[BACK]
 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 N semi-open set, N semi-closed set, N regularly open set, N regularly closed set, N continuous mapping, N open mapping, N closed mapping, N semi-continuous mapping, N semi-open mapping, N semi-closed mapping. Additionally, we attempt to demonstrate a portion of their properties and furthermore referred to some examples.

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

1  Introduction

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.

1.1 Motivations

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 [2326] 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  Methodologies

2.1 Definition [8]

A neutrosophic set (NS) AN on X can be expressed as AN={<xX,TAN(x),IAN(x),FAN(x)>}, where T,I,F:X]-0, 1+[. Note that 0TAN(x)+IAN(x)+FAN(x)3.

2.2 Definition [8]

Complement of AN is expressed as

ANc(x)={<xX,TANc(x)=FAN(x),IANc(x)=1IAN(x),FANc(x)=TAN(x)>}.

2.3 Definition [8]

Let Xϕ and AN={<xX,TAN(x),IAN(x),FAN(x)>} and BN={<xX,TBN(x),IBN(x),FBN(x)>} are NSs. Then

(i) ANBN={<x,min(TAN(x),TBN(x)),min(IAN(x),IBN(x)),max(FAN(x),FBN(x))>}

(ii) ANBN={<x,max(TAN(x),TBN(x)),max(IAN(x),IBN(x)),min(FAN(x),FBN(x))>}

(iii) ANBN  if  TAN(x)TBN(x),IAN(x)IBN(x),FAN(x)FBN(x), for  xX.

2.4 Definition [12]

Let Xϕ, then neutrosophic topology space (NTS) on X is a family TXN of neutrosophic subsets of X satisfying the following axiom:

(i) 0XN,1XNTXN

(ii) GN1GN2TXN; for GN1,GN2TXN

(iii) GNiTXN,{GNi:iJ}TXN.

Then the pair (X,TXN) is called a NTS.

2.5 Definition [12]

Let (X,TXN) be NTS. Then for a NS AN={<x,μNi,σNi,δNi>:xX}, neutrosophic interior of AN can be defined as NInt(AN)={<x, μNi, σNi, δNi>:xX}.

2.6 Definition [12]

Let (X,TXN) be NTS. Then for a NS AN={<x,μNi,σNi,δNi>:xX}, neutrosophic closure of AN can be defined as NCl(AN)={<x, μNi, σNi, δNi>:xX}.

3  Results and Discussion

3.1 Definition

Let A be a NS of NTS (X,TXN), then A is called a N semi-open set (NSOS) of X if a BTXN such that ANCl(NInt(B)).

3.2 Definition

Let A be a NS of NTS (X,TXN), then A is called a N semi-closed set (NSCoS) of X if a BcTXN such that NInt(NCl(B))A.

3.3 Lemma

Let ϕ:XY be a mapping and {Aα} be a family of NSs of Y, then

i)  ϕ1( Aα)= ϕ1(Aα) and (ii) ϕ1( Aα)= ϕ1(Aα).

Prove is Straightforward.

3.4 Lemma

Let A,B be NSs of X and Y, then 1XNA×B=(Ac×1XN)(1XN×Bc).

Proof:

Let (p,q) be any element of X×Y, (1XNA×B)(p,q)=max(1XNA(p),1XNB(q))=max{(Ac×1XN)(p,q),(Bc×1XN)(p,q)}={(Ac×1XN)(1XN×Bc)}(p,q), for each (p,q)X×Y.

3.5 Lemma

Let ϕi:XiYi and Ai be NSs of Yi, i=1,2; we have (ϕ1×ϕ2)1(A1×A2)=ϕ11(A1)×ϕ21(A2).

Proof:

For each (p1,p2)X1×X2, we have

(ϕ1×ϕ2)1(A1×A2)(p1,p2)=(A1×A2)(ϕ1(p1),ϕ2(p2))=min{A1ϕ1(p1),A2ϕ2(p2)}=min{ϕ11(A1)(p1),ϕ21(A2)(p2)}=(ϕ11(A1)×ϕ21(A2))(p1,p2)

3.6 Lemma

Let ψ:XX×Y be the graph of a mapping ϕ:XY. Then, if A,B be NSs of X and Y, ψ1(A×B)=A  ϕ1(B).

