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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2022.018518

ARTICLE

A New Attempt to Neutrosophic Soft Bi-Topological Spaces

Arif Mehmood1, Muhammad Aslam2, Muhammad Imran Khan3, Humera Qureshi3, Choonkil Park4,* and Jung Rye Lee5

1Department of Mathematics & Statistics, Riphah International University, Sector I-14, Islamabad, 44000, Pakistan
2Department of Mathematics, College of Sciences, King Khalid University, Abha, 61413, Saudi Arabia
3Department of Pure and Applied Mathematics (Statistics), University of Haripur, Haripur, 22630, Pakistan
4Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Korea
5Department of Data Science, Daejin University, Kyunggi, 11159, Korea
*Corresponding Author: Choonkil Park. Email: baak@hanyang.ac.kr
Received: 30 July 2021; Accepted: 12 September 2021

Abstract: In this article, new generalized neutrosophic soft * b open set is introduced in neutrosophic soft bi-topological structurers (NSBTS) concerning soft points of the space. This new set is produced by making the marriage of soft semi-open set with soft pre-open set in neutrosophic soft topological structure. An ample of results are investigated in NSBTS on the basis of this new neutrosophic soft * b open set. Proper examples are settled for justification of these results. The non-validity of some results is vindicated with examples.

Keywords: Neutrosophic soft set (NSS); neutrosophic soft points (NSP); neutrosophic soft bi-topological structurers (NSBTS); neutrosophic soft * b-open set and neutrosophic soft * b-separation axioms

1  Introduction

Fuzzy set theory[1] is the most importantly effective way to deal with vagueness and incomplete data and it is being developed and used in various fields of science. Fuzzy set (FS), which is directly an extension of the crisp sets. However, it has a shortcoming, i.e., it only addresses membership value and is unable to address the non-membership value. Since fuzzy set theory (FST) was too young at that time and researchers were working actively. Finally, Atanassov[2] addressed the deficiency in fuzzy set theory in sophisticated way and resulted in bouncing up a new idea with new wording “intusionistic fuzzy set theory (IFST)”. This approach supposes membership value and non-membership value.

Molodtsov[3] opened a new window of research and inaugurated the concept of soft set theory (SST) to address the uncertainty, which links a crisp set with another set of parameters. Soft set theory has a big hand as an application in many fields like function smoothness, Riemann integration, measurement theory, game theory, etc.[4]. The second leading cause of cancer death among men in most industrialized countries is prostate cancer which depends upon various factors, such as family history, age, ethnicity, and prostate-specific blood level (PSA).

PSA levels in the blood are an important way of diagnosing patients initially[5–7]. Yuksel etal.[8] discussed prostate cancer (PSA), prostrate volume (PV) and age factors of patients on the basis of fuzzy set and soft sets.

Wei etal.[9] leaked out the concept of VSS which is an extension to Huang etal.[10] deeply studied[9] and pointed out some incorrect results. They verified the incorrect result with examples and gave some more new definitions. Wang etal.[11] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. They authors discussed the basic concepts related to vague soft topological and studied the results in vague soft topology. The soft set to the hyper soft set was generalized by Smarandache[12]. In addition to this, the author introduced the hybrids of crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic hyper-soft set. Bera etal.[13] opened the door to a new world of mathematics and inaugurated the conception of new structure (neutrosophic soft topology) on the basis of neutrosophic soft set (NSS). He discussed all the fundamentals in polite way and on the basis of these fundamentals he moved to the fundamentals results and for better understanding suitable examples were put forwarded. Ozturk etal.[14] are pioneered in new operations on neutrosophic soft sets and then new approach to neutrosophic soft topology on the basis of these new operations. Ozturk etal.[15] leaked out concept of neutrosophic soft mapping, neutrosophic soft open mapping and neutrosophic soft homeomorphism on the basis of operation defined in[14]. Mehmood etal.[16] generated new open set in NSTS, known as left b star open sets. Neutrosophic soft separation axioms in NSTS are switched on to different results which are countability theorems, linking of these theorems with Hausdorff spaces, convergency of sequences, continuity, product of spaces, Bolzano Weirstrass property, compactness and sequentially compactness, etc. AL-Nafee[17] introduced new family of NSS. The author defined new operation on neutrosophic set. These operations are union, intersection etc. On the basis of these new operations the author defined NSTS. AL-Nafee etal.[18] continued his work and extended the NSTS to NSBTS on the basis of operations defined in[18]. The authors regenerated all the fundamental results of NSBTS on these basic operations. Hasan etal.[19] inaugurated NSBTS on the basis of the operations defined in[13]. The authors introduced pair-wise neutrosophic soft (closed) sets in NSBTS. These references[13–19] became source of motivation for my new research.

In our study, we worked with the operations given in references[14–16] which are entirely different from references[13,17]. Then unlike[18,19], neutrosophic soft bi-topological structure is reconstructed. In Section 2, some basic recipes are inaugurated. In Section 3, Neutrosophic soft bi-topology (NSBT) is addressed with examples. Some results, union and intersection are also studied in (NSBT). In Section 4, some main results are addressed. Section 5, more main results are addressed. In Section 6, some concluding remarks and future work are announced.

2  Related Work

In this section, fundamental concepts are addressed.

Definition 2.1.[20] A neutrosophic set (NS) symbolized by A on the key set π is defined as: A={⟨κ,ℸ⟨κ⟩,iA⟨κ⟩,FA⟨κ⟩:κ∈π⟩}where,[ℸ:π⟶(0−,1+)i:π⟶(0−,1+),F:π⟶(0−,1+)0−≼ℸA⟨κ⟩+iA⟨κ⟩+FA⟨κ⟩≼3+.,].

Definition 2.2.[3] π be key set, θ be a set of all parameters, and L(π) symbolizes the key set of π. A pair ⟨f,θ⟩ is referred to as a soft set (SS) over π, where f is a mapping given by f:θ→L(π).

Definition 2.3.[21,22] Let π be KS and θ set of parameters. Let L(π) signifies power set of all neutrosophic sets on π. Then a neutrosophic soft set (f~,θ) over π is a set defined by a set valued function f~ representing a mapping f~:θ→L(π) where f~ is called approximate function of the neutrosophic soft set (f~,θ). It can be written as a set of ordered pairs: (f~,θ)={((𝓃,[[κ,Tf~(n)(κ),If~(n)(κ),Ff~(n)(κ):κ∈π]]):𝓃∈θ}.

Definition 2.4.[13] Let (f~,θ) be a neutrosophic soft over the key set π then the complement of (f~,θ) is signified (f~,θ)c and is defined as follows:

(f~,θ)c={((𝓃,[[κ,ℸf~(n)(v),1−if~(n)(κ),Ff~(n)(κ):κ∈π]]):𝓃∈θ}. It’s clear that

((f~,θ)c)c=(f~,θ).

Definition 2.5.[22] Let (f~,θ), (ρ~,θ) two neutrosophic soft over the key set π. (f~,θ) is supposed to be neutrosophic soft sub-set of (ρ~,θ) if ℸf~(n)(κ)≼ℸρ~(n)(κ), if~(n)(κ)≼iρ~(n)(κ), Ff~(n)(κ)≽Fρ~(n)(κ), ∀𝓃∈θ&∀ κ∈π. It is denoted by (f~,θ)⊑(ρ~,θ). (f~,θ) is said to be neutrosophic soft equal to (ρ~,θ) if (f~,θ) is neutrosophic soft sub-set of (ρ~,θ) and (ρ~,θ) is neutrosophic soft sub-set of (f~,θ). It is denoted by (f~,θ)=(ρ~,θ).

Definition 2.6.[16] Let (f~1,θ),(f~2,θ) be two neutrosophic soft sub-sets over the key set π so that (f~1,θ)≠(f~2,θ). Then their union is denoted by (f~1,θ)⨆~(f~2,θ)=(f~3,θ) and is defined as (f~3,θ)={((𝓃,[[κ,ℸf~3(n)(κ),if~3(n)(κ),Ff~3(n)(κ):κ∈π]]):𝓃∈θ}. where

[ℸf~3(n)(κ)=max[ℸf~1(n)⟨κ⟩,ℸf~2(n)⟨κ⟩,if~3(n)(κ)=max[if~1(n)⟨κ⟩,if~2(n)⟨κ⟩,Ff~3(n)(κ)=min[Ff~1(n)⟨κ⟩,Ff~2(n)⟨κ⟩.]

Definition 2.7.[16] Let (f~1,θ), (f~2,θ) be two neutrosophic soft sub-sets over key set π such that (f~1,θ)≠(f~2,θ). Then their intersection is denoted by (f~1,θ)⊓~(f~2,θ)=(f~3,θ), is defined as [(f~3,𝓃)={((𝓃,[[[κ,ℸf~3(n)⟨κ⟩,if~3(n)⟨κ⟩,Ff~3(n)⟨κ⟩:κ∈π]]):𝓃∈θ}.] where,

where[ℸf~3(n)⟨κ⟩=min[ℸf~1(n)⟨κ⟩,ℸf~2(n)⟨κ⟩,if~3(n)⟨κ⟩=min[if~1(n)⟨κ⟩,if~2(n)⟨κ⟩,Ff~3(n)⟨κ⟩=max[Ff~1(n)⟨κ⟩,Ff~2(n)⟨κ⟩.]

Definition 2.8.[14] Neutrosophic soft set (f~,θ) overkey set π is said to be a neutrosophic soft null set If ℸf(n)⟨κ⟩=0,if(n)⟨κ⟩=0,Ff(n)⟨κ⟩=1;,∀n∈θ,∀κ∈π.

It is signified as 0(π,θ).

Definition 2.9.[14] Neutrosophic soft set (f~,θ) over key set π is an absolute neutrosophic soft set if Tf(n)⟨κ⟩=1,if(n)⟨κ⟩=1,Ff(n)⟨κ⟩=0;,∀n∈θ&∀κ∈π.

It is signified as 1(π,𝓃). Clearly, 0(π,𝓃)c=1(π,𝓃), 1(π,𝓃)c=0(π,𝓃).

Definition 2.10.[14] Let neutrosophic soft set (π~,θ) be the family of all NS soft sets and τ⊂NSS(π~,θ). Then τ is said to be a neutrosophic soft topology on π~ if (1). 0(⟨π⟩,𝓃),1(⟨π⟩,𝓃)∈τ,(2). The union of any number of neutrosophic set soft sets in τ∈τ,(3). The intersection of a finite number of neutrosophic soft sets in τ∈τ, then (π~,τ,θ) is said to be neutrosophic soft topological space over π~.

Definition 2.11.[14] Let neutrosophic soft set (π,θ)~ be the family of all neutrosophic set over π~, κ∈π~, then NSκ⟨y1,y2,y3⟩ is NS point, for 0<y1,y2,y3≤1 and is defined as follows:

κ(⟨y1,y2,y3⟩(𝓎)=⌊{⟨y1,y2,y3⟩if𝓎=κ}(0,0,1)if𝓎≠κ}⌋

Definition 2.12.[14] Let neutrosophic soft set (π,θ)~ be the family of all neutrosophic soft sets over key set π Then NSS(κ⟨y1,y2,y3⟩)e is called a neutrosophic soft point, for every κ∈π,~0≺{y1,y2,y3}≼1,e∈θ, and is defined as follows:

κe⟨y1,y2,y3⟩e‘(𝓎)=[⟨y1,y2,y3⟩if e‘=e∧𝓎=κ(0,0,1)if e‘≠e∧𝓎≠κ]

Definition 2.13.[14] Let (f~,θ) be a NSS over key set π. κe⟨y1,y2,y3⟩∈NSS(f~,θ) if y1≼ℸf~(n)⟨κ⟩, y2≼if~(n)⟨κ⟩, y3≽Ff~(n)⟨κ⟩.

