn=0~,r∈ζ0,E2c,if αn≤[εn]1,r≤12,1−r≥12,E1c,if αn≤[εn]2,r≤13,1−r≥23,0~,otherwise.

If r≤13 and 1−r≥23, then [εn]2c and [εn]2 are not r-single-valued neutrosophic θ£s-separated for r≤13 and 1−r≥23. If r>13 and 1−r<23, we have [εn]2c and [εn]2 are r-single-valued neutrosophic separated.

Theorem 12. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS. For r∈ζ0 and αn,εn∈ζF~.

(a)   If αn and εn are single-valued neutrosophic θ£s-separated, and [αn]1,[εn]1∈ζF~ such that [αn]1≤αn [εn]1≤εn, then [αn]1 and [εn]1 are also single-valued neutrosophic θ£s-separated,

(b)   If αnq¯εn either both are r-θ£s-open or r-δ£s-closed, then αn and εn are single-valued neutrosophic θ£s-separated,

(c)   If αn and εn either both are r-θ£s-open or r-δ£s-closed and if [ωn]1=αn∩[εn]c and ω2=εn∩[αn]c, then [ωn]1 and [ωn]1 are single-valued neutrosophic θ£s-separated.

Proof. (a) Since [αn]1≤αn we have [[αn]1]θ£s≤[αn]θ£s. Then, εn≤[αn]θ£s⇒[εn]1≤[αn]θ£s⇒[εn]1≤[[αn]1]θ£s. Similarly [αn]1≤[[εn]1]θ£s. Hence [αn]1 and [εn]1 are single-valued neutrosophic θ£s-separated.

(b) When αn and εn are r-δ£s-closed, then αn=[αn]θ£s and εn=[εn]θ£s. Since αnq¯εn we have [αn]θ£sq¯εn and [εn]θ£sq¯αn.

When αn and εn are r-θ£s-open, [αn]c and [εn]c are r-θ£s-closed. Then αnq¯εn⇒αn≤[εn]c⇒[αn]θ£s≤[[εn]c]θ£s=[εn]c⇒[αn]θ£sq¯εn. Similarly,[εn]θ£sq¯αn. Hence αn and εn are single-valued neutrosophic θ£s-separated.

(c) When αn and εn are r-θ£s-open, [αn]c and [εn]c are r-θ£s-closed. Since [ωn]1≤[εn]c, [[ωn]1]θ£s≤[[εn]c]θ£s=[εn]c and so [[ωn]1]θ£sq¯εn. Thus [ωn]2q¯[[ωn]1]θ£s. Similarly,[ωn]1q¯[[ωn]2]θ£s. Hence [ωn]1 and [ωn]1 are single-valued neutrosophic θ£s-separated.

When αn and εn are r-θ£s-closed, αn=[αn]θ£s and εn=[εn]θ£s. Since [ωn]1≤[εn]c, [εn]θ£sq¯[ωn]1 and hence [[ωn]2]θ£sq¯[ωn]1. Similarly,[[ωn]1]θ£sq¯[ωn]2. Hence [ωn]1 and [ωn]1 are single-valued neutrosophic θ£s-separated.

Theorem 13. Two non-null αn,εn∈ζF~ are single-valued neutrosophic θ£s-separated if and only if there exist two r-θ£s-open sets ωn and πn such that αn≤ωn, εn≤πn, αnq¯πn and εnq¯ωn.

Proof. Let αn and εn be single-valued neutrosophic θ£s-separated. Putting πn=[[αn]θ£s]c and ωn=[[εn]θ£s]c, then ωn and πn are r-θ£s-open such that αn≤ωn, εn≤πn, αnq¯πn and εnq¯ωn.

Conversely, let ωn and πn be r-θ£s-open sets such that αn≤ωn, εn≤πn, αnq¯πn and εnq¯ωn. Since [πn]c and [ωn]c are r-θ£s-closed, we have [αn]θ£s≤[πn]c≤[εn]c and [εn]θ£s≤[ωn]c≤[αn]c. Thus [αn]θ£sq¯εn and [εn]θ£sq¯αn. Hence αn and εn are single-valued neutrosophic θ£s-separated.

Definition 19. An SVNS which cannot be expressed as the union of two single-valued neutrosophic θ£s-separated is said to be single-valued neutrosophic θ£s-connected.

Definition 20. An SVNS αn in a SVNITS (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is said to be single-valued neutrosophic δ£s-connected if αn cannot be expressed as the union of two single-valued neutrosophic δ£s-separated.

For an SVNS αn in a SVNITS(F~,τϱ~σ~ς~,£sϱ~σ~ς~), the following implications hold: single-valued neutrosophic connected ⇒ single-valued neutrosophic δ£s-connected ⇒ single-valued neutrosophic θ£s-connected. If τϱ~(αn)≥r,τσ~(αn)≤1−r,τς~(αn)≤1−r, then these three properties are equivalent.

