Softb-SeparationAxiomsinNeutrosophicSoft
TopologicalStructures
ArifMehmoodKhattak,NaziaHanif,FawadNadeem,Muhammad Zamir,ChoonkilPark,GiorgioNordo,ShamoonaJabeen
Received8February2019; Revised14March2019; Accepted27May2019
Abstract. TheideaofneutrosophicsetwasfloatedbySmarandache bysupposingatruthmembership,anindeterminacymembershipanda falsehoodorfalsitymembershipfunctions.Neutrosophicsoftsetsbonded byMajihavebeenutilizedsuccessfullytomodeluncertaintyinseveral areasofapplicationsuchascontrol,reasoning,patternrecognitionand computervision.Thefirstaimofthisarticlebouncestheideaofneutrosophicsoftb-openset,neutrosophicsoftb-closedsetsandtheirproperties. Alsotheideaofneutrosophicsoftb-neighborhoodandneutrosophicsoft b-separationaxiomsinneutrosophicsofttopologicalstructuresarealsoreflectedhere.Laterontheimportantresultsarediscussedrelatedtothese newlydefinedconceptswithrespecttosoftpoints.Theconceptofneutrosophicsoftb-separationaxiomsofneutrosophicsofttopologicalspacesis diffusedindifferentresultswithrespecttosoftpoints.Furthermore,propertiesofneutrosophicsoftb-T i -space(i =0, 1, 2, 3, 4)andsomelinkage betweenthemarediscussed.
2010AMSClassification: 03E72,04A72,18B05
Keywords: Neutrosophicsoftset,Neutrosophicsoftpoint,Neutrosophicsoftbopenset,Neutrosophicsoftb-neighborhood,Neutrosophicsoftb-separationaxioms.
CorrespondingAuthor: ArifMehmoodKhattak(mehdaniyal@gmail.com)
1. Introduction
Thetraditionalfuzzysetsischaracterisedbythemembershipvalueorthegrade ofmembershipvalue.Sometimesitmaybeverydifficulttoassignthemembership valueforafuzzysets.Consequentlytheconceptofintervalvaluedfuzzysetswas proposedtocaptureandgriptheuncertaintyofgradeofmembershipvalue.In somereallifeproblemsinexpertsystem,beliefsystem,informationfusionandso on,wemustconsiderthetruth-membershipaswellasthefalsity-membershipfor properdescriptionofanobjectinuncertain,ambiguousenvironment.Neitherthe fuzzysetsnortheintervalvaluedfuzzysetsisheretoshoulderuptheresponsibilitiesforsuchasituation.Theimportanceofintuitionisticfuzzysetsisautomaticallycomeinplayinsuchadangeroussituation.Theintuitionisticfuzzysetscan onlyhandletheincompleteinformationconsideringboththetruth-membershipor association(orsimplymembership)andnon-membershipvalues.Itdoesnothandletheindeterminateandinconsistentinformationwhichexistsinbeliefsystem.
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Smarandache[17]introducednewconceptofneutrosophicsetwhichisamathematicaltoolforhandlingproblemsinvolvingimprecise,indeterminacyandinconsistentdata.Thewordsˆoneutrosophy¨oandˆoneutrosophic¨owereintroducedby Smarandache.ˆoNeutrosophy¨o(noun)meansknowledgeofneutralthought,while ˆoneutrosophic¨o(adjective),meanshavingthenatureoforhavingthecharacteristic ofneutrosophy.Thistheoryisstraightforwardgeneralizationofcrispsets,fuzzyset theory[18],intuitionisticfuzzysettheory[1]etc.Someworkhavebeensupposedon neutrosophicsetsbysomeresearchersinmanyareaofmathematics[4, 15].Many practicalproblemsineconomics,engineering,environment,socialscience,medical science,etc.cannotbedealtwithbyclassicaltechniques,becauseclassicalmethodshaveheritablecomplexities.Thesecomplexitiesmaybetakingbirthdueto theinsufficiencyofthetheoriesofparamertrizationtools.Eachofthesetheories hasitstransmissibledifficulties,aswasnakedbyMolodtsov[14].Molodtsovoriginatedanabsolutelymodernaccesstocopewithuncertaintyandvaguenessand applieditprogressivelyindifferentdirectionssuchassmoothnessoffunctions,game theory,operationsresearch,Riemannintegration,perronintegration,andsoon. Honestly,Theoryofsoftsetisfreefromtheparameterizationmeagernesssyndrome offuzzysettheory,roughsettheory.probabilitytheoryfordealingwithuncertaintyShabirandNaz[16]firstfloatedthenotionofsofttopologicalspaces,which aredefinedoveraninitialuniverseofdiscoursewithafixedsetofparameters,and showedthatasofttopologicalspacegivesaparameterizedfamilyoftopological structures.Theoreticalstudiesofsofttopologicalspaceswerealsodonebysome authorsin[2, 3, 6, 8].ThecombinationofNeutrosophicsetwithsoftsetswas firstintroducedbyMaji[13].ThiscombinationmakesentirelyanewmathematicalmodelæNeutrosophicsoftsetÆandlateronthisnotionwasimprovedbyDeli andBroumi[7].Workwasprogressivelycontinue,lateronmathematiciancamein actionanddefinedanewmathematicalstructureknownasneutrosophicsofttopologicalspaces.NeutrosophicsofttopologicalspaceswerepresentedbyBerain[5]. Guzide.S[9]attemptedtobringtogethertheareasofspheres,softrealnumbersand softpoints.Relatingspherestosoftrealnumbersandsoftpointsprovidesanatural andintrinsicconstructionofsoftspheres.Guzide[10]discussedthetheoryofsoft topologicalspacegeneratedbyL-softsetsisintroduced.Asacontinuationofthe studyofoperationsonL-softsets,theaimofthispaperistointroducenewsoft topologiesusingrestrictedandextendedintersectionsonL-softsetsandtostudy thedifferencesofthesesofttopologies.Guzide[11]discussedsoftpointÆsandsoft matrixformwhichwerenotdescribedbeforeisdefinedforeachsetofparameters. Thematrixrepresentationofsoftpointsisusefulforstoringallsoftpointsthatcan beobtainedinalldifferentparameters.Theproposedsoftmatrixprovidesevery softpointthatchangeswitheachparameterthattakesplaceinasoftsetisproved andshowedthatitenablesdetailedexaminationinapplicationofsoftsettheory. Guzide[12]discussedcomparativeresearchonthedefinitionofSoftPoint.