Definition 2.14.[14] Let (π~,τ,θ) be a NSTS over π. Neutrosophic soft set (f~,θ) in (π,τ,θ) is called a neutrosophic soft neighbourhood of the neutrosophic soft point κ⋋⟨y1,y2,y3⟩∈(f~,θ), if there exists a neutrosophic soft open set (𝓆~,θ) such that κ⋋⟨y1,y2,y3⟩∈(𝓆~,θ).

3  Neutrosophic Soft Bi-Topology

In this part, the concept of NSBTS is defined. Furthermore, new types of open and closed sets have been introduced in neutrosophic soft bitopological spaces.

Definition 3.1. If⟨π,τ1,θ⟩, ⟨π,τ2,θ⟩ are two NSTS, then ⟨π,τ1,τ2,θ⟩ is called NSBTS. If ⟨π,τ1,τ2,θ⟩ be NSBTS. Neutrosophic soft sub-set (f,θ) is said to be open in ⟨π,τ1,τ2,θ⟩ if there exists a neutrosophic soft open set (f1,θ) in τ1 and neutrosophic soft open set (f2,θ) in τ2 such that (f,θ)=(f1,θ)∪(f2,θ).

Example 3.2. Let π={κ1,κ2,κ3}, θ={e1,e2} and τ1={0(π,θ),1(π,θ),(ω1,θ),(ω2,θ)},

τ1={0(π,θ),1(π,θ),(⋌1,θ),(⋌2,θ)}, where (ω1,θ),(ω2,θ),(⋌1,θ) ve (⋌2,θ) being neutrosophic soft sub-sets as following:

f(ω1,θ)(e1)=[⟨κ1,0210,0310,0810⟩,⟨κ2,0410,0410,0410⟩,⟨κ3,0210,0410,0310⟩]f(ω1,θ)(e2)=[⟨κ1,0310,0210,0610⟩,⟨κ2,0110,0510,0510⟩,⟨κ3,0410,0310,0410⟩]f(ω2,θ)(e1)=[⟨κ1,0410,0310,0610⟩,⟨κ2,0410,0510,0310⟩,⟨κ3,0310,0510,0210⟩]f(ω2,θ)(e2)=[⟨κ1,0310,0410,0510⟩,⟨κ2,0210,0610,0410⟩,⟨κ3,0410,0610,0310⟩]f(⋌1,θ)(e1)=[⟨κ1,0510,0410,0410⟩,⟨κ2,0610,0610,0210⟩,⟨κ3,0410,0610,0110⟩]f(⋌1,θ)(e2)=[⟨κ1,0410,0610,0310⟩,⟨κ2,0310,0710,0310⟩,⟨κ3,0510,0710,0110⟩]f(⋌2,θ)(e1)=[⟨κ1,0110,0210,0710⟩,⟨κ2,0310,0310,0310⟩,⟨κ3,0110,0210,0210⟩]f(⋌2,θ)(e2)=[{⟨κ1,0110,0210,0710⟩,⟨κ2,0310,0310,0310⟩,⟨κ3,0110,0210,0210⟩}].

Then (ω1,θ)∪(ω2,θ)=(ω2,θ), (ω1,θ)∪(⋌1,θ)=(⋌1,θ), (ω1,θ)∪(⋌2,θ)=(ω2,θ), (⋌1,θ)∪(⋌2,θ)=(⋌1,θ), (ω2,θ)∪(⋌2,θ)=(ω2,θ) and (ω1,θ)∩(ω2,θ)=(ω2,θ), (ω1,θ)∩⋌=(⋌1,θ), (ω1,θ)∩(⋌2,θ)=(ω2,θ), (⋌1,θ)∩(⋌2,θ)=(⋌1,θ), (ω2,θ)∩(⋌2,θ)=(⋌2,θ).

Therefore, τ1, τ2 are NSBTS on π so (π,τ1,τ2,θ) is a NSBTS.

Theorem 3.3. Let (π,τ1,τ2,θ) be a NSBTS. Then τ1∩τ2 is a NSBTS on π.

Proof:T1 and NST3 are clear. For NST2, let {(ωi,θ);i∈I}∈τ1∩τ2. Then (ωi,θ)∈τ1, (ωi,θ)∈τ2. As τ2, τ2 are Neutrosophic soft topologies on π, then ∪i(ωi,θ)∈τ1, ∪i(ωi,θ)∈τ2. Therefore ∪i(ωi,θ)∈τ1∩τ2.

Remark 3.4. Let (π,τ1,τ2,θ) be a NSBTS, then τ1∪τ2 need not be a NSBTS on π.

Example 3.5. Let π={κ1,κ2,κ3}, θ={e1,e2} and τ1={0(π,θ),1(π,θ),(ω1,θ),(ω2,θ),(ω3,θ)}, τ2={0(π,θ),1(π,θ),(⋌1,θ),(⋌2,θ) where (ω1,θ),(ω2,θ),(⋌1,θ) ve (⋌2,θ) being NSSs are as following:

f(ω1,θ)(e1)=[⟨κ1,0210,0310,0810⟩,⟨κ2,0410,0410,0410⟩,⟨κ3,0210,0410,0310⟩]f(ω1,θ)(e2)=[⟨κ1,0310,0210,0610⟩,⟨κ2,0110,0510,0510⟩,⟨κ3,0410,0310,0510⟩]f(ω2,θ)(e1)=[⟨κ1,0410,0310,0610⟩,⟨κ2,0410,0510,0310⟩,⟨κ3,0310,0510,0210⟩]f(ω2,θ)(e2)=[⟨κ1,0310,0410,0510⟩,⟨κ2,0210,0610,0410⟩,⟨κ3,0410,0610,0310⟩]f(ω3,θ)(e1)=[⟨κ1,0510,0410,0410⟩,⟨κ2,0610,0610,0210⟩,⟨κ3,0410,0610,010⟩]f(ω3,θ)(e2)=[⟨κ1,0410,0610,0310⟩,⟨κ2,0310,0710,0310⟩,⟨κ3,0510,0710,0110⟩]f(⋌1,θ)(e1)=[⟨κ1,0510,0410,0410⟩,⟨κ2,0610,0610,0210⟩,⟨κ3,0410,0610,0110⟩]f(⋌1,θ)(e2)=[⟨κ1,0410,0610,0310⟩,⟨κ2,0310,0710,0310⟩,⟨κ3,0510,0710,0110⟩]f(⋌2,θ)(e1)=[⟨κ1,0110,0210,0710⟩,⟨κ2,0310,0310,0310⟩,⟨κ3,0110,0210,0210⟩]f(⋌2,θ)(e2)=[⟨κ1,0110,0210,0710⟩,⟨κ2,0310,0310,0310⟩,⟨κ3,0110,0210,0210⟩].

Here τ1∪τ2={0(Y,θ),1(π,θ),(ω1,θ),(ω2,θ),(ω3,θ),(⋌1,θ),(⋌2,θ)} is not aneutrosophic soft topology on π because (ω3,θ)∪(⋌2,θ)∉τ1∪τ2.

Definition 3.6. Let (π,τ1,τ2,θ) be a NSBTS. Then aneutrosophic soft set

(⋉,θ)={(e,{⟨κ,T⋉(e)(κ),I⋉(e)(κ),F⋉(e)(κ)⟩}:κ∈π,e∈θ} is called as a pairwise neutrosophic soft open set (PNSOS) if there exist a NSOS τ1 and a NSOS τ2 such that for all x∈U (⋉,θ)=(⋉1,θ)∪(⋉2,θ)={(e,{⟨x,max{T⋉(e)(κ),TG(e)(κ)},max{I⋉(e)(x),IG(e)(κ)},min{F⋉(e)(κ),FG(e)(κ)}⟩}):e∈θ}.

Definition 3.7. Let (π,τ1,τ2,θ) be a NSBTS. Then a neutrosophic soft set

(⋉,θ)={(e,{⟨κ,T⋉(e)(κ),I⋉(e)(κ),F⋉(e)(κ)⟩}):κ∈π,e∈θ} is called as a PNSOS if there exist a NSOS τ1 and a τ2 such that for all κ∈π (⋉,θ)=(⋉1,θ)∪(⋉2,θ)={(e,{⟨x,max{TH(e)(κ),TG(e)(κ)},max{I⋉(e)(κ),IG(e)(κ)},min{F⋉(e)(κ),FG(e)(κ)}⟩}):e∈θ}.

The set of all pairwise neutrosophic open sets in (π,τ1,τ2,θ) is denoted by PNSO (π,τ1,τ2,θ).

Definition 3.8. Let (π,τ1,τ2,θ) be a NSBTS. Then a NSS

(⋉,θ)={(e,{⟨κ,T⋉(e)(κ),I⋉(e)(κ),F⋉(e)(x)⟩}):κ∈π,e∈θ} is called as a pairwise neutrosophic soft closed set (PNSC) if (⋉,θ)c is a PNSO. It is clear that τ1 and a NSOC τ2 such that for all κ∈π (⋉,θ)=(⋉1,θ)∩(⋉2,θ)={(e,{⟨κ,min{⋉(κ),TG(e)(κ)},min{I⋉(e)(κ),IG(e)(κ)},max{F⋉(e)(κ),FG(e)(κ)}⟩}):e∈θ}

The set of all PNSC in (π,τ1,τ2,θ) is denoted by PNSC (π,τ1,τ2,θ).

Example 3.9. Let π={κ1,κ2,κ3}, θ={e1,e2}, τ1={0(π,θ),1(π,θ),(ω1,θ)}, τ2={0(π,θ),1(π,θ),(ω2,θ)} where (ω1,θ), (ω2,θ) are defined as

f(ω1,θ)(e1)=[⟨κ1,0210,0310,0810⟩,⟨κ2,0410,0410,0410⟩,⟨κ3,0210,0410,0310⟩]f(ω1,θ)(e2)=[⟨κ1,0310,0210,0610⟩,⟨κ2,0110,0510,0510⟩,⟨κ3,0410,0310,0510⟩]f(ω2,θ)(e1)=[⟨κ1,0410,0210,0610⟩,⟨κ2,0110,0510,0510⟩,⟨κ3,0410,0310,0510⟩]f(ω2,θ)(e2)=[⟨κ1,0410,0310,0610⟩,⟨κ2,0410,0510,0310⟩,⟨κ3,0310,0510,0210⟩]

Then (ω1,θ)∪(ω2,θ)={(e1,{⟨κ1,0210,0210,0810⟩,⟨κ2,0110,0410,0510⟩,⟨κ3,0210,0310,0510⟩}), (e2,{⟨κ1,0310,0210,0610⟩,⟨κ2,0110,0510,0510⟩,⟨κ3,0310,0310,0510⟩})} is a PNSOS. Also

f(ω1,θ)c(e1)={[⟨κ1,0810,0710,0210⟩,⟨κ2,0610,0610,0610⟩,⟨κ3,0310,0610,0210⟩]}f(ω1,θ)c(e2)={[⟨κ1,0610,0810,0310⟩,⟨κ2,0510,0510,0110⟩,⟨κ3,0510,0710,0410⟩]}f(ω2,θ)c(e2)={[⟨κ1,0610,0810,0310⟩,⟨κ2,0510,0510,0110⟩,⟨κ3,0310,0710,0410⟩]}f(ω2,θ)c(e1)={[⟨κ1,0610,0710,0410⟩,⟨κ2,0310,0510,0410⟩,⟨κ3,0210,0510,0310⟩]}.

