TopologicalStructuresviaInterval-Valued NeutrosophicCrispSets
DongsikJo 1 ,S.Saleh 2,Jeong-GonLee 3,* ,KulHur 3 andChenXueyou 4
1 DepartmentofDigitalContentsEngineering,WonkwangUniversity,Iksan54538,Korea; dongsik1005@wku.ac.kr
2 DepartmentofMathematics,FacultyofEducation-Zabid,HodeidahUniversity,HodeidahP.O.Box3114, Yemen;s_wosabi@hoduniv.net.ye
3 DivisionofAppliedMathematics,WonkwangUniversity,Iksan-Si54538,Korea;kulhur@wku.ac.kr
4 SchoolofMathematics,ShandongUniversityofTechnology,Zibo255049,China;xueyou-chen@163.com
* Correspondence:jukolee@wku.ac.kr
Received:13November2020;Accepted:7December2020;Published:10December2020
Abstract: Inthispaper,weintroducethenewnotionofinterval-valuedneutrosophiccrispsets providingatoolforapproximatingundefinableorcomplexconceptsinrealworld.First,wedeal withsomeofitsalgebraicstructures.Wealsodefineaninterval-valuedneutrosophiccrisp(vanishing) pointandobtainsomeofitsproperties.Next,wedefineaninterval-valuedneutrosophiccrisp topology,base(subbase),neighborhood,andinterior(closure),respectivelyandinvestigatesome ofeachproperty,andgivesomeexamples.Finally,wedefineaninterval-valuedneutrosophiccrisp continuityandquotienttopologyandstudysomeofeachproperty.
Keywords: interval-valuedneutrosophiccrispset;interval-valuedneutrosophiccrisp(vanishing) point;interval-valuedneutrosophiccrisptopologicalspace;interval-valuedneutrosophiccrispbase (subbase);interval-valuedneutrosophiccrispneighborhood;interval-valuedneutrosophiccrisp closure(interior);interval-valuedneutrosophiccrispcontinuity;interval-valuedneutrosophiccrisp quotienttopology
MSC: 54A10
1.Introduction
Numerousmathematicianshavebeentryingtofindamathematicalexpressionofthe complexationanduncertaintyinrealworldforalongtime.Forexample,Zadeh[1]definedafuzzyset asageneralizationofaclassicalsetin1965.Zadeh[2](1975),Pawlak[3](1982),Atanassov[4](1983), AtanassovandGargov[5](1989),GauandBuchrer[6](1993),Smarandache[7](1998),Molodtsov[8] (1999),Lee[9],Torra[10],Junetal.[11](2012),andLeeetal.[12](2020)introducedtheconceptof interval-valuedfuzzysets,roughsets,intuitionisticfuzzysets,interval-valuedintuitionisticfuzzysets, vaguesets,neutrosophicsets,softsets,bipolarfuzzysets,hessitantfuzzysets,cubicsetscombined byinterval-valuedfuzzysetsandfuzzysets,andoctahedronsetscombinedbyinterval-valued fuzzysets,intuitionisticfuzzysets,andfuzzysets,inturn,inordertosolvevariouscomplexand uncertainproblems.
In1996,cCoker[13]proposedtheconceptofanintuitionisticsetasthegeneralizationofaclassical setandthespecialcaseofanintuionisticfuzzysetandhestudiedtopologicalstructuresbasedon intuitionisticsetsin[14].Kimetal.[15]dealtwithcategoricalstructuresbasedonintuitionisticsets. Theyalsoobtainedfurtherpropertiesofintuionistictopologyin[16].In2014, Salamaetal.[17] defined neutrosophiccrispsetsasthegeneralizationofclassicalsetsandthespecialcaseofneutrosophic
Symmetry 2020, 12,2050;doi:10.3390/sym12122050 www.mdpi.com/journal/symmetry
setsproposedbySmarandache[7,18,19],andstudiedsomeofitsproperties.Moreover,theydealt withtopologicalstructuresbasedontheneutrosophiccrispsetsin[17]. Huretal.[20] investigated categoricalstructuresvianeutrosophiccrispsets.Manyresearchers[21–29]havediscussedtopological structuresvianeutrosophiccrispsets.Recently,Kimetal.[30]introducedtheconceptofan interval-valuedsetasthegeneralizationofaclassicalsetandthespecializationofaninterval-valued fuzzyset,andappliedittotopologicalstructures.
Thispaperconsiderstwoperspectives.First,wedefinetheinterval-valuedneutrosophic crispset,anewconceptthatcombinestheinterval-valuedsetandneutrosophiccrispset.Asan example,supposeacountryconductsapollduringanelectionthatdeterminesthehighesthead ofadministration.Atthistime,thepreferenceforCandidateAisdividedintothreegroups:Favor, neutral,andrejectionamongitscitizensfromtheviewpointofneutrosophiccrispset,buttheminimum andmaximumforeachofafavor,neutral,andrejectionfromtheviewpointofinterval-valued neutrosophiccrispset.Thegroupisconsidered.Then,itisbelievedthattheresultsofthepoll bythenewconceptaremoreaccuratethanthosebytheneutrosophiccrispset.Thus,thisnew conceptisneeded.Second,sincethetopologycanbeappliedtohighdimensionaldatasets,bigdata, andcomputationalevaluations(see[31–33],respectively),westudytopologicalstructuresbasedon interval-valuedneutrosophiccrispsets.Inordertoaccomplishsuchresearch,first,werecallsome definitionsrelatedtointuitionisticsets,interval-valuedsets,andneutrosophiccrispsets.Secondly, weintroducethenewconceptofinterval-valuedneutrosophiccrispsetandobtainsomeofits algebraicstructures,andgivesomeexamples.Wealsodefineinterval-valuedneutrosophiccrisp pointsoftwotypesanddiscussthecharacterizationsoftheinclusion,equality,intersection,andunion ofinterval-valuedneutrosophiccrispsets.Thirdly,wedefineaninterval-valuedneutrosophiccrisp topology,aninterval-valuedneutrosophiccrispbaseandsubbase,andstudysomeoftheirproperties. Fourthly,weintroducetheconceptsofinterval-valuedneutrosophiccrispneighborhoodsoftwo typesandfindsomeoftheirproperties.Inparticular,weprovethatthereisanIVNCTunderthe hypothesissatisfyingsomepropertiesofinterval-valuedneutrosophiccrispneighborhoods.Moreover, wedefineaninterval-valuedneutrosophiccrispinteriorandclosureanddealwithsomeoftheir properties.Inparticular,weshowthatthereisauniqueIVNCTforinterval-valuedneutrosophiccrisp interior[resp.closure]operators.Finally,weintroducetheconceptsofinterval-valuedneutrosophic crispcontinuous[resp.openandclosed]mappingsandquotienttopologiesandobtainsomeof theirproperties.
Throughoutthispaper,weassumethat X, Y arenon-emptysets,unlessotherwisestated.
2.Preliminaries
Inthissection,werecalltheconceptofanintuitionisticsetproposedin[13].Wealsorecallsome conceptsandresultsintroducedandstudiedin[30,34,35],respectively.
Definition1 ([13]). Theform A =(A∈ , A∈) suchthat A∈ , A∈ ⊂ X,and A∈ ∩ A∈ = ∅ iscalled anintuitionisticset(briefly,IS)of X,where A∈ [resp. A∈]representsthesetofmemberships[resp. non-memberships]ofelementsof X to A.Infact, A∈ [resp. A∈]isasubsetof X agreeingorapproving [resp.refusingoropposing]foracertainopinion,suggestion,orpolicy.
Theintuitionisticemptyset[resp.theintuitionisticwholeset]of X,denotedby ∅ [resp. X],isdefined by ∅ =(∅, X) [resp. X =(X, ∅)].ThesetofallISsof X willbedenotedby IS(X).Itisalsoclearthatfor each A ∈ IS(X), χA =(χA∈ , χA∈ ) isanintuitionisticfuzzysetin X proposedbyAtanassov[4].Thuswecan considertheintuitionisticsetAinXasanintuitionisticfuzzysetinX.
Furthermore,wecaneasilycheckthatforeach A ∈ IS(X), A∈ ∪ A∈ = X (infact, A∈c ∩ A∈c = ∅) ingeneral(seeExample 1)butif A∈ ∪ A∈ = X,then A∈c ∩ A∈c = ∅.Wedenotethefamily {A ∈ IS(X) : A∈ ∪ A∈ = X} asIS∗(X).
Example1. LetX = {a, b, c, d, e} beasetandconsidertheISAinXgivenby: A =({a, b, c}, {d})
Thenclearly,A∈ ∪ A∈ = X.Infact,A∈c ∩ A∈c = ∅.
Fortheinclusion,equality,union,andintersectionofintuitionisticsets,andthecomplementofan intuitionisticset,theoperations [] and <> on IS(X),referto[13].
Definition2 ([34,36]). Theform A = AT , AI , AF suchthat AT , AI , AF ⊂ X iscalledaneutrosophic crispset(briefly,NCS)in X,where AT , AI ,and AF representthesetofmemberships,indeterminacies, andnon-membershipsrespectivelyofelementsofXtoA.
Weconsiderneutrosophiccrispempty[resp.whole]setsoftwotypesin X,denotedby ∅1,N , ∅2,N [resp.X1,N , X2,N ]anddefinedby(seeRemark1.1.1in[34]): ∅1,N = ∅, ∅, X , ∅2,N = ∅, X, X [resp.X1,N = X, X, ∅ , X2,N = X, ∅, ∅ ].
WewilldenotethesetofallNCSsinXdenotedbyNC(X)
Itisobviousthat A = A, ∅, Ac ∈ NC(X) foreachordinarysubset A of X.Thenwecanconsider anNCSin X asthegeneralizationofanordinarysubsetof X.Itisalsoclearthat A = A∈ , ∅, A∈ is anNCSin X foreach A ∈ IS(X).ThusanNCSin X canbeconsideredasthegeneralizationofan intuitionisticsetin X.Furthermore,wecaneasilyseethatforeach A ∈ N(X), χA = χAT , χAI , χAF
isaneutrosophicsetin X introducedbySalamaandSmarandache[7,18,19].SoanNCSisaspecial caseofaneutrosophicset.
Definition3 ([34]). LetA ∈ NC(X).ThenthecomplementofA,denotedbyAi,c (i = 1,2)anddefinedby: A1,c = AF , AI c , AT , A2,c = AF , AI , AT
Definition4 ([34]) LetA, B ∈ NC(X).ThenAissaidtobe:
(i) A1-typesubsetofB,denotedbyA ⊂1 B,ifitsatisfiesthefollowingconditions: AT ⊂ BT , AI ⊂ BI , AF ⊃ BF , (ii) A2-typesubsetofB,denotedbyA ⊂2 B,ifitsatisfiesthefollowingconditions: AT ⊂ BT , AI ⊃ BI , AF ⊃ BF
Definition5 ([34]). LetA, B ∈ NC(X)
(i) Thei-intersectionofAandB,denotedbyA ∩i B(i = 1,2)anddefinedby: A ∩1 B = AT ∩ BT , AI ∩ BI , AF ∪ BF , A ∩2 B = AT ∩ BT , AI ∪ BI , AF ∪ BF .
(ii) Thei-unionofAandB,denotedbyA ∪i B(i = 1,2)anddefinedby: A ∪1 B = AT ∪ BT , AI ∪ BI , AF ∩ BF , A ∪2 B = AT ∪ BT , AI ∩ BI , AF ∩ BF
(iii) []A = AT , AI , AT c , A = AF c , AI , AF
ThefollowingsareimmediateresultsofDefinitions 3, 4,and 5
Proposition1 (SeeProposition3.3in[20]andalsocompareitwithProposition3.5in[15]) Let A, B, C ∈ NC(X) andleti = 1,2.Thenwehave:
(1) (SeeProposition1.1.1in[34]) ∅i,N ⊂i A ⊂i Xi,N , (2) IfA ⊂i BandB ⊂i C,thenA ⊂i C, (3) A ∩i B ⊂i AandA ∩i B ⊂i B, (4) A ⊂i A ∪i BandB ⊂i A ∪i B, (5) A ⊂i BifandonlyifA ∩i B = A, (6) A ⊂i BifandonlyifA ∪i B = B.
