Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.018066

ARTICLE

On Some Properties of Neutrosophic Semi Continuous and Almost Continuous Mapping

1Department of Mathematical Sciences, Bodoland University, Kokrajhar, 783370, India

2Central Institute of Technology, Kokrajhar, 783370, India

*Corresponding Author: Bhimraj Basumatary. Email: brbasumatary14@gmail.com

Received: 26 June 2021; Accepted: 17 August 2021

Abstract: The neutrality’s origin, character, and extent are studied in the Neutrosophic set. The neutrosophic set is an essential issue to research since it opens the door to a wide range of scientific and technological applications. The neutrosophic set can find its spot to research because the universe is filled with indeterminacy. Neutrosophic set is currently being developed to express uncertain, imprecise, partial, and inconsistent data. Truth membership function, indeterminacy membership function, and falsity membership function are used to express a neutrosophic set in order to address uncertainty. The neutrosophic set produces more rational conclusions in a variety of practical problems. The neutrosophic set displays inconsistencies in data and can solve real-world problems. We are directed to do our work in semi-continuous and almost continuous mapping on the basis of the neutrosophic set by observing these. Since we are going to study the properties of semi continuous and almost continuous mapping, we present the meaning of

Keywords: 𝓝∽ regularly open set; 𝓝∽ regularly closed set; 𝓝∽ semi-continuous mapping; 𝓝∽ almost continuous mapping

After Zadeh [1] created fuzzy set theory (FST), FST was used to define the idea of membership value and explain the concept of uncertainty. Many researchers attempted to apply FST to a variety of other fields of science and technology. Atanassov [2] expanded on the concept of fuzzy set theory and introduced the concept of degree of non-membership, as well as proposing intuitionistic fuzzy set theory (IFST). Chang [3] introduced fuzzy topology (FT), and Coker [4] generalized the concept of FT to intuitionistic fuzzy topology (IFT). Rosenfeld [5] introduced the concept of fuzzy groups and Foster [6] proposed the idea of fuzzy topological groups. Azad [7] went through fuzzy semi-continuity (FSC), fuzzy almost continuity (FAC), and fuzzy weakly continuity (FWC). Smarandache [8,9] suggested neutrosophic set theory (NST) by generalizing FST and IFST and valuing indeterminacy as a separate component. Many researchers have attempted to apply NST to a variety of scientific and technological fields. Kandil et al. [10] studied the fuzzy bitopological spaces. Mwchahary et al. [11] did their work in neutrosophic bitopological space. Neutrosophic topology was proposed by Salama et al. [12,13]. The semi-continuous mapping was investigated by Noiri [14] and the term almost continuous mappings were coined by Singal et al. [15]. The idea of fuzzy neutrosophic groups and a topological group of the neutrosophic set was studied by Sumathi et al. [16,17]. NST was used as a tool in a group discussion framework by Abdel-Basset et al. [18]. Abdel-Basset et al. [19] investigated the use of the base-worst technique to solve chain problems using a novel plithogenic model.

In this current decade, neutrosophic environments are mainly interested by different fields of researchers. In Mathematics also much theoretical research has been observed in the sense of neutrosophic environment. It will be necessary to carry out more theoretical research to establish a general framework for decision-making and to define patterns for complex network conceiving and practical application. Salama et al. [13] studied neutrosophic closed set and neutrosophic continuous functions. The idea of almost continuous functions is done in 1968 [15] in topology. Similarly, the notion of fuzzy almost contra continuous and fuzzy almost contra α-continuous functions was discussed in [20]. Recently, Al-Omeri et al. [21,22] introduced and studied a number of the definitions of neutrosophic closed sets, neutrosophic mapping, and obtained several preservation properties and some characterizations about neutrosophic of connectedness and neutrosophic connectedness continuity. More recently, in [23–26] authors have given how a new trend of Neutrosophic theory is applicable in the field of Medicine and multimedia with a novel and powerful model. From the literature survey, it is noticed that exactly the properties of neutrosophic semi-continuous and almost continuous mapping are not done. To update this research gap, in this research article, we attempt to investigate the neutrosophic semi-continuous and almost continuous mapping and its properties. Also, we study properties of the neutrosophic semi-open set (NSOS), neutrosophic semi-closed set (NSCoS), neutrosophic regularly open set (NROS), neutrosophic regularly closed set (NRCoS), neutrosophic semi-continuous (NSC), and neutrosophic almost continuous mapping (NACM).