Therefore,

(ω1,θ)c∩(ω2,θ)c={(e1,{⟨κ1,0810,0710,0210⟩,⟨κ2,0610,0610,0610⟩,⟨κ3,0810,0610,0710⟩})(e2,{⟨κ1,0710,0810,0410⟩,⟨κ2,0910,0510,0510⟩,⟨κ3,0610,0710,0110⟩})}

is a PNSC.

Theorem 3.10. Let (π,τ1,τ2,θ) be a NSBTS. In this case

1.    0(π,θ),1(π,θ)∈PNSO(π,τ1,τ2,θ).

2.    If {(⋉i,θ)|i∈I}⊆PNSO(π,τ1,τ2,θ) then ⋃i∈I(⋉i,θ)∈PNSO(π,τ1,τ2,θ).

3.    If {(Gi,θ)|i∈I}⊆PNSC(π,τ1,τ2,θ) then ⋂i∈I(Gi,θ)∈PNSC(π,τ1,τ2,θ).

Proof.

1.    Since 0(π,θ)∪0(π,θ)=0(π,θ), 1(π,θ)∪1(π,θ)=1(π,θ) then 0(π,θ), 1(π,θ) are PNSC.

2.    Since (⋉i,θ)∈PNSO(π,τ1,τ2,θ), there exist (⋉i1,θ)∈τ1, (⋉i1,θ)∈τ2 such that (⋉i,θ)=(⋉i1,θ)∪(⋉i1,θ) for all i∈I. Then

⋃i∈I(⋉i,θ)=⋃i∈I((⋉i1,θ)∪(⋉i1,θ))=(⋃i∈I(⋉i1,θ))∪(⋃i∈I(⋉i2,θ)).

As τ1, τ2 are NST on π, ⋃i∈I(⋉i1,θ)∈τ1, ⋃i∈I(⋉i2,θ)∈τ1.

Therefore, ⋃i∈I(⋉i,θ)∈PNSO(π,τ1,τ2,θ).

3.    Since (Gi,θ)∈PNSC(π,τ1,τ2,θ), there exist (Gi1,θ)c∈τ1 and (Gi2,θ)c∈τ2 such that (Gi,θ)=(Gi1,θ)∩(Gi2,θ) for all i∈I. Then

⋂i∈I(Gi,θ)=⋂i∈I((Gi1,θ)∩(Gi2,θ))=(⋂i∈I(Gi1,θ))∩(⋂i∈I(Gi2,θ)).

Then ⋂i∈I(Gi,θ)∈PNSC(π,τ1,τ2,θ) as (⋂i∈I(Gi1,θ))c∈τ1, (⋂i∈I(Gi2,θ))c∈τ1.

Theorem 3.11. Let (π,τ1,τ2,θ) be a NSBTS. Then every neutrosophic soft i-open set is a PNSOS.

Proof. Let (⋉,θ)∈τI or (⋉,θ)∈τ2. Since (⋉,θ)=(⋉,θ)∪0(X,θ), then (⋉,θ)∈PNSO(π).

Theorem 3.12. Let (π,τ1,τ2,θ) be a NSBTS and (⋌,θ),(ω,θ)∈NSS(π). Then,

1.    clPNSS(0(π,θ))=0(π,θ) and clPNSS(1(π,θ))=1(π,θ)

2.    (⋌,θ)⊆clPNSS(⋌,θ)

3.    (⋌,θ) is a PNSCS if clPNSS(⋌,θ)=(⋌,θ)

4.    clPNSS(⋌,θ)⊆clPNSS(ω,θ) if(⋌,θ)⊆(ω,θ)

5.    clPNSS(⋌,θ)∪clPNSS(ω,θ)⊂clPNSS((⋌,θ)∪(ω,θ))

6.    clPNSS(clPNSS(⋌,θ))=clPNSS(⋌,θ), i.e., clPNSS(⋌,θ) is a PNSCS.

Proof. It is obvious.

Theorem 3.13. Let (π,τ1,τ2,θ) be a NSBTS over π and (ω,θ)∈NSS(π). Then, κe(y1,y2,y3)∈clPNSS(ω,θ) if and only if for all Uxe(y1,y2,y3)∈τ12(κe(y1,y2,y3)) where Uκe(y1,y2,y3) is any PNSOS contains κe(y1,y2,y3) and τ12(κe(y1,y2,y3)) is the family of all PNSOS contains κe(y1,y2,y3), Uxe(y1,y2,y3)∩(⋌,θ)≠0(π,θ).

Proof. Let κe(y1,y2,y3)∈clPNSS(ω,θ) and suppose that there exists Uκe(y1,y2,y3)∈τ12(xe(y1,y2,y3)) such that Uκe(y1,y2,y3)∩(⋌,θ)=0(π,θ). Then (⋌,θ)⊂(Uκe(y1,y2,y3))c. Thus clPNSS(⋌,θ)⊂clPNSS(Uκe(y1,y2,y3))c=(Uκe(y1,y2,y3))c which implies clPNSS(ω,θ)∩Uκe(y1,y2,y3)c=0(π,θ), a contradiction.

Conversely, that κe(y1,y2,y3)∉clPNSS(⋌,θ), then κe(y1,y2,y3)∈(clPNSS(⋌,θ))c∈τ12(xe(y1,y2,y3)). Therefore, by hypothesis, (clPNSS(⋌,θ))∩(⋌,θ)≠0(π,θ), a contradiction.

4  Main Results

In this section, some new definitions are introduced which are necessary for the up-coming sections.

Definition 4.1. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS over π,(f~,θ) be a neutrosophic soft set overπ. Then (f~,θ) is

(1) Neutrosophic soft semi-open if (f~,θ)⊆VScl(VSint(f~,θ)).

(2) Neutrosophic soft pre-open if (f~,θ)⊆VSint(VScl(f~,θ)).

(3) Neutrosophic soft * b open if (f~,θ)⊆VScl(VSint(f~,θ))⊔VSint(VScl(f~,θ)) and neutrosophic soft * b close if (f~,θ)⊇VScl(VSint(f~,θ))⊓VSint(VScl(f~,θ)).

Definition 4.2. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS over π, κ1⋋⟨y1,y2,y3⟩≠κ2⋋/⟨y1/,y2/,y3/⟩ are neutrosophic soft points. If there exist NS * b open sets (f~,θ)&(𝓆~,θ) such that κ1⋋⟨y1,y2,y3⟩∈(f~,θ),κ1⋋⟨y1,y2,y3⟩⊓(𝓆~,θ)=0(⟨π⟩~,θ) or κ2⋋/⟨y1/,y2/,y3/⟩∈(𝓆~,θ),κ2⋋/⟨y1/,y2/,y3/⟩⊓(f~,θ)=0(⟨π⟩~,θ), Then ⟨π,τ1,τ2,θ⟩ is called a NSB * b0 space.

Definition 4.3. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS over π, κ1⋋⟨y1,y2,y3⟩≠κ2⋋/⟨y1/,y2/,y3/⟩ are neutrosophic soft points. If there exist NS * b-open sets (f~,θ),(𝓆~,θ) such that κ1⋋⟨y1,y2,y3⟩∈(f~,θ),κ1⋋⟨y1,y2,y3⟩⊓(𝓆~,θ)=0(⟨π⟩~,θ) and κ2⋋/⟨Δ1/,Δ2/,Δ3/⟩∈(𝓆~,θ),κ2⋋/⟨y1/,y2/,y3/⟩⊓(f~,θ)=0(⟨π⟩~,θ), Then ⟨π,τ1,τ2,θ⟩ is called a NSB * b1 space.

Definition 4.4. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS over π,κ1⋋⟨y1,y2,y3⟩≠κ2⋋/⟨y1/,y2/,y3/⟩ are neutrosophic soft points. If there exists NS * b open set (f~,θ), such that κ1⋋⟨y1,y2,y3⟩∈(f~,θ), κ2⋋/⟨y1/,y2/,y3/⟩∈(𝓆~,θ) & (f~,θ)⊓(𝓆~,θ)=0(⟨π⟩~,θ). Then ⟨π,τ1,τ2,θ⟩ is called a NSB * b2 space.

Example 4.5. let π~={𝓍1,𝓍2}, the set of conditions by θ={e1,e2}. Let us consider neutrosophic set and 𝓍e11(0.1,0.4,0.7), 𝓍e21(0.2,0.5,0.6), 𝓍e12(0.3,0.3,0.5) and 𝓍e12(0.4,0.4,0.4) be neutrosophic soft points. Then the family τ1={0(⟨π⟩~,θ),1(⟨π⟩~,θ),(f1~,θ),(f2~,θ), (f3~,θ), (f4~,θ),(f5~,θ),(f6~,θ), (f7~,θ),(f8~,θ),…(f15~,θ)}, where (f1~,θ)=𝓍e11(0.1,0.4,0.7),(f2~,θ)=𝓍e21(0.2,0.5,0.6), (f3~,θ)=𝓍e12(0.3,0.3,0.5), (f4~,θ)=𝓍e22(0.4,0.4,0.4)(f5~,θ)=(f1~,θ)∪(f2~,θ),(f6~,θ)=(f1~,θ)∪(f3~,θ),(f7~,θ)=(f2~,θ)∪(f4~,θ),(f8~,θ)=(f2~,θ)∪(f3~,θ),(f9~,θ)=(f2~,θ)∪(f4~,θ),(f10~,θ)=(f3~,θ)∪(f4~,θ),(f11~,θ)=(f1~,θ)∪(f2~,θ)∪(f3~,θ),(f12~,θ)=(f1~,θ)∪(f2~,θ)∪(f4~,θ),(f13~,θ)=(f2~,θ)∪(f3~,θ)∪(f4~,θ),(f14~,θ)=(f1~,θ)∪(f3~,θ)∪(f4~,θ), (f15~,θ)={𝓍e11(0.1,0.4,0.7),𝓍e21(0.2,0.5,0.6),𝓍e12(0.3,0.3,0.5),𝓍e22(0.4,0.4,0.4)} is a NSTS. Also τ2={0(⟨π⟩~,θ),1(⟨π⟩~,θ),(f1~,θ)}NSTS. Thus (π, τ1,τ2, θ) be a NSBTS. Also (π, τ1,τ2, θ) is VSB * b2 structure.

Theorem 4.6. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS. Then ⟨π,τ1,τ2,θ⟩ be a NSB * b1 structure if and only if each neutrosophic soft point is a NS * b-close.

