On Some Properties of Neutrosophic Semi Continuous and Almost Continuous Mapping

Bhimraj Basumatary; Nijwm Wary; Jeevan Khaklary; Usha Basumatary

doi:10.32604/cmes.2022.018066

Bhimraj Basumatary1,*, Nijwm Wary1, Jeevan Krishna Khaklary2 and Usha Rani Basumatary1

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

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).

A neutrosophic set (NS)

$AN$ on

$X$ can be expressed as

$AN={<x∈X,TAN(x),IAN(x),FAN(x)>}$ , where

$T,I,F:X⟶$ ]-0, 1+[. Note that

$0≼TAN(x)+IAN(x)+FAN(x)≼3$ .

Let

$X≠ϕ$ and

$AN={<x∈X,TAN(x),IAN(x),FAN(x)>}$ and

$BN={<x∈X,TBN(x),IBN(x),FBN(x)>}$ are NSs. Then

(iii)

$AN≼BN$ if

$TAN(x)≼TBN(x),IAN(x)≼IBN(x),FAN(x)≽FBN(x)$ , for

$x∈X$ .

Let

$X≠ϕ$ , then neutrosophic topology space (NTS) on

$X$ is a family

$TXN$ of neutrosophic subsets of

$X$ satisfying the following axiom:

Let

$(X,TXN)$ be NTS. Then for a NS

$AN={<x,μNi,σNi,δNi>:x∈X}$ , neutrosophic interior of

$AN$ can be defined as

$N∽Int(AN)={<x,⋓ μNi,⋒ σNi,⋒ δNi>:x∈X}$ .

Let

$(X,TXN)$ be NTS. Then for a NS

$AN={<x,μNi,σNi,δNi>:x∈X}$ , neutrosophic closure of

$AN$ can be defined as

$N∽Cl(AN)={<x,⋒ μNi,⋓ σNi,⋓ δNi>:x∈X}$ .

Let

$A$ be a NS of NTS

$(X,TXN)$ , then

$A$ is called a

$N∽$ semi-open set (

$NSOS$ ) of

$X$ if

$∃$ a

$B∈TXN$ such that

$A≼N∽Cl(N∽Int(B))$ .

Let

$A$ be a NS of NTS

$(X,TXN)$ , then

$A$ is called a

$N∽$ semi-closed set (

$N$ SCoS) of

$X$ if

$∃$ a

$Bc∈TXN$ such that

$N∽Int(N∽Cl(B))≼A$ .

$ϕ−1(⋓ Aα)=⋓ ϕ−1(Aα)$ and (ii)

$ϕ−1(⋒ Aα)=⋒ ϕ−1(Aα)$ .

Let

$(p,q)$ be any element of

$X×Y$ ,

$(1XN−A×B)(p,q)=max(1XN−A(p),1XN−B(q))=max{(Ac×1XN)(p,q),(Bc×1XN)(p,q)}={(Ac×1XN)⋓(1XN×Bc)}(p,q)$ , for each

$(p,q)∈X×Y$ .

Let

$ϕi:Xi⟶Yi$ and

$Ai$ be NSs of

$Yi$ ,

$i=1,2$ ; we have

$(ϕ1×ϕ2)−1(A1×A2)=ϕ1−1(A1)×ϕ2−1(A2)$ .

Let

$ψ:X⟶X×Y$ be the graph of a mapping

$ϕ:X⟶Y$ . Then, if

$A,B$ be NSs of

$X$ and

$Y$ ,

$ψ−1(A×B)=A ⋒ ϕ−1(B)$ .

For a family

${A}α$ of NSs of NTS

$(X,TXN)$ ,

$⋓ N∽Cl(Aα)≼N∽Cl(⋓ (Aα))$ . In case

$B$ is a finite set,

$⋓ N∽Cl(Aα)≼N∽Cl(⋓ (Aα))$ . Also,

$⋓ N∽Int(Aα)≼N∽Int(⋓ (Aα))$ , 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

$(X,TXN)$ .