Proposition2 (SeeProposition3.4in[20]andalsocompareitwithProposition3.6in[15]). Let A, B, C ∈ NC(X) andleti = 1,2.Thenwehave:
(1) (Idempotentlaws): A ∪i A = A,A ∩i A = A, (2) (Commutativelaws): A ∪i B = B ∪i A,A ∩i B = B ∩i A, (3) (Associativelaws): A ∪i (B ∪i C)=(A ∪i B) ∪i C,A ∩i (B ∩i C)=(A ∩i B) ∩i C, (4) (Distributivelaws): A ∪i (B ∩i C)=(A ∪i B) ∩i (A ∪i C), A ∩i (B ∪i C)=(A ∩i B) ∪i (A ∩i C), (5) (Absorptionlaws): A ∪i (A ∩i B)= A,A ∩i (A ∪i B)= A, (6) (DeMorgan’slaws): (A ∪1 B)1,c = A1,c ∩1 B1,c , (A ∪1 B)2,c = A2,c ∩2 B2,c , (A ∪2 B)1,c = A1,c ∩2 B1,c , (A ∪2 B)2,c = A2,c ∩1 B2,c , (A ∩1 B)1,c = A1,c ∪1 B1,c , (A ∩1 B)2,c = A2,c ∪2 B2,c , (A ∩2 B)1,c = A1,c ∪2 B1,c , (A ∩2 B)2,c = A2,c ∪1 B2,c , (7) (Ai,c )i,c = A, (8) (8a) A ∪i ∅i,N = A,A ∩i ∅i,N = ∅i,N , (8b) A ∪i Xi,N = Xi,N ,A ∩i Xi,N = A, (8c) Xi,N i,c = ∅i,N , ∅i,N i,c = Xi,N , (8d) A ∪i Ai,c = Xi,N i,c,A ∩i Ai,c = ∅i,N , ingeneral.
Definition6 (See[34,37]). Let a ∈ X.Thentheform aN = {a}, ∅, {a}c [resp. aNV = ∅, {a}, {a}c ]is calledaneutrisophiccrisp[resp.vanishing]pointinX. Wedenotethesetofallneutrisophiccrisppointsandallneutrisophiccrispvanishingpointsin X by NP (X)
Definition7 (See[34,37]). Leta ∈ XandletA ∈ NC(X).Then, (i) aN saidtobelongtoA,denotedbyaN ∈ A,ifa ∈ AT , (ii) aNV saidtobelongtoA,denotedbyaNV ∈ A,ifa ∈ AF Result1 ([34],Proposition1.2.6). Let A ∈ NC(X).Then, A = AN ∪1 ANV , where AN = 1 aN ∈A aN , ANV = 1 aNV ∈A aNV .Infact, AN = AT , ∅, AT c and ANV = ∅, AI , AF
Definition8 ([30,35]). Theform [A , A+ ]= {B ⊂ X : A ⊂ B ⊂ A+ } suchthat A , A+ ⊂ X iscalled aninterval-valuedsets(briefly,IVS)in X,where A [resp. A+]representsthesetofminimum[resp.maximum] membershipsofelementsof X to A.Infact, A [resp. A+]isaminimum[resp.maximum]subsetof X agreeing orapprovingforacertainopinion,suggestion,orpolicy.
[∅, ∅] [resp. [X, X]]iscalledtheinterval-valuedempty[resp.whole]setin X anddenotedby ∅ [resp. X]. ThesetofallIVSsinXwillbedenotedbyIVS(X)
Foranyclassicalsubset A of X, [A, A] ∈ IVS(X) isobvious.ThenwecanconsideranIVSin X as thegeneralizationofaclassicalsubsetof X.Also,if A =[A , A+ ] ∈ IVS(X),then χA =[χA , χA+ ] is aninterval-valuedfuzzysetin X introducedbyZadeh[2].Thusaninterval-valuedfuzzysetcanbe consideredasthegeneralizationofanIVS.
Furthermore,wecaneasilycheckthatforeach A ∈ IVS(X), A = A+ (infact, A+ ∩ A c = ∅) ingeneral(seeExample 2)butif A = A+,then A+ ∩ A c = ∅.Wedenotethefamily {A ∈ IVS(X) : A = A+ } as IVS∗(X)
Example2. LetX = {a, b, c, d, e} andconsidertheIVSAinXgivenby: A =[{a, b}, {a,, b, c}].
ThenwecaneasilycalculatethatA = A+ andA+ ∩ A c = ∅
Fortheinclusion,equality,union,andintersectionofintuionisticsets,andthecomplementofan intuitionisticsetreferto[30,35].
3.Interval-ValuedNeutrosophicCrispSets
Inthissection,weintroducetheconceptofaninterval-valuedneutrosophiccrispsetcombinedby aneutrosophiccrispsetandaninterval-valuedset,andobtainsomeofitsproperties. Definition9. Theform [AT, , AT,+ ], [AI, , AI,+ ], [AF, , AF,+ ] iscalledaninterval-valuedneutrosophic crispset(briefly,IVNCS)inX,where [AT, , AT,+ ], [AI, , AI,+ ], [AF, , AF,+ ] ∈ IVS(X)
Inthiscase, [AT, , AT,+ ], [AI, , AI,+ ],and [AF, , AF,+ ] representtheIVSofmemberships, indeterminacies,andnon-membershipsrespectivelyofelementsofXtoA.
Inparticular,anIVNCSisdefinedasthreetypesbelow.
AnIVNCSA = [AT, , AT,+ ], [AI, , AI,+ ], [AF, , AF,+ ] inXissaidtobeof:
(i) Type1,ifitsatisfiesthefollowingconditions:
[AT, , AT,+ ] ∩ [AI, , AI,+ ]= ∅, [AT, , AT,+ ] ∩ [AF, , AF,+ ]= ∅, [AI, , AI,+ ] ∩ [AF, , AF,+ ]= ∅,
equivalently,AT,+ ∩ AI,+ = ∅, AT,+ ∩ AF,+ = ∅, AI,+ ∩ AF,+ = ∅, (ii) Type2,ifitsatisfiesthefollowingconditions:
[AT, , AT,+ ] ∩ [AI, , AI,+ ]= ∅, [AT, , AT,+ ] ∩ [AF, , AF,+ ]= ∅, [AI, , AI,+ ] ∩ [AF, , AF,+ ]= ∅, [AT, , AT,+ ] ∪ [AI, , AI,+ ] ∪ [AF, , AF,+ ]= X, equivalently,AT,+ ∩ AI,+ = ∅, AT,+ ∩ AF,+ = ∅, AI,+ ∩ AF,+ = ∅, AT, ∪ AI, ∪ AF, = X, (iii) Type3,ifitsatisfiesthefollowingconditions:
[AT, , AT,+ ] ∩ [AI, , AI,+ ] ∩ [AF, , AF,+ ]= ∅, [AT, , AT,+ ] ∪ [AI, , AI,+ ] ∪ [AF, , AF,+ ]= X, equivalently,AT,+ ∩ AI,+ ∩ AF,+ = ∅, AT, ∪ AI, AF, = X
ThesetofallIVNCSsofType1[resp.Type2andType3]in X isdenotedby IVN1(X) [resp. IVN2(X) and IVN3(X)],and IVNCS(X)= IVN1(X) ∪ IVN2(X) ∪ IVN3(X),where IVNCS(X) isthesetofall IVNCSsinX.
Foranyclassicalsubset A of X, [A, A], ∅, [Ac , Ac ] ∈ IVNCS(X) isclear.Thenwecanconsider anINCSin X canbeconsideredasthegeneralizationofaclassicalsubsetof X.Moreover,if A = [AT, , AT,+ ], [AI, , AI,+ ], [AF, , AF,+ ] ∈ IVNCS(X),then: χA =([χAT, , χAT,+ ], [χAI, , χAI,+ ], [χAF, , χAF,+ ])
isanintervalneutrosophicsetin X proposedbyYe[38].ThuswecanconsideranIVSasthe generalizationofanIVNCS.
Remark1.
(1) IVN2(X) ⊂ IVN1(X), IVN2(X) ⊂ IVN3(X), (2) IVN1(X) ⊂ IVN2(X), IVN1(X) ⊂ IVN3(X) ingeneral, (3) IVN3(X) ⊂ IVN1(X), IVN3(X) ⊂ IVN2(X) ingeneral.
Example3. LetX = {a, b, c, d, e, f , g, h, i}.ConsidertwoIVNCSsinXgivenby:
A = [{a, b, c}, {a, b, c, d}], [{e}, {e, f }], [{g, h}, {g, h, i}] , B = [{a, b, c}, {a, b, c}], [{a, e, f }, {a, e, f }], [{g, h, i}, {g, h, i}] (i) [AT, , AT,+ ] ∩ [AI, , AI,+ ]= ∅, [AT, , AT,+ ] ∩ [AF, , AF,+ ]= ∅,
[AI, , AI,+ ] ∩ [AF, , AF,+ ]= ∅ But
[AT, , AT,+ ] ∪ [AI, , AI,+ ] ∪ [AF, , AF,+ ]=[{a, b, c, d, e, f , g, h}, X}] = X. Then A ∈ IVN1(X) but A ∈ IVN2(X).Moreover,wehave:
[AT, , AT,+ ] ∩ [AI, , AI,+ ] ∩ [AF, , AF,+ ]= ∅
ThusA ∈ IVN3(X).SowecanconfirmthatRemark 1 (2)holds. (ii) [BT, , BT,+ ] ∩ [BI, , BI,+ ] ∩ [BF, , BF,+ ]= ∅, [BT, , BT,+ ] ∪ BCI, , BI,+ ] ∪ [BF, , BF,+ ]= X But [BT, , BT,+ ] ∩ [BI, , BI,+ ]=[{a}, {a}] = ∅
ThenB ∈ IVN3(X) butB ∈ IVN1(X), B ∈ IVN2(X).ThuswecanconfirmthatRemark 1 (3)holds.
Definition10. Wemaydefinetheinterval-valuedneutrosophiccrispemptysetsandtheinterval-valued neutrosophiccrispwholesets,denotedby ∅i,IVN andXi,IVN (i = 1,2,3,4),respectivelyasfollows: (i) ∅1,IVN = ∅, ∅, X , ∅2,IVN = ∅, X, X , ∅3,IVN = ∅, X, ∅ , ∅4,IVN = ∅, ∅, ∅ , (ii) X1,IVN = X, X, ∅ , X2,IVN = X, ∅, ∅ , X3,IVN = X, ∅, X , X4,IVN = X, X, X
Definition11. Let A ∈ IVNCS(X).Thenthecomplementsof A,denotedby Ai,c (i = 1,2,3),isanIVNCS inX,respectivelyasfollows:
A1,c = [AT, , AT,+ ]c , [AI, , AI,+ ]c , [AF, , AF,+ ]c ,
A2,c = [AF, , AF,+ ], [AI, , AI,+ ], [AT, , AT,+ ] ,
A3,c = [AF, , AF,+ ], [AI, , AI,+ ]c , [AT, , AT,+ ] .
Example4. Let A = [{a, b, c}, {a, b, c, d}], [{e}, {e, f }], [{g, h}, {g, h, i}] betheIVNCSin X givenin Example 3.Thenwecaneasilycheckthat:
A1,c =< [{e, f , g, h, i}, {d, e, f , g, h, i}], [{a, b, c, d, g, h, i}, {a, b, c, d, f , g, h, i}], [{a, b, c, d, e, f }, {a, b, c, d, e, f , i}] >,
A2,c = [{g, h}, {g, h, i}], [{e}, {e, f }], [{a, b, c}, {a, b, c, d}] ,
A3,c =< [{g, h}, {g, h, i}], [{a, b, c, d, g, h, i}, {a, b, c, d, f , g, h, i}], [{a, b, c}, {a, b, c, d}] >
Definition12. Let A, B ∈ IVNCS(X).Thenwemaydefinetheinclusionsbetween A and B,denotedby A ⊂i B(i = 1,2),asfollows:
A ⊂1 Biff [AT, , AT,+ ] ⊂ [BT, , BT,+ ], [AI, , AI,+ ] ⊂ [BI, , BI,+ ], [AF, , AF,+ ] ⊃ [BF, , BF,+ ], A ⊂2 Biff [AT, , AT,+ ] ⊂ [BT, , BT,+ ], [AI, , AI,+ ] ⊃ [BI, , BI,+ ], [AF, , AF,+ ] ⊃ [BF, , BF,+ ]
Proposition3. ForanyA ∈ IVNCS(X),thefollowingshold: (1) ∅1,IVN ⊂1 A ⊂1 X1,IVN , ∅2,IVN ⊂2 A ⊂2 X2,IVN , (2) ∅i,IVN ⊂j ∅i,IVN ,Xi,IVN ⊂j Xi,IVN , (i = 1,2,3,4, j = 1,2).