Proof. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS over π.κ1⋋⟨y1,y2,y3⟩ be an arbitrary neutrosophic soft point. We establish κ1⋋⟨y1,y2,y3⟩ is a neutrosophic soft * b-open set. Let κ2⋋/⟨y1/,y2/,y3/⟩∈κ1⋋⟨y1,y2,y3⟩. Then either κ2⋋/⟨y1/,y2/,y3/⟩≻κ1⋋⟨y1,y2,y3⟩ or κ2⋋/⟨y1/,y2/,y3/⟩≺κ2⋋/⟨y1/,y2/,y3/⟩ or κ2⋋/⟨y1/,y2/,y3/⟩≻≻κ1⋋⟨y1,y2,y3⟩ or κ2⋋/⟨y1/,y2/,y3/⟩≺≺κ1⋋⟨y1,y2,y3⟩. This means that κ2⋋/⟨y1/,y2/,y3/⟩ and κ1⋋⟨y1,y2,y3⟩ are two are distinct NS points. Thus κ1≻κ2 or κ1≺κ2 or ⋋/≻⋋ or ⋋/≺⋋ or κ1≻≻κ2 or κ1≺≺κ2 or ⋋/≻≻⋋ or ⋋/≺≺⋋. Since ⟨π,τ1,τ2,θ⟩ be a NS * b1 structure, ∃ a NS * b-open set (𝓆~,θ) so that κ2⋋/⟨y1/,y2/,y3/⟩∈(𝓆~,θ) and κ1⋋⟨y1,y2,y3⟩⊓(𝓆~,θ)=0(⟨π⟩~,θ). Since, κ1⋋⟨y1,y2,y3⟩⊓(𝓆~,θ)=0(⟨π⟩~,θ). So κ2⋋/⟨y1/,y2/,y3/⟩∈(𝓆~,θ)⊂κ1⋋⟨y1,y2,y3⟩. Thus κ1⋋⟨y1,y2,y3⟩ is a NS * b-open set, i.e., κ1⋋⟨y1,y2,y3⟩ is a NS * b-closed set. Suppose that each neutrosophic soft point κ1⋋⟨y1,y2,y3⟩ is a NS * b-closed. Then (κ1⋋⟨y1,y2,y3⟩)c is a NS * b-open set. Let κ1⋋⟨y1,y2,y3⟩⊓κ2⋋/⟨y1/,y2/,y3/⟩=0(⟨π⟩~,θ). Thus (κ2⋋/⟨y1/,y2/,y3/⟩,θ)∈(κ1⋋⟨y1,y2,y3⟩)c, κ1⋋⟨y1,y2,y3⟩⊓(κ1⋋⟨y1,y2,y3⟩)c=0(⟨π⟩~,θ). So ⟨π,τ1,τ2,θ⟩ be a NSB- * b1 space.

Theorem 4.7. Let ⟨π,τ1,τ2,θ⟩ be a NSTBS over the father set π Then (π,τ,θ) is NS- * b2 space if and only if for distinct neutrosophic soft points κ1⋋⟨y1,y2,y3⟩, κ2⋋/⟨y1/,y2/,y3/⟩, there exists a NS * b-open set (f~,θ) containing but not κ2⋋/⟨y1/,y2/,y3/⟩ such that κ2⋋/⟨y1/,y2/,y3/⟩∉(f~,θ).¯

Proof. Let κ1⋋⟨y1,y2,y3⟩≻κ2⋋/⟨y1/,y2/,y3/⟩ be two neutrosophic soft points in NS * b2 space. Then there exists disjoint NS * b open sets (f~,θ),(𝓆~,θ) such that κ1⋋⟨y1,y2,y3⟩∈(f~,θ)&κ2⋋/⟨y1/,y2/,y3/⟩∈(𝓆~,θ). Since ⟨κ1⋋⟨y1,y2,y3⟩,θ⟩⊓κ2⋋/⟨y1/,y2/,y3/⟩=0(⟨π⟩~,θ) and (f~,θ)⊓(𝓆~,θ)=0(⟨π⟩~,θ).κ2⋋/⟨y1/,y2/,y3/⟩∉(f~,θ)⟹κ2⋋/⟨y1/,y2/,y3/⟩∉(f~,θ).¯ Next suppose that, κ1⋋⟨y1,y2,y3⟩≻κ2⋋/⟨y1/,y2/,y3/⟩, there exists a NS * b open set (f~,θ) containing κ1⋋⟨y1,y2,y3⟩ but not κ2⋋/⟨y1/,y2/,y3/⟩ such that κ2⋋/⟨y1/,y2/,y3/⟩∉(f~,θ)¯c that is (f~,θ) and (f~,θ)¯c are mutually exclusive NS * b open sets supposing κ1⋋⟨y1,y2,y3⟩ and κ2⋋/⟨y1/,y2/,y3/⟩ in turn.

Theorem 4.8. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS. Then ⟨π,τ1,τ2,θ⟩ is NS * b1 space if every neutrosophic soft point κ1⋋⟨y1,y2,y3⟩∈(f~,θ)∈(π,τ,θ). If there exists NS * b open set (𝓆~,θ) such that κ1⋋⟨y1,y2,y3⟩∈(𝓆~,θ)⊂(𝓆~,θ)¯⊂(f~,θ), Then ⟨π,τ1,τ2,θ⟩ a NS * b2 space.

Proof. Suppose κ1⋋⟨y1,y2,y3⟩⊓κ2⋋/⟨y1/,y2/,y3/⟩=0(⟨π⟩~,θ). Since ⟨π,τ1,τ2,θ⟩ is NS * b1 space. κ1⋋⟨y1,y2,y3⟩ and κ2⋋/⟨y1/,y2/,y3/⟩ are NS∗b closed sets in ⟨π,τ1,τ2,θ⟩. Then κ1⋋⟨y1,y2,y3⟩∈(κ2⋋/⟨y1/,y2/,y3/⟩)c∈⟨π,τ1,τ2,θ⟩. Thus there exists a NS * b open set (𝓆~,θ)∈⟨π,τ1,τ2,θ⟩ such that κ1⋋⟨y1,y2,y3⟩∈(𝓆~,θ)⊂(𝓆~,θ)¯⊂(κ2⋋/⟨y1/,y2/,y3/⟩)c. So we have κ2⋋/⟨y1/,y2/,y3/⟩∈(𝓆~,θ) and (𝓆~,θ)⊓((𝓆~,θ))c=0(⟨π⟩~,θ), that is ⟨π,τ1,τ2,θ⟩ is a neutrosophic soft * b2 space.

Definition 4.9. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS. (f~,θ) be a NS * b closed set and κ1⋋⟨y1,y2,y3⟩⊓(f~,θ)=0(⟨X⟩~,θ). If there exists NS * b-open sets (𝓆1~,θ) and (𝓆2~,θ) such that κ1⋋⟨y1,y2,y3⟩∈(𝓆1~,θ),(f~,θ)⊂(𝓆2~,θ) & κ1⋋⟨y1,y2,y3⟩⊓(𝓆1~,θ)=0(⟨π⟩~,θ), then ⟨π,τ1,τ2,θ⟩ is called a NBS * b-regular space. ⟨π,τ1,τ2,θ⟩ is said to be NS * b3 space, if it is both a NS regular and NS B * b1 space.

Theorem 4.10. Let ⟨π,τ1,τ2,θ⟩ be a NSBTS. (π,τ,θ) is soft * b3 space if and only if for every κ1⋋⟨y1,y2,y3⟩∈(f~,θ) that is (𝓆~,θ)∈ (⟨π,τ1,τ2,θ⟩ such that κ1⋋⟨y1,y2,y3⟩∈(𝓆~,θ)⊂(𝓆~,θ)¯⊂(f~,θ).

Proof. Let ⟨π,τ1,τ2,θ⟩ is NS * b3 space and κ1⋋⟨y1,y2,y3⟩∈(f~,θ)∈⟨π,τ1,τ2,θ⟩. Since ⟨π,τ1,τ2,θ⟩ is NS * b3 space for the neutrosophic soft point κ1⋋⟨y1,y2,y3⟩ and NS * b closed set (f~,θ)c, there exists (𝓆1~,θ) and (𝓆2~,θ) such that κ1⋋⟨y1,y2,y3⟩∈(𝓆1~,θ),(f~,θ)c⊂(𝓆2~,θ) and (𝓆1~,θ)∩(𝓆2~,θ)=0(⟨π⟩~,θ). Then we have ⟨κ1⋋⟨y1,y2,y3⟩,θ⟩∈(𝓆1~,θ)⊂(𝓆2~,θ)c⊂(f~,θ). Since (𝓆2~,θ)cNS∗b closed set (𝓆1~,θ)¯⊂(𝓆2~,θ)c. Conversely, let ⟨κ1⋋⟨y1,y2,y3⟩,θ⟩∩(H~,θ)=0(⟨π⟩~,θ) and (H~,θ) be a NS * b closed set. κ1⋋⟨y1,y2,y3⟩∈(H~,θ)c and from the condition of the theorem, we have κ1⋋⟨y1,y2,y3⟩∈(𝓆~,θ)⊂(𝓆~,θ)¯⊂(H~,θ)c. Thus κ1⋋⟨y1,y2,y3⟩∈(𝓆~,θ),(H~,θ)⊂(𝓆~,θ)¯c and (𝓆~,θ)∩(𝓆~,θ)¯c=0(⟨π⟩~,θ). So (⟨π,τ1,τ2,θ⟩ is NSB * b3 space.

Proof. Suppose 𝒻(κ1⋋⟨y1,y2,y3⟩),𝒻(κ2⋋/⟨y1/,y2/,y3/⟩) be two points of ⟨Y⟩~ such that either 𝒻(κ1⋋⟨y1,y2,y3⟩)≻𝒻(κ2⋋/⟨y1/,y2/,y3/⟩) or ≺𝒻(κ1⋋⟨y1,y2,y3⟩)≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩). Then (κ1⋋⟨y1,y2,y3⟩, κ2⋋/⟨y1/,y2/,y3/⟩)∉{(κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≻(κ2⋋/⟨y1/,y2/,y3/⟩):𝒻(κ1⋋⟨y1,y2,y3⟩)≻ 𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)} or (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∉{κ1⋋⟨y1,y2,y3⟩ with κ1⋋⟨y1,y2,y3⟩≺κ2⋋/⟨y1/,y2/,y3/⟩: 𝒻(κ1⋋⟨y1,y2,y3⟩)≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}, that is (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∈{(κ1⋋⟨y1,y2,y3⟩, κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≻κ2⋋/⟨y1/,y2/,y3/⟩:𝒻(κ1⋋⟨y1,y2,y3⟩)≻𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c or (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∈{(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≺κ2⋋/⟨y1/,y2/,y3/⟩: 𝒻(κ1⋋⟨y1,y2,y3⟩)≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c. Since, (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∈{(κ1⋋⟨y1,y2,y3⟩, κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≻κ2⋋/⟨y1/,y2/,y3/⟩:𝒻(κ1⋋⟨y1,y2,y3⟩)≻𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c or (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∈{(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≺κ2⋋/⟨y1/,y2/,y3/⟩: 𝒻(κ1⋋⟨y1,y2,y3⟩)≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c is soft set in ⟨π~,τ1,τ2,θ⟩×⟨⟨Y⟩~,F1,F2,θ⟩, then there exists NS * b open sets ⟨G,θ⟩ and ⟨𝓀,θ⟩ in ⟨π~,τ1,τ2,θ⟩ such that (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∈⟨G,θ⟩×⟨𝓀,θ⟩⋐{((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≻κ2⋋/⟨y1/,y2/,y3/⟩:𝒻((κ1⋋⟨y1,y2,y3⟩))≻ 𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c or (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∈⟨G,θ⟩×⟨𝓀,θ⟩⋐{((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≺κ2⋋/⟨y1/,y2/,y3/⟩:𝒻(κ1⋋⟨y1,y2,y3⟩)≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c. Then, since 𝒻 is NS * b open, 𝒻(⟨G,θ⟩) and 𝒻(⟨𝓀,θ⟩) are NS * b open sets in (⟨Y⟩~,F1,F2,θ) containing 𝒻(κ1⋋⟨y1,y2,y3⟩) and 𝒻(κ2⋋/⟨y1/,y2/,y3/⟩) respectively, and 𝒻(⟨G,θ⟩) ∩~𝒻(⟨𝓀,θ⟩)=0(⟨π⟩~,θ)~ otherwise 𝒻(⟨G,θ⟩) ×𝒻(⟨𝓀,θ⟩)∩~{((κ1⋋⟨y1,y2,y3⟩),) with (κ1⋋⟨y1,y2,y3⟩)≻κ2⋋/⟨y1/,y2/,y3/⟩:𝒻((κ1⋋⟨y1,y2,y3⟩))≻𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)} or (((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩))∉{((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩)with(κ1⋋⟨y1,y2,y3⟩)≺κ2⋋/⟨y1/,y2/,y3/⟩: 𝒻((κ1⋋⟨y1,y2,y3⟩))≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}=0(⟨π⟩~,θ)~. It follows that ⟨⟨Y⟩~,F1,F2,θ⟩ is NSB * b Hausdorff space.