Proof:

For each pX, we have

ψ1(A×B)(p)=(A×B)ψ(p)=(A×B)(p,ϕ(p))=min{A(p),B(ϕ(p))}=(A  ϕ1(B))(p)

3.7 Lemma

For a family {A}α of NSs of NTS (X,TXN),  NCl(Aα)NCl( (Aα)). In case B is a finite set,  NCl(Aα)NCl( (Aα)). Also,  NInt(Aα)NInt( (Aα)), where a subfamily B of (X,TXN) is said to be subbase for (X,TXN) if the collection of all intersections of members of B forms a base for (X,TXN).

3.8 Lemma

For a NS A of NTS (X,TXN), (a) 1NInt(A)=NCl(1A), and (b) 1NCl(A)=NInt(1A).

Prove is Straightforward.

3.9 Theorem

The statements below are equivalent:

i)  A is a NSCoS,

ii)  Ac is a NSOS,

iii)  NInt(NCl(A))A, and

iv)  NCl(NInt(Ac))Ac.

Proof:

(i) and (ii) are equivalent follows from Lemma 3.8, since for a NS A of NTS (X,TXN) such that 1NInt(A)=NCl(1A) and 1NCl(A)=NInt(1A).

(i)(iii). By definition a NCoS B such that NInt(B)AB and hence NInt(B)ANCl(A)B. Since NInt(B) is the greatest NOS contained in B, we have NInt(NCl(B))NInt(B)A.

(iii)(i) follows by taking B=NCl(A).

(ii)(iv) can similarly be proved.

3.10 Theorem

i)  Arbitrary union of NSOSs is a NSOS, and

ii)  Arbitrary intersection of NSCoSs is a NSCoS.

Proof:

(i) Let {Aα} be a collection of NSOSs of NTS (X,TXN). Then a BαTXN such that BαAαNCl(Bα), for each α. Thus, BαAαNCl(Bα)NCl((Bα)) [Lemma 3.7], and BαTXN, this shows that Bα is a NSOS.

(ii) Let {Aα} be a collection of NSCoSs of NTS (X,TXN). Then a BαTXN such that NInt(Bα)AαBα, for each α. Thus, NInt((Bα))NInt(Bα)AαBα [Lemma 3.7], and BαTXN, this shows that Bα is a NSCoS.

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 X={a,b} and A,B be neutrosophic subsets of X such that

A={<a(0.6,0.3,0.2)>,<b(0.5,0.2,0.3)>} B={<a(0.5,0.4,0.3)>,<b(0.4,0.2,0.3)>}

Then, TXN={1XN,0XN,A,B,AB,AB} is a NTS on X.

Let P={<a(0.8,0.2,0.1)>,<b(0.7,0.2,0.3)>} be any neutrosophic set XN, then NInt(P)={G:G is open set,GP}=AB=A and NCl(P)={KP:K is closed set inTXN}=1XN. Therefore, P is a NSOS which is not a NOS and also by Theorem 3.9, Pc is a NSCoS which is not an NCoS.

3.13 Theorem

If (X,TXN) and (Y,TYN) are NTSs and X is product related to Y. Then the product A×B of a NSOS A of X and a NSOS B of Y is NSOS of the neutrosophic product space X×Y.

Proof:

Let PANCl(P) and QBNCl(Q), where PTXN and QTYN. Then P×QA×BNCl(P)×NCl(Q). For NSs P’s of X and Q’s of Y, we have

a) inf{P,Q}=min{infP,infQ},

b) inf{P×1XN}=(infP)×1XN, and

c) inf{1XN×Q}=1XN×(infQ).

It is sufficient to prove NCl(A×B)NCl(A)×NCl(B). Let PTNX and QTNY.

Then

NCl(A×B)=inf{(P×Q)c|(P×Q)cA×B}=inf{(Pc×1XN)(1XN×Qc)|(Pc×1XN)(1XN×Qc)A×B}=inf{(Pc×1XN)(1XN×Qc)|PcAorQcB}=min[inf{(Pc×1XN)(1XN×Qc)|PcA},inf{(Pc×1XN)(1XN×Qc)|QcB}]

Since, inf{(Pc×1XN)(1XN×Qc)|PcA}inf{(Pc×1XN)|PcA}=inf{Pc|PcA}×1XN=NCl(A)×1XN

and inf{(Pc×1XN)(1XN×Qc)|QcB}inf{(1XN×Qc)|QcB}=1XN×inf{Qc|QcB}=1XN×NCl(B)

We have, NCl(A×B)min{NCl(A)×1XN,1XN×NCl(B)}=NCl(A)×NCl(B). Hence the result.