Proof. Straightforward. Definition13. LetA, B ∈ IVNCS(X), (Aj )j∈J ⊂ IVNCS(X)
(i) TheintersectionofAandB,denotedbyA ∩i B(i = 1,2),isanIVNCSinXdefinedby: A ∩1 B =< [AT, , AT,+ ] ∩ [BT, , BT,+ ], [AI, , AI,+ ] ∩ [BI, , BI,+ ], [AF, , AF,+ ] ∪ [BF, , BF,+ ] >, A ∩2 B =< [AT, , AT,+ ] ∩ [BT, , BT,+ ], [AI, , AI,+ ] ∪ [BI, , BI,+ ], [AF, , AF,+ ] ∪ [BF, , BF,+ ] >
(i ) Theintersectionof (Aj )j∈J ,denotedby i j∈J Aj (i = 1,2),isanIVNCSinXdefinedby: 1 j∈J Aj = j∈J [AT, j , AT,+ j ], j∈J [AI, j , AI,+ j ], j∈J [AF, j , AF,+ j ] , 2 j∈J Aj = j∈J [AT, j , AT,+ j ], j∈J [AI, j , AI,+ j ], j∈J [AF, j , AF,+ j ]
(ii) TheunionofAandB,denotedbyA ∪i B(i = 1,2),isanIVNCSinXdefinedby: A ∪1 B =< [AT, , AT,+ ] ∪ [BT, , BT,+ ], [AI, , AI,+ ] ∪ [BI, , BI,+ ], [AF, , AF,+ ] ∩ [BF, , BF,+ ] >, A ∪2 B =< [AT, , AT,+ ] ∪ [BT, , BT,+ ], [AI, , AI,+ ] ∩ [BI, , BI,+ ], [AF, , AF,+ ] ∩ [BF, , BF,+ ] > .
(ii ) Theunionof (Aj )j∈J ,denotedby i j∈J Aj (i = 1,2),isanIVNCSinXdefinedby: 1 j∈J Aj = j∈J [AT, j , AT,+ j ], j∈J [AI, j , AI,+ j ], j∈J [AF, j , AF,+ j ] , 2 j∈J Aj = j∈J [AT, j , AT,+ j ], j∈J [AI, j , AI,+ j ], j∈J [AF, j , AF,+ j ] .
(iii) []A = [AT, , AT,+ ], [AI, , AI,+ ], [AT, , AT,+ ]c . (iv) <> A = [AF, , AF,+ ]c , [AI, , AI,+ ], [AF, , AF,+ ] .
FromDefinitions 10–13,wegetsimilarresultsfromPropositions3.5and3.6in[30].
Proposition4. LetA, B, C ∈ IVNCS(X),i = 1,2.Then, (1) IfA ⊂i BandB ⊂i C,thenA ⊂i C, (2) A ⊂i A ∪i BandB ⊂i A ∪i B, (3) A ∩i B ⊂i AandA ∩i B ⊂i B, (4) A ⊂i BifandonlyifA ∩i B = A, (5) A ⊂i BifandonlyifA ∪i B = B.
Proposition5. LetXA, B, C ∈ IVNCS(X), (Aj )j∈J ⊂ IVNCS(X),andleti = 1,2;k = 1,2,3.Then
(1) (Idempotentlaws)A ∪i A = A, A ∩i A = A, (2) (Commutativelaws)A ∪i B = B ∪i A, A ∩i B = B ∩i A, (3) (Associativelaws)A ∪i (B ∪i C)=(A ∪i B) ∪i C, A ∩i (B ∩i C)=(A ∩i B) ∩i C, (4) (Distributivelaws)A ∪i (B ∩i C)=(A ∪i B) ∩i (A ∪i C), A ∩i (B ∪i C)=(A ∩i B) ∪i (A ∩i C), (4 ) (Generalizeddistributivelaws) ( i j∈J Aj ) ∪i A = i j∈J (Aj ∪i A), ( i j∈J Aj ) ∩i A = i j∈J (Aj ∩i A), (5) (Absorptionlaws)A ∪i (A ∩i B)= A, A ∩i (A ∪i B)= A, (6) (DeMorgan’slaws) (A ∪i B)k,c = Ak,c ∩i Bk,c , (A ∩i B)k,c = Ak,c ∪i Bk,c , (6 ) (GeneralizedDeMorgan’slaws) ( i j∈J Aj )k,c = i j∈J Ak,c j , (7) (Ak,c )k,c = A, (8) (8a) A ∪i ∅i,IVN = A, A ∩i ∅i,IVN = ∅i,IVN , (8b) A ∪i Xi,IVN = Xi,IVN , A ∩i Xi,IVN = A, (8c) X1,IVN 1,c = ∅1,IVN , X1,IVN 2,c = ∅2,IVN , X1,IVN 3,c = ∅1,IVN , X2,IVN 1,c = ∅2,IVN , X2,IVN 2,c = ∅1,IVN , X2,IVN 3,c = ∅2,IVN , X3,IVN 1,c = ∅3,IVN , X3,IVN 2,c = X3,IVN , X3,IVN 3,c = X4,IVN , X4,IVN 1,c = ∅4,IVN , X4,IVN 2,c = X4,IVN , X4,IVN 3,c = X3,IVN , ∅1,IVN 1,c = X1,IVN , ∅1,IVN 2,c = X2,IVN , ∅1,IVN 3,c = X1,IVN , ∅2,IVN 1,c = X2,IVN , ∅2,IVN 2,c = X1,IVN , ∅2,IVN 3,c = X2,IVN , ∅3,IVN 1,c = X3,IVN , ∅3,IVN 2,c = ∅3,IVN , ∅3,IVN 3,c = ∅4,IVN , ∅4,IVN 1,c = X4,IVN , ∅4,IVN 2,c = ∅4,IVN , ∅4,IVN 3,c = ∅3,IVN , (8d) A ∪i Ak,c = Xj,IVN , A ∩i Ak,c = ∅j,IVN ingeneral (seeExample 5), wherej = 1,2,3,4
Example5. ConsidertheIVNCSAinXgiveninExample 4.Then, A ∩1 A1,c = [{a, b, c}, {a, b, c, d}], [{e}, {e, f }], [{g, h}, {g, h, i}] ∩1 < [{e, f , g, h, i}, {d, e, f , g, h, i}], [{a, b, c, d, g, h, i}, {a, b, c, d, f , g, h, i}], [{a, b, c, d, e, f }, {a, b, c, d, e, f , i}] > = [∅, {d}], [∅, { f }], [{a, b, c, d, e, f , g, h}, X}] = ∅j,IVN
Similarly,wecancheckthat: A ∪1 A1,c = Xj,IVN , A ∩1 A2,c = ∅j,IVN , A ∪1 A2,c = Xj,IVN
Additionally,wecaneasilychecktheremainders.
Aneighborhoodsystemofapointisveryimportantinaclassicaltopology.Thenweproposean interval-valuedneutrosophiccrisppointtodefinetheconceptofaninterval-valuedneutrosophiccrisp neighborhood.Moreover,whenwedealwithseparationaxiomsinaninterval-valuedneutrosophic crisptopology,thenotionofinterval-valuedneutrosophiccrisppointsisused.Thenwedefineitbelow.
Definition14. Let a ∈ X, A ∈ IVNCS(X).Thentheform [{a}, {a}], ∅, [{a}c , {a}c ] [resp. ∅, [{a}, {a}], [{a}c , {a}c ] ]iscalledaninterval-valuedneutrosophic[resp.vanishing]pointin X and denotedby aIVN [resp. aIVNV ].Wewilldenotethesetofallinterval-valuedneutrosophicpointsin X as IVNP (X)
(i) WesaythataIVN belongstoA,denotedbyaIVN ∈ A,ifa ∈ AT,+ . (ii) WesaythataIVNV belongstoA,denotedbyaIVNV ∈ A,ifa ∈ AF,+
Proposition6. LetA ∈ IVNCS(X).ThenA = AIVN ∪1 AIVNV , whereAIVN = 1 aIVN ∈A aIVN , AIVNV = 1 aIVNV ∈A aIVNV . Infact,
AIVN = [AT, , AT,+ ], ∅, [AT, , AT,+ ]c and
AIVNV = ∅, [AI, , AI,+ ], [AF, , AF,+ ]
Proof. AIVN = 1 aIVN ∈A aIVN = 1 aIVN ∈A [{a}, {a}], ∅, [{a}c , {a}c ]
= aIVN ∈A [{a}, {a}], aIVN ∈A ∅, aIVN ∈A [{a}c , {a}c ]
= [ a∈AT, {a}, a∈AT,+ {a}], ∅, [ a∈AT,+ {a}c , a∈AT, {a}c ]
= [AT, , AT,+ ], ∅, [AT,+ c , AT, c ]
= [AT, , AT,+ ], ∅, [AT, , AT,+ ]c ,
AIVNV = 1 aIVNN ∈A aIVNV = 1 aIVNV ∈A ∅, [{a}, {a}], [{a}c , {a}c ] = aIVNV ∈A ∅, aIVNV ∈A [{a}, {a}], aIVNV ∈A [{a}c , {a}c ]
= ∅, [ a∈AI, {a}, a∈AI,+ {a}, [ a∈AF,+ {a}c , a∈AF, {a}c ]
= ∅, [AI, , AI,+ ], [AF, , AF,+ ] .
Thenwehave,
AIVN ∪1 AIVNV = [AT, , AT,+ ], ∅, [AT, , AT,+ ]c ∪1 ∅, [AI, , AI,+ ], [AF, , AF,+ ] = [AT, , AT,+ ] ∪ ∅, ∅ ∪ [AI, , AI,+ ], [AT, , AT,+ ]c ∩ [AF, , AF,+ ] = [AT, , AT,+ ], [AI, , AI,+ ], [AT,+ c ∩ AF, , AT, c ∩ AF,+ = [AT, , AT,+ ], [AI, , AI,+ ], [AF, , AF,+ = A.
Thiscompletestheproof. Example6. LetX = {a, b, c, d, e, f , g, h, i} andconsidertheIVNCSinXgivenby: A = [{a, b}, {a, b, c}], [{d}, {d, e}], [{ f , g}, { f , g, h}]
Thenclearly,wehave: AIVN = 1 aIVI ∈A [{a}, {a}], ∅, [{a}c , {a}c ] = [{a, b}, {a, b, c}], ∅, [{a}c ∩{b}c ∩{c}c , {a}c ∩{b}c ]
= [{a, b}, {a, b, c}], ∅, [{d, e, f , g, h, i}, {c, d, e, f , g, h, i}] = [AT, , AT,+ ], ∅, [AT, , AT,+ ]c , AIVNV = 1 aIVNV ∈A ∅, [{a}, {a}], [{a}c , {a}c ] =< ∅, [{d}, {d, e}], [{a}c ∩{b}c ∩{c}c ∩{d}c ∩{e}c ∩{h}c ∩{i}c , {a}c ∩{b}c ∩{c}c ∩{d}c ∩{e}c ∩{i}c ] > = ∅, [{d}, {d, e}], [{ f , g}, { f , g, h}] = ∅, [AI, , AI,+ ], [AF, , AF,+ ] .
Thus AIVN ∪1 AIVNV = [{a, b}, {a, b, c}], [{d}, {d, e}], [{ f , g}, { f , g, h}] = A Sowecanconfirmthat Proposition 6 holds.
Proposition7. Let (Aj )j∈J ⊂ IVNCS(X) andleta ∈ X. (1)aIVN ∈ 1 j∈J Aj [resp.aIVNV ∈ 1 j∈J Aj] ⇔ aIVN ∈ Aj [resp.aIVNV ∈ Aj]foreachj ∈ J. (2)aIVN ∈ 1 j∈J Aj [resp.aIVNV ∈ 1 j∈J Aj] ⇔ thereexistsj ∈ JsuchthataIVN ∈ Aj [resp.aIVNV ∈ Aj.
Proof. (1)Suppose aIVN ∈ 1 j∈J Aj andlet A = 1 j∈J Aj.Since AT,+ = j∈J AT,+ j , a ∈ j∈J AT,+ j Then a ∈ AT,+ j foreach j ∈ J.Thus aIVN ∈ Aj foreach j ∈ J.Theconverseisprovedsimilarly. Theproofofthesecondpartisomitted. (2)Suppose aIVNV ∈ 1 j∈J Aj andlet A = 1 j∈J Aj.Since AF,+ = j∈J AT,+ j , a ∈ j∈J AT,+ j . Then a ∈ AT,+ j forsome j ∈ J.Thus aIVNV ∈ Aj forsome j ∈ J.Theconverseisshownsimilarly. Theproofofthefirstpartisomitted.