Theorem 5.5. Let ⟨π~,τ1,τ2,θ⟩ be a NSB second countable space and let ⟨𝒻,θ⟩ be NS uncountable sub set of ⟨π~,τ1,τ2,θ⟩. Then, at least one point of ⟨𝒻,θ⟩ is a soft limit point of ⟨𝒻,θ⟩.

Proof. Let W=⟨B1,B2,B3,B4,…Bn:n∈N⟩for⟨π~,τ1,τ2,θ⟩.

Let, if possible, no point of ⟨𝒻,θ⟩ be a soft limit point of ⟨𝒻,θ⟩. Then, for each κ1⋋⟨y1,y2,y3⟩∈⟨𝒻,θ⟩ there exists NS * b open set ⟨ρ,θ⟩(κ1⋋⟨y1,y2,y3⟩) such that κ1⋋⟨y1,y2,y3⟩∈⟨ρ,θ⟩(κ1⋋⟨y1,y2,y3⟩) and ⟨ρ,θ⟩(κ1⋋⟨y1,y2,y3⟩)∩~⟨𝒻,θ⟩={(κ1⋋⟨y1,y2,y3⟩)}. Since W is soft base there exists Bn(κ1⋋⟨y1,y2,y3⟩)∈W such that (κ1⋋⟨y1,y2,y3⟩)∈Bn(κ1⋋⟨y1,y2,y3⟩)⋐⟨ρ,θ⟩(κ1⋋⟨y1,y2,y3⟩). Therefore Bn(κ1⋋⟨y1,y2,y3⟩)∩~⟨𝒻,θ⟩⋐⟨ρ,θ⟩(κ1⋋⟨y1,y2,y3⟩)∩~⟨𝒻,θ⟩={(κ1⋋⟨y1,y2,y3⟩)}. More-over, if κ1⋋⟨y1,y2,y3⟩1 and κ1⋋⟨y1,y2,y3⟩2 be any two NS points such that κ1⋋⟨y1,y2,y3⟩1≠κ1⋋⟨y1,y2,y3⟩2 which means either κ1⋋⟨y1,y2,y3⟩1≻κ1⋋⟨y1,y2,y3⟩2 or κ1⋋⟨y1,y2,y3⟩1≺κ1⋋⟨y1,y2,y3⟩2 then there exists Bn(κ1⋋⟨y1,y2,y3⟩)1 and Bn(κ1⋋⟨y1,y2,y3⟩)2 in W such that Bn(κ1⋋⟨y1,y2,y3⟩)1∩~⟨𝒻,θ⟩={(κ1⋋⟨y1,y2,y3⟩)1} and Bn(κ1⋋⟨y1,y2,y3⟩)2∩~⟨𝒻,θ⟩={(κ1⋋⟨y1,y2,y3⟩)2}. Now, (κ1⋋⟨y1,y2,y3⟩)1≠(κ1⋋⟨y1,y2,y3⟩)2 which guarantees that {(κ1⋋⟨y1,y2,y3⟩)1}≠{(κ1⋋⟨y1,y2,y3⟩)2} which implies that Bn(κ1⋋⟨y1,y2,y3⟩)1∩~⟨𝒻,θ⟩≠Bn(κ1⋋⟨y1,y2,y3⟩)2∩~⟨𝒻,θ⟩ which implies Bn(κ1⋋⟨y1,y2,y3⟩)1≠Bn(κ1⋋⟨y1,y2,y3⟩)2. Thus, there exists a one to one soft correspondence of ⟨𝒻,θ⟩ on to {Bn(κ1⋋⟨y1,y2,y3⟩):(κ1⋋⟨y1,y2,y3⟩)∈⟨𝒻,θ⟩}. Now, ⟨𝒻,θ⟩ being NS uncountable, it follows that {Bn(κ1⋋⟨y1,y2,y3⟩):(κ1⋋⟨y1,y2,y3⟩)∈⟨𝒻,θ⟩} is NS uncountable. But, this is purely a contradiction.

5  More Main Results

In this section, more main results are addressed. The structures in one space can be switched over to another space through soft functions satisfying some more conditions. These structures are separation axioms and other separation axioms, compactness and countably compactness.

Theorem 5.1. Let ⟨π,τ1,τ2,θ⟩ be NSBTS such that it is NSB * b Hausdorff space and (⟨Y⟩~,F1,F2,θ) be an-other NSBTS. Let ⟨𝒻,θ⟩: (π,τ1,τ2,θ)⟶(⟨Y⟩~,F1,F2,θ) be an-other NSBTS be a soft function such that it is soft monotone and continuous. Then (⟨Y⟩~,F1,F2,θ) is also of characteristic of NSB * b Hausdorffness.

Proof. Suppose κ1⋋⟨y1,y2,y3⟩1,κ1⋋⟨y1,y2,y3⟩2∈⟨π⟩~ such that either κ1⋋⟨y1,y2,y3⟩1≻κ1⋋⟨y1,y2,y3⟩2 or κ1⋋⟨y1,y2,y3⟩1≺κ1⋋⟨y1,y2,y3⟩2. Since ⟨𝒻,θ⟩ is soft monotone. Let us suppose the monotonically increasing case. So, κ1⋋⟨y1,y2,y3⟩1≻κ1⋋⟨Δ1,Δ2,Δ3⟩2 or κ1⋋⟨y1,y2,y3⟩1≺κ1⋋⟨y1,y2,y3⟩2 implies that 𝒻⟨κ1⋋⟨y1,y2,y3⟩⟩1≻𝒻⟨κ1⋋⟨y1,y2,y3⟩⟩2 or 𝒻⟨κ1⋋⟨y1,y2,y3⟩⟩1≺𝒻⟨κ1⋋⟨y1,y2,y3⟩⟩2 respectively. Suppose κ2⋋/⟨y1/,y2/,y3/⟩1, κ2⋋/⟨y1/,y2/,y3/⟩2∈⟨Y⟩~ such that κ2⋋/⟨y1/,y2/,y3/⟩1≻κ2⋋/⟨y1/,y2/,y3/⟩2 or κ2⋋/⟨y1/,y2/,y3/⟩1≺κ2⋋/⟨y1/,y2/,y3/⟩2 so, κ2⋋/⟨y1/,y2/,y3/⟩1≻κ2⋋/⟨y1/,y2/,y3/⟩2 or κ2⋋/⟨y1/,y2/,y3/⟩1≺κ2⋋/⟨y1/,y2/,y3/⟩2 respectively such that κ2⋋/⟨y1/,y2/,y3/⟩1=𝒻⟨κ1⋋⟨y1,y2,y3⟩⟩1,κ2⋋/⟨y1/,y2/,y3/⟩2=𝒻κ1⋋⟨y1,y2,y3⟩2, since, (π,τ1,τ2,θ) is NSB * b Hausdorff space so there exists mutually disjoint NS * b open sets ⟨𝓀1,θ⟩ and ⟨𝓀2,θ⟩∈(π,τ1,τ2,θ)⟹𝒻(⟨𝓀1,θ⟩) and 𝒻(⟨𝓀2,θ⟩)∈⟨Y⟩~,F,θ). We claim that 𝒻(⟨𝓀1,θ⟩)∩~𝒻(⟨𝓀2,θ⟩)=0(⟨π⟩~,θ~. Otherwise 𝒻(⟨𝓀1,θ⟩)∩~𝒻(⟨𝓀2,θ⟩)≠0(⟨π⟩~,θ)~. Suppose there exists κ3⋋//⟨y1//,y2//,y3//⟩1∈𝒻(⟨𝓀1,θ⟩)∩~𝒻(⟨𝓀2,θ⟩)⟹κ3⋋//⟨y1//,y2//,y3//⟩1∈𝒻(⟨𝓀1,θ⟩)κ3⋋//⟨y1//,y2//,y3//⟩1∈𝒻(⟨𝓀2,θ⟩),κ3⋋//⟨y1//,y2//,y3//⟩1∈𝒻(⟨𝓀1,θ⟩),𝒻 is soft one-one and there exists κ3⋋//⟨y1//,y2//,y3//⟩2∈⟨𝓀1,θ⟩ such that κ3⋋//⟨y1//,y2//,y3//⟩1=𝒻(κ3⋋//⟨y1//,y2//,y3//⟩2),κ3⋋//⟨y1//,y2//,y3//⟩1∈𝒻(⟨𝓀2,θ⟩)⟹∃κ3⋋//⟨y1//,y2//,y3//⟩3∈⟨𝓀2,θ⟩ such that 𝒻(κ3⋋//⟨y1//,y2//,y3//⟩3)⟹𝒻(κ3⋋//⟨y1//,y2//,y3//⟩2)=𝒻(κ3⋋//⟨y1//,y2//,y3//⟩3). Since, 𝒻 is soft one-one ⟹κ3⋋//⟨y1//,y2//,y3//⟩2=κ3⋋//⟨y1//,y2//,y3//⟩3 implies that κ3⋋//⟨y1//,y2//,y3//⟩2∈𝒻(⟨𝓀1,θ⟩),κ3⋋//⟨y1//,y2//,y3//⟩2∈𝒻(⟨𝓀2,θ⟩) implies that κ3⋋//⟨y1//,y2//,y3//⟩2∈𝒻(⟨𝓀1,θ⟩)∩~𝒻(⟨𝓀2,θ⟩). This is contradiction because ⟨𝓀1,θ⟩∩~⟨𝓀2,θ⟩=0(⟨π⟩~,θ). Therefore, 𝒻(⟨𝓀1,θ⟩)∩~𝒻(⟨𝓀2,θ⟩)=0(⟨π⟩~,θ). Finally, κ1⋋⟨y1,y2,y3⟩1≻κ1⋋⟨y1,y2,y3⟩2orκ1⋋⟨y1,y2,y3⟩1≺κ1⋋⟨y1,y2,y3⟩2⟹κ1⋋⟨y1,y2,y3⟩1≠κ1⋋⟨y1,y2,y3⟩2⟹𝒻(κ1⋋⟨y1,y2,y3⟩1)≻𝒻(κ1⋋⟨y1,y2,y3⟩2) or 𝒻(κ1⋋⟨y1,y2,y3⟩1)≺𝒻(κ1⋋⟨y1,y2,y3⟩2)⟹𝒻(κ1⋋⟨y1,y2,y3⟩1)≠𝒻(κ1⋋⟨y1,y2,y3⟩2). Given a pair of points κ2⋋/⟨y1/,y2/,y3/⟩1,κ2⋋/⟨y1/,y2/,y3/⟩2∈⟨Y⟩~∋κ2⋋/⟨y1/,y2/,y3/⟩1≠κ2⋋/⟨y1/,y2/,y3/⟩2 We are able to find out mutually exclusive NS * b open sets 𝒻(⟨𝓀1,θ⟩),𝒻(⟨𝓀2,θ⟩)∈(⟨Y⟩~,F1,F2,θ) such that κ2⋋/⟨y1/,y2/,y3/⟩1∈𝒻(⟨𝓀1,θ⟩),κ2⋋/⟨y1/,y2/,y3/⟩2∈𝒻(⟨𝓀2,θ⟩) this proves that (⟨Y⟩~,F1,F2,θ) is NSB * b Husdorff space.