Thefirstaimofthisarticlebouncestheideaofneutrosophicsoftb-openset, neutrosophicsoftb-neighborhoodandneutrosophicsoftb-separationaxiomsinneutrosophicsofttopologywhichisdefinedonneutrosophicsoftsets.Lateronthe importantresultsarediscussedrelatedtothesenewlydefinedconceptswithrespect
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tosoftpoints.Finally,theconceptofb-separationaxiomsofneutrosophicsofttopologicalspacesisdiffusedindifferentresultswithrespecttosoftpoints.Furthermore, propertiesofneutrosophicsoftb-T i-space(i =0, 1, 2, 3, 4)andsomelinkagebetween themarediscussed.Wehopethattheseresultswillbestfitforfuturestudyonneutrosophicsofttopologytocarryoutageneralframeworkforpracticalapplications.
2. Preliminaries
Inthissectionwenowstatecertainusefuldefinitions,theorems,andseveral existingresultsforneutrosophicsoftsetsthatwerequireinthenextsections. Definition2.1 ([17]). AneutrosophicsetAontheuniversesetXisdefinedas: A = { x,T A(x),I A(x), A(x) : x ∈ X}, whereT,I, : X →] 0,1+[and 0 T A(x)+ I A(x)+ A(x) 3+ Definition2.2. [14]LetXbeaninitialuniverse,Ebeasetofallparameters,and P(x)denotethepowersetofX.Thenapair( ,E)iscalledasoftsetoverX,where isamappinggivenby : E → P (X).
Inotherwords,thesoftsetisaparameterizedfamilyofsubsetsofthesetX.For λ ∈ E, (λ)maybeconsideredasthesetof λ-elementsofthesoftset( ,E),oras thesetof λ-approximateelementoftheset,i.e.,
( ,E)= {(λ, (λ)): λ ∈ E, : E → P (X)}
AftertheneutrosophicsoftsetwasdefinedbyMaji[13],thisconceptwasmodified byDeliandBroumi[7]asgivenbelow.
Definition2.3 ([7]). LetXbeaninitialuniversesetandEbeasetofparameters. LetP(X)denotethesetofallneutrosophicsetsofX.Thenaneutrosophicsoftset ( ,E)overXisasetdefinedbyasetvaluedfunction representingamapping : E → P (X),where iscalledtheapproximatefunctionoftheneutrosophicsoft set( ,E).Inotherwords,theneutrosophicsoftsetisaparameterizedfamilyof someelementsofthesetP(X)andthereforeitcanbewrittenasasetofordered pairs:
( ,E)= {(λ, x,T (λ)(x),I (λ)(x), (λ)(x) : x ∈ X): λ ∈ E}, where T (λ)(x), I (λ)(x), (λ)(x) ∈ [0,1]arerespectivelycalledthetruth-membership, indeterminacy-membership,andfalsity-membershipfunctionof (λ).Sincethe supremumofeachT,I,Fis1,theinequality
0 T (λ)(x)+ I (λ)(x)+ (λ)(x) 3
isobvious.
Definition2.4 ([5]). Let( ,E)beaneutrosophicsoftsetovertheuniversesetX. Thecomplementof( ,E)isdenotedby( ,E)c andisdefinedby: ( ,E)c = {(λ, x, (λ)(x), 1 I (λ)(x),T (λ)(x) : x ∈ X): λ ∈ E}
Itisobviousthat(( ,E)c)c =( ,E). 3
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Definition2.5 ([13]). Let( ,E)and(G,E)betwoneutrosophicsoftsetsoverthe universesetX.( ,E)issaidtobeaneutrosophicsoftsubsetof(G,E),if T (λ)(x) T G(λ)(x),I (λ)(x) I G(λ)(x), (λ)(x) G(λ)(x), ∀λ ∈ E, ∀x ∈ X. Itisdenotedby( ,E) (G,E).( ,E)issaidtobeneutrosophicsoftequalto (G,E)if( ,E)isaneutrosophicsoftsubsetof(G,E)and(G,E)isaneutrosophic softsubsetof( ,E).Itisdenotedby( ,E)=(G,E).
3. Neutrosophicsoftpointandrelatedcharacteristics
Definition3.1. Let( 1,E)and( 2,E)betwoneutrosophicsoftsetsoveruniverse setX.Thentheirunion,denotedby( 1,E) ( 2,E)=( 3,E),isdefinedby: ( 3,E)= {(λ, x,T 3 (λ)(x),I 3 (λ)(x), 3 (λ)(x) : x ∈ X): λ ∈ E}, where T 3 (λ)(x)=max {T 1 (λ)(x),T 2 (λ)(x)}, I 3 (λ)(x)=min {I 1 (λ)(x),I 2 (λ)(x)}, 3 (λ)(x)=min { 1 (λ)(x), 2 (λ)(x)}
Definition3.2. Let( 1,E)and( 2,E)betwoneutrosophicsoftsetsoverthe universesetX.Thentheirintersection,denotedby( 1,E) ( 2,E)=( 3,E),is definedby: ( 3,E)= {(λ, x,T 3 (λ)(x),I 3 (λ)(x), 3 (λ)(x) : x ∈ X): λ ∈ E}, where T 3 (λ)(x)=min {T 1 (λ)(x),T 2 (λ)(x)}, I 3 (λ)(x)=max {I 1 (λ)(x),I 2 (λ)(x)}, 3 (λ)(x)=max { 1 (λ)(x), 2 (λ)(x)}.
