On Some Properties of Neutrosophic Semi Continuous and Almost Continuous Mapping

Bhimraj Basumatary,Nijwm Wary,Jeevan Krishna Khaklary and Usha Rani Basumatary

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

2Central Institute of Technology,Kokrajhar,783370,India

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 toaddress 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 contraα-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 Methodologies

2.1 Definition[8]

A neutrosophic set (NS)ANonXcan be expressed asAN= {< 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.

2.2 Definition[8]

Complement ofANis expressed as

ANc(x)={}.

2.3 Definition[8]

LetX≠φandAN={}andBN={}are NSs.Then

2.4 Definition[12]

LetX≠φ,then neutrosophic topology space (NTS)onXis a familyTXNof neutrosophic subsets ofXsatisfying the following axiom:

(i)0XN,1XN∈TXN

(ii)GN1GN2∈TXN; forGN1,GN2∈TXN

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

2.5 Definition[12]

Let(X,TXN)be NTS.Then for a NSAN=｛:x∈X},neutrosophic interior ofANcan be defined as

2.6 Definition[12]

Let(X,TXN)be NTS.Then for a NSAN={:x∈X},neutrosophic closure ofANcan be defined as

3 Results and Discussion

3.1 Definition

LetAbe a NS of NTSthenAis called aN∽semi-open set (NSOS)ofXif ∃aB∈TXNsuch thatAN∽Cl(N∽Int(B)).

3.2 Definition

LetAbe a NS ofthenAis called aN∽semi-closed set (NSCoS)ofXif∃aBc∈TXNsuch thatN∽Int(N∽Cl(B))A.

3.3 Lemma

Letφ:X−→Ybe a mapping and {Aα}be a family of NSs ofY,then

(i)φ−1(Aα)=φ−1(Aα)and (ii)φ−1(Aα)=φ−1(Aα).

Prove is Straightforward.

3.4 Lemma

LetA,Bbe NSs ofXandY,then 1XN−A×B=(Ac×1XN)(1XN×Bc).

Proof:

Let(p,q)be any element ofX×Y,maxfor each(p,q)∈X×Y.

3.5 Lemma

Letφi:Xi−→YiandAibe NSs ofYi,i=1,2; we have(φ1×φ2)−1(A1×A2)=φ1−1(A1)×φ2−1(A2).

Proof:

For each(p1,p2)∈X1×X2,we have

3.6 Lemma

Letψ:X−→X×Ybe the graph of a mappingφ:X−→Y.Then,ifA,Bbe NSs ofXandY,ψ−1(A×B)=Aφ−1(B).

Proof:

For eachp∈X,we have

3.7 Lemma

For a family {A}αof NSs ofN∽Cl(Aα)N∽Cl((Aα)).In caseBis a finite set,N∽Cl(Aα)N∽Cl((Aα)).Also,N∽Int(Aα)N∽Int((Aα)),where a subfamilyBofis said to be subbase forif the collection of all intersections of members ofBforms a base for

3.8 Lemma

For a NSAof(a)1 −N∽Int(A)=N∽Cl(1 −A),and (b)1 −N∽Cl(A)=N∽Int(1 −A).

Prove is Straightforward.

3.9 Theorem

The statements below are equivalent:

(i)Ais a NSCoS,

(ii)Acis a NSOS,

(iii)N∽Int(N∽Cl(A))A,and

(iv)N∽Cl(N∽Int(Ac))Ac.

Proof:

(i)and (ii)are equivalent follows from Lemma 3.8,since for a NSAof NTSsuch that 1 −N∽Int(A)=N∽Cl(1 −A)and 1 −N∽Cl(A)=N∽Int(1 −A).

(i)⇒(iii).By definition ∃a NCoSBsuch thatN∽Int(B)ABand henceN∽Int(B)AN∽Cl(A)B.SinceN∽Int(B)is the greatest NOS contained inB,we haveN∽Int(N∽Cl(B))N∽Int(B)A.

