Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.018518

ARTICLE

A New Attempt to Neutrosophic Soft Bi-Topological Spaces

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

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.

In this section, fundamental concepts are addressed.

Definition 2.1.[20] A neutrosophic set (NS) symbolized by A on the key set

Definition 2.2.[3]

Definition 2.3.[21,22] Let

Definition 2.4.[13] Let

Definition 2.5.[22] Let

Definition 2.6.[16] Let

Definition 2.7.[16] Let

Definition 2.8.[14] Neutrosophic soft set

It is signified as

Definition 2.9.[14] Neutrosophic soft set

It is signified as

Definition 2.10.[14] Let neutrosophic soft set

Definition 2.11.[14] Let neutrosophic soft set

Definition 2.12.[14] Let neutrosophic soft set

Definition 2.13.[14] Let

Definition 2.14.[14] Let

3 Neutrosophic Soft Bi-Topology

In this part, the concept of

Definition 3.1.

Example 3.2. Let

Then

Therefore,

Theorem 3.3. Let

Proof:T1 and NST3 are clear. For NST2, let

Remark 3.4. Let

Example 3.5. Let

Here

Definition 3.6. Let

Definition 3.7. Let

The set of all pairwise neutrosophic open sets in

Definition 3.8. Let

The set of all PNSC in

Example 3.9. Let

Then

Therefore,

is a PNSC.

Theorem 3.10. Let

1.

2. If

3. If

Proof.

1. Since

2. Since

As

Therefore,

3. Since

Then

Theorem 3.11. Let

Proof. Let

Theorem 3.12. Let

1.

2.

3.

4.

5.

6.

Proof. It is obvious.

Theorem 3.13. Let

Proof. Let

Conversely, that

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

Definition 4.1. Let

(1) Neutrosophic soft semi-open if

(2) Neutrosophic soft pre-open if

(3) Neutrosophic soft * b open if

Definition 4.2. Let

Definition 4.3. Let

Definition 4.4. Let

Example 4.5. let

Theorem 4.6. Let

Proof. Let

Theorem 4.7. Let

Proof. Let

Theorem 4.8. Let

Proof. Suppose

Definition 4.9. Let

Theorem 4.10. Let

Proof. Let

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

Proof. Suppose

Theorem 5.2. Let

Proof. Suppose

Theorem 5.3. Let

Proof. Given that

Theorem 5.4. Let

Proof. Suppose

Theorem 5.5. Let

Proof. Let

Let, if possible, no point of

Theorem 5.6. Let

Proof: Suppose

Theorem5.7. Let

Proof. Given

Theorem5.8. If

Proof. Let

Theorem 5.9. Let

Proof. Given

Theorem5.10. Let

Proof: Let If