Definition3.3. Aneutrosophicsoftset( ,E)overtheuniversesetXissaidtobe anullneutrosophicsoftset,ifforeach λ ∈ E andeach x ∈ X,
T (λ)(x)=0,I (λ)(x)=0, (λ)(x)=1.
Itisdenotedby0(X,E)
Definition3.4. Aneutrosophicsoftset( ,E)overtheuniversesetXissaidtobe anabsoluteneutrosophicsoftset,ifforeach λ ∈ E andeach x ∈ X,
T (λ)(x)=1,I (λ)(x)=1, (λ)(x)=0
Itisdenotedby1(X,E)
Clearly,0c(X,E)=1(X,E) and1c(X,E)=0(X,E)
Definition3.5. LetNSS(X,E)bethefamilyofallneutrosophicsoftsetsoverthe universesetXand ❁ NSS(X,E).Then issaidtobeaneutrosophicsofttopology onX,ifitsatisfiesthefollowingaxioms:
(i)0(X,E) and1(X,E) belongto ,
(ii)theunionofanynumberofneutrosophicsoftsetsin belongsto , (iii)theintersectionofafinitenumberofneutrosophicsoftsetsin belongsto .
Then(X, ,E)issaidtobeaneutrosophicsofttopologicalspaceoverX.Each memberof issaidtobeaneutrosophicsoftopenset.
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Definition3.6. Let(X, ,E)beaneutrosophicsofttopologicalspaceoverXand ( ,E)beasubsetofneutrosophicsofttopologicalspaceoverX.Then( ,E)issaid tobeaneutrosophicsoftclosedsetiffitscomplementisaneutrosophicsoftopen set.
Definition3.7. Let(X, ,E)beaneutrosophicsofttopologicalspaceoverXand ( ,E)beasubsetofneutrosophicsofttopologicalspaceoverX.Then( ,E)issaid tobeaneutrosophicsoftb-open(NSBO)set,if
( ,E) ⊆ NSint(NScl(( ,E))) NScl(NSint(( ,E))).
Definition3.8. Let(X, ,E)beaneutrosophicsofttopologicalspaceoverXand ( ,E)beasubsetofneutrosophicsofttopologicalspaceoverX.Then( ,E)issaid tobeaneutrosophicsoftb-open(NSBC)set,if
( ,E) ⊇ NSint(NScl(( ,E))) NScl(NSint(( ,E))).
Definition3.9. LetNSbethefamilyofallneutrosophicsetsovertheuniverseset X,let x ∈ X andlet0 <α,β,γ 1.Thentheneutrosophicset x(α,β,γ) iscalleda neutrosophicpointandisdefinedasfollow:foreach y ∈ X,
(3.1)
x(α,β,γ)(y)= (α,β,γ),ify = x (0, 0, 1),ify = x.
Itisclearthateveryneutrosophicsetistheunionofitsneutrosophicpoints. Definition3.10. SupposethatX= {x1,x2}.Thenneutrosophicset
A = { x 1 , 0.1, 0.3, 0.5 , x 2 , 0.5, 0.4, 0.7 } istheunionofneutrosophicpoints x1(0 1,0 3,0 5) and x2(0 1,0 3,0 5)
Nowwedefinetheconceptofneutrosophicsoftpointsforneutrosophicsoftsets. Definition3.11. LetNSS(X,E)bethefamilyofallneutrosophicsoftsetsoverthe universesetXandletx ∈ X,0 <α,β,γ 1, λ ∈ E.Thentheneutrosophicsoft set xλ(α,β,γ) iscalledaneutrosophicsoftpointandisdefinedasfollows:foreach y ∈ X, (3.2) x λ(α,β,γ)(λ )(y)= (α,β,γ) ifλ = λandy = x (0, 0, 1),ifλ = λory = x. Definition3.12. SupposethattheuniversesetXisgivenbyX= {x1,x2} andthe setofparametersbyE= {λ1,λ2}.Letusconsiderneutrosophicsoftsets( ,E) overtheuniverseXasfollows: (3.3)( ,E)= λ1 = { x1 , 0.3, 0.7, 0.6 , x2 , 0.4, 0.3, 0.8 } λ2 = { x1 , 0 4, 0 6, 0 8 , x2 , 0 3, 0 7, 0 2 } Itisclearthat( ,E)isunionofitsneutrosophicsoftpoints x1λ1(0 3,0 7,0 6) , x1λ2(0 4,0 6,0 8) , x2λ1(0 4,0 3,0 8),and x2λ2(0 3,0 7,0 2).Here (3.4) x 1λ1(0 3,0 7,0 6) = λ1 = { x1 , 0 3, 0 7, 0 6 , x2 , 0, 0, 1 } λ2 = { x1 , 0, 0, 1 , x2 , 0, 0, 1 } , 5
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(3.5) x 1λ2(0 4,0 6,0 8) = λ1 = { x1 , 0.3, 0.7, 0.6 , x2 , 0, 0, 1 } λ2 = { x1 , 0.4, 0.6, 0.8 , x2 , 0, 0, 1 } ,
(3.6) x 2λ1(0 4,0 3,0 8) = λ1 = { x1 , 0, 0, 1 , x2 , 0.4, 0.3, 0.8 } λ2 = { x1 , 0, 0, 1 , x2 , 0, 0, 1 } , (3.7) x 2λ2(0 3,0 7,0 2) = λ1 = { x1 , 0, 0, 1 , x2 , 0, 0, 1 } λ2 = { x1 , 0, 0, 1 , x1 , 0.3, 0.7, 0.2 }
Definition3.13. Let( ,E)beaneutrosophicsoftsetovertheuniversesetX.We saythat xλ(α,β,γ) ∈ ( ,E)readasbelongingtotheneutrosophicsoftset( ,E), whenever α T (λ)(x),β I (λ)(x)and γ (λ)(x).