3.14 Definition

A NS A of NTS X is called a N regularly open set (NROS) of (X,TXN) if NInt(NCl(A))=A.

3.15 Definition

A NS A of NTS (X,TXN) is called a N regularly closed set (NRCoS) of X if NCl(NInt(A))=A.

3.16 Theorem

A NS A of NTS (X,TXN) is a NRO iff Ac is NRCo.

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 A is NOS. Now NCl(A)=1XN and NInt(NCl(A))=1XN. Therefore, NInt(NCl(A))A, hence A is not NROS.

3.19 Remark

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

3.20 Example

Let X={a,b,c} and TXN={0XN,1XN,A,B,C} be NTS on X, where

A={<a(0.4,0.5,0.6)>,<b(0.7,0.5,0.3)>,<c(0.5,0.5,0.5)>}

B={<a(0.6,0.5,0.4)>,<b(0.3,0.5,0.7)>,<c(0.5,0.5,0.5)>},

C={<a(0.6,0.5,0.4)>,<b(0.7,0.5,0.3)>,<c(0.5,0.5,0.5)>}.

Then Cl(A)=Bc,Int(Bc)=A

Clearly, Int(Cl(A))=A.

Similarly, Int(Cl(B))=B.

Now, AB=C.

But Cl(AB)=1XN and Int(Cl(AB))=1XN.

Hence, A and B are two NROSs but AB is not NROS.

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 A1 and A2 be any two NROSs of NTS (X,TXN). Since A1A2 is NOS [from Remark 3.17], we have A1A2NInt(NCl(A1A2)). Now, NInt(NCl(A1A2))NInt(NCl(A1))=A1 and NInt(NCl(A1A2))NInt(NCl(A2))=A2 implies that NInt(NCl(A1A2))A1A2. Hence the theorem.

(ii) Let A1 and A2 be any two NROSs of NTS (X,TXN). Since A1A2 is NOS [from Remark 3.17], we have A1A2NCl(NInt(A1A2)). Now, NCl(NInt(A1A2))NCl(NInt(A1))=A1 and NCl(NInt(A1A2))NCl(NInt(A2))=A2 implies that A1A2NCl(NInt(A1A2)). Hence the theorem.

3.22 Theorem

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

(ii) The interior of a NCoS is NROS.

Proof:

(i) Let A be a NOS of NTS (X,TXN), clearly, NInt(NCl(A))NCl(A)NCl(NInt(NCl(A)))NCl(A). Now, A is NOS implies that ANInt(NCl(A)) and hence NCl(A)NCl(NInt(NCl(A))). Thus, NCl(A) is NRCoS.

(ii) Let A be a NCoS of a NTS (X,TXN), clearly, NCl(NInt(A))NInt(A)NInt(NCl(NInt(A)))NInt(A). Now, A is NCoS implies that ANCl(NInt(A)) and hence NInt(A)NInt(NCl(NInt(A))). Thus, NInt(A) is NROS.

3.23 Definition

Let ϕ:(X,TXN)(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is called a N continuous mapping (NCM), if ϕ1(A)TXN for each ATYN; or equivalently ϕ1(B) is a NCoS of X for each NCoS B of Y.

3.24 Definition

Let ϕ:(X,TXN)(Y,TXN) be a mapping from NTS (X,TXN) to another NTS (Y,TYN), then ϕ is said to be a N open mapping (NOM), if ϕ(A)TYN for each ATXN.

3.25 Definition

Let ϕ:(X,TXN)(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (Y,TYN), then ϕ is said to be a N closed mapping (NCoM) if ϕ(B) is a NCoS of Y for each NCoS B of X.

3.26 Definition

Let ϕ:(X,TXN)(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N semi-continuous mapping (NSCM), if ϕ1(A) is a neutrosophic semi-open set of X, for each ATYN.

3.27 Definition

Let ϕ:(X,TXN)(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N semi-open mapping (NSOM), if ϕ(A) is a NSOS for each ATXN.

3.28 Definition

Let ϕ:(X,TXN)(Y,TYN) be a mapping from NTS (X,TXN) to another NTS (X,TYN), then ϕ is said to be a N semi-closed mapping (NSCoM), if ϕ(B) is a NSCoS for each NCoS B of X.