Proposition8. LetA, B ∈ IVNCS(X).Then,
(1) A ⊂1 BifandonlyifaIVN ∈ A ⇒ aIVN ∈ B[resp.aIVNV ∈ A ⇒ aIVNV ∈ B]foreacha ∈ X. (2) A = BifandonlyifaIVN ∈ A ⇔ aIVN ∈ B[resp.aIVNV ∈ A ⇔ aIVNV ∈ B]foreacha ∈ X.
Proof. Straightforward.
Whenwediscusswithcontinuitiesinaclassicaltopology,theconceptsofthepreimageandimage ofaclassicalsubsetunderamappingareused.ThenwedefineonesofanIVNCSunderamapping asfollows.
Definition15. Letf : X → Ybeamapping,A ∈ IVNCS(X), B ∈ IVNCS(Y)
(i) TheimageofAunderf,denotedbyf (A),isanIVNCSinYdefinedas: f (A)= [ f (AT, ), f (AT,+ )], [ f (AI, ), f (AI,+ )], [ f (AF, ), f (AF,+ )]
(ii) ThepreimageofBunderf,denotedbyf 1(B),isanintervalsetinXdefinedas: f 1(B)= [ f 1(BT, ), f 1(BT,+ )], [ f 1(BI, ), f 1(BI,+ )], [ f 1(BF, ), f 1(BF,+ )] .
Itisclearthatf (aIVN )= f (a)IVN andf (aIVNV )= f (a)IVNV foreacha ∈ X.
Fromtheabovedefinition,wehavesimilarresultsoftheimageandthepreimageofclassical subsetsunderamapping.
Proposition9. Let f : X → Y beamapping, A, A1, A2 ∈ IVNCS(X), (Aj )j∈J ⊂ IVNCS(X) andlet B, B1, B2 ∈ IVNCS(Y), (Aj )j∈J ⊂ IVNCS(Y).Leti = 1,2;k = 1,2,3;l = 1,2,3,4.Then,
(1) IfA1 ⊂i A2,thenf (A1) ⊂i f (A2), (2) IfB1 ⊂i B2,thenf 1(B1) ⊂i f 1(B1), (3) A ⊂i f 1( f (A)) andiffisinjective,thenA = f 1( f (A)), (4) f ( f 1(B)) ⊂i Bandiffissurjective,f ( f 1(B))= B, (5) f 1( i j∈J Bj )= i j∈J f 1(Bj ), (6) f 1( i j∈J Bj )= i j∈J f 1(Bj ), (7) f ( i j∈J Aj )i ⊂i i j∈J f (Aj ) andiffissurjective,thenf ( i j∈J Aj )i = i j∈J f (Aj ), (8) f ( i j∈J Aj ) ⊂i i j∈J f (Aj ) andiffisinjective,thenf ( i j∈J Aj )= i j∈J f (Aj ), (9) Iffissurjective,thenf (A)k,c ⊂i f (Ak,c ), (10) f 1(Bk,c )= f 1(B)k,c , (11) f 1(∅l,IVN )= ∅l,IVN ,f 1(Xl,IVN )= Xl,IVN , (12) f (∅l,IVN )= ∅l,IVN andiffissurjective,thenf (Xl,IVN )= Xl,IVN , (13) Ifg : Y → Zisamapping,then (g ◦ f ) 1(C)= f 1(g 1(C)),foreachC ∈ [Z].
Proof. Theproofsarestraightforward.
4.Interval-ValuedTopologicalSpaces
Inthissection,wedefineaninterval-valuedneutrosophiccrisptopologyon X andstudysome ofitsproperties,andgivesomeexamples.Wealsointroducetheconceptsofaninterval-valued neutrosophiccrispbaseandsubbase,andafamilyofIVNCSsgetsthenecessaryandsufficient conditionstobecomeIVNCBandgivessomeexamples.
Fromthissectiontotherestsections, ⊂1, ∪1 , ∩1 , 3,c , ∅1,IVN ,and X1,IVN aredenotedby ⊂, ∩, ∪, c , ∅IVN ,and XIVN ,respectively.
Definition16. Let ∅ = τ ⊂ IVNCS(X).Then τ iscalledaninterval-valuedneutrosophiccrisptopology (briefly,IVNCT)onX,ifitsatisfiesthefollowingaxioms:
(IVNCO1) ∅IVN , XIVN ∈ τ, (IVNCO2) A ∩ B ∈ τ foranyA, B ∈ τ, (IVNCO3) j∈J Aj ∈ τ foranyfamily (Aj )j∈J ofmembersof τ
Inthiscase,thepair (X, τ) iscalledaninterval-valuedneutrosophiccrisptopologicalspace(briefly, IVNCTS)andeachmemberof τ iscalledaninterval-valuedneutrosophiccrispopenset(briefly,IVNCOS)in X. AnIVNCSAiscalledaninterval-valuedneutrosophiccrispclosedset(briefly,IVNCCS)inX,ifAc ∈ τ
Itisobviousthat {∅IVN , XIVN } [resp. IVNC(X)]isanIVNCTon X,andcalledtheinterval-valued neutrosophiccrispindiscretetopology(briefly,IVNCIT)[resp.theinterval-valuedneutrosophiccrispdiscrete topology(briefly,IVNCDT)]on X.Thepair (X, τIVN,0 ) [resp. (X, τIVN,1 )]iscalledaninterval-valued neutrosophiccrispindiscrete[resp.discrete]space(briefly,IVNCITS)[resp.(briefly,IVNCDTS)].
IVNCT(X) representsthesetofallIVNCTson X.ForanIVNCTS X,thesetofallIVNCOs[resp. IVNCCSs]inXisdenotedbyIVNCO(X) [resp.IVNCC(X)].
Remark2. (1)Foreach τ ∈ IVNCT(X),considerthreefamiliesofIVSsinX: τT = {[AT, , AT,+ ] ∈ IVS(X) : A ∈ τ}, τ I = {[AI, , AI,+ ] ∈ IVS(X) : A ∈ τ}, τ F = {[AF,+ c , AF, c ] ∈ IVS(X) : A ∈ τ}
Thenwecaneasilycheckthat τT , τ I and τ F areIVTsonX.
Inthiscase, τT [resp. τ I and τ F]iscalledthemembership[resp.indeterminacyandnon-membership] topologyof τ andwewrite τ = τT , τ I , τ F .Infact,wecanconsider (X, τT , τ I , τ F ) asaninterval-valued tri-topologicalspaceonX(seetheconceptofbitopologyintroducedbyKelly[39]).
Furthermore,wecanconsiderthreeintuitionistictopologyonXproposedbycCoker[14]:
τT = {(AT, , AT,+ c ) ∈ IS(X) : A ∈ τ}, τI = {(AI, , AI,+ c ] ∈ IS(X) : A ∈ τ},
τF = {AF,+ c , AF, ) ∈ IS(X) : A ∈ τ}.
LetusalsoconsidersixfamiliesofordinarysubsetsofX:
τT, = {AT, ⊂ X : A ∈ τ}, τT,+ = {AT,+ ⊂ X : A ∈ τ},
τ I, = {AI, ⊂ X : A ∈ τ}, τ I,+ = {AI,+ ⊂ X : A ∈ τ},
τ F, = {AT,+ c ⊂ X : A ∈ τ}, τ F,+ = {AI, c ⊂ X : A ∈ τ}.
Thenclearly, τT, , τT,+ , τ I,+ , τ I, , τ F, , τ F,+ areordinarytopologiesonX. (2)Let (X, τo ) beanordinarytopologicalspace.ThentherearefourIVNCTsonXgivenby:
τ1 = { [G, G], ∅, [Gc , Gc ] ∈ IVNC(X) : G ∈ τo } ifG = X {∅IVN , XIVN } ifG = X,
τ2 = { [G, G], X, [Gc , Gc ] ∈ IVNC(X) : G ∈ τo } ifG = X {∅IVN , XIVN } ifG = X,
τ3 = { [∅, G], ∅, [∅, Gc ] ∈ IVNC(X) : G ∈ τo } ifG = ∅ {∅IVN , XIVN } ifG = ∅,
τ4 = { [∅, G], X, [∅, Gc ] ∈ IVNC(X) : G ∈ τo } ifG = ∅ {∅IVN , XIVN } ifG = ∅
(3)Let (X, τIV ) beanIVTSintroducedbyKimetal.[30].Thenclearly, τ = { [A , A+ ], ∅, [A+ c , A c ] ∈ IVNC(X) : A ∈ τIV }∈ IVNCT(X)
(4)Let (X, τI ) beanITSintroducedbycCoker[14].Thenclearly, τ = { [A∈ , A∈c ], ∅, [A∈ , A∈c ] ∈ IVNC(X) : A ∈ τI }∈ IVNCT(X)
(5)Let (X, τNC ) beaneutrosophiccrisptopologicalspaceintroducedbySalamaandSmarandache[34]. Thenclearly, τ = { [AT , AT ], [AI , AI ], [AF , AF ] ∈ IVN∗(X)) : A ∈ τNC }∈ IVNCT(X)
FromRemark 2,wecaneasilyseethatanIVNCTisageneralizationofaclassicaltopology,an IVT,anIT,andneutrosophiccrisptopology.ThenwehavethefollowingFigure 1: Example7. (1)LetX = {a, b}.Thenwecaneasilycheckthat: τ
,1 = {∅
, a
, b
, aIVNV , bIVNV , ∅, ∅, [{b}, {b}] , [{a}, {a}], [{a}, {a}], [{b}, {b}] , XIVN } (2)Let A ∈ IVNCS(X).Then A issaidtobefinite,if AT,+ , AI,+,and AF,+ arefinite.Considerthefamily τ = {U ∈ IVNCS(X) : U = ∅IVN or Uc isfinite}.
Thenwecaneasilyprovethat τ ∈ IVNCT(X) Inthiscase, τ iscalledaninterval-valuedneutrosiophiccrispcofinitetopology(briefly,IVNCCFT)onX.
(3)Let A ∈ IVNCS(X).Then A issaidtobecountable,if AT,+ , AI,+,and AF,+ arecountable. Considerthefamily:
τ = {U ∈ IVNCS(X) : U = ∅IVN or Uc iscountable}.
Thenwecaneasilyshowthat τ ∈ IVNCT(X)
Interval-valuedneutrosophiccrisptopology Neutrosophiccrisptopology
Interval-valued topology Intuitionistic topology Classical topology
Figure1. Therelationshipsamongfivetopologies.
Inthiscase, τ iscalledaninterval-valuedneutrosiophiccrispcocountabletopology(briefly,IVNCCCT) onX.
(4)LetX = {a, b, c, d, e, f , g, h, i} andthefamily τ ofIVNCSsonXgivenby:
τ = {∅IVN , A1, A2, A3, A4, XIVN },
whereA1 = [{a, b}, {a, b, c}], [{e}, {e, f }], [{g}, {g, i}] ,
A2 = [{a, d}, {a, c, d}], [{e}, {e}], [{g, h}, {g, h, i}] , A3 = [{a}, {a, c}], [{e}, {e}], [{g, h}, {g, h, i}] , A4 = [{a, b, d}, {a, b, c, d}], [{e}, {e, f }], [{g}, {g, i}]
Thenwecaneasilycheckthat τ ∈ IVNCT(X) (5)LetX = {0,1} Considerthefamily τ ofIVNCSsonXgivenby: τ = {∅IVN , [{0}, {0}], ∅, [{1}, {1}] , XIVN }
Thenwecaneasilyprovethat τ ∈ IVNCT(X).Inthiscase, (X, τ) iscalledtheinterval-valuedneutrosophic crispSierpin skispace.
FromDefinition 16,wehavethefollowing.
Proposition10. LetXbeanIVNCTS.Then:
Todiscuss IVNCT(X) withaview-pointoflatticetheory,wedefineanorderbetweentwoIVCTs. Definition17. Let τ1, τ2 ∈ IVNCT(X).Thenwesaythat τ1 iscontainedin τ2 or τ1 iscoarserthan τ2 or τ2 isfinerthan τ1,if τ1 ⊂ τ2, i.e.,A ∈ τ2 foreachA ∈ τ1
Foreach τ ∈ IVNCT(X), τIVN,0 ⊂ τ ⊂ τIVN,1 isclear.
FromDefinitions 14 and 16,wegetthefollowing.