(iii)⇒(i)follows by takingB=N∽Cl(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 ofThen ∃aBα∈TXNsuch thatBαAαN∽Cl(Bα),for eachα.Thus,BαAαN∽Cl(Bα)N∽Cl((Bα))[Lemma 3.7],andBα∈TXN,this shows thatBαis a NSOS.

(ii)Let {Aα} be a collection of NSCoSs ofThen ∃aBα∈TXNsuch thatN∽Int(Bα)AαBα,for eachα.Thus,N∽Int((Bα))N∽Int(Bα)AαBα[Lemma 3.7],andBα∈TXN,this shows thatBαis 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 inExample 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

LetX={a,b}andA,Bbe neutrosophic subsets of X such that

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

3.13 Theorem

If(X,TXN)and(Y,TYN)are NTSs andXis product related toY.Then the productA×Bof a NSOSAofXand a NSOSBofYis NSOS of the neutrosophic product spaceX×Y.

Proof:

LetPAN∽Cl(P)andQBN∽Cl(Q),whereP∈TXNandQ∈TYN.ThenP×QA×BN∽Cl(P)×N∽Cl(Q).For NSsP’s ofXandQ’s ofY,we have

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

It is sufficient to proveN∽Cl(A×B)N∽Cl(A)×N∽Cl(B).LetP∈TNXandQ∈TNY.

Then

We have,N∽Cl(A×B)minN∽Cl(A)×1XN,1XN×N∽Cl(B)=N∽Cl(A)×N∽Cl(B).Hence the result.

3.14 Definition

A NSAof NTSXis called aN∽regularly open set (NROS)of(X,TXN)ifN∽Int(N∽Cl(A))=A.

3.15 Definition

A NSAof NTS(X,TXN)is called aN∽regularly closed set (NRCoS)ofXifN∽Cl(N∽Int(A))=A.

3.16 Theorem

A NSAofis a NRO iffAcis NRCo.

Proof:It follows fromLemma 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

FromExample 3.12,it is clear thatAis NOS.NowN∽Cl(A)= 1XNandN∽Int(N∽Cl(A))=1XN.Therefore,N∽Int(N∽Cl(A))≠A,henceAis not NROS.

3.19 Remark

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

3.20 Example

LetX={a,b,c}andbe NTS onX,where

ThenCl(A)=Bc,Int(Bc)=A

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

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

Now,A∪B=C.

Hence,AandBare two NROSs butA∪Bis 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)LetA1andA2be any two NROSs of NTSSinceA1A2is NOS [from Remark 3.17],we haveA1A2N∽Int(N∽Cl(A1A2)).Now,N∽Int(N∽Cl(A1A2))N∽Int(N∽Cl(A1))=A1andN∽Int(N∽Cl(A1A2))N∽Int(N∽Cl(A2))=A2implies thatN∽Int(N∽Cl(A1A2))A1A2.Hence the theorem.

(ii)LetA1andA2be any two NROSs of NTS(X,TXN).SinceA1A2is NOS [from Remark 3.17],we haveA1A2N∽Cl(N∽Int(A1A2)).Now,N∽Cl(N∽Int(A1A2))N∽Cl(N∽Int(A1))=A1andN∽Cl(N∽Int(A1A2))N∽Cl(N∽Int(A2))=A2implies thatA1A2N∽Cl(N∽Int(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)LetAbe a NOS ofclearly,N∽Int(N∽Cl(A))N∽Cl(A)⇒N∽Cl(N∽Int(N∽Cl(A)))N∽Cl(A).Now,Ais NOS implies thatAN∽Int(N∽Cl(A))and henceN∽Cl(A)N∽Cl(N∽Int(N∽Cl(A))).Thus,N∽Cl(A)is NRCoS.

(ii)LetAbe a NCoS of aclearly,N∽Cl(N∽Int(A))N∽Int(A)⇒N∽Int(N∽Cl(N∽Int(A)))N∽Int(A).Now,Ais NCoS implies thatAN∽Cl(N∽Int(A))and henceN∽Int(A)N∽Int(N∽Cl(N∽Int(A))).Thus,N∽Int(A)is NROS.