Definition3.14. Let(X, ,E)beaneutrosophicsofttopologicalspaceoverX.A neutrosophicsoftset( ,E)in(X, ,E)iscalledaneutrosophicsoftb-neighborhood oftheneutrosophicsoftpoint xλ(α,β,γ) ∈ ( ,E),ifthereexistsaneutrosophicsoft b-openset(G,E)suchthat xλ(α,β,γ) ∈ (G,E) ❁ ( ,E).
Theorem3.15. Let (X, ,E) beaneutrosophicsofttopologicalspaceand ( ,E) be aneutrosophicsoftsetoverX.Then ( ,E) isaneutrosophicsoftb-opensetifand onlyif ( ,E) isaneutrosophicsoftb-neighborhoodofitsneutrosophicsoftpoints.
Proof. Let( ,E)beaneutrosophicsoftb-opensetand xλ(α,β,γ) ∈ ( ,E).Then xλ(α,β,γ) ∈ ( ,E) ❁ ( ,E).Thus( ,E)isaneutrosophicsoftb-neighborhoodof xλ(α,β,γ) .
Conversely,let( ,E)beaneutrosophicsoftb-neighborhoodofitsneutrosophic softpoints.Let xλ(α,β,γ) ∈ ( ,E).Since( ,E)isaneutrosophicsoftb-neighborhood oftheneutrosophicsoftpoint xλ(α,β,γ),thereexists(G,E) ∈ suchthat x λ(α,β,γ) ∈ (G,E) ❁ ( ,E)
Since( ,E)= {xλ(α,β,γ) : xλ(α,β,γ) ∈ ( ,E)},itfollowsthat( ,E)isaunionof neutrosophicsoftb-opensets.Then( ,E)isaneutrosophicsoftb-openset.
Theb-neighborhoodsystemofaneutrosophicsoftpoint xλ(α,β,γ),denotedbyU (xλ(α,β,γ),E),isthefamilyofallitsb-neighborhoods.
Theorem3.16. TheneighborhoodsystemU (xλ(α,β,γ),E) at xλ(α,β,γ) inaneutrosophicsofttopologicalspace (X, ,E) hasthefollowingproperties.
(1) If ( ,E) ∈ U (xλ(α,β,γ),E),then xλ(α,β,γ) ∈ ( ,E)
(2) If ( ,E) ∈ U (xλ(α,β,γ),E) and ( ,E) ❁ (H,E),then (H,E) ,then (H,E) ∈ U (xλ(α,β,γ),E)
(3) If ( ,E),If (G,E) ∈ U (xλ(α,β,γ),E) ,then ( ,E) (G,E) ∈ U (xλ(α,β,γ),E)
(4) If ( ,E) ∈ U (xλ(α,β,γ),E),thenthereexistsa (G,E) ∈ U (xλ(α,β,γ),E) such that (G,E) ∈ U (yλ (α ,β ,γ ),E),foreach yλ (α ,β ,γ ) ∈ (G,E) 6
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Proof. Theproofsof(1),(2)and(3)isobviousfromDefinition 3.12. (4)Suppose( ,E) ∈ U(xλ(α,β,γ),E).Thenthereexistsaneutrosophicsoft b-openset(G,E)suchthat xλ(α,β,γ) ∈ (G,E) ❁ ( ,E).ThusbyProposition 3.1,(G,E) ∈ U(xλ(α,β,γ),E).Soforeach yλ (α ,β ,γ ) ∈ (G,E),(G,E) ∈ U (yλ (α ,β ,γ ),E).
Definition3.17. Let xλ(α,β,γ) and yλ (α ,β ,γ ) betwoneutrosophicsoftpoints.For theneutrosophicsoftpoints xλ(α,β,γ) and yλ (α ,β ,γ ) overacommonuniverseX,we saythatneutrosophicsoftpointsaredistinctpoints,if x λ(α,β,γ) y λ (α ,β ,γ ) =0(X,E) .
Itisclearthat xλ(α,β,γ) and yλ (α ,β ,γ ) aredistinctneutrosophicsoftpointsif andonlyifx =yor λ = λ
4. Neutrosophicsoftb-separationstructures
Inthissection,weconsiderneutrosophicsoftb-separationaxiomsandneutrosophicsofttopologicalsubspaceconsistingofdistinctneutrosophicsoftpointsof neutrosophicsofttopologicalspaceoverX.
Definition4.1. (i)Let(X, ,E)beaneutrosophicsofttopologicalspaceoverX, and xλ(α,β,γ) and yλ (α ,β ,γ ) aredisticntneutrosophicsoftpoints.Ifthereexist neutrosophicsoftb-opensets(F,E)and(G,E)suchthat xλ(α,β,γ) ∈ (F,E)and xλ(α,β,γ) (G,E)=0(X,E) or yλ (α ,β ,γ ) ∈ (G,E)and yλ (α ,β ,γ ) (F,E)=0(X,E) , then(X, ,E)iscalledaneutrosophicsoftb-T o-space.
(ii)Let(X, ,E)beaneutrosophicsofttopologicalspaceoverXand xλ(α,β,γ) and yλ (α ,β ,γ ) aredisticntneutrosophicsoftpoints.Ifthereexistneutrosophicsoft b-opensets(F,E)and(G,E)suchthat xλ(α,β,γ) ∈ (F,E), xλ(α,β,γ) (G,E)=0(X,E) or yλ (α ,β ,γ ) ∈ (G,E), yλ (α ,β ,γ ) (F,E)=0(X,E) , then(X, ,E)iscalledaneutrosophicsoftb-T 1-space.