For a NS

$A$ of NTS

$(X,TXN)$ , (a)

$1−N∽Int(A)=N∽Cl(1−A)$ , and (b)

$1−N∽Cl(A)=N∽Int(1−A)$ .

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

$A$ of NTS

$(X,TXN)$ such that

$1−N∽Int(A)=N∽Cl(1−A)$ and

$1−N∽Cl(A)=N∽Int(1−A)$ .

(i)

$⇒$ (iii). By definition

$∃$ a NCoS

$B$ such that

$N∽Int(B)≼A≼B$ and hence

$N∽Int(B)≼A≼N∽Cl(A)≼B$ . Since

$N∽Int(B)$ is the greatest NOS contained in

$B$ , we have

$N∽Int(N∽Cl(B))≼N∽Int(B)≼A$ .

(i) Let

${Aα}$ be a collection of NSOSs of NTS

$(X,TXN)$ . Then

$∃$ a

$Bα∈TXN$ such that

$Bα≼Aα≼N∽Cl(Bα)$ , for each α. Thus,

$⋒Bα≼⋓Aα≼⋓N∽Cl(Bα)≼N∽Cl(⋓(Bα))$ [Lemma 3.7], and

$⋓Bα∈TXN$ , this shows that

$⋓Bα$ is a NSOS.

(ii) Let

${Aα}$ be a collection of NSCoSs of NTS

$(X,TXN)$ . Then

$∃$ a

$Bα∈TXN$ such that

$N∽Int(Bα)≼Aα≼Bα$ , for each α. Thus,

$N∽Int(⋒(Bα))≼⋒N∽Int(Bα)≼⋒Aα≼⋒Bα$ [Lemma 3.7], and

$⋓Bα∈TXN$ , this shows that

$⋒Bα$ is a NSCoS.

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.

Let

$P={<a(0.8,0.2,0.1)>,<b(0.7,0.2,0.3)>}$ be any neutrosophic set

$XN$ , then

$N∽Int(P)=⋓{G:G is open set,G≼P}=A⋓B=A$ and

$N∽Cl(P)=⋒{K≽P:K is closed set inTXN}=1XN$ . 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.

$(X,TXN)$ 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

$X×Y$ .

Let

$P≼A≼N∽Cl(P)$ and

$Q≼B≼N∽Cl(Q)$ , where

$P∈TXN$ and

$Q∈TYN$ . Then

$P×Q≼A×B≼N∽Cl(P)×N∽Cl(Q)$ . For NSs

$P$ ’s of

$X$ and

$Q$ ’s of

$Y$ , we have

It is sufficient to prove

$N∽Cl(A×B)≽N∽Cl(A)×N∽Cl(B)$ . Let

$P∈TNX$ and

$Q∈TNY$ .

$Since, inf{(Pc×1XN)⋓(1XN×Qc)|Pc≽A}≽inf{(Pc×1XN)|Pc≽A}=inf{Pc|Pc≽A}×1XN=N∽Cl(A)×1XN$

We have,

$N∽Cl(A×B)≽min{N∽Cl(A)×1XN,1XN×N∽Cl(B)}=N∽Cl(A)×N∽Cl(B)$ . Hence the result.

A NS

$A$ of NTS

$X$ is called a

$N∽$ regularly open set (NROS) of

$(X,TXN)$ if

$N∽Int(N∽Cl(A))=A$ .

A NS

$A$ of NTS

$(X,TXN)$ is called a

$N∽$ regularly closed set (NRCoS) of

$X$ if

$N∽Cl(N∽Int(A))=A$ .

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

From Example 3.12, it is clear that

$A$ is NOS. Now

$N∽Cl(A)=1XN$ and

$N∽Int(N∽Cl(A))=1XN$ . Therefore,

$N∽Int(N∽Cl(A))≠A$ , hence

$A$ is not NROS.