2.1 Definition [8]

A neutrosophic set (NS)

2.2 Definition [8]

Complement of

2.3 Definition [8]

Let

(i)

(ii)

(iii)

2.4 Definition [12]

Let

(i)

(ii)

(iii)

Then the pair

2.5 Definition [12]

Let

2.6 Definition [12]

Let

Let

Let

3.3 Lemma

Let

i)

Prove is Straightforward.

3.4 Lemma

Let

Proof:

Let

3.5 Lemma

Let

Proof:

For each

3.6 Lemma

Let

Proof:

For each

3.7 Lemma

For a family

3.8 Lemma

For a NS

Prove is Straightforward.

3.9 Theorem

The statements below are equivalent:

i)

ii)

iii)

iv)

Proof:

(i) and (ii) are equivalent follows from Lemma 3.8, since for a NS

(i)

(iii)

(ii)

3.10 Theorem

i) Arbitrary union of NSOSs is a NSOS, and

ii) Arbitrary intersection of NSCoSs is a NSCoS.

Proof:

(i) Let

(ii) Let

3.11 Remark

It is clear that every neutrosophic open set (NOS) (neutrosophic closed set (NCoS)) is a NSOS (NSCoS). The converse is false, it is seen in Example 3.12. It also shows that the intersection (union) of any two NSOSs (NSCoSs) need not be a NSOS (NSCoS). Even the intersection (union) of a NSOS (NSCoS) with a NOS (NCoS) may fail to be a NSOS (NSCoS). It should be noted that the ordinary topological setting the intersection of a NSOS with an NOS is a NSOS.

Further, the closure of NOS is a NSOS and the interior of NCoS is a NSCoS.

3.12 Example

Let

Then,

Let

3.13 Theorem

If

Proof:

Let

a)

b)

c)

It is sufficient to prove

Then

We have,

3.14 Definition

A NS

3.15 Definition

A NS

3.16 Theorem

A NS

Proof: It follows from Lemma 3.8.

3.17 Remark

It is obvious that every NROS (NRCoS) is NOS (NCoS). The converse need not be true. For this we cite an example.

3.18 Example

From Example 3.12, it is clear that

3.19 Remark

The union (intersection) of any two NROSs (NRCoS) need not be a NROS (NRCoS).

3.20 Example

Let

Then

Clearly,

Similarly,

Now,

But

Hence,

3.21 Theorem

(i) The intersection of any two NROSs is a NROS, and

(ii) The union of any two NRCoSs is a NRCoS.

Proof:

(i) Let

(ii) Let

3.22 Theorem

(i) The closure of a NOS is NRCoS, and

(ii) The interior of a NCoS is NROS.

Proof:

(i) Let

(ii) Let

3.23 Definition

Let

3.24 Definition

Let

3.25 Definition

Let

3.26 Definition

Let

3.27 Definition

Let

3.28 Definition

Let

3.29 Remark

From Remark 3.11, a NCM (NOM, NCoM) is also a NSCM (NSOM, NSCoM). But the converse is not true.

3.30 Example

Let

Then

Let

Then

3.31 Theorem

Let

Proof:

Let

That

3.32 Theorem

Let

Proof:

For a NOS

3.33 Theorem

Let

Proof:

From Lemma 3.6,

3.34 Remark

The converse of Theorem 3.33 is not true.

3.35 Definition

A mapping

3.36 Theorem

Let

a)

b)

c)

d)

Proof:

Consider that

(a)

(c)

(b)

3.37 Remark

Clearly, a NCM is NACM. But the converse needs not be true.

3.38 Example

Let

Then

Now, let

Here,

3.39 Theorem

3.40 Definition

A NTS

3.41 Theorem

Let

Proof:

From Remark 3.37, it suffices to prove that if

which shows that

3.42 Theorem

Let

Proof:

Let

Now,

Thus, by Theorem 3.36(c),

3.43 Theorem

Let

Proof:

Since

3.44 Theorem

Let

Proof:

Consider that

Thus, by Theorem 3.36(c),

Conversely, let

Since

and hence using Lemmas 3.3(i), 3.6 and 3.7 and Theorems 3.36(c), we have

Thus, by Theorem 3.36(c),

The truth membership function, indeterminacy membership function, and falsity membership function are all employed in the Neutrosophic Set to overcome uncertainty. First, we developed the definitions of

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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