(iii)Let(X, ,E)beaneutrosophicsofttopologicalspaceoverX,and xλ(α,β,γ) and yλ (α ,β ,γ ) aredisticntneutrosophicsoftpoints.Ifthereexistneutrosophicsoft b-opensets(F,E)and(G,E)suchthat xλ(α,β,γ) ∈ (F,E), yλ (α ,β ,γ ) ∈ (G,E)and(F,E) (G,E)=0(X,E) , then(X, ,E)iscalledaneutrosophicsoftb-T 2-space.
Example4.2. LetX= {x1,x2} beauniverseset,E= {λ1,λ2} beaparametersset, and(x1)λ1(0 1,0 4,0 7),(x1)λ2(0 2,0 5,0 6),(x2)λ1(0 3,0 3,0 5),and(x2)λ2(0 4,0 4,0 4) be neutrosophicsoftpoints.Thenthefamily = {0(X,E) , 1(X,E) , (F 1,E), (F 2,E), (F 3,E), (F 4,E), (F 5,E), (F 6,E), (F 7,E), (F 8,E)} ,where 7
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(F 1,E)= {x1λ1 (0 1,0 4,0 7) } ,
(F 2,E)= {(x1)λ2(0 2,0 5,0 6)} ,
(F 3,E)= {(x2)λ1(0 3, 0 3, 0 5)} ,
(F 4,E)=(F 1,E) (F 2,E),
(F 5,E)=(F 1,E) (F 3,E),
(F 6,E)=(F 2,E) (F 3,E),
(F 7,E)=(F 1,E) (F 2,E) (F 3,E),
(F 8,E)= {(x1)λ1(0 1,0 4,0 7) , (x1)λ2(0 2,0 5,0 6) , (x2)λ2(0 3,0 3,0 5) , (x2)λ2(0 4, 0 4, 0 4)} isaneutrosophicsofttopologyoverX.Thus(X, ,E)isaneutrosophicsofttopologicalspaceoverX.Also,(X, ,E)isaneutrosophicsoftb- T 0-spacebutnota neutrosophicsoftb-T 1-spacebecauseforneutrosophicsoftpoints(x1)λ1(0 1, 0 4, 0 7) and(x2)λ2(0.4, 0.4, 0.4),(X, ,E)isnotaneutrosophicsoftb-T 1-space.
Example4.3. LetX= N beanaturalnumberssetandE= {λ} beaparameters set.Here nλ(αn,βn,γn) areneutrosophicsoftpoints.Herewecangive(αn,βn,γn) appropriatevaluesandtheneutrosophicsoftpoints nλ(αn,βn,γn) ,mλ(αn,βn,γn) are distinctneutrosophicsoftpointsifandonlyifn =m.Itisclearthatthereisoneto-onecompatibitilybetweenthesetofnaturalnumbersandthesetofneutrosophic softpoints N λ = {nλ(αn,βn,γn)}
Thenwegivecofinitetopologyonthisset.Thenneutrosophicsoftset(F,E) isaneutrosophicsoftb-opensetifandonlyifthefiniteneutrosophicsoftpointis discardedfrom N λ.Hence,(X, ,E)isaneutrosophicsoftb-T 1-spacebutnota neutrosophicsoftb-T 2-space.
Example4.4. LetX= {x1,x2} beauniverseset,E= {λ1,λ2} beaparametersset, and x1λ1(0 1,0 4,0 7) ,x1λ2(0 2,0 5,0 6) ,x2λ1(0 3,0 3,0 5),and x2λ2(0 4,0 4,0 4),beneutrosophicsoftpoints.Thenthefamily = {0(X,E) , 1(X,E) , ( 1,E), ( 2,E),..., ( 15,E)} ,where
( 1,E)= {x1λ1(0 1,0 4,0 7)} ,
( 2,E)= {(x1)λ2(0 2,0 5,0 6)} ,
( 3,E)= {(x2)λ1(0 3,0 3,0 5)} ,
( 4,E)= {(x2)λ2(0 4,0 4,0 4)} ,
(F 5,E)=(F 1,E) (F 2,E), (F 6,E)=(F 1,E) (F 3,E), (F 7,E)=(F 1,E) (F 4,E), (F 8,E)=(F 2,E) (F 3,E), (F 9,E)=(F 2,E) (F 4,E), (F 10,E)=(F 3,E) (F 4,E), (F 11,E)=(F 1,E) (F 2,E) (F 3,E), (F 12,E)=(F 1,E) (F 2,E) (F 4,E), (F 13,E)=(F 2,E) (F 3,E) (F 4,E), (F 14,E)=(F 1,E) (F 3,E) (F 4,E), 8
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(F 15,E)= {(x1)λ1(0 1,0 4,0 7) , (x1)λ2(0 2,0 5,0 6) , (x2)λ2(0 3,0 3,0 5) , (x2)λ2(0 4,0 4,0 4)}
isaneutrosophicsofttopologyoverX.Thus(X, ,E)isaneutrosophicsofttopologicalspaceoverX.Also,(X, ,E)isaneutrosophicsoftb-T 2-space. Theorem4.5. Let (X, ,E) beaneutrosophicsofttopologicalspaceoverX.Then (X, ,E) isaneutrosophicsoftb-T 1-spaceifandonlyifeachneutrosophicsoftpoint isaneutrosophicsoftb-closedset.
Proof. Let(X, ,E)beaneutrosophicsoftb-T 1-spaceand xλ(α,β,γ) beanarbitrary neutrosophicsoftpoint.Weshowthat(xλ(α,β,γ)) λ isaneutrosophicsoftb-open set.Let yλ (α ,β ,γ ) ∈ (xλ(α,β,γ)) λ .Then xλ(α,β,γ) and yλ (α ,β ,γ ) aredistinct neutrosophicsoftpoints.Thusx =yor λ = λ.