Theorem 5.2. Let (π~,τ1,τ2,θ) be NSBTS and (⟨Y⟩~,F1,F2,θ) be an-other NSBTS which satisfies one more condition of NSB * b Hausdorffness. Let ⟨𝒻,θ⟩: (π~,τ1,τ2,θ)⟶(⟨Y⟩~,F1,F2,θ) a soft function such that it is soft monotone and continuous. Then (π~,τ1,τ2,θ) is also of characteristics of NSB * b Hausdorffness.

Proof. Suppose κ1⋋⟨y1,y2,y3⟩1,κ1⋋⟨y1,y2,y3⟩2∈π~ such that either κ1⋋⟨y1,y2,y3⟩1≻κ1⋋⟨y1,y2,y3⟩2 or κ1⋋⟨y1,y2,y3⟩1≺κ1⋋⟨y1,y2,y3⟩2. Let κ1⋋⟨y1,y2,y3⟩1≻κ1⋋⟨y1,y2,y3⟩2 or κ1⋋⟨y1,y2,y3⟩1≺κ1⋋⟨y1,y2,y3⟩2 implies that 𝒻(κ1⋋⟨y1,y2,y3⟩1)≻𝒻(κ1⋋⟨y1,y2,y3⟩2)or𝒻(κ1⋋⟨y1,y2,y3⟩1)≺𝒻(κ1⋋⟨y1,y2,y3⟩2) respectively. Suppose κ2⋋/⟨y1/,y2/,y3/⟩1,κ2⋋/⟨y1/,y2/,y3/⟩2∈⟨Y⟩~ such that κ2⋋/⟨y1/,y2/,y3/⟩1≻κ2⋋/⟨y1/,y2/,y3/⟩2 or κ2⋋/⟨y1/,y2/,y3/⟩1≺κ2⋋/⟨y1/,y2/,y3/⟩2. So, κ2⋋/⟨y1/,y2/,y3/⟩1≻κ2⋋/⟨y1/,y2/,y3/⟩2orκ2⋋/⟨y1/,y2/,y3/⟩1≺κ2⋋/⟨y1/,y2/,y3/⟩2 respectively such that κ2⋋/⟨y1/,y2/,y3/⟩2=𝒻((κ1⋋⟨y1,y2,y3⟩)1),κ2⋋/⟨y1/,y2/,y3/⟩2=𝒻(κ1⋋⟨y1,y2,y3⟩2) such that κ1⋋⟨y1,y2,y3⟩1=𝒻−1(κ2⋋/⟨y1/,y2/,y3/⟩1) and κ1⋋⟨y1,y2,y3⟩2=𝒻−1(κ2⋋/⟨y1/,y2/,y3/⟩2). Since κ2⋋/⟨y1/,y2/,y3/⟩1,κ2⋋/⟨y1/,y2/,y3/⟩2∈⟨Y⟩~ but (⟨Y⟩~,F1,F2,θ) is NSB * b Hausdorff space. So according to definition κ2⋋/⟨y1/,y2/,y3/⟩1≻κ2⋋/⟨y1/,y2/,y3/⟩2 or κ2⋋/⟨y1/,y2/,y3/⟩1≺κ2⋋/⟨y1/,y2/,y3/⟩2. There definitely exists NS * b open sets ⟨𝓀1,θ⟩ and ⟨𝓀2,θ⟩∈(⟨Y⟩~,F1,F2,θ) such that κ2⋋/⟨y1/,y2/,y3/⟩1∈⟨𝓀1,θ⟩ and κ2⋋/⟨y1/,y2/,y3/⟩2∈⟨𝓀2,θ⟩ and these two NS * b open sets which are disjoint. Since 𝒻−1(⟨𝓀1,θ⟩) and 𝒻−1(⟨𝓀2,θ⟩) are NS * b open in (π~,τ1,τ2,θ). Now, 𝒻−1(⟨𝓀1,θ⟩)∩~𝒻−1(⟨𝓀1,θ⟩)=𝒻−1(⟨𝓀1,θ⟩∩~⟨𝓀2,θ⟩)=𝒻−1(∅~)=0(⟨π~⟩~,θ)~ and (κ2⋋/⟨y1/,y2/,y3/⟩)1∈⟨𝓀1,θ⟩⇒𝒻−1((κ2⋋/⟨y1/,y2/,y3/⟩)1)∈𝒻−1(⟨𝓀1,θ⟩)⇒κ1⋋⟨y1,y2,y3⟩1∈(⟨𝓀1,θ⟩), κ2⋋/⟨y1/,y2/,y3/⟩2∈⟨𝓀2,θ⟩⇒𝒻−1(κ2⋋/⟨y1/,y2/,y3/⟩2)∈𝒻−1(⟨𝓀2,θ⟩) implies that κ1⋋⟨y1,y2,y3⟩2∈(⟨𝓀2,θ⟩). (⟨𝓀1,θ⟩), κ1⋋⟨y1,y2,y3⟩2∈𝒻−1(⟨𝓀2,θ⟩). Accordingly, NSBTS is * b Hausdorff space.

Theorem 5.3. Let (π~,τ1,τ2,θ) be NSBTS and (⟨Y⟩~,F1,F2,θ) be an-other NSBTS. Let ⟨𝒻,θ⟩: (π~,τ1,τ2,θ)⟶(⟨Y⟩~,F1,F2,θ) be a soft mapping. Let (⟨Y⟩~,F1,F2,θ) is NSB * b Hausdorff space then it is guaranteed that {(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩):𝒻(κ1⋋⟨y1,y2,y3⟩)=𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)} is a NS * b closed sub-set of (π~,τ1,τ2,θ)×(⟨Y⟩~,F1,F2,θ).

Proof. Given that (π~,τ1,τ2,θ) be NSBTS and (⟨Y⟩~,F1,F2,θ) be an-other NSBTS. Let ⟨𝒻,θ⟩: (π~,τ1,τ2,θ)⟶(⟨Y⟩~,F1,F2,θ) be a soft mapping such that it is continuous mapping (⟨Y⟩~,F1,F2,θ) is NSB * b Hausdorff space Then we will prove that {(((κ1⋋⟨y1,y2,y3⟩)),κ2⋋/⟨y1/,y2/,y3/⟩): 𝒻((κ1⋋⟨y1,y2,y3⟩))=𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)} is a NS * b closed sub-set of (π~,τ1,τ2,θ)×(⟨Y⟩~,F1,F2,θ). Equavilintly, we will prove that {(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩):𝒻((κ1⋋⟨y1,y2,y3⟩))=(κ2⋋/⟨y1/,y2/,y3/⟩)}c is NS * b open sub-set of (π~,τ1,τ2,θ)⟶(⟨Y⟩~,F1,F2,θ). Let ((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩)∈{(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩) with κ1⋋⟨y1,y2,y3⟩≻κ2⋋/⟨y1/,y2/,y3/⟩:𝒻(κ1⋋⟨y1,y2,y3⟩)≻𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c or (κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)∈{(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩)withκ1⋋⟨y1,y2,y3⟩≺κ2⋋/⟨y1/,y2/,y3/⟩: 𝒻(κ1⋋⟨y1,y2,y3⟩)≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c. Then, 𝒻(κ1⋋⟨y1,y2,y3⟩)≻𝒻(κ2⋋/⟨y1/,y2/,y3/⟩) or ≺𝒻(κ1⋋⟨y1,y2,y3⟩)≺𝒻(κ2⋋/⟨y1/,y2/,y3/⟩) accordingly. Since, (⟨Y⟩~,F1,F2,θ) is * b Hausdorff space. Certainly, 𝒻(κ1⋋⟨y1,y2,y3⟩),𝒻(κ2⋋/⟨y1/,y2/,y3/⟩) are points of (⟨Y⟩~,F1,F2,θ), there exists NS * b open sets ⟨G,θ⟩,⟨𝓀,θ⟩∈(⟨Y⟩~,F1,F2,θ) such that 𝒻((κ1⋋⟨y1,y2,y3⟩))∈⟨G,θ⟩, 𝒻(κ1⋋⟨y1,y2,y3⟩)∈⟨𝓀,θ⟩ provided ⟨G,θ⟩∩~⟨𝓀,θ⟩=0(⟨π~⟩~,θ)Y~. Since, ⟨𝒻,θ⟩ is soft continuous, 𝒻−1(⟨G,θ⟩ & 𝒻−1(⟨𝓀,θ⟩ are NS * b open sets in ⟨π~,τ1,τ2,θ⟩ supposing κ1⋋⟨y1,y2,y3⟩ and κ2⋋/⟨y1/,y2/,y3/⟩ respectively and so 𝒻−1(⟨G,θ⟩×𝒻−1(⟨H,θ⟩ is basic NS * b open set in (π~,τ1,τ2,θ)⟶(⟨Y⟩~,F1,F2,θ) containing ((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩). Since ⟨G,θ⟩∩~⟨𝓀,θ⟩=0θY~, it is clear by the definition of {((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩):𝒻(κ1⋋⟨y1,y2,y3⟩)=𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)} that is {𝒻−1(⟨G,θ⟩&𝒻−1(⟨𝓀,θ⟩}∩~{((κ1⋋⟨y1,y2,y3⟩),κ2⋋/⟨y1/,y2/,y3/⟩):𝒻(𝓍)=𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}=0(π,θ), that is 𝒻−1(⟨G,θ⟩×𝒻−1(⟨𝓀,θ⟩⋐{(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩):𝒻(κ1⋋⟨y1,y2,y3⟩)=𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c. Hence, {(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩):𝒻(κ1⋋⟨y1,y2,y3⟩)=𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)}c implies that {(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩):𝒻(κ1⋋⟨y1,y2,y3⟩)=𝒻(κ2⋋/⟨y1/,y2/,y3/⟩)} is NS * b closed.

Theorem 5.4. Let (π~,τ1,τ2,θ) be NSBTS and (⟨Y⟩~,F1,F2,θ) be an-other NSBTS. Let ⟨𝒻,θ⟩: (π~,τ1,τ2,θ)⟶(⟨Y⟩~,F1,F2,θ) be a NS * b open mapping such that it is onto. If the soft set {(κ1⋋⟨y1,y2,y3⟩,κ2⋋/⟨y1/,y2/,y3/⟩):𝒻((κ1⋋⟨y1,y2,y3⟩))=𝒻((κ2⋋/⟨y1/,y2/,y3/⟩))} is NS * b closed in ⟨π~,τ1,τ2,θ⟩×(⟨Y⟩~,F1,F2,θ), then ⟨π~,τ1,τ2,θ⟩ will behave as NSB * b Hausdorff space.