Proposition11. Let (τj )j∈J ⊂ IVNCT(X).Then j∈J τj ∈ IVNCT(X) Infact, j∈J τj isthecoarsestIVNCTonXcontainingeach τj
Proposition12. Let τ, γ ∈ IVNCT(X).Wedefine τ ∧ γ and τ ∨ γ asfollows: τ ∧ γ = {W : W ∈ τ, W ∈ γ}, τ ∨ γ = {W : W = U ∪ V, U ∈ τ, V ∈ γ}
Thenwehave: (1) τ ∧ γ isanIVNCTonXwhichisthefinestIVNCTcoarserthanboth τ and γ, (2) τ ∨ γ isanIVNCTonXwhichisthecoarsestIVNCTfinerthanboth τ and γ, Proof. (1)Clearly, τ ∧ γ ∈ IVNCT(X) Let η beanyIVNCTon X whichiscoarserthanboth τ and γ, andlet W ∈ η.Then W ∈ τ and W ∈ γ.Thus W ∈ τ ∧ γ.So η iscoarserthan τ ∧ γ (2)Theproofissimilarto(1).
FromDefinition 17,Propositions 11 and 12,wecaneasilyseethat (IVNCT(X), ⊂) formsa completelatticewiththeleastelement τIVN,0 andthegreatestelement τIVN,1 .
Atopologyonasetcanbeacomplicatedcollectionofsubsetsofsubsetsofaset,anditcanbe difficulttodescribetheentirecollection.Inmostcases,onedescribesasubcollection(calledabaseand asubbase)that“generates”thetopology.ThenwedefineabaseandasubbaseinanIVNCT.Moreover, weintroducethevariousintervalsviaIVNCSsinrealline R.
Definition18. Let (X, τ) beanIVNCTS.
(i)Asubfamily β of τ iscalledaninterval-valuedneutrosophiccrispbase(briefly,IVNCB)for τ,ifforeach A ∈ τ,A = ∅IVN orthereis β ⊂ β suchthatA = β
(ii)Asubfamily σ of τ iscalledaninterval-valuedneutrosophiccrispsubbase(briefly,IVNCSB)for τ,if thefamily β = { σ : σ isafinitesubsetof σ} isanIVNCBfor τ
Remark3. (1)Let β beanIVNCBforanIVNCT τ onanon-emptyset X andconsiderthreefamiliesofIVSs inX: βT = {[AT, , AT,+ ] ∈ IVS(X) : A ∈ β}, βI = {[AI, , AI, ] ∈ IVS(X) : A ∈ β}, βF = {[AF,+ c , AF, c ] ∈ IVS(X) : A ∈ β}.
Thenwecaneasilyseethat βT , βI ,and βF areaninterval-valuedbase(see[30])for τT , τ I ,and τ F,respectively. Furthermore,wecanconsiderthreeintuitionisticbaseonXdefinedbycCoker[14]: βT = {(AT, , AT,+ c ) ∈ IS(X) : A ∈ β}, β I = {(AI, , AI,+ c ] ∈ IS(X) : A ∈ β}, βF = {AF,+ c , AF, ) ∈ IS(X) : A ∈ β}.
LetalsousconsidersixfamiliesofordinarysubsetsofX: βT, = {AT, ⊂ X : A ∈ β}, βT,+ = {AT,+ ⊂ X : A ∈ β}, βI, = {AI, ⊂ X : A ∈ β}, βI,+ = {AI,+ ⊂ X : A ∈ β}, βF, = {AT,+ c ⊂ X : A ∈ β}, βF,+ = {AI, c ⊂ X : A ∈ β}
Thenclearly, βT, , βT,+ , βI,+ , βI, , βF, , βF,+ areordinarybasesforordinarytopologies τT, , τT,+ , τ I,+ , τ I, , τ F, , τ F,+ onX,respectively.
(2)Let σ beanIVNCSBforanIVNCT τ onanon-emptysetXandconsiderthreefamiliesofIVSsinX:
σT = {[AT, , AT,+ ] ∈ IVS(X) : A ∈ σ}, σ I = {[AI,+ , AI, ] ∈ IVS(X) : A ∈ σ},
σF = {[AF,+ c , AF, c ] ∈ IVS(X) : A ∈ σ}.
Thenwecaneasilyseethat σT , σ I ,and σF areaninterval-valuedsubbases(see[30])for τT , τ I ,and τ F , respectively.
Furthermore,wecanconsiderthreeintuitionisticbaseonXdefinedbycCoker[14]:
σT = {(AT, , AT,+ c ) ∈ IS(X) : A ∈ σ}, σI = {(AI, , AI,+ c ] ∈ IS(X) : A ∈ σ}, σF = {AF,+ c , AF, ) ∈ IS(X) : A ∈ σ}.
LetalsousconsidersixfamiliesofordinarysubsetsofX:
σT, = {AT, ⊂ X : A ∈ σ}, σT,+ = {AT,+ ⊂ X : A ∈ σ}, σ I, = {AI, ⊂ X : A ∈ σ}, σ I,+ = {AI,+ ⊂ X : A ∈ σ}, σF, = {AT,+ c ⊂ X : A ∈ σ}, σF,+ = {AF, c ⊂ X : A ∈ σ}.
Thenclearly, σT, , σT,+ , σ I,+ , σ I, , σF, , σF,+ areordinarysubbasesforordinarytopologies τT, , τT,+ , τ I,+ , τ I, , τ F, , τ F,+ onX,respectively.
Example8. (1)Let σ = { [(a, b), (a, ∞)], [∅, ∅], [∅, ( ∞, a]] : a, b ∈ R} bethefamilyofIVNCsin R Then σ generatesanIVNCT τ on R whichiscalledthe“usualleftinterval-valuedneutrosophiccrisptopology (briefly,ULIVNCT)”on R.Infact,theIVNCB β for τ canbewrittenintheform: β = {RIVN }∪{∩γ∈Γ Sγ : Sγ ∈ σ, Γ isfinite} and τ consistsofthefollowingIVNCSsin R: τ = {∅IVN , RIVN }∪{ [∪(aj, bj ), (c, ∞)], ∅, ∅ } or τ = {∅IVN , RIVN }∪{ [∪(ak, bk ), R], ∅, ∅ }, where aj, bj, c ∈ R, {aj : j ∈ J} isboundedfrombelow, c < inf {aj : j ∈ J} and ak, bk ∈ R, {ak : k ∈ K} is notboundedfrombelow.
Similarly,onecandefinethe“usualrightinterval-valuedneutrosophiccrisptopology(briefly,URIVNCT)” on R usingananalogueconstruction.
(2)Considerthefamily σ ofIVNCSsin R: σ = { [(a, b), (a1, ∞) ∩ ( ∞, b1)], ∅, [∅, ( ∞, a1] ∪ [b1, ∞] : a, b, a1, b1 ∈ R, a1 ≤ a, b1 ≥ b} Then σ generatesanIVNCT τ on R whichiscalledthe“usualinterval-valuedneutrosophiccrisptopology (briefly,UIVNCT)”on R.Infact,theIVNCB β for τ canbewrittenintheform: β = {RIVN }∪{∩γ∈Γ Sγ : Sγ ∈ σ, Γ isfinite} andtheelementsof τ canbeeasilywrittendownasin(1).
(3)Considerthefamily σ[0,1] ofIVNCSsin R:
σ[0,1] = { [[a, b], [a, ∞) ∩ ( ∞, b]], ∅, [∅, ( ∞, a] ∪ [b, ∞] : a, b ∈ R and0 ≤ a ≤ b ≤ 1}. Then σ[0,1] generatesanIVNCT τ[0,1] on R whichiscalledthe“usualunitclosedintervalinterval-valued neutrosophiccrisptopology”on R.Infact,theIVNCB β[0,1] for τ[0,1] canbewrittenintheform: β[0,1] = {RIVN }∪{∩γ∈Γ Sγ : Sγ ∈ σ[0,1], Γ isfinite} andtheelementsof τ canbeeasilywrittendownasin(1).
Inthiscase, ([0,1], τ[0,1] ) iscalledthe“interval-valuedneutrosophiccrispnusualunitclosedinterval”and denotedby [0,1]IVNCI .Infact, [0,1]IVNCI = [[0,1], [0, ∞) ∪ ( ∞,1]], ∅, ∅
(4)Let β = {aIVN : a ∈ X}∪{aIVNV : a ∈ X}.Then β isanIVNCBfortheinterval-valuedneutrosophic crispdiscretetopology τ1 onX.
(5)LetX = {a, b, c, d, e, f , g, h, i} andconsiderthefamily β ofIVNCSsinXgivenby: β = {A, B, XIVN }, whereA = [{a, b}, {a, b, c}], [{e}, {e, f }], [{g}, {g, i}] , B = [{a, d}, {a, c, d}], [{e}, {e}], [{g, h}, {g, h, i}] . Assumethat β isanIVNCBforanIVNCT τ on X.Thenbythedefinitionofbase, β ⊂ τ.Thus A, B ∈ τ.So A ∩ B = [{a}, {a, c}], [{e}, {e}], [{g, h}, {g, h, i}] ∈ τ.Howeverforany β ⊂ β, A ∩ B = β .Hence β isnotanIVNCBforanIVNCTonX.
From(1),(2),and(3)inExample 8,wecandefineinterval-valuedneutrosophiccrispintervals asfollowing.
Definition19. Leta, b ∈ R suchthata ≤ b.Then:
(i) (Theclosedinterval) [a, b]IVNCI = [[a, b], [a, ∞) ∩ ( ∞, b]], ∅, ∅ , (ii) (Theopeninterval) (a, b)IVNCI = [(a, b), (a, ∞) ∩ ( ∞, b)], ∅, ∅ , (iii) (Thehalfopenintervalorthehalfclosedinterval)
(a, b]IVNCI = [(a, b], (a, ∞) ∩ ( ∞, b]], ∅, ∅ , [a, b)IVI = [[a, b), [a, ∞) ∩ ( ∞, b)], ∅, ∅ ,
(iv) (Thehalfinterval-valuedrealline)
( ∞, a]IVNCI = [( ∞, a], ( ∞, a]], ∅, ∅ ,
( ∞, a)IVNCI = [( ∞, a), ( ∞, a)], ∅, ∅ ,
[a, ∞)IVNCI = [[a, ∞), [a, ∞)], ∅, ∅ ,
(a, ∞)IVNCI = [(a, ∞), (a, ∞)], ∅, ∅ ,
(v) (Theinterval-valuedrealline)
( ∞, ∞)IVMCI = [( ∞, ∞), ( ∞, ∞)], ∅, ∅ = RIVN
ThefollowingprovideanecessaryandsufficientconditionwhichacollectionofIVNCSsinaset X isanIVNCBforsomeIVNCTon X.
Theorem1. Let β ⊂ IVNCS(X).Then β isanIVNCBforanIVNCT τ on X ifandonlyifitsatisfiesthe followingproperties:
(1) XIVN = β, (2) If B1, B2 ∈ β and aIVN ∈ B1 ∩ B2 [resp. aIVNV ∈ B1 ∩ B2],thenthereexists B ∈ β suchthat aIVN ∈ B ⊂ B1 ∩ B2 [resp.aIVNV ∈ B ⊂ B1 ∩ B2].
Proof. Theproofisthesameasoneinclassicaltopologicalspaces.
Example9. LetX = {a, b, c} andconsiderthefamilyofIVNCSsinXgivenby: β = {A1, A2, A3, A3}, whereA1 = [{b}, {a, b}], [{b}, {b}], [{c}, {c}] ,
A2 = [{b, c}, {b, c}], [{a}, {a}], ∅ ,
A3 = [{a}, {a}], [{c}, {c}], [{b}, {b}] ,
A4 = [{b}, ∅, [{c}, {c}] ,
Thenclearly, β satisfiestwoconditionsofTheorem 1.Thus β isanIVNCBforanIVNCT τ on X.Infact, wehave:
τ = {∅IVN , A1, A2, A3, A4, A5, A6, A7, XIVN }, whereA5 = [{b, c}, X], [{a, b}, {a, b}], ∅ ,
A6 = [{a, b}, {a, b}], [{b, c}, {b, c}], ∅ ,
A7 = X, [{a, c}, {a, c}], ∅
ThefollowingprovideasufficientconditionwhichacollectionofIVNCSsinaset X isanIVNCB forsomeIVNCTon X.