3.23 Definition

Letφ:(X,TXN)−→(Y,TYN)be a mapping from NTS(X,TXN)to another NTS(X,TYN),thenφis called aN∽continuous mapping (NCM),ifφ−1(A)∈TXNfor eachA∈TYN; or equivalentlyφ−1(B)is a NCoS ofXfor each NCoSBofY.

3.24 Definition

Letφ:(X,TXN)−→(Y,TXN)be a mapping from NTS(X,TXN)to another NTS(Y,TYN),thenφis said to be aN∽open mapping (NOM),ifφ(A)∈TYNfor eachA∈TXN.

3.25 Definition

Letφ:(X,TXN)−→(Y,TYN)be a mapping from NTS(X,TXN)to another NTS(Y,TYN),thenφis said to be aN∽closed mapping (NCoM)ifφ(B)is a NCoS ofYfor each NCoSBofX.

3.26 Definition

Letφ:(X,TXN)−→(Y,TYN)be a mapping from NTS(X,TXN)to another NTS(X,TYN),thenφis said to be aN∽semi-continuous mapping (NSCM),ifφ−1(A)is a neutrosophic semiopen set ofX,for eachA∈TYN.

3.27 Definition

Letφ:(X,TXN)−→(Y,TYN)be a mapping from NTS(X,TXN)to another NTS(X,TYN),thenφis said to be aN∽semi-open mapping (NSOM),ifφ(A)is a NSOS for eachA∈TXN.

3.28 Definition

Letφ:(X,TXN)−→(Y,TYN)be a mapping from NTS(X,TXN)to another NTS(X,TYN),thenφis said to be aN∽semi-closed mapping (NSCoM),ifφ(B)is a NSCoS for each NCoSBofX.

3.29 Remark

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

3.30 Example

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

Letφ:(X,TXN)−→(Y,TYN)be a mapping defined asφ(a)=y,φ(b)=x.

Thenφ:(X,TXN)−→(Y,TYN)is NSCM but not NCM.

3.31 Theorem

LetX1,X2,Y1andY2be NTSs such thatX1is product related toX2.Then,the productφ1×φ2:X1×X2−→Y1×Y2of NSCMsφ1:X1−→Y1andφ2:X2−→Y2is NSCM.

Proof:

LetA≡(Aα×Bβ),whereAα’s andBβ’s are NOSs ofY1andY2,respectively,be a NOS ofY1×Y2.By usingLemma 3.3(i)andLemma 3.5,we have

That(φ1×φ2)−1(A)is a NSOS follows fromTheorem 3.13andTheorem 3.10(i).

3.32 Theorem

LetX,X1andX2be NTSs andpi:X1×X2−→Xi(i=1,2)be the projection ofX1×X2ontoXi.Then,ifφ:X−→X1×X2is a NSCM,piφis also NSCM.

Proof:

For a NOSAofXi,we have(piφ)−1(A)=φ−1(pi−1(A)).Thatpiis a NCM andφis a NSCM imply that(piφ)−1(A)is a NSOS ofX.

3.33 Theorem

Letφ:X−→Ybe a mapping from NTSXto another NTSY.Then if the graphψ:X−→X×Yofφis NSCM,thenφis also NSCM.

Proof:

FromLemma 3.6,φ−1(A)=1XNφ−1(A)=ψ−1(1XN×A),for each NOSAofY.Sinceψis a NSCM and 1XN×Ais a NOSX×Y,φ−1(A)is a NSOS ofXand henceφis a NSCM.

3.34 Remark

The converse ofTheorem 3.33is not true.

3.35 Definition

A mappingφ:(X,TXN)−→(Y,TYN)from NTSXto another NTSYis said to be aN∽almost continuous mapping (NACM),ifφ−1(A)∈TXNfor each neutrosophic regularly open setAofY.

