Computer Modeling in Engineering & Sciences

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

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

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

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,

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:

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:

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:

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:

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

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

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

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

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

Therefore,

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).