Proposition13. Let σ ⊂ IVNCS(X) suchthat XIVN = σ.ThenthereexistsauniqueIVNCT τ on X suchthat σ isanIVNCSBfor τ Proof. Let β = {B ∈ IVNCS(X) : B = n i=1 Si and Si ∈ σ}.Let τ = {U ∈ IVNCS(X) : U = ∅ orthereisasubcollection β of β suchthat U = β }.Thenwecanshowthat τ istheunique IVNCTon X suchthat σ isanIVNCSBfor τ
InProposition 13, τ iscalledtheIVNCTon X generatedby σ Example10. LetX = {a, b, c, d, e} andconsiderthefamily σ ofIVNCSsinXgivenby: σ = {A1, A2, A3, A4}, whereA1 = [{a}, {a}], [{b}, {b}], [{c, d}, {c, d}] ,
A2 = [{a, b, c}, {a, b, c}], [{b, d}, {b, d}], [{e}, {e}] ,
A3 = [{b, c, e}, {b, c, e}], [{c, e}, {c, d, e}], [{d}, {d}] ,
A4 = [{c, d}, {c, d}], [{a, c}, {a, c}], [{a, b}, {a, b}]
Thenclearly, σ = XIVN .Let β bethecollectionofallfiniteintersectionsofmembersof σ.Thenwehave: β = {A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12},
whereA5 = [{a}, {a}], [{b}, {b}], [{c, d, e}, {c, d, e}] ,
A6 = ∅, [{b}, {b}], [{c, d}, {c, d}] ,
A7 = ∅, ∅, [{a, b, c, d}, {a, b, c, d}] ,
A8 = [{b, c}, {b, c}], [∅, {d}], [{d, e}, {d, e}] ,
A9 = [{c}, {c}], ∅, [{a, b, e}, {a, b, e}] ,
A10 = [{c}, {c}], [{c}, {c}], [{a, b, d}, {a, b, d}] ,
A11 = ∅, ∅, [{c, d, e}, {c, d, e}] ,
A12 = [{c}, {c}], ∅, [{a, b, d, e}, {a, b, d, e}] . ThuswehavethegeneratedIVNCT τ by σ:
τ = {∅IVN , A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15, A16, A17, A18, XIVN }, whereA13 = [{a, b, c}, {a, b, c}], [{b, d}, {b, d}], ∅ ,
A14 = [{a, b, c, e}, {a, b, c, e}], [{b, c, e}, {b, c, d, e}], [{d}, {d}] ,
A15 = [{a, c, d}, {a, c, d}], [{a, b, c}, {a, b, c}], ∅ ,
A16 = [{a, b, c, e}, {a, b, c, e}], [{b, c, d, e}, {b, c, d, e}], ∅ ,
A17 = [{a, b, c, d}, {a, b, c, d}], [{a, b, c, d}, {a, b, c, d}], ∅ ,
A18 = X, [{a, c, e}, {a, c, e}], ∅ .
Remark4. Byusing“⊂2, ∪2, ∩2, i,c (i = 1,2,3), ∅2,IN , X2,IN ,and INC(X),wecanhavethedefinitions correspondingtoDefinitions 16 and 18,respectively.
5.Interval-ValuedNeutrosophicCrispNeighborhoods
Inthissection,weintroducetheconceptofinterval-valuedneutrosophiccrispneighborhoodsof IVNPsoftwotypes,andfindtheirvariouspropertiesandgivesomeexamples.
Definition20. LetXbeanIVNCTS,a ∈ X,N ∈ IVNCS(X).Then:
(i) N iscalledaninterval-valuedneutrosophiccrispneighborhood(briefly,IVNCN)of aIVN ,ifthereexists aU ∈ IVNCO(X) suchthat: aIVN ∈ U ⊂ N,i.e., a ∈ UT, ⊂ NT, ,
(ii) N iscalledaninterval-valuedneutrosophiccrispvanishingneighborhood(briefly,IVNCVN)of aIVNV , ifthereexistsaU ∈ IVNCO(X) suchthat: aIVNV ∈ U ⊂ N,i.e., a ∈ N F,+ ⊂ UF,+
ThesetofallIVNCNs[resp.IVNCVNs]of aIVN [resp. aIVNV ]isdenotedby N(aIVN ) [resp. N(IVNV )]and willbecalledanIVNCneighborhoodsystemofaIVN [resp.aIVNV ].
Example11. Let X = {a, b, c, d, e, f , g, h, i} andlet τ betheIVNCTon X giveninExample 7 (4).Consider theIVNCSN = [{a, b, d}, {a, b, c, d}], [{e}, {e}], [{g}, {g}] inX.Thenwecaneasilycheckthat: N ∈ N(aIVN ) ∩ N(aIVNV ), N ∈ N(bIVN ) ∩ N(bIVNV ), N ∈ N(dIVN ) ∩ N(dIVNV ), N ∈ N(cIVNV )
AnIVNCneighborhoodsystemof aIVN hasasimilarpropertyforaneighborhoodsystemofa pointinaclassicaltopologicalspace.
Proposition14. LetXbeanIVNCTS,a ∈ X.
[IVNCN1] IfN ∈ N(aIVN ),thenaIVN ∈ N. [IVNCN2] IfN ∈ N(aIVN ) andN ⊂ M,thenM ∈ N(aIVN ). [IVNCN3] IfN, M ∈ N(aIVN ),thenN ∩ M ∈ N(aIVN ) [IVNCN4] IfN ∈ N(aIVN ),thenthereexistsM ∈ N(aIVN ) suchthatN ∈ N(bIVN ) foreachbIVN ∈ M.
Proof. Theproofsof[IVNCN1],[IVNCN2],and[IVNCN4]areeasy.
[IVNCN3]Suppose N, M ∈ N(aIVN ).Thenthereare U, V ∈ IVNCO(X) suchthat aIVN ∈ U ⊂ N and aIVN ∈ V ⊂ M. Let W = U ∩ V.Thenclearly, W ∈ IVNCO(X) and aIVN ∈ W ⊂ N ∩ M.Thus N ∩ M ∈ N(aIVN ).
Inaddition,anIVNCneighborhoodsystemof aIVNV hasthesimilarproperty.
Proposition15. LetXbeanIVNCTS,a ∈ X. [IVNCVN1] IfN ∈ N(aIVNV ),thenaIVNV ∈ N. [IVNCVN2] IfN ∈ N(aIVNV ) andN ⊂ M,thenM ∈ N(aIVNV ). [IVNCVN3] IfN, M ∈ N(aIVNV ),thenN ∩ M ∈ N(aIVNV ). [IVNCVN4] If N ∈ N(aIVNV ),thenthereexists M ∈ N(aIVNV ) suchthat N ∈ N(bIVNV ) foreach bIVNV ∈ M.
Proof. TheproofissimilartooneofProposition 15. FromDefinition 20,wehavetwoIVNCTscontainingagivenIVNCT. Proposition16. Let (X, τ) beanIVNCTSandletusdefinetwofamilies: τIVN = {U ∈ IVNCS(X) : U ∈ N(aIVN ) foreach aIVN ∈ U} and τIVNV = {U ∈ IVNCS(X) : U ∈ N(aIVNV ) foreach aIVNV ∈ U} Thenwehave: (1) τIVN , τIVNV ∈ IVNCT(X), (2) τ ⊂ τIVN and τ ⊂ τIVNV . Proof. (1)Weonlyprovethat τIVNV ∈ IVNCT(X) (IVNCO1)Fromthedefinitionof τIVNV ,wehave ∅IVN , XIVN ∈ τIVNV (IVNCO2)Let U, V ∈ IVN∗(X) suchthat U , V ∈ τIVNV andlet aIVNV ∈ U ∩ V Thenclearly, U, V ∈ N(aIVNV ).Thusby[IVNCVN3], U ∩ V ∈ N(aIVNV ).So U ∩ V ∈ τIVNV (IVNCO3)Let (Uj )j∈J beanyfamilyofIVNCSsin τIVNV ,let U = j∈J Uj andlet aIVNV ∈ U ThenbyProposition 7 (2),thereis j0 ∈ J suchthat aIVNV ∈ Uj0 .Since Uj0 ∈ τIVNV , Uj0 ∈ N(aIVNV ) by thedefinitionof τIVNV .Since Uj0 ⊂ U, U ∈ N(aIVNV ) by[IVNCVN2].Sobythedefinitionof τIVNV , U ∈ τIVNV (2)Let U ∈ τ.Thenclearly, U ∈ N(aIVN ) and U ∈ N(aIVNV ) foreach aIVN ∈ G and aIVNV ∈ G, respectively.Thus U ∈ τIVN and U ∈ τIVNV .Sotheresultshold. Remark5. (1)Fromthedefinitionsof τIVN and τIVNV ,wecaneasilyhave: τIVN = τ ∪{U ∈ IVNCS(X) : V T, ⊂ UT, , V ∈ τ} and τIVNV = τ ∪{U ∈ IVNCS(X) : UF,+ ⊂ V F,+ , V ∈ τ}
(2)ForanyIVNCT τ onasetX,wecanhavesixIVTsonXgivenby:
τT IVN = {[UT, , UT,+ ] ∈ IVS(X) : U ∈ τIVN },
τ I IVN = {[U I, , U I,+ ] ∈ IVS(X) : U ∈ τIVN
τ F IVN = {[UF,+ c , UF, c ] ∈ IVS(X) : U ∈ τIVN },
τT IVNV = {[UT, , UT, ] ∈ IVS(X) : U ∈ τIVNV },
τ I IVNV = {[U I, , U I,+ ] ∈ IVS(X) : U ∈ τIVNV },
τ F IVNV = {[UF,+ c , UF,+ ] ∈ IVS(X) : U ∈ τIVNV }.
Furthermore,wehave12ordinarytopologiesonX:
τT, IVN = {UT, ⊂ X : U ∈ τIVN }, τT,+ IVN = {UT,+ ] ⊂ X : U ∈ τIVN },
τ I, IVN = {U I, ⊂ X : U ∈ τIVN }, τ I,+ IVN = {U I,+ ] ⊂ X : U ∈ τIVN },
τ F, IVN = {UF,+ c ⊂ X : U ∈ τIVN }, τ F,+ IVN = {UF, c ⊂ X : U ∈ τIVN },
τT, IVNV = {UT, ⊂ X : U ∈ τIVNV }, τT,+ IVNV = {UT, ⊂ X : U ∈ τIVNV },
τ I, IVNV = {U I, ⊂ X : U ∈ τIVNV }, τ I,+ IVNV = {U I,+ ⊂ X : U ∈ τIVNV },
τ F, IVNV = {UF,+ c ⊂ X : U ∈ τIVNV }, τ F,+ IVNV = {UF,+ ⊂ X : U ∈ τIVNV }.
Example12. LetX = {a, b, c, d, e, f , g, h, i} andconsiderIVNCT τ onXgiveninExample 7 (4).Thenfrom Remark 5 ((1),wehave:
τIVN = τ ∪{A5, A6, A7},
whereA5 = [{a, b, c}, {a, b, c}], [{e}, {e, f }], [{g}, {g, i}] ,
A6 = [{a, c, d}, {a, c, d}], [{e}, {e}], [{g, h}, {g, h, i}] , A7 = [{a, b, c, d}, {a, b, c, d}], [{e}, {e, f }], [{g}, {g, i}]
Additionally,wehave:
τIVNV = τ ∪{A8, A9, A10, A11}, whereA8 = [{a, b}, {a, b, c}], [{e}, {e, f }], [{g}, {g}] , A9 = [{a, d}, {a, c, d}], [{e}, {e}], [{g, h}, {g, h}] ,
A10 = [{a}, {a, c}], [{e}, {e}], [{g}, {g, h}] ,
A11 = [{a, b, d}, {a, b, c, d}], [{e}, {e, f }], [{g}, {g}] . SowecanconfirmthatProposition 16 holds.
ThefollowingistheimmediateresultofProposition 16 (2).