Since(X, ,E)isaneutrosophicsoftb-T 1-space,thereexistsaneutrosophicsoft b-openset(G,E)suchthat y λ (α ,β ,γ ) ∈ (G,E)and x λ(α,β,γ) (G,E)=0(X,E) Since xλ(α,β,γ) (G,E)=0(X,E),wehave yλ (α ,β ,γ ) ∈ (G,E) ❁ (xλ(α,β,γ)) λ .Thus (xλ(α,β,γ)) λ isaneutrosophicsoftb-openset,i.e., xλ(α,β,γ) isaneutrosophicsoft b-closedset. Supposethateachneutrosophicsoftpoint xλ(α,β,γ) isaneutrosophicsoftb-closed set.Then(xλ(α,β,γ)) λ isaneutrosophicsoftopenset.Let xλ(α,β,γ) yλ (α ,β ,γ ) =0(X,E).Thus yλ (α ,β ,γ ) ∈ (xλ(α,β,γ)) λ and xλ(α,β,γ) (xλ(α,β,γ)) λ =0(X,E).So (X, ,E)isaneutrosophicsoftb-T 1-spaceoverX.
Theorem4.6. Let (X, ,E) beaneutrosophicsofttopologicalspaceoverX.Then (X, ,E) isaneutrosophicsoftb-T 2-spaceifffordistinctneutrosophicsoftpoints xλ(α,β,γ) and yλ (α ,β ,γ ),thereexistsaneutrosophicsoftb-openset ( ,E) containing xλ(α,β,γ) butnot yλ (α ,β ,γ ) suchthat yλ (α ,β ,γ ) doesnotbelongto ( ,E).
Proof. Let xλ(α,β,γ) and yλ (α ,β ,γ ) betwoneutrosophicsoftpointsinneutrosophic softb-T 2-space(X, ,E).Thenthereexistdisjointneutrosophicsoftb-openset ( ,E),(G,E)suchthat xλ(α,β,γ) ∈ ( ,E), yλ (α ,β ,γ ) ∈ (G,E).Since xλ(α,β,γ) yλ (α ,β ,γ ) =0(X,E) and( ,E) (G,E)=0(X,E) , yλ (α ,β ,γ ) doesnotbelong to( ,E).Itimpliesthat yλ (α ,β ,γ ) doesnotbelongto( ,E).
Nextsupposethat,fordistinctneutrosophicsoftpoints xλ(α,β,γ) , yλ (α ,β ,γ ) , thereexistsaneutrosophicsoftb-openset( ,E)containing xλ(α,β,γ) butnot yλ (α ,β ,γ ) suchthat yλ (α ,β ,γ ) doesnotbelongto( ,E).Then yλ (α ,β ,γ ) ∈ (( ,E))c ,i.e.,( ,E)and(( ,E))c aredisjointneutrosophicsoftb-opensetscontaining xλ(α,β,γ) , yλ (α ,β ,γ ),respectively.
Theorem4.7. Let (X, ,E) beaneutrosophicsoftb-T 1-spaceforeveryneutrosophicsoftpoint xλ(α,β,γ) ∈ ( ,E) ∈ .Ifthereexistsaneutrosophicsoftb-open 9
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set (G,E) suchthat xλ(α,β,γ) ∈ (G,E) ❁ (G,E) ❁ ( ,E),then (X, ,E) isaneutrosophicsoftb-T 2-space.
Proof. Supposethat xλ(α,β,γ) yλ (α ,β ,γ ) =0(X,E).Since(X, ,E)isaneutrosophicsoft T 1-space, xλ(α,β,γ) and yλ (α ,β ,γ ) areneutrosophicsoftb-closedsets in .Then xλ(α,β,γ) ∈ (yλ (α ,β ,γ ))c ∈ .Thusthereexistsaneutrosophicsoft b-openset(G,E)in suchthat xλ(α,β,γ) ∈ (G,E) ❁ (G,E) ❁ (yλ (α ,β ,γ ))c . Sowehave yλ (α ,β ,γ ) ∈ ((G,E))c , xλ(α,β,γ) ∈ (G,E)and(G,E) ((G,E))c = 0(X,E),i,e.,(X, ,E)isaneutrosophicsoftb-T 2-space.
Remark4.8. Let(X, ,E)beaneutrosophicsoftb-T 1-spacefori=0,1,2. Foreachx =y,neutrosophicpoints x(α,β,γ)and y(α ,β ,γ )haveneighborhoods satisfyingconditionsofb-T i-spaceinneutrosophictopologicalspace(X, λ)foreach λ ∈ E because xλ(α,β,γ) and yλ (α ,β ,γ ) aredistinctneutrosophicsoftpoints.
Definition4.9. Let(X, ,E)beaneutrosophicsofttopologicalspaceoverX, ( ,E)beaneutrosophicsoftb-closedsetand xλ(α,β,γ) ( ,E)=0(X,E).Ifthere existneutrosophicsoftb-openopensets(1,E)and(G2,E)suchthat xλ(α,β,γ) ∈ (G1,E),( ,E) ❁ (G2,E),and(G1,E) (G2,E)=0(X,E),then(X, ,E)iscalled aneutrosophicsoftb-regularspace.(X, ,E)issaidtobeaneutrosophicsoftbT 3-spaceifisbothaneutrosophicsoftb-regularandneutrosophicsoftb-T 1-space. Theorem4.10. Let (X, ,E) beaneutrosophicsofttopologicalspaceoverX, (X, ,E) isaneutrosophicsoftb-T 3-spaceifandonlyifforevery xλ(α,β,γ) ∈ ( ,E) ∈ ,thereexists (G,E) ∈ suchthat xλ(α,β,γ) ∈ (G,E) ❁ (G,E) ❁ ( ,E).