(i) Let

$A1$ and

$A2$ be any two NROSs of NTS

$(X,TXN)$ . Since

$A1⋒A2$ is NOS [from Remark 3.17], we have

$A1⋒A2≼N∽Int(N∽Cl(A1⋒A2))$ . Now,

$N∽Int(N∽Cl(A1⋒A2))≼N∽Int(N∽Cl(A1))=A1$ and

$N∽Int(N∽Cl(A1⋒A2))≼N∽Int(N∽Cl(A2))=A2$ implies that

$N∽Int(N∽Cl(A1⋒A2))≼A1⋒A2$ . Hence the theorem.

(ii) Let

$A1$ and

$A2$ be any two NROSs of NTS

$(X,TXN)$ . Since

$A1⋓A2$ is NOS [from Remark 3.17], we have

$A1⋓A2≽N∽Cl(N∽Int(A1⋓A2))$ . Now,

$N∽Cl(N∽Int(A1⋓A2))≽N∽Cl(N∽Int(A1))=A1$ and

$N∽Cl(N∽Int(A1⋓A2))≽N∽Cl(N∽Int(A2))=A2$ implies that

$A1⋓A2≼N∽Cl(N∽Int(A1⋓A2))$ . Hence the theorem.

(i) Let

$A$ be a NOS of NTS

$(X,TXN)$ , clearly,

$N∽Int(N∽Cl(A))≼N∽Cl(A)⇒N∽Cl(N∽Int(N∽Cl(A)))≼N∽Cl(A)$ . Now,

$A$ is NOS implies that

$A≼N∽Int(N∽Cl(A))$ and hence

$N∽Cl(A)≼N∽Cl(N∽Int(N∽Cl(A)))$ . Thus,

$N∽Cl(A)$ is NRCoS.

(ii) Let

$A$ be a NCoS of a NTS

$(X,TXN)$ , clearly,

$N∽Cl(N∽Int(A))≽N∽Int(A)⇒N∽Int(N∽Cl(N∽Int(A)))≽N∽Int(A)$ . Now,

$A$ is NCoS implies that

$A≽N∽Cl(N∽Int(A))$ and hence

$N∽Int(A)≽N∽Int(N∽Cl(N∽Int(A)))$ . Thus,

$N∽Int(A)$ is NROS.

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

$ϕ−1(A)∈TXN$ for each

$A∈TYN$ ; or equivalently

$ϕ−1(B)$ is a NCoS of

$X$ for each NCoS

$B$ of

$Y$ .

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

$A∈TXN$ .

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

$X$ .

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

$ϕ−1(A)$ is a neutrosophic semi-open set of

$X$ , for each

$A∈TYN$ .

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

$A∈TXN$ .

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

$X$ .

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

Let

$X1$ ,

$X2$ ,

$Y1$ and

$Y2$ be NTSs such that

$X1$ is product related to

$X2$ . Then, the product

$ϕ1×ϕ2:X1×X2⟶Y1×Y2$ of NSCMs

$ϕ1:X1⟶Y1$ and

$ϕ2:X2⟶Y2$ is NSCM.

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

That

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

Let

$X$ ,

$X1$ and

$X2$ be NTSs and

$pi:X1×X2⟶Xi(i=1,2)$ be the projection of

$X1×X2$ onto

$Xi$ . Then, if

$ϕ:X⟶X1×X2$ is a NSCM,

$piϕ$ is also NSCM.

For a NOS

$A$ of

$Xi$ , we have

$(piϕ)−1(A)=ϕ−1(pi−1(A))$ . That

$pi$ is a NCM and

$ϕ$ is a NSCM imply that

$(piϕ)−1(A)$ is a NSOS of

$X$ .

Let

$ϕ:X⟶Y$ be a mapping from NTS

$X$ to another NTS

$Y$ . Then if the graph

$ψ:X⟶X×Y$ of

$ϕ$ is NSCM, then

$ϕ$ is also NSCM.

From Lemma 3.6,

$ϕ−1(A)=1XN⋒ϕ−1(A)=ψ−1(1XN×A)$ , for each NOS

$A$ of

$Y$ . Since

$ψ$ is a NSCM and

$1XN×A$ is a NOS

$X×Y$ ,

$ϕ−1(A)$ is a NSOS of

$X$ and hence

$ϕ$ is a NSCM.