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 X={a,b}, Y={x,y}, and

A={<a(0.6,0.3,0.2)>,<b(0.5,0.2,0.3)>} B={<x(0.5,0.4,0.3)>,<y(0.4,0.2,0.3)>}, C={<x(0.8,0.2,0.1)>,<y(0.7,0.2,0.3)>}.

Then TXN={0XN,1XN,A} and TYN={0XN,1XN,B,C} are NTSs on X and Y.

Let ϕ:(X,TXN)(Y,TYN) be a mapping defined as ϕ(a)=y,ϕ(b)=x.

Then ϕ:(X,TXN)(Y,TYN) is NSCM but not NCM.

3.31 Theorem

Let X1, X2, Y1 and Y2 be NTSs such that X1 is product related to X2. Then, the product ϕ1×ϕ2:X1×X2Y1×Y2 of NSCMs ϕ1:X1Y1 and ϕ2:X2Y2 is NSCM.

Proof:

Let A(Aα×Bβ), where Aα’s and Bβ’s are NOSs of Y1 and Y2, respectively, be a NOS of Y1×Y2. By using Lemma 3.3(i) and Lemma 3.5, we have

(ϕ1×ϕ2)1(A)=[ϕ11(Aα)×ϕ21(Aβ)].

That (ϕ1×ϕ2)1(A) is a NSOS follows from Theorem 3.13 and Theorem 3.10(i).

3.32 Theorem

Let X, X1 and X2 be NTSs and pi:X1×X2Xi(i=1,2) be the projection of X1×X2 onto Xi. Then, if ϕ:XX1×X2 is a NSCM, piϕ is also NSCM.

Proof:

For a NOS A of Xi, we have (piϕ)1(A)=ϕ1(pi1(A)). That pi is a NCM and ϕ is a NSCM imply that (piϕ)1(A) is a NSOS of X.

3.33 Theorem

Let ϕ:XY be a mapping from NTS X to another NTS Y. Then if the graph ψ:XX×Y of ϕ is NSCM, then ϕ is also NSCM.

Proof:

From Lemma 3.6, ϕ1(A)=1XNϕ1(A)=ψ1(1XN×A), for each NOS A of Y. Since ψ is a NSCM and 1XN×A is a NOS X×Y, ϕ1(A) is a NSOS of X and hence ϕ is a NSCM.

3.34 Remark

The converse of Theorem 3.33 is not true.

3.35 Definition

A mapping ϕ:(X,TXN)(Y,TYN) from NTS X to another NTS Y is said to be a N almost continuous mapping (NACM), if ϕ1(A)TXN for each neutrosophic regularly open set A of Y.

3.36 Theorem

Let ϕ:(X,TXN)(Y,TYN) be a mapping. Then the statements below are equivalent:

a) ϕ is a NACM,

b) ϕ1(F) is a NCoS, for each NRCoS F of Y,

c) ϕ1(A)NInt(ϕ1(NInt(NCl(A))), for each NOS A of Y,

d) NCl(ϕ1(NCl(NInt(F))))ϕ1(F), for each NCoS F of Y.

Proof:

Consider that ϕ1(Ac)=(ϕ1(A))c, for any NS A of Y, (a) (b) follows from Theorem 3.16.

(a) (c). Since A is a NOS of Y, ANInt(Cl(A)) and hence ϕ1(A)ϕ1(NInt(NCl(A))). From Theorem 3.22(ii), NInt(NCl(A)) is a NROS of Y, hence ϕ1(NInt(NCl(A))) is a NOS of X. Thus, ϕ1(A)ϕ1(NInt(NCl(A)))=NInt(ϕ1(NInt(NCl(A)))).

(c) (a). Let A be a NROS of Y, then we have ϕ1(A)NInt(ϕ1(NInt(NCl(A))))=NInt(ϕ1(A)). Thus, have ϕ1(A)=NInt(ϕ1(A)). This shows that ϕ1(A) is a NOS of X.

(b) (d) similarly can be proved.

3.37 Remark

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

3.38 Example

Let X={a,b}, Y={x,y}, and

A={<a(0.6,0.5,0.3)>,<b(0.4,0.5,0.5)>}

B={<a(0.2,0.5,0.7)>,<b(0.4,0.5,0.5)>},

C={<x(0.6,0.5,0.3)>,<y(0.4,0.5,0.5)>},

D={<x(0.2,0.5,0.7)>,<y(0.4,0.5,0.5)>},

E={<x(0.2,0.5,0.5)>,<y(0.3,0.5,0.7)>}.