Theorem 5.6. Let ⟨π~,τ1,τ2,θ⟩ and ⟨⟨Y⟩~,F1,F2,θ⟩ be two NSBTS and suppose ⟨f,θ⟩ be a NS continuous function such that ⟨f,θ⟩:⟨π~,τ1,τ2,θ⟩⟶⟨⟨Y⟩~,F1,F2,θ⟩ is NS continuous function and let ⟨L,θ⟩⋐⟨π~,τ1,τ2,θ⟩ supposes the B.V.P. then safely f(⟨L,θ⟩) has the B.V.P.

Proof: Suppose ⟨L,θ⟩ be an infinite NS sub-set of ⟨f,θ⟩, so that ⟨L,θ⟩ contains an enumerable NS set ⟨(κ1⋋⟨y1,y2,y3⟩)n:n∈N⟩ then there exists enumerable NS set ⟨(κ2⋋/⟨y1/,y2/,y3/⟩)n:n∈N⟩⋐⟨L,θ⟩ s.t. f((κ2⋋/⟨y1/,y2/,y3/⟩)n)=(κ1⋋⟨y1,y2,y3⟩)n.⟨L,θ⟩ has B.V.P implies that every infinite soft subset of ⟨L,θ⟩ supposes soft accumulation point belonging to ⟨L,θ⟩ this implies that ⟨(κ2⋋/⟨y1/,y2/,y3/⟩)n:n∈N⟩ has soft neutrosophic limit poit, say, (κ2⋋/⟨y1/,y2/,y3/⟩)0∈⟨L,θ⟩ implies that the limit of soft sequence ⟨(κ2⋋/⟨y1/,y2/,y3/⟩)n:n∈N⟩ is (κ2⋋/⟨y1/,y2/,y3/⟩)0∈⟨L,θ⟩⟹(κ2⋋/⟨y1/,y2/,y3/⟩)n⟶(κ2⋋/⟨y1/,y2/,y3/⟩)0∈⟨L,θ⟩. f is soft continuous implies that it is soft continuous. Furthermore (κ2⋋/⟨y1/,y2/,y3/⟩)n⟶(κ2⋋/⟨y1/,y2/,y3/⟩)0∈⟨L,θ⟩⟹f(((κ2⋋/⟨y1/,y2/,y3/⟩)n))⟶f(((κ2⋋/⟨y1/,y2/,y3/⟩)0))∈f(⟨L,θ⟩)⟹(κ1⋋⟨y1,y2,y3⟩)n⟶f((κ2⋋/⟨y1/,y2/,y3/⟩)0)∈f(⟨L,θ⟩) implies that limit of a soft sequence ⟨(κ1⋋⟨y1,y2,y3⟩)n:n∈N⟩ is f((κ2⋋/⟨y1/,y2/,y3/⟩)0)∈f(⟨L,θ⟩) implies that limit of a soft sequence ⟨(κ1⋋⟨y1,y2,y3⟩)n:n∈N⟩ is f(⟨y1/,y2/,y3/⟩)∈⟨f,θ⟩(⟨L,θ⟩). Finally we have shown that there exists infinite soft subset ⟨(κ1⋋⟨y1,y2,y3⟩)n:n∈N⟩ of f(⟨L,θ⟩) containing a limit point f((κ2⋋/⟨y1/,y2/,y3/⟩)0)∈f(⟨L,θ⟩). This guarantees that f(⟨L,θ⟩) has B.V.P.

Theorem 5.7. Let ⟨π~,τ1,τ2,θ⟩ NSBTS and let ⟨(κ1⋋⟨y1,y2,y3⟩)n~⟩ be a NS sequence in ⟨π~,τ1,τ2,θ⟩ such that it converges to a point (κ1⋋⟨y1,y2,y3⟩)0 then the soft set ⟨𝓆,θ⟩ consisting of the points (κ1⋋⟨y1,y2,y3⟩)n0 and (κ1⋋⟨y1,y2,y3⟩)n(n=1,2,3,…) is soft NSB compact.

Proof. Given ⟨π~,τ1,τ2,θ⟩NSBTS and let ⟨(κ1⋋⟨y1,y2,y3⟩)n~⟩ be a NS sequence in ⟨π~,τ1,τ2,θ⟩ such that it converges to a point (κ1⋋⟨y1,y2,y3⟩)n0 that is (κ1⋋⟨y1,y2,y3⟩)n~⟶(κ1⋋⟨y1,y2,y3⟩)n0∈ π~. Let ⟨𝓆,θ⟩=[κ1⋋⟨y1,y2,y3⟩~,(κ1⋋⟨y1,y2,y3⟩)2~,(κ1⋋⟨y1,y2,y3⟩)3~,(κ1⋋⟨y1,y2,y3⟩)4~,(κ1⋋⟨y1,y2,y3⟩)5~,(κ1⋋⟨y1,y2,y3⟩)6~,(κ1⋋⟨y1,y2,y3⟩)7~,…]. Let {⟨S,θ⟩α:α∈Δ} be NS * b open covering of ⟨𝓆,θ⟩ so that ⟨𝓆,θ⟩⋐∪~{⟨S,θ⟩α:α∈Δ},(κ1⋋⟨y1,y2,y3⟩)n0∈⟨𝓆,θ⟩ implies that there exists α0∈Δ such that (κ1⋋⟨y1,y2,y3⟩)n0∈⟨S,θ⟩α0. According to the definition of soft convergence, (κ1⋋⟨y1,y2,y3⟩)n0∈⟨S,θ⟩α0∈(π~,τ1,τ2,θ) implies thatthere exists n0∈Ns.t.n≥n0 and (κ1⋋⟨y1,y2,y3⟩)n∈⟨S,θ⟩α0. Evidently, ⟨S,θ⟩α0 contains the points (κ1⋋⟨y1,y2,y3⟩)n0,(κ1⋋⟨y1,y2,y3⟩)n0+1, (κ1⋋⟨y1,y2,y3⟩)n0+2, (κ1⋋⟨y1,y2,y3⟩)n0+3,(κ1⋋⟨y1,y2,y3⟩),… (κ1⋋⟨y1,y2,y3⟩)n0+n,… Look carefully at the points and train them in a way as, (κ1⋋⟨y1,y2,y3⟩)1,(κ1⋋⟨y1,y2,y3⟩)2,(κ1⋋⟨y1,y2,y3⟩)3, (κ1⋋⟨y1,y2,y3⟩)4,… (κ1⋋⟨y1,y2,y3⟩)n,… generating a finite soft set. Let 1≤n0−1. Then (κ1⋋⟨y1,y2,y3⟩)i∈⟨𝓆,θ⟩. For this i,(κ1⋋⟨y1,y2,y3⟩)i∈⟨𝓆,θ⟩. Hence there exists αi∈Δ such that (κ1⋋⟨y1,y2,y3⟩)i∈⟨S,θ⟩αi. Evidently ⟨𝓆,θ⟩⋐⋃r=0n0−1⟨S,θ⟩αi~. This shows that {⟨S,θ⟩αi:0≤n0−1} is NS * b open cover of ⟨𝓆,θ⟩. Thus an arbitrary NS * b open cover {⟨S,θ⟩α:α∈Δ} of ⟨𝓆,θ⟩ is reducible to a finite NS cub-cover {⟨S,θ⟩αi:i=0,1,2,3,…n0−1}, it follows that ⟨𝓆,θ⟩ is soft NSB * b compact.

Theorem 5.8. If (π~,τ1,τ2,θ)NSBTS such that it has the characteristics of NS * b sequentially compactness. Then ⟨π~,τ1,τ2,θ⟩ is safely NSB * b countably compact.

Proof. Let ⟨π~,τ1,τ2,θ⟩NSBTS and let ⟨ρ,θ⟩ be finite soft sub-set of ⟨π~,τ1,τ2,θ⟩. Let [(κ1⋋⟨y1,y2,y3⟩)1~,(κ1⋋⟨y1,y2,y3⟩)2~,(κ1⋋⟨y1,y2,y3⟩)3~,(κ1⋋⟨y1,y2,y3⟩)4~,(κ1⋋⟨y1,y2,y3⟩)5~,(κ1⋋⟨y1,y2,y3⟩)6~,(κ1⋋⟨y1,y2,y3⟩)7~,…] be a soft sequence of soft points of ⟨ρ,θ⟩. Then, ⟨ρ,θ⟩ being finite, at least one of the elements in ⟨ρ,θ⟩ say (κ1⋋⟨y1,y2,y3⟩)0~ must be duplicated an in-finite number of times in the NS sequence. Hence, [(κ1⋋⟨y1,y2,y3⟩)0~,(κ1⋋⟨y1,y2,y3⟩)0~,(κ1⋋⟨y1,y2,y3⟩)0~,(κ1⋋⟨y1,y2,y3⟩)0~,(κ1⋋⟨y1,y2,y3⟩)0~,(κ1⋋⟨y1,y2,y3⟩)0~,(κ1⋋⟨y1,y2,y3⟩)0~,…] is soft sub-sequence of ⟨(κ1⋋⟨y1,y2,y3⟩)n~⟩ such that it is soft constant sequence and repeatedly constructed by single soft number (κ1⋋⟨y1,y2,y3⟩)0~ and we know that a soft constant sequence converges on its self. So it converges to (κ1⋋⟨y1,y2,y3⟩)0~ which belongs to ⟨ρ,θ⟩. Hence, ⟨ρ,θ⟩ is soft sequentially NSB * b compact.

Theorem 5.9. Let ⟨π~,τ1,τ2,θ⟩NSBTS and (⟨Y⟩~,F1,F2,θ) be another NSBTS. Let ⟨𝒻,θ⟩ be a soft continuous mapping of a soft neutrosophic sequentially compact NS * b space ⟨π~,τ1,τ2,θ⟩ into (⟨Y⟩~,F1,F2,θ), then ⟨𝒻,θ⟩⟨π~,τ1,τ2,θ⟩ is NSB * b sequentially compact.