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

$ϕ−1(A)∈TXN$ for each neutrosophic regularly open set

$A$ of

$Y$ .

Let

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

$N∽Cl(ϕ−1(N∽Cl(N∽Int(F))))≼ϕ−1(F)$ , for each NCoS

$F$ of

$Y$ .

Consider that

$ϕ−1(Ac)=(ϕ−1(A))c$ , for any NS

$A$ of

$Y$ , (a)

$⇔$ (b) follows from Theorem 3.16.

(a)

$⇒$ (c). Since

$A$ is a NOS of

$Y$ ,

$A≼N∽Int(Cl(A))$ and hence

$ϕ−1(A)≼ϕ−1(N∽Int(N∽Cl(A)))$ . From Theorem 3.22(ii),

$N∽Int(N∽Cl(A))$ is a NROS of

$Y$ , hence

$ϕ−1(N∽Int(N∽Cl(A)))$ is a NOS of

$X$ . Thus,

$ϕ−1(A)≼ϕ−1(N∽Int(N∽Cl(A)))=N∽Int(ϕ−1(N∽Int(N∽Cl(A))))$ .

(c)

$⇒$ (a). Let

$A$ be a NROS of

$Y$ , then we have

$ϕ−1(A)≼N∽Int(ϕ−1(N∽Int(N∽Cl(A))))=N∽Int(ϕ−1(A))$ . Thus, have

$ϕ−1(A)=N∽Int(ϕ−1(A))$ . This shows that

$ϕ−1(A)$ is a NOS of

$X$ .

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

$ϕ−1(E)$ is not open set in

$TXN$ and hence NACM is not NCM.

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

$TXN$ .

Let

$ϕ:(X,TXN)⟶(Y,TYN)$ be a mapping from NTS

$X$ to a NSRS

$Y$ . Then

$ϕ$ is NACM iff

$ϕ$ is NCM.

From Remark 3.37, it suffices to prove that if

$ϕ$ is NACM then it is NCM. Let

$A∈TNY$ , 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

Let

$X1$ ,

$X2,Y1$ and

$Y2$ be the NTSs such that

$Y1$ is product related to

$Y2$ . Then the product

$ϕ1×ϕ2:X1×X2⟶Y1×Y2$ of NACMs

$ϕ1:X1⟶Y1$ and

$ϕ2:X2⟶Y2$ is NACM.

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

Let

$X$ ,

$X1$ and

$X2$ be NTSs and

$pi:X1×X2⟶Xi(i=1,2)$ be the projection of

$X1×X2$ onto

$Xi$ . Then if

$ϕ:X⟶X1×X2$ is a NACM,

$piϕ$ is also a NACM.

Since

$pi$ is NCM Definition 3.23, for any NS

$A$ of

$Xi$ , we have (i)

$N∽Cl(pi−1(A))≼pi−1(N∽Cl(A))$ and (ii)

$N∽Int(pi−1(A))≽pi−1(N∽Int(A))$ . Again, since (i) each

$pi$ is a NOM, and (ii) for any NS

$A$ of

$Xi$ (a)

$A≼pi−1pi(A)$ , and (b)

$pi−1pi(A)≼A$ , we have

$pi(N∽Int(pi−1(A)))≼pipi−1(A)≼A$ and hence

$pi(N∽Int(pi−1(A)))≼N∽Int(A)$ . Thus,

$N∽Int(pi−1(A))≼pi−1pi(N∽Int(pi−1(A)))≼(pi−1(N∽Int(A))$ establishes that

$N∽Int(pi−1(A))≼pi−1(N∽Int(A))$ . Now, for any NOS

$A$ of

$Xi$ ,

Let

$X$ and

$Y$ be NTSs such that

$X$ is product related to

$Y$ and let

$ϕ:X⟶Y$ be a mapping. Then, the graph

$ψ:X⟶X×Y$ of

$ϕ$ is NACM iff

$ϕ$ is NACM.

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

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

$Bα⋒N∽Int(ϕ−1(N∽Int(N∽Cl(Aβ))))$ is a NOSs of

$X$ contained in

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.

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

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