Then TXN={0XN,1XN,A,B} and TYN={0XN,1XN,C,D,E} are NTSs on X and Y.

Now, let ϕ:(X,TXN)(Y,TYN) be a mapping defined as ϕ(a)=y,ϕ(b)=x and clearly ϕ is NACM.

Here, 0XN,1XN,C,D are open sets in TYN but ϕ1(E) is not open set in TXN and hence NACM is not NCM.

3.39 Theorem

N semi-continuity and N almost continuity are independent notions.

3.40 Definition

A NTS (X,TXN) is said to be a N semi-regularly space (NSRS) iff the collection of all NROSs of X forms a base for NT TXN.

3.41 Theorem

Let ϕ:(X,TXN)(Y,TYN) be a mapping from NTS X to a NSRS Y. Then ϕ is NACM iff ϕ is NCM.

Proof:

From Remark 3.37, it suffices to prove that if ϕ is NACM then it is NCM. Let ATNY, then A= Aα, where Aα’s are NROSs of Y. Now, from Lemma 3.3(i), 3.7 and Theorem 3.36(c), we get

ϕ1(A)= ϕ1(Aα) NInt(ϕ1(NCl(Aα)))= NInt(ϕ1(Aα)). NInt(ϕ1(Aα))=NInt(ϕ1(Aα)).

which shows that ϕ1(Aα)TXN.

3.42 Theorem

Let X1, X2,Y1 and Y2 be the NTSs such that Y1 is product related to Y2. Then the product ϕ1×ϕ2:X1×X2Y1×Y2 of NACMs ϕ1:X1Y1 and ϕ2:X2Y2 is NACM.

Proof:

Let A= (Aα×Bβ), where Aα’s and Bβ’s are NOSs of Y1 and Y2 respectively, be a NOS of Y1×Y2. Following Lemma 3.5, for (p1,p2)X1×X2, we have

(ϕ1×ϕ2)1(A)(p1,p2)=(ϕ1×ϕ2)1{(Aα×Bβ)}(p1,p2)={(Aα×Bβ)(ϕ1(p1),ϕ2(p2))}=[min{Aαϕ1(p1),Bβϕ2(p2)}]=[min{ϕ11(Aα)(p1),ϕ21(Bβ)(p2)}]=[(ϕ11(Aα)×ϕ21(Bβ))](p1,p2)i.e.,(ϕ1×ϕ2)1(A)={ϕ11(Aα)×ϕ21(Bβ)}

Now, (ϕ1×ϕ2)1(A)={ϕ11(Aα)×ϕ21(Bβ)}

[NInt(ϕ11(NInt(NCl(Aα))))×NInt(ϕ21(NInt(NCl(Bβ))))][NInt{ϕ11(NInt(NCl(Aα)))×ϕ21(NInt(NCl(Bβ)))}][NInt{ϕ11(NInt(NCl(Aα)))×ϕ21(NInt(NCl(Bβ)))}]=NInt[(ϕ1×ϕ2)1{NInt(NCl(Aα×Bβ))}]NInt[(ϕ1×ϕ2)1{NInt(NCl((Aα×Bβ)))}]=NInt[(ϕ1×ϕ2)1(NInt(NCl(A)))]

Thus, by Theorem 3.36(c), ϕ1×ϕ2 is NACM.

3.43 Theorem

Let X, X1 and X2 be NTSs and pi:X1×X2Xi(i=1,2) be the projection of X1×X2 onto Xi. Then if ϕ:XX1×X2 is a NACM, piϕ is also a NACM.