Theorem 14. Let αn be a non-null single-valued neutrosophic θ£s-connected in a SVNITS (F~,τϱ~σ~ς~,£sϱ~σ~ς~). If αn is contained in the union of two single-valued neutrosophic θ£s-separated εn and ωn, then exactly one of the following conditions (a) or (b) holds:

(a)   αn≤εn and αn∩ωn=0~,

(b)   αn≤ωn and αn∩εn=0~.

Proof. We first note that when αn∩ωn=0~, then αn≤εn, since αn≤εn∪ωn. Similarly, when αn∩εn=0~, we have αn≤ωn. Since αn≤εn∪ωn, both αn∩εn=0~ and αn∩ωn=0~ cannot hold simultaneously. Again, if αn∩εn≠0~ and αn∩ωn≠0~, then, by Theorem 12 (1), αn∩ωn and αn∩εn are single-valued neutrosophic θ£s-separated such that αn=(αn∩εn)∪(αn∩ωn), contradicting the single-valued neutrosophic θ£s-connectedness of αn. Hence, exactly one of the conditions (1) or (2) above must hold.

Theorem 15. Let {[αn]j|j∈J} be a collection of single-valued neutrosophic θ£s-connected in (F~,τϱ~σ~ς~,£sϱ~σ~ς~). If there exists i∈J such that [αn]j∩[αn]i≠0~ for each j∈J, then αn=∪{[αn]j|j∈J} is single-valued neutrosophic θ£s-connected.

Proof. Suppose that αn is not single-valued neutrosophic θ£s-connected. Then there exist single-valued neutrosophic θ£s-separated εn and ωn such that αn=εn∩ωn. By Theorem 14, we have either (a) [αn]j≤εn with [αn]j∩ωn=0~ or (b) [αn]j≤ωn with [αn]j∩εn=0~ for each j∈J. Similarly, either (a′) [αn]i≤εn with [αn]i∩ωn=0~ or (b′) [αn]i≤ωn with [αn]i∩εn=0~ for each i∈J . We may assume, without loss of generality, that [αn]j is non-null for each j∈J, and hence exactly one of the conditions (a) and (b), and exactly one of (a′) and (b′) will hold.

Since [αn]j∩[αn]i≠0~ for each j∈J, the conditions (a) and (b′) cannot happen, and similarly (b) and (1′) cannot hold simultaneously. If (a) and (a′) hold, then [αn]j≤εn with [αn]j∩ωn=0~. Then αn≤εn with αn∩ωn=0~ and thus ωn=0~ a contradiction. Similarly, if (b) and (b′) hold, then we have εn=0~ again a contradiction.

Lemma 4. An SVNITS (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVNA£s-regular iff [αn]δ£s=[αn]θ£s for every αn∈ζF~.

Proof. Obvious.

Theorem 16. Let (F~,τϱ~σ~ς~,£sϱ~σ~ς~) be an SVNITS, αn∈ζF~, r∈ζ0. If (F~,τϱ~σ~ς~,£sϱ~σ~ς~) is SVNA£s-regular and αn is single-valued neutrosophic θ£s-connected set, then αn is single-valued neutrosophic δ£s-connected set.

Proof. Follows easily by virtue of Lemma 4.

Corollary 1. For a αn∈ζF~ of SVNA£s-regular space (F~,τϱ~σ~ς~,£sϱ~σ~ς~), the following are equivalent:

(a)   αn is r-single-valued neutrosophic connected,

(b)   αn is r-single-valued neutrosophic δ£s-connected,

(c)   αn is r-single-valued neutrosophic θ£s-connected.

Proof. Follows easily by virtue of Theorem 16.

6  Conclusion

The neutrosophic set theory has been established and applied extensively to many problems involving uncertainties. Herein, we provided clear definitions of single-valued neutrosophic operators CIτ~ϱ~σ~ς~θ£s and CIτ~ϱ~σ~ς~δ£s created from an SVNI topological space (F~,τϱ~σ~ς~,£sϱ~σ~ς~) and we established that CIτ~ϱ~σ~ς~δ£s(αn,r)=CIτ~ϱ~σ~ς~(αn,r) when £sϱ~σ~ς~=£s0ϱ~σ~ς~. In addition, we presented the idea of r-single-valued neutrosophic θ£s-connectedness based on a single-valued neutrosophic ideal £sϱ~σ~ς~ which has kindred with a preceding r-single-valued neutrosophic connectedness and the relationships among them are inspected. Moreover, we introduced an r-single-valued neutrosophic δ£s-connectedness connected to a single-valued neutrosophic δ on the set F~ and analyzed some of their properties. This study not only provides a hypothetical basis for additional requests in neutrosophic topology, but also for the expansion of other methodical aspects.

Discussion for further works:

The current concept can be extended by

•   Investigating neutrosophic metric topological spaces;

•   Investigating the products of connected and Hausdorff spaces for (F~,τϱ~σ~ς~,£sϱ~σ~ς~).

Acknowledgement: The authors acknowledge Majmaah University for supporting this study. They are also grateful to the referees for their valuable comments and suggestions, which have improved the contents of this article.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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