Proof. Given ⟨π~,τ1,τ2,θ⟩NSBTS and (⟨Y⟩~,F1,F2,θ) be another NSBTS. Let ⟨𝒻,θ⟩ be a soft continuous mapping of a NSB sequentially compact space (π~,τ1,τ2,θ) into (⟨Y⟩~,F1,F2,θ) then we have to prove ⟨𝒻,θ⟩⟨π~,τ1,τ2,θ⟩ NS sequentially. For this we proceed as. Let ⟨[(κ2⋋/⟨y1/,y2/,y3/⟩)1~,(κ2⋋/⟨y1/,y2/,y3/⟩)2~,,(κ2⋋/⟨y1/,y2/,y3/⟩)5~,(κ2⋋/⟨y1/,y2/,y3/⟩)6~,(κ2⋋/⟨y1/,y2/,y3/⟩)7~,…(κ2⋋/⟨y1/,y2/,y3/⟩)n~,….]⟩ be a soft sequence of neutrosophic soft points in ⟨𝒻,θ⟩(π~,τ1,τ2,θ), Then for each n∈N there exists ⟨(κ1⋋⟨y1,y2,y3⟩)1~,(κ1⋋⟨y1,y2,y3⟩)2~,(κ1⋋⟨y1,y2,y3⟩)4~,(κ1⋋⟨y1,y2,y3⟩)5~,(κ1⋋⟨y1,y2,y3⟩)7~,…(κ1⋋⟨y1,y2,y3⟩)n~,….∈(π~,τ1,τ2,θ)⟩such that ⟨𝒻,θ⟩[((κ1⋋⟨y1,y2,y3⟩)1~,(κ1⋋⟨y1,y2,y3⟩)2~,(κ1⋋⟨y1,y2,y3⟩)3~,(κ1⋋⟨y1,y2,y3⟩)7~,…(κ1⋋⟨y1,y2,y3⟩)n~,….)]=⟨[(κ2⋋/⟨y1/,y2/,y3/⟩)1~,(κ2⋋/⟨y1/,y2/,y3/⟩)2~,(κ2⋋/⟨y1/,y2/,y3/⟩)3~,κ2⋋/⟨y1/,y2/,y3/⟩4~,(κ2⋋/⟨y1/,y2/,y3/⟩)6~,(κ2⋋/⟨y1/,y2/,y3/⟩)7~,…(κ2⋋/⟨y1/,y2/,y3/⟩)n~,….]⟩. Thus we obtain a soft sequence [(κ1⋋⟨y1,y2,y3⟩)1~,(κ1⋋⟨y1,y2,y3⟩)2~,(κ1⋋⟨y1,y2,y3⟩)3~,(κ1⋋⟨y1,y2,y3⟩)4~,(κ1⋋⟨y1,y2,y3⟩)5,(κ1⋋⟨y1,y2,y3⟩)6~,(κ1⋋⟨y1,y2,y3⟩)7~,…(κ1⋋⟨y1,y2,y3⟩)n~,….] in ⟨π~,τ1,τ2,θ⟩. But ⟨π~,τ1,τ2,θ⟩ being soft sequentially NSB * b compact, there is a NS sub-sequence ⟨(κ1⋋⟨y1,y2,y3⟩)ni~⟩of

⟨(κ1⋋⟨y1,y2,y3⟩)n~⟩ such that ⟨(κ1⋋⟨y1,y2,y3⟩)ni~⟩⟶(κ1⋋⟨y1,y2,y3⟩)~∈⟨π~,τ1,τ2,θ⟩. So, by NS * b contiuity of ⟨𝒻,θ⟩,⟨(κ1⋋⟨y1,y2,y3⟩)ni~⟩⟶(κ1⋋⟨y1,y2,y3⟩θ)~ implies that ⟨𝒻,θ⟩(⟨(κ1⋋⟨y1,y2,y3⟩)ni~⟩)⟶⟨𝒻,θ⟩(⟨(κ1⋋⟨y1,y2,y3⟩)n~⟩)∈⟨𝒻,θ⟩⟨π~,τ1,τ2,θ⟩. Thus, ⟨𝒻,θ⟩(⟨(κ2⋋/⟨y1/,y2/,y3/⟩)ni~⟩) is a soft sub-sequence of ⟨[(κ2⋋/⟨y1/,y2/,y3/⟩)1~,(κ2⋋/⟨y1/,y2/,y3/⟩)2~,(κ2⋋/⟨y1/,y2/,y3/⟩)3,(κ2⋋/⟨y1/,y2/,y3/⟩)5~,(κ2⋋/⟨y1/,y2/,y3/⟩)5~,(κ2⋋/⟨y1/,y2/,y3/⟩)6~,(κ2⋋/⟨y1/,y2/,y3/⟩)7~,…(κ2⋋/⟨y1/,y2/,y3/⟩)n~,….]⟩ converges to (⟨𝒻,θ⟩)(κ1~) in ⟨𝒻,θ⟩ ⟨π~,τ1,τ2,θ⟩. Hence, ⟨𝒻,θ⟩⟨π~,τ1,τ2,θ⟩ is NS * b sequentially compact.

Theorem5.10. Let ⟨π~,τ1,τ2,θ⟩NSBTS and suppose ⟨𝒻,θ⟩,⟨𝓆,θ⟩ be two NS continuous function on a NS BTS ⟨π~,τ1,τ2,θ⟩ in to NSBTS ⟨⟨Y⟩~,F1,F2,θ⟩ which is NSB * b Hausdorff. Then, soft set {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(𝓆)((κ1⋋⟨y1,y2,y3⟩))} is NS * b closed of ⟨π~,τ1,τ2,θ⟩.

Proof: Let If {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(𝓆)((κ1⋋⟨y1,y2,y3⟩))} is a NS set of function. If {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(𝓆)(κ1)}c=∅~, it is clearly NS * b open and therefore, {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(𝓆)((κ1⋋⟨y1,y2,y3⟩))} is * b closed, that is nothing is proved in this case. Let us consider the case when {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))= (𝓆)((κ1⋋⟨y1,y2,y3⟩))}c≠(κ1⋋⟨y1,y2,y3⟩) and let ρ∈{(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))= (𝓆)((κ1⋋⟨y1,y2,y3⟩))}c. Then ρ does not belong {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(𝓆)(κ1)}. Result in (𝒻)(ρ)≠(𝓆)(ρ). Now, (⟨Y⟩~,F1,F2,θ) being NSB * b Hausdorff space so there exists NS * b open sets ⟨𝓆,θ⟩, ⟨H,θ⟩ of (𝒻)(ρ) and (𝓆)(ρ) respectively such that ⟨𝓆,θ⟩ and ⟨H,θ⟩ such that these NS sets such that the possibility of one rules out the possibility of other. By soft continuity of ⟨𝒻,θ⟩,⟨𝓆,θ⟩, ⟨𝒻,θ⟩−1 as well as ⟨𝓆,θ⟩−1 is NS * b open nhd of ρ and therefore, so is ⟨𝒻,θ⟩−1∩~⟨𝓆,θ⟩−1 is contained in {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(𝓆)((κ1⋋⟨y1,y2,y3⟩))}, for, (κ1⋋⟨y1,y2,y3⟩)∈(⟨𝒻,θ⟩−1∩~⟨𝓆,θ⟩−1)⟹(𝒻)((κ1⋋⟨y1,y2,y3⟩))∈⟨𝓆,θ⟩ and (𝓆)((𝒻)((κ1⋋⟨y1,y2,y3⟩))≠(𝓆)((κ1⋋⟨y1,y2,y3⟩)) because ⟨𝓆,θ⟩ and ⟨H,θ⟩ are mutually exclusive. This implies that κ1 does not belong to {(κ1⋋⟨y1,y2,y3⟩)∈π~:(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(𝓆)((κ1⋋⟨y1,y2,y3⟩))}. Therefore, ρ∈(𝒻)−1(⟨𝓆,θ⟩)∩~(𝓆)−1(⟨𝓆,θ⟩)⋐{κ1⋋⟨y1,y2,y3⟩∈π~:(𝒻)(κ1⋋⟨y1,y2,y3⟩)=(𝓆)(κ1⋋⟨y1,y2,y3⟩)}c. This shows that {κ1⋋⟨y1,y2,y3⟩∈π~:(𝒻)(κ1⋋⟨y1,y2,y3⟩)=(𝓆)(κ1⋋⟨y1,y2,y3⟩)}c is neighborhood of each of its points. So, {κ1⋋⟨y1,y2,y3⟩∈π~:(𝒻)(κ1⋋⟨y1,y2,y3⟩)=(𝓆)(κ1⋋⟨y1,y2,y3⟩)}cNS∗b open and hence {κ1⋋⟨y1,y2,y3⟩∈π~: (𝒻)(κ1⋋⟨y1,y2,y3⟩)=(𝓆)(κ1⋋⟨y1,y2,y3⟩)} is NS * b closed.

Theorem 5.11. Let ⟨π~,τ1,τ2,θ⟩NSBTS such that it is NSB * b Hausdorff space and let (𝒻) be soft continuous function of (π~,τ1,τ2,θ) into itself. Then, the NS set of fixed points under (𝒻) is a NSB * b closed set.

Proof. Let δ={(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)}. If δc=∅~, Then is NS * b open and therefore {(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)}NS∗b closed. So, let {(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)}c≠∅~ and let κ2⋋/⟨y1/,y2/,y3/⟩∈{(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)}c. Then, κ2⋋/⟨y1/,y2/,y3/⟩ does not belong to {(𝒻)(κ1⋋⟨y1,y2,y3⟩)=κ1⋋⟨y1,y2,y3⟩} and therefore (𝒻)(κ2⋋/⟨y1/,y2/,y3/⟩) ≠κ2⋋/⟨y1/,y2/,y3/⟩. Now, κ2⋋/⟨y1/,y2/,y3/⟩, (𝒻)(κ2⋋/⟨y1/,y2/,y3/⟩) being two distinct points of the NSB * b Hausdorff space ⟨π~,τ1,τ2,θ⟩, so there exists N S * b open sets ⟨𝓆,θ⟩, ⟨H,θ⟩ such that κ2⋋/⟨y1/,y2/,y3/⟩∈⟨𝓆,θ⟩,(𝒻)(κ2⋋/⟨y1/,y2/,y3/⟩)∈⟨H,θ⟩ and ⟨𝓆,θ⟩,⟨H,θ⟩ are disjoint. Also, by the NS continuity of (𝒻), (𝒻)−1(⟨H,θ⟩ is NS * b open set containing 𝓎. We pretend that ⟨𝓆,θ⟩∩~(𝒻)−1(⟨H,θ⟩)⋐{(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)}c. Since μ∈⟨𝓆,θ⟩∩~(𝒻)−1(⟨H,θ⟩)⟹μ∈⟨𝓆,θ⟩,μ∈(𝒻)−1⟹μ∈⟨𝓆,θ⟩,(𝒻)(μ)∈⟨H,θ⟩⟹μ≠(𝒻)(μ). As ⟨𝓆,θ⟩∩~⟨H,θ⟩=∅~ implies that μ does not belong to {(𝒻)(κ1⋋⟨y1,y2,y3⟩)=(κ1⋋⟨y1,y2,y3⟩)}⟹μ∈{(𝒻)(κ1⋋⟨y1,y2,y3⟩)=κ1⋋⟨y1,y2,y3⟩}c. Therefore, κ2⋋/⟨y1/,y2/,y3/⟩∈⟨𝓆,θ⟩∩~(𝒻)−1(⟨H,θ⟩)⋐{(𝒻)(κ1⋋⟨y1,y2,y3⟩)=(κ1⋋⟨y1,y2,y3⟩)}c. Thus, {(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)}c is the NS neighbourhood of each of its points. So, {(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)}c is NS * b open and hence {(𝒻)((κ1⋋⟨y1,y2,y3⟩))=(κ1⋋⟨y1,y2,y3⟩)} is NSB * b closed.

6  Conclusion

Neutrosophic soft topology (NST) is extension of vague soft topology (VST). Vague soft topology gives two types of information. One is true and the second one is false. It does not give information about the indeterminacy (doubtful) case. Neutrosophic soft topology is dominant over vague soft topology because it supposes all the information that is true, false and indeterminacy at the same time. Neutrosophic soft topology has a narrow domain as compared to neutrosophic soft bi-topology (NSBT). In our work, we regenerated some structures in neutrosophic soft bi-topological spaces with a new definition that is * b open sets concerning soft points. We worked with the operations given in references[14–16] which are entirely different from those in references[13,17]. Then unlike[18,19], neutrosophic soft bi-topological structure is reconstructed. In future, we would try to see the validity of the given structures in neutrosophic soft tri-topological structures.

Funding Statement: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through research groups program under Grant No. R.G. P-2/29/42.

Availability of Supporting Data: No data were used to support this study.

Author Contributions: All authors read and approved the final manuscript.

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

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