Proof. Let(X, ,E)beaneutrosophicsoftb-T 3-spaceand xλ(α,β,γ) ∈ ( ,E) ∈ .Since(X, ,E)isaneutrosophicsoftb-T 3-spacefortheneutrosophicsoftpoint xλ(α,β,γ) andneutrosophicsoftb-closedset( ,E)c,thereexist(G1,E),(G2,E) ∈ suchthat xλ(α,β,γ) ∈ (G1,E),( ,E)c ❁ (G2,E),and(G1,E) (G2,E)= 0(X,E).Thenwehave xλ(α,β,γ) ∈ (G1,E) ❁ (G2,E)c ❁ ( ,E).Since(G2,E)c isa neutrosophicsoftb-closedset,(G1,E) ❁ (G2,E)c .
Conversely,let xλ(α,β,γ) (H,E)=0(X,E) and(H,E)beaneutrosophicsoft b-closedset.Then xλ(α,β,γ) ∈ (H,E)c andfromtheconditionofthetheorem,we have xλ(α,β,γ) ∈ (G,E) ❁ (G,E) ❁ (H ,E)c Thus xλ(α,β,γ) ∈ (G,E),(H,E) ❁ ((G,E))c,and(G,E) (G,E)c =0(X,E).So (X, ,E)isaneutrosophicsoftb-T 3-space.
Definition4.11. Aneutrosophicsofttopologicalspace(X, ,E)overXiscalled aneutrosophicsoftb-normalspace,ifforeverypairofdisjointneutrosophicsoft b-closedset( 1,E),( 2,E),thereexistsdisjointneutrosophicsoftb-opensets (G1,E),(G2,E)suchthat( 1,E) ❁ (G1,E)and( 2,E) ❁ (G2,E). (X, ,E)issaidtobeaneutrosophicsoftb-T 4-space,ifitisbothaneutrosophic softb-normalandneutrosophicsoftb-T 1-space. 10
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Theorem4.12. Let (X, ,E) beaneutrosophicsofttopologicalspaceoverX.Then (X, ,E) isaneutrosophicsoftb-T 4-spaceifandonlyif,foreachneutrosophicsoft b-closedset ( ,E) andneutrosophicsoftb-openset (G,E) with ( ,E) ❁ (G,E), thereexistsaneutrosophicsoftb-openset (D,E) suchthat
( ,E) ❁ (D,E) ❁ (D,E) ❁ (G,E)
Proof. Let(X, ,E)beaneutrosophicsoftb-T 4-space,let( ,E)beaneutrosophic softb-closedsetandlet( ,E) ❁ (G,E) ∈ .Then(G,E)c isaneutrosophicsoft b-closedsetand( ,E) (G,E)c =0(X,E).Since(X, ,E)isaneutrosophicsoft b-T 4-space,thereexistneutrosophicsoftb-opensets(D1,E)and(D2,E)suchthat ( ,E) ❁ (D1,E), (G,E)c ❁ (D2,E)and(D1,E) (D2,E)=0(X,E) Thus( ,E) ❁ (D1,E) ❁ (D2,E)c ❁ (G,E),(D2,E)c isaneutrosophicsoftb-closed setand(D1,E) ❁ (D2,E)c.So( ,E) ❁ (D1,E) ❁ (D1,E) ❁ (G,E). Conversely,let( 1,E),( 2,E)betwodisjointneutrosophicsoftb-closedsets. Then( 1,E) ❁ ( 2,E)c.Fromtheconditionoftheorem,thereexistsaneutrosophic softb-openset(D,E)suchthat( 1,E) ❁ (D,E) ❁ (D1,E) ❁ ( 2,E)c.Thus (D,E),((D,E))c areneutrosophicsoftb-opensetsand( 1,E) ❁ (D,E),( 2,E) ❁ ((D,E))c and(D,E) ((D,E))c =0(X,E).So(X, ,E)isaneutrosophicsoft b-T 4-space.
Definition4.13. Let(X, ,E)beaneutrosophicsofttopologicalspaceoverX and( ,E)beanarbitraryneutrosophicsoftset.Then ( ,E) = {( ,E) (H,E): (H,E) ∈ } issaidtobeneutrosophicsofttopologyon( ,E)and(( ,E), ( ,E),E) iscalledaneutrosophicsofttopologicalsubspaceof(X, ,E).
Theorem4.14. Let (X, ,E) beaneutrosophicsofttopologicalspaceoverX.If (X, ,E) isaneutrosophicsoftb-T i-space,thentheneutrosophicsofttopological subspace (( ,E), ( ,E),E) isaneutrosophicsoftb-T i-spacefori=0,1,2,3.
Proof. Let xλ(α,β,γ) , yλ (α ,β ,γ ) ∈ (( ,E), ( ,E),E)suchthat xλ(α,β,γ) yλ (α ,β ,γ ) =0(X,E).Thenthereexistneutrosophicsoftb-openset( 1,E)and( 2,E)satisfyingtheconditionsofneutrosophicsoftb-T i-spacesuchthat xλ(α,β,γ) ∈ ( 1,E), yλ (α ,β ,γ ) ∈ ( 2,E).Thus xλ(α,β,γ) ∈ ( 1,E) ( ,E)and yλ (α ,β ,γ ) ∈ ( 2,E) ( ,E).Also,theneutrosophicsoftb-openset( 1,E) ( ,E),( 2,E) ( ,E) in ( ,E) satisfytheconditionsofneutrosophicsoftb-T i-spacefori=0,1,2,3.