Corollary1. Let (X, τ) beanIVNCTSandletIVNCCτ [resp.IVNCCτIVN andIVNCCτIVNV ]bethesetof allIVNCCSsw.r.t. τ [resp. τIVN and τIVNV ].Then, IVNCCτ ⊂ IVNCCτIVN ,and IVNCCτ ⊂ IVNCCτIVNV
Example13. Let (X, τ) betheIVNCTSgiveninExample 12.Thenwehave:
IVNCCτ = {∅IVN , XIVN , Ac 1, Ac 2, Ac 3, Ac 4},
IVNCCτIVN = IVNCCτ ∪{Ac 5, Ac 6, Ac 7}, IVCτIVNV = IVCτ ∪{Ac 8, Ac 9, Ac 10, Ac 11}, whereAc 1 = [{g}, {g, i}], [{a, b, c, d, h}, {a, b, c, d, f , h}], [{a, b}, {a, b, c}] ,
Ac 2 = [{g, h}, {g, h, i}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a, d}, {a, c, d}] ,
Ac 3 = [{g, h}, {g, h, i}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a}, {a, c}] ,
Ac 4 = [{g}, {g, i}], [{a, b, c, d, h}, {a, b, c, d, f , h}], [{a, b, d}, {a, b, c, d}] ,
Ac 5 = [{g}, {g, i}], [{a, b, c, d, h}, {a, b, c, d, f , h}], [{a, b, c}, {a, b, c}] ,
Ac 6 = [{g, h}, {g, h, i}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a, c, d}, {a, c, d}] ,
Ac 7 = [{g}, {g, i}], [{a, b, c, d, h}, {a, b, c, d, f , h}], [{a, b, c, d}, {a, b, c, d}] ,
Ac 8 = [{g}, {g}], [{a, b, c, d, h}, {a, b, c, d, f , h}], [{a, b}, {a, b, c}] ,
Ac 9 = [{g, h}, {g, h}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a, d}, {a, c, d}] ,
Ac 10 = [{g}, {g, h}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a}, {a, c}] ,
Ac 11 = [{g}, {g}], [{a, b, c, d, h}, {a, b, c, d, f , h}], [{a, b, d}, {a, b, c, d}]
ThuswecanconfirmthatCorollary 1 holds.
NowletusconsidertheconversesofPropositions 14 and 15
Proposition17. Supposetoeach a ∈ X,therecorrespondsaset N∗(aIVNV ) ofIVNCSsin X satisfyingthe conditions[IVNCVN1],[IVNCVN2],[IVNCVN3],and[IVNCVN4]inProposition 15.Thenthereisan IVNCTonXsuchthatN∗(aIVNV ) isthesetofallIVNCVNsofaIVNV inthisIVNCTforeacha ∈ X.
Proof. Let, τIVNV = {U ∈ IVNCS(X) : U ∈ N(aIVNV ) foreach aIVNV ∈ U}, where N(aIVNV ) denotesthesetofallIVNCVNsin τ Thenclearly, τIVNV ∈ IVNCT(X) byProposition 16.Wewillprovethat N∗(aIVNV ) isthesetofall IVNCVNsof aIVNV) in τIVNV foreach a ∈ X Let V ∈ IVN∗(X) suchthat V ∈ N∗(aIVNV ) andlet U betheunionofalltheIVNCVPs bIVNV in X suchthat U ∈ N∗(aIVNV ).Ifwecanprovethat: aIVNV ∈ U ⊂ V and U ∈ τIVNV , thentheproofwillbecomplete.
Since V ∈ N∗(aIVNV ), aIVNV ∈ U bythedefinitionof U.Moreover, U ⊂ V.Suppose bIVNV ∈ U Thenby[IVNCVN4],thereisanIVNCS W ∈ N∗(bIVNV ) suchthat V ∈ N∗(cIVNV) ) foreach cIVNV ∈ W Thus cIVNV ∈ U.ByProposition 9, W ⊂ U.Soby[IVNCVN2], U ∈ N∗(bIVNV ) foreach bIVNV ∈ U Hencebythedefinitionof τIVNV , U ∈ τIVNV .Thiscompletestheproof.
Proposition18. Supposetoeach a ∈ X,therecorrespondsaset N∗(aIVN ) ofIVNCSsin X satisfyingthe conditions[IVNCN1],[IVNCN2],[IVNCN3],and[IVNCN4]inProposition 14.ThenthereisanIVNCTon XsuchthatN∗(aIVN ) isthesetofallIVNCNsofaIVN) inthisIVNCTforeacha ∈ X.
Proof. TheproofissimilartoProposition 17
ThefollowingprovideanecessaryandsufficientconditionwhichanIVNCSsisanIVNCOSin anIVNCTS.
Theorem2. Let (X, τ) beanIVNCTS, A ∈ IVNCS(X).Then A ∈ τ ifandonlyif A ∈ N(aIVN ) and A ∈ N(aIVNV ) foreachaIVN , aIVNV ∈ A.
Proof. Suppose A ∈ N(aIVN ) and A ∈ N(aIVNV ) foreach aIVN , aIVNV ∈ A.Thenthereare UaIVN , VaIVNV ∈ τ suchthat aIVN ∈ UaIVN ⊂ A and aIVNV ∈ VaIVNV ⊂ A.Thus, A =( aIVN ∈A aIVN ) ∪ ( aIVNV ∈A aIVNV ) ⊂ ( aIVN ∈A UaIVN ) ∪ ( aIVNV ∈A VaIVNV ) ⊂ A
So A =( aIVN ∈A UaIVN ) ∪ ( aIVNV ∈A VaIVNV ).Since UaIVN , VaIVNV ∈ τ, A ∈ τ. Theproofofthenecessaryconditioniseasy.
NowwewillgivetherelationamongthreeIVNCTs, τ, τIVN and τIVNV Proposition19. τ = τIVN ∩ τIVNV . Proof. FromProposition 16 (2),itisclearthat τ ⊂ τIVN ∩ τIVNV . Conversely,let U ∈ τIVN ∩ τIVNV Thenclearly, U ∈ τIVN and U ∈ τIVNV Thus U isanIVNCNof eachofitsIVNCPs aIVN andanIVNCVNofeachofitsIVNCVPs aIVNV .Thus,thereare UaIVN , UaIVNV ∈ τ suchthat aIVN ∈ UaIVN ⊂ U and aIVNV ∈ UaIVNV ⊂ U.Sowehave: UIVN = aIVN ∈U aIVN ⊂ aIVN ∈U UaIVN ⊂ U and UIVNV = aIVNV ∈U aIVNV ⊂ aIVNV ∈U UaIVNV ⊂ U.
ByProposition 5,weget: U = UIVN ∪ UIVNV ⊂ ( aIVN ∈U UaIVN ) ∪ ( aIVNV ∈U UaIVNV ) ⊂ U,i.e., U =( aIVN ∈U UaIVN ) ∪ ( aIVNV ∈U UaIVNV ) Itisobviousthat ( aIVN ∈U UaIVN ) ∪ ( aIVNV ∈U UaIVNV ) ∈ τ Hence U ∈ τ Therefore τIVN ∩ τIVNV ⊂ τ Thiscompletestheproof.
FromProposition 19,wegetthefollowing. Corollary2. Let (X, τ) beanIVNCTS.Then, IVNCCτ = IVNCCτIVN ∩ IVNCCτIVNV . Example14. InExample 12,wecaneasilycheckthatCorollary 2 holds.
6.InteriorsandClosuresofIVNCSs
Inthissection,wedefineinterval-valuedneutrosophiccrispinteriorsandclosures,and investigatesomeoftheirpropertiesandgivesomeexamples.Inparticular,wewillshowthatthereis
auniqueIVNCTonaset X fromtheinterval-valuedneutrosophiccrispclosure[resp.interior]operator.
InanIVNCTS,wecandefineaclosureandaninterioraswellastwoothertypesofclosuresand interiorsbyProposition 16.
Definition21. Let (X, τ) beanIVNCTS,A ∈ IVNCS(X)
(i) Theinterval-valuedneutrosophiccrispclosureof A w.r.t. τ,denotedby IVNcl(A),isanIVNCSin X definedas:
IVNcl(A)= {K : Kc ∈ τ and A ⊂ K}.
(ii) Theinterval-valuedneutrosophiccrispinteriorof A w.r.t. τ,denotedby IVNint(A),isanIVSin X definedas:
IVNint(A)= {G : G ∈ τ and G ⊂ A}.
(iii) Theinterval-valuedneutrosophiccrispclosureof A w.r.t. τIVN ,denotedby clIVN (A),isanIVNCSin X definedas: clIVN (A)= {K : Kc ∈ τIVN and A ⊂ K}.
(iv) Theinterval-valuedneutrosophiccrispinteriorof A w.r.t. τIVN ,denotedby intIVN (A),isanIVSin X definedas: intIVN (A)= {G : G ∈ τIVN and G ⊂ A}.
(v) Theinterval-valuedneutrosophiccrispclosureof A w.r.t. τIVNV ,denotedby clIVNV (A),isanIVNCSin X definedas: clIVNV (A)= {K : Kc ∈ τIVNV and A ⊂ K}.
(vi) Theinterval-valuedneutrosophiccrispinteriorof A w.r.t. τIVNV ,denotedby intIVNV (A),isanIVNCSin Xdefinedas: intIVNV (A = {G : G ∈ τIVNV and G ⊂ A}.
Remark6. Fromtheabovedefinition,itisobviousthatthefollowingshold:
IVNint(A) ⊂ intIVN (A), IVNint(A) ⊂ intIVNV (A) and clIVN (A) ⊂ IVNcl(A), clIVNV (A) ⊂ IVNcl(A).
Example15. Let (X, τ) betheIVNCTSgiveninExamples 12 and 13.ConsidertwoIVNCSsinX: A = [{a, b, c}, {a, b, c, d}], [{a, e}, {a, e, f }], [{g}, {g}] , B = [{g, h}, {g, h, i}], [{a, b, c, d, f }, {a, b, c, d, e, f }], [{a}, {a, c}] . Then, IVNint(A)= {G ∈ τ : G ⊂ A} = A1 ∪ A3 = [{a, b}, {a, b, c}], [{e}, {e, f }], [{g}, {g, i}] , intIVN (A)= {G ∈ τIVN : G ⊂ A} = A1 ∪ A3 ∪ A5 = [{a, b, c}, {a, b, c}], [{e}, {e, f }], [{g}, {g, i}] , intIVNV (A)= {G ∈ τIVNV : G ⊂ A} = A1 ∪ A3 ∪ A8 ∪ A10 = [{a, b}, {a, b, c}], [{e}, {e, f }], [{g}, {g}] and IVNcl(B)= {F : Fc ∈ τ, B ⊂ F} = Ac 2 ∩ Ac 3 = [{g, h}, {g, h, i}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a, d}, {a, c, d}] , clIVN (B)= {F : Fc ∈ τIVN , B ⊂ F} = Ac 2 ∩ Ac 3 ∩ Ac 6 ∩ Ac 10 = [{g, h}, {g, h, i}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a, c, d}, {a, c, d}] ,
clIVNV (B)= {F : Fc ∈ τIVNV , B ⊂ F} = Ac 2 ∩ Ac 3 ∩ Ac 9 ∩ Ac 10 = [{g}, {g, h}], [{a, b, c, d, f }, {a, b, c, d, f }], [{a, d}, {a, c, d}] .
ThuswecanconfirmthatRemark 6 holds.
Proposition20. Let (X, τ) beanIVNCTS,A ∈ IVNCS(X).Then, IVNint(Ac )=(IVNcl(A))c andIVNcl(Ac )=(IVNint(A))c
Proof. IVNint(Ac )= {U ∈ τ : U ⊂ Ac } = {U ∈ τ : U ⊂ AF , AI c , AT } = {U ∈ τ : UT ⊂ AF , U I ⊂ AI c , UF ⊃ AT } = {U ∈ τ : UT ⊂ AF , U I c ⊂ AI , UF ⊃ AT } =( {Uc : U ∈ τ, A ⊂ Uc })c =(IVNcl(A))c Similarly,wecanshowthat IVNcl(Ac )=(IVNint(A))c
Proposition21. Let (X, τ) beanIVNCTS,A ∈ IVNCS(X).Then, IVNint(A)= intIVN (A) ∩ intIVNV (A).
Proof. TheproofisstraightforwardfromProposition 19 andDefinition 21 ThefollowingistheimmediateresultofDefinition 21,andPropositions 20 and 21
Corollary3. Let (X, τ) beanIVNCTSandletA ∈ IVNCS(X).Then, IVNcl(A)= clIVN (A) ∪ clIVNV (A)
Example16. LetAandBbetwoIVNCSsinXgiveninExample 15.Thenwecaneasilycheckthat: intIVN (A) ∩ intIVNV (A)= IVNint(A), clIVN (B) ∪ clIVNV (B)= IVNcl(B).
Theorem3. LetXbeanIVNCTS,A ∈ IVNCS(X).Then: (1) A ∈ IVNCC(X) ⇔ ifA = IVNcl(A), (2) A ∈ IVNCO(X) ⇔ A = IVNint(A)
Proof. Straightforward.
Proposition22 (KuratowskiClosureAxioms). LetXbeanIVNCTS,A, B ∈ IVNCS(X).Then, [IVNCK0] IfA ⊂ B,thenIVNcl(A) ⊂ IVNcl(B), [IVNCK1] IVNcl(∅IVN )= ∅IVN , [IVNCK2] A ⊂ IVNcl(A), [IVNCK3] IVNcl(IVNcl(A))= IVNcl(A), [IVNCK4] IVcl(A ∪ B)= IVNcl(A) ∪ IVNcl(A).