3.36 Theorem

Letφ:(X,TXN)−→(Y,TYN)be a mapping.Then the statements below are equivalent:

(a)φis a NACM,

(b)φ−1(F)is a NCoS,for each NRCoSFofY,

(c)φ−1(A)N∽Int(φ−1(N∽Int(N∽Cl(A))),for each NOSAofY,

Proof:

(a)⇒ (c).SinceAis a NOS ofY,AN∽Int(Cl(A))and henceφ−1(A)φ−1(N∽Int(N∽Cl(A))).FromTheorem 3.22(ii),N∽Int(N∽Cl(A))is a NROS ofY,henceφ−1(N∽Int(N∽Cl(A)))is a NOS ofX.Thus,φ−1(A)φ−1(N∽Int(N∽Cl(A)))=N∽

(c)⇒(a).LetAbe a NROS ofY,then we haveThus,haveThis shows thatφ−1(A)is a NOS ofX.

(b)⇔(d)similarly can be proved.

3.37 Remark

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

3.38 Example

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

Now,letφ:(X,TXN)−→(Y,TYN)be a mapping defined asφ(a)=y,φ(b)=xand clearlyφis NACM.

Here,0XN,1XN,C,Dare open sets inTYNbutφ−1(E)is not open set inTXNand hence NACM is not NCM.

3.39 Theorem

N∽semi-continuity andN∽almost continuity are independent notions.

3.40 Definition

A NTS(X,TXN)is said to be aN∽semi-regularly space (NSRS)iffthe collection of all NROSs ofXforms a base for NTTXN.

3.41 Theorem

Letφ:(X,TXN)−→(Y,TYN)be a mapping from NTSXto a NSRSY.Thenφis NACM iffφis NCM.

Proof:

FromRemark 3.37,it suffices to prove that ifφis NACM then it is NCM.LetA∈TNY,thenA=Aα,whereAα’s are NROSs ofY.Now,fromLemma 3.3(i),3.7 andTheorem 3.36(c),we get

which shows thatφ−1(Aα)∈TXN.

3.42 Theorem

LetX1,X2,Y1andY2be the NTSs such thatY1is product related toY2.Then the productφ1×φ2:X1×X2−→Y1×Y2of NACMsφ1:X1−→Y1andφ2:X2−→Y2is NACM.

Proof:

LetA=(Aα×Bβ),whereAα’s andBβ’s are NOSs ofY1andY2respectively,be a NOS ofY1×Y2.FollowingLemma 3.5,for(p1,p2)∈X1×X2,we have

Thus,byTheorem 3.36(c),φ1×φ2is NACM.

3.43 Theorem

LetX,X1andX2be NTSs andpi:X1×X2−→Xi(i=1,2)be the projection ofX1×X2ontoXi.Then ifφ:X−→X1×X2is a NACM,piφis also a NACM.

Proof:

Sincepiis NCMDefinition 3.23,for any NSAofXi,we have (i)pi−1(N∽Cl(A))and (ii)Again,since (i)eachpiis a NOM,and (ii)for any NSAofXi(a)and (b)we haveand henceThus,establishes thatN∽Now,for any NOSAofXi,

3.44 Theorem

LetXandYbe NTSs such thatXis product related toYand letφ:X−→Ybe a mapping.Then,the graphψ:X−→X×Yofφis NACM iffφis NACM.

Proof:

Consider thatψis a NACM andAis a NOS ofY.Then usingLemma 3.6andTheorem 3.36(c),we have

Thus,byTheorem 3.36(c),φis NACM.

Conversely,letφbe a NACM andB=(Bα×Aβ),whereBα’s andAβ’s are NOSs ofXandY,respectively,be a NOS ofX×Y.

and hence usingLemmas 3.3(i),3.6 and3.7andTheorems 3.36(c),we have

indentThus,byTheorem 3.36(c),ψis 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 ofN∽semi-open set,N∽semi-closed,N∽regularly open set,N∽regularly closed set,N∽continuous mapping,N∽open mapping,N∽closed mapping,N∽semicontinuous mapping,N∽semi-open mapping,N∽semi-closed mapping,set in order to propose the definition ofN∽almost continuous mapping.Some properties ofN∽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.