Theorem4.15. Let (X, ,E) beaneutrosophicsofttopologicalspaceoverX.If (X, ,E) isaneutrosophicsoftb-T 4-spaceand ( ,E) isaneutrosophicsoftb-closed setin (X, ,E),then (( ,E), ( ,E),E) isaneutrosophicsoftb-T 4-space. Proof. Let(X, ,E)beaneutrosophicsoftb-T 4-spaceand( ,E)beaneutrosophic softb-closedsetin(X, ,E).Let( 1,E)and( 2,E)betwoneutrosophicsoftbclosedsetsin(( ,E), ( ,E),E)suchthat( 1,E) ( 2,E)=0(X,E).When( ,E) isaneutrosophicsoftb-closedsetin(X, ,E),( 1,E)and( 2,E)areneutrosophic 11
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softb-closedsetsin(X, ,E).Since(X, ,E)isaneutrosophicsoftb-T 4-space, thereexistneutrosophicsoftb-opensets(G1,E)and(G2,E)suchthat( 1,E) ❁ (G1,E),( 2,E) ❁ (G2,E)and(G1,E) (G2,E)=0(X,E).Then( 1,E)=(G1,E) ( ,E),( 2,E)=(G2,E) ( ,E)and((G1,E) ( ,E)) ((G2,E) ( ,E))= 0(X,E).Thisimpliesthat(( ,E), ( ,E),E)isaneutrosophicsoftb-T 4-space.
5. Conclusion
Neutrosophicsoftb-separationstructuresarethemostimperativeandfascinatingnotionsinneutrosophicsofttopologywehaveintroducedneutrosophicsoftbseparationaxiomsinneutrosophicsofttopologicalstructureswithrespecttosoft points,whicharedefinedoveraninitialuniverseofdiscoursewithafixedsetof variables.Wefurtherinvestigatedandscrutinizedsomeessentialfeaturesofthe initiatedneutrosophicsoftb-separationstructures.Itissupposedthattheseresults willbeveryveryusefulforfuturestudiesonneutrosophicsofttopologytocarryout ageneralframeworkforpracticalapplications.Applicationsofneutrosophicsoft b-separationstructuresinneutrosophicsofttopologicalspacescanbetracedoutin decisionmakingproblems.
Acknowledgements. Theauthorsexpresstheirbasketfulofthankstothe refereeforbringingthemanuscriptinmeaningfulform.
References
[1] K.Atanassov,Intuitionisticfuzzysets,FuzzysetsSyst.20(1986)87–96.
[2] S.BayramovandC.Gunduz,Onintuitionisticfuzzysofttopologicalspaces,TWMSJPure ApplMath.5(2014)66–79.
[3] S.BayramovandC.Gunduz,Anewapproachtoseparabilityandcompactnessinsofttopologicalspaces,TWMSJPureApplMath.9(2018)82–93.
[4] T.BeraandN.K.Mahapatra,Onneutrosophicsoftfunction,AnnfuzzyMathInform.12 (1)(2016)101–119.
[5] T.BeraandN.K.Mahapatra,Introductiontoneutrosophicsofttopologicalspace,Opsearch. 54(2017)841–867.
[6] N.Cagman,S.KaratasandS.Enginoglu,Softtopology,ComputMathAppl.62(2011) 351–358.
[7] I.DeliandS.Broumi,Neutrosophicsoftrelationsandsomeproperties,AnnFuzzyMath Inform.9(1)(2015)169–182.
[8] A.C.GunduzandS.Bayramov,OntheTietzeextensiontheoreminsofttopologicalspaces, ProceedingsoftheInstituteofMathematicsandMechanicsoftheNationalAcademyofSciencesofAzerbaijan43(2017)105–115.
[9] S.Guzide,ANewConstructionofSpheresViaSoftRealNumbersandSoftPoints,MathematicsLetters.4(3)(2018)39–43.
[10] S.Guzide,SoftTopologyGeneratedbyL-SoftSets,JournalofNewTheory.4(24)(2018) 88–100.
[11] S.Guzide,TheParameterizationsReductionofSoftPointanditsApplicationswithSoft Matrix,InternationalJournalofComputerApplications.164(1)(2017)1–6.
[12] S.Guzide,AComparativeResearchonthedefinitionofSoftPoint,InternationalJournalof ComputerApplications.163(2)(2017)1–4.
[13] P.K.MajI,Neutrosophicsoftset,AnnFuzzyMathInform.5(1)(2013)157–168.
[14] D.Molodtsov,Softsettheory-firstresults,ComputMathAppl.37(1999)19–31.
[15] A.A.SalmaandS.A.Alblowi,NeutrosophicsetandNeutrosophictopologicalspaces,IOSR JMath.3(2012)31–35.
12
ArifMehmoodKhattaketal./Ann.FuzzyMath.Inform. x (201y),No.x,xx–xx
[16] M.ShabirandM.Naz,Onsofttopologicalspaces,IOSRJMath.61(2011)1786–1799.
[17] F.Smarandache,Neutrosophicset,ageneralisationoftheintuitionisticfuzzysets,IntJPure ApplMath.24(2005)287–297.
[18] L.A.Zadeh,Fuzzysets,InformControl.8(1965)338–353.
ArifMehmoodKhattak (mehdaniyal@gmail.com)
DepartmentofMathematicsandStatistics,RiphahInternationalUniversity,Sector, I-14,Islamabad,Pakistan
NaziaHanif (aliangul333@gmail.com)
DepartmentofMathematics,UniversityofScienceandTechnology,Bannu,Khyber Pakhtunkhwa,Pakistan
FawadNadeem (fawadnadeem2@gmail.com )
DepartmentofMathematics,UniversityofScienceandTechnology,Bannu,Khyber Pakhtunkhwa,Pakistan
MuhammadZamir (zamirburqi@ustb.edu.pk)
DepartmentofMathematics,UniversityofScienceandTechnology,Bannu,Khyber Pakhtunkhwa,Pakistan
ChoonkilPark (baak@hanyang.ac.kr)
DepartmentofMathematics,ResearchinstituteforNaturalSciences,HanyangUniversity,Seoul133-791,RepublicofKorea
GiorgioNordo (giorgio.nordo@unime.it)
MIFT-DipartimentodiDcienzeMatematicheeInformatiche,ScienzeFisicheescienze dellaTerra,MessinaUniversity,Messina,Italy
ShamoonaJabeen (shamoonaafzal@yahoo.com )
SchoolofMathematicsandsystemScience,BeihangUniversityBeijingChina
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