Proof:

Since pi is NCM Definition 3.23, for any NS A of Xi, we have (i) NCl(pi1(A))pi1(NCl(A)) and (ii) NInt(pi1(A))pi1(NInt(A)). Again, since (i) each pi is a NOM, and (ii) for any NS A of Xi (a) Api1pi(A), and (b) pi1pi(A)A, we have pi(NInt(pi1(A)))pipi1(A)A and hence pi(NInt(pi1(A)))NInt(A). Thus, NInt(pi1(A))pi1pi(NInt(pi1(A)))(pi1(NInt(A)) establishes that NInt(pi1(A))pi1(NInt(A)). Now, for any NOS A of Xi,

(piϕ)1(A)=ϕ1(pi1(A))NInt{ϕ1(NInt(NCl(pi1(A))))}NInt{ϕ1(NInt(pi1(NCl(A))))}=NInt{ϕ1(pi1(NInt(NCl(A))))}=NInt(piϕ)1(NInt(NCl(A)))

3.44 Theorem

Let X and Y be NTSs such that X is product related to Y and let ϕ:XY be a mapping. Then, the graph ψ:XX×Y of ϕ is NACM iff ϕ is NACM.

Proof:

Consider that ψ is a NACM and A is a NOS of Y. Then using Lemma 3.6 and Theorem 3.36(c), we have

ϕ1(A)=1ϕ1(A)=ψ1(1×A)NInt(ψ1(NInt(NCl(1×A))))=NInt(ψ1(1×NInt(NCl(A))))=NInt(ψ1(NInt(1×NCl(A))))=NInt(ψ1(NInt(NCl(A))))

Thus, by Theorem 3.36(c), ϕ is NACM.

Conversely, let ϕ be a NACM and B= (Bα×Aβ), where Bα’s and Aβ’s are NOSs of X and Y, respectively, be a NOS of X×Y.

Since BαNInt(ϕ1(NInt(NCl(Aβ)))) is a NOSs of X contained in

NInt(NCl(Bα))ϕ1(NInt(NCl(Aβ))), BαNInt(ϕ1(NInt(NCl(Aβ)))) NInt[NInt(NCl(Bα))ϕ1(NInt(NCl(Aβ)))]

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

ϕ1(B)=ϕ1((Bα×Aβ))=[Bαϕ1(Aβ)][BαNInt(ϕ1(NInt(NCl(Aβ))))][NInt(NInt(NCl(Bα)))ϕ1(NInt(NCl(Aβ)))]NInt[ψ1(NInt(NCl(Bα)))×NInt(NCl(Aβ))]NInt[ψ1(NInt(NCl(Bα)))×NInt(NCl(Aβ))]NInt[ψ1(NInt(NCl((Bα×Aβ))))]=NInt[ψ1(NInt(NCl(B)))].

Thus, by Theorem 3.36(c), ψ is NACM.

4  Conclusion

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 N semi-open set, N semi-closed, N regularly open set, N regularly closed set, N continuous mapping, N open mapping, N closed mapping, N semi-continuous mapping, N semi-open mapping, N semi-closed mapping, set in order to propose the definition of N almost continuous mapping. Some properties of N almost continuous mapping have been demonstrated. We expect that our study may spark some new ideas for the construction of the neutrosophic almost continuous mapping. 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. In the future, we would like to extend our work to study some properties in the neutrosophic semi and almost topological group with the help of the neutrosophic semi and almost continuous mapping.

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.

## References

1.  Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI 10.1016/S0019-9958(65)90241-X. [Google Scholar] [CrossRef]

2.  Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI 10.1016/S0165-0114(86)80034-3. [Google Scholar] [CrossRef]

3.  Chang, C. L. (1968). Fuzzy topological space. Journal of Mathematical Analysis and Applications, 24(1), 182–190. DOI 10.1016/0022-247X(68)90057-7. [Google Scholar] [CrossRef]

4.  Coker, D. (1997). An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and System, 88(1), 81–89. DOI 10.1016/S0165-0114(96)00076-0. [Google Scholar] [CrossRef]

5.  Rosenfeld, A. (1971). Fuzzy groups. Journal of Mathematical Analysis and Applications, 35(3), 512–517. DOI 10.1016/0022-247X(71)90199-5. [Google Scholar] [CrossRef]

6.  Foster, D. H. (1979). Fuzzy topological groups. Journal of Mathematical Analysis and Applications, 67(2), 549–564. DOI 10.1016/0022-247X(79)90043-X. [Google Scholar] [CrossRef]

7.  Azad, K. K. (1981). On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity. Journal of Mathematical Analysis and Applications, 82(1), 14–32. DOI 10.1016/0022-247X(81)90222-5. [Google Scholar] [CrossRef]

8.  Smarandache, F. (2002). Neutrosophy and neutrosophic logic, first international conference on neutrosophy. Neutrosophic logic, set, probability, and statistics. USA: American Research Press. [Google Scholar]