Proof. Straightforward.
Let IVNcl∗ : IVNCS(X) → IVNCS(X) bethemappingsatisfyingtheproperties[IVNCK1], [IVNCK2],[IVNCK3],and[IVNCK4].Thenthemapping IVcl∗ iscalledtheinterval-valued neutrosophiccrispclosureoperator(briefly,IVNCCO)on X.
Proposition23. Let IVNcl∗ betheIVNCCOon X.ThenthereexistsauniqueIVNCT τ on X such that IVNcl∗(A)= IVNcl(A),foreach A ∈ IVNCS(X),where IVNcl(A) denotestheinterval-valued neutrosophiccrispclosureofAintheIVNCTS (X, τ).Infact, τ = {Ac ∈ IVNCS(X) : IVNcl∗(A)= A}
Proof. Theproofisalmostsimilartothecaseofclassicaltopologicalspaces.
Proposition24. ⇔LetXbeanIVNCTS,A, B ∈ IVNCS(X).Then, [IVNCI0] IfA ⊂ B,thenIVNint(A) ⊂ IVNint(B), [IVNCI1] IVNint(XIVN )= XIVN , [IVNCI2] IVNint(A) ⊂ A, [IVNCI3] IVNint(IVNint(A))= IVNint(A), [IVNCI4] IVNint(A ∩ B)= IVNint(A) ∩ IVNint(A).
Proof. Straightforward.
Let IVNint∗ : IVNCS(X) → IVNCS(X) bethemappingsatisfyingtheproperties[IVNCI1], [IVNCI2],[IVNCI3],and[IVNCI4].Thenthemapping IVNint∗ iscalledtheinterval-valued neutrosophiccrispinterioroperator(briefly,IVNCIO)on X.
Proposition25. Let IVNint∗ betheIVNCIOon X.ThenthereexistsauniqueIVNCT τ on X suchthat IVNint∗(A)= IVNint(A) foreach A ∈ IVNCS(X),where IVNint(A) denotestheinterval-valued neutrosophiccrispinteriorofAintheIVNCTS (X, τ).Infact, τ = {A ∈ IVNCS(X) : IVNint∗(A)= A}
Proof. TheproofissimilartooneofProposition 23.
7.Interval-ValuedNeutrosophicCrispContinuousMappings
Inthissection,wedefineaninterval-valuedneutrosophiccrispcontinuityandquotienttopology, andstudysomeoftheirproperties.
Definition22. Let X, τ), (Y, δ) betwoIVTSsproposedin[30].Thenamapping f : X → Y issaidtobe interval-valuedcontinuous,iff 1(V) ∈ τ foreachV ∈ δ.
Definition23. Let X, τ), (Y, δ) betwoIVNCTSs.Thenamapping f : X → Y issaidtobeinterval-valued neutrosophiccrispcontinuous,iff 1(V) ∈ τ foreachV ∈ δ
FromRemark 2 (1),andDefinitions 22 and 23,wecaneasilyhavethefollowing. Theorem4. Let (X, τ), (Y, δ) betwoIVNCTSsandlet f : X → Y beamapping.Then f isinterval-valued neutrosophiccrispcontinuousifandonlyif f : (X, τT ) → (Y, δT ), f : (X, τ I ) → (Y, δI ),and f : (X, τ F ) → (Y, δF ) areinterval-valuedcontinuous,respectively.
ThefollowingsareimmediateresultsofProposition 9 (13)andDefinition 23.
Proposition26. LetX, Y, ZbeIVNCTSs.
(1) Theidentitymappingid : X → Xiscontinuous. (2) Iff : X → Yandg : Y → Zarecontinuous,theng ◦ f : X → Ziscontinuous.
Remark7. FromProposition 26,wecaneasilyseethattheclassofallIVNCTSsandcontinuousmappings, denotedby IVNCTop,formsaconcretecategory.
ThefollowingsareimmediateresultsofDefinition 23
Proposition27. LetX, YbeINCTSs.
(1) IfXisanIVNCDTS,thef : X → Yiscontinuous, (2) IfYisanIVNCITS,thenf : X → Yiscontinuous.
Theorem5. LetX, YbeIVNCTSsandletf : X → Ybeamapping.Thenthefollowingsareequivalent:
(1) fiscontinuous, (2) f 1(C) ∈ IVNCC(X) foreachC ∈ IVNCC(Y), (3) f 1(S) ∈ IVNCO(X) foreachmemberSofthesubbasefortheIVNCTonY, (4) IVNcl( f 1(B)) ⊂ f 1(IVNcl(B)) foreachB ∈ IVNC(Y), (5) f (IVNcl(A)) ⊂ IVNcl( f (A)) foreachA ∈ IVNC(X).
Proof. Theproofsof(1)⇒(2)⇒(3)⇒(1)areobvious. (2)⇒(4):Supposethecondition(2)holdsandlet B ∈ INC(Y).ByProposition 22 [IVNCK2], B ⊂ IVNcl(B).ThenbyProposition 9 (2), f 1(B) ⊂ f 1(IVNcl(B)).ThusbyProposition 22 [INCK0], IVNcl( f 1(B)) ⊂ IVNcl( f 1(IVNcl(B))).
Since IVNcl(B) ∈ IVNCC(Y), f 1(IVNcl(B)) ∈ IVNCC(X) bythecondition(2).SobyTheorem 3 (1), IVNcl( f 1(IVNcl(B)))= f 1(IVNcl(B)).Hence IVNcl( f 1(B)) ⊂ f 1(IVNcl(B)) (4)⇒(5):Supposethecondition(4)holdsandlet B = f (A) foreach A ∈ IVNC(X).Thenwehave IVNcl( f 1( f (A))) ⊂ f 1(IVNcl( f (A))).ThusbyProposition 9 (3), IVNcl(A) ⊂ f 1(IVNcl( f (A))) SobyProposition 9 (1)and(4), f (IVNcl(A)) ⊂ IVNcl( f (A))
(5)⇒(4):Theproofissimilarto(4)⇒(5).
Theorem6. Let X, Y beIVNCTSsandlet f : X → Y beamapping.Then f iscontinuousifandonlyif f 1(IVNint(B)) ⊂ IVNint( f 1(B)) foreachB ∈ INC(Y).
Proof. Theproofisstraightforward. Definition24. Let (X, τ), (Y, δ) betwoIVNCTSs.Thenamappingf : X → Yissaidtobe: (i) Interval-valuedneutrosophiccrispopen,iff (U) ∈ δ foreachU ∈ τ, (ii) Interval-valuedneutrosophiccrispclosed,iff (C) ∈ IVNCC(Y) foreachC ∈ IVNCC(X)
Proposition28. Let X, Y, Z beIVNCTSs,let f : X → Y and g : Y → Z bemappings.If f , g areopen[resp. closed],theng ◦ gisopen[resp.closed].
Proof. Theproofisstraightforward.
Theorem7. Let X, Y beIVNCTSsandlet f : X → Y beamapping.Then f isopenifandonlyif IVNint( f (A)) ⊂ f (IVNint(A)) foreachA ∈ IVNC(X)
Proof. Theproofisstraightforward.
Proposition29. Let X, Y beIVNCTSsandlet f : X → Y beinjective.If f iscontinuous,then f (IVNint(A)) ⊂ IVNint( f (A)) foreachA ∈ IVNC(X).
Proof. Theproofisstraightforward.
ThefollowingistheimmediateresultofTheorem 7 andProposition 29
Corollary4. Let X, Y beIVNCTSsandlet f : X → Y becontinuous,open,andinjective.Then f (IVNint(A))= IVNint( f (A)) foreachA ∈ IVNC(X).
Theorem8. Let X, Y beIVNCTSsandlet f : X → Y beamapping.Then f iscloseifandonlyif IVNcl( f (A)) ⊂ f (IVNcl(A)) foreachA ∈ IVNC(X).
Proof. Theproofisstraightforward.
ThefollowingistheimmediateresultofTheorems 5 and 8.
Corollary5. Let X, Y beIVNCTSsandlet f : X → Y beamapping.Then f iscontinuousandclosedifand onlyiff (VINcl(A))= IVNcl( f (A)) foreachA ∈ IVNC(X)
Definition25. Let (X, τ), (Y, δ) betwoIVNCTSs.Thenamapping f : X → Y iscalledaninterval-valued neutrosophiccrisphomeomorphism,iffisbijective,continuous,andopen.
Theorem9. Let X, Y beIVNCDTSsandlet f : X → Y beamapping.Then f isahomeomorphismifand onlyiffisbijective.
Proof. Theproofisstraightforward.
Definition26. Let (X, τ) beanIVNCTS,let Y beasetandlet f : X → Y beasurjectivemapping.Let δ be thefamilyofIVNCSsinYgivenby: δ = {B ∈ IVNC(Y) : f 1(B) ∈ τ}
Then δ iscalledtheinterval-valuedneutrosophiccrispquotienttopology(briefly,IVNCQT)onY. Itcaneasilybeseenthat δ ∈ IVNCT(Y).Itisalsoobviousthatforeach B ∈ IVNC(Y), B isclosedin δ ifandonlyiff 1(B) isclosedinX.
Proposition30. Let (X, τ), (Y, δ) betwoIVNCTSs,where δ istheIVNCQTon Y.Thenasurjection f : X → Yiscontinuousandopen.Moreover, δ isthefinesttopologyonYwhichfiscontinuous.
Proof. Theproofissimilartotheclassicalcase.
ThefollowingistheimmediateresultofProposition 30
Corollary6. Let (X, τ), (Y, δ) betwoIVNCTSs.Ifamapping f : X → Y iscontinuous,open,andsujective, then δ istheIVNCQTonY.Buttheconversedoesnotholdingeneral (SeeExample 17).
Example17. Let ([0,1], τ) beanIVNCTSandlet A =[ 1 2 ,1].Considerthecharacteristicfunction χA : [0,1] →{0,1},where {0,1} istheinterval-valuedneutrosophiccrispSierpin skispace(seeExample 7 (5)). Thenwecaneasilyseethatthetopologyon {0,1} istheIVNCQT.Ontheotherhand, ( 1 2 ,1)IVNCI ∈ τ but χA (( 1 2 ,1)IVNCI ) isnotopenin {0,1}.Thus χA isnotanopenmapping.
Theorem10. Let (X, τ), (Y, δ), (Z, σ) beIVNCTSs,where δ istheIVNCQTon Y.Let f : X → Y and g : Y → Zbemappings.Thengiscontinuousifandonlyifg ◦ fiscontinuous.
Proof. Theproofissimilartotheclassicalcase.
8.Conclusions
WeobtainedvariouspropertiesofIVNCSsanddiscussedwithIVNCTSswhichcanbeconsidered asaninterval-valuedtri-opologicalspace.Moreover,wedefinedaninterval-valuedneutrosophic crispbaseandsubbaseandprovedthecharacterizationofaninterval-valuedneutrosophiccrisp base.Next,weintroducedtheconceptofinterval-valuedneutrosophiccrispneighborhoodsand obtainedsomesimilarpropertiestoclassicalneighborhoods.Furthermore,wedefinedinterval-valued neutrosophiccrispclosuresandinteriors,andfoundsomeproperties.Wealsointroducedtheconcept ofinterval-valuedneutrosophiccrispcontinuitiesandobtaineditsvariousproperties.
Infuture,weexpectthatonecanapplytheconceptofIVNCSstogroupandringtheory, BCK-algebra,andcategorytheory,etc.Wealsoexpectthatonecandefinethenotionsofinterval-valued softsetsandinterval-valuedneutrosophiccrispsoftsets.Besides,thetheoremsdevelopedinthis manuscriptwillpromotefuturestudiesonthegeometrycalibrationformulti-cameras.
AuthorContributions: Createandconceptualizeideas,J.-G.L.andK.H.;Writing—originaldraftpreparation, J.-G.L.andK.H.;Writing—reviewandediting,D.J.;S.S.andC.X.;Fundingacquisition,D.J.Allauthorshaveread andagreedtothepublishedversionofthemanuscript.
Funding: ThisworkwassupportedbytheInstituteofInformation&communicationsTechnologyPlanning &Evaluation(IITP),grantfundedbytheKoreangovernment(MSIT)(No.2020-0-00226,Developmentof High-Definition,UnstructuredPlenopticvideoAcquisitionTechnologyforMediumandLargeSpace).
Acknowledgments: Wewouldliketothanktheanonymousreviewersfortheirverycarefulreadingandvaluable comments/suggestions.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
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