9.  Smarandache, F. (2005). Neutrosophic set–A generalization of the intuitionistic fuzzy set. International Journal of Pure and Applied Mathematics, 24(3), 287–297. [Google Scholar]

10. Kandil, A., Nouth, A. A., El-Sheikh, S. A. (1995). On fuzzy bitopological spaces. Fuzzy Sets and System, 74(3), 353–363. DOI 10.1016/0165-0114(94)00333-3. [Google Scholar] [CrossRef]

11. Mwchahary, D. D., Basumatary, B. (2020). A note on neutrosophic bitopological space. Neutrosophic Sets and Systems, 33, 134–144. DOI 10.5281/zenodo.3782868. [Google Scholar] [CrossRef]

12. Salama, A. A., Alblowi, S. A. (2012). Neutrosophic set and neutrosophic topological spaces. IOSR Journal of Mathematics, 3(4), 31–35. DOI 10.9790/5728-0343135. [Google Scholar] [CrossRef]

13. Salama, A. A., Smarandache, F., Kroumov, V. (2014). Neutrosophic closed sets and neutrosophic continuous functions. Neutrosophic Sets and Systems, 4, 4–8. DOI 10.5281/zenodo.30213. [Google Scholar] [CrossRef]

14. Noiri, T. (1973). On semi-continuous mapping. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche. Matematiche e Naturali, 54(8), 132–136. [Google Scholar]

15. Singal, M. K., Singal, A. R. (1968). Almost continuous mappings. Yokohama Mathematical Journal, 16, 63–73. [Google Scholar]

16. Sumathi, I. R., Arockiarani, I. (2015). Fuzzy neutrosophic groups. Advanced in Fuzzy Mathematics, 10(2), 117–122. [Google Scholar]

17. Sumathi, I. R., Arockiarani, I. (2016). Topological group structure of neutrosophic set. Journal of Advanced Studies in Topology, 7(1), 12–20. DOI 10.20454/jast.2016.1003. [Google Scholar] [CrossRef]

18. Abdel-Baset, M., Mohamed, R., Zaied, A. E. N. H., Smarandache, F. (2019). A hybride plitogenic decision-making approach with quality function deployment for selecting supply chain sustainability metrics. Symmetry, 11(7), 903. DOI 10.3390/sym11070903. [Google Scholar] [CrossRef]

19. Abdel-Baset, M., Manogaran, G., Gamaland, A., Smarandache, F. (2019). A group decision making frame work based on neutrosophic TOPSIS approach for smart medical device selection. Journal of Medical System, 43(2), 38. DOI 10.1007/s10916-019-1156-1. [Google Scholar] [CrossRef]

20. Nandhini, M., Palanisamy, M. (2017). Fuzzy almost contra α-continuous function. International Journal of Advance Research and Innovative Ideas in Education, 3(4), 1964–1971. [Google Scholar]

21. Al-Omeri, W., Smarandache, F. (2017). New neutrosophic sets via neutrosophic topological spaces. Neutrosophic operational research, vol. 1, pp. 189–209. Brussels (BelgiumPons. [Google Scholar]

22. Al-Omeri, W. (2016). Neutrosophic crisp sets via neutrosophic crisp topological spaces. Neutrosophic Sets and Systems, 13, 96–104. DOI 10.5281/zenodo.570855. [Google Scholar] [CrossRef]

23. Abdel-Basset, M., El-hoseny, M., Gamal, A., Smarandache, F. (2019). A novel model for evaluation hospital medical care systems based on plithogenic sets. Artificial Intelligence in Medicine, 100, 101710. DOI 10.1016/j.artmed.2019.101710. [Google Scholar] [CrossRef]

24. Abdel-Basset, M., Chang, V., Mohamed, M., Smarandche, F. (2019). A refined approach for forecasting based on neutrosophic time series. Symmetry, 11(4), 457. DOI 10.3390/sym11040457. [Google Scholar] [CrossRef]

25. Smarandache, F. (1998). Neutrosophy: Neutrosophic probability, set, and logic: Analytic synthesis & synthetic analysis, vol. 105. Michigan, USA: Ann Arbor. [Google Scholar]

26. Smarandache, F. (2013). N-valued refined neutrosophic logic and its applications in physics. Progress in Physics, 4, 143–146. DOI 10.5281/zenodo.30245. [Google Scholar] [CrossRef]