A New Approach to Operations on Neutrosophic Soft Sets and to Neutrosophic Soft Topological Spaces

Vol.10,No.3,pp.481–493,2019

ISSN0975-8607(online);0976-5905(print)

PublishedbyRGNPublications http://www.rgnpublications.com

DOI:10.26713/cma.v10i3.1068 ResearchArticle

ANewApproachtoOperationsonNeutrosophic SoftSetsandtoNeutrosophicSoftTopological Spaces

TahaYasinOzturk1,*,CigdemGunduzAras2 andSadiBayramov3

1DepartmentofMathematics,KafkasUniversity,Kars36100,Turkey

2DepartmentofMathematics,KocaeliUniversity,Kocaeli41380,Turkey

3DepartmentofAlgebraandGeometry,BakuStateUniversity,Baku,AZ1148,Azerbaican

*Correspondingauthor: taha36100@kafkas.edu.tr

Abstract. Inthisstudy,were-definesomeoperationsonneutrosophicsoftsetsdifferentlyfromthe studies[3,9].Onthisoperationsaregiveninterestingexamplesandthembasicproperties.Inthe directionofthesenewlydefinedoperations,weconstructtheneutrosophicsofttopologicalspaces differentlyfromthestudy[3].Finally,weintroducebasicdefinitionsandtheoremsonneutrosophic softtopologicalspaces.

Keywords. Neutrosophicsoftset;Neutrosophicsofttopologicalspace;Neutrosophicsoftinterior; Neutrosophicsoftclosure

MSC. 54A40;54E55;54D10

Received: August17,2018 Accepted: February7,2019

Copyright © 2019TahaYasinOzturk,CigdemGunduzArasandSadiBayramov. Thisisanopenaccessarticle distributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,and reproductioninanymedium,providedtheoriginalworkisproperlycited

1.Introduction

Thecontributionofmathematicstothepresent-daytechnologyinreachingtoafasttrendcannot beignored.Thetreoriespresenteddifferentlyfromclassicalmethodsinstudiessuchasfuzzy set[15],intuitionisticset[7],softset[11],neutrosophicset[14],etc.Havegreatimportance inthiscontributionofmathematicsinrecentyears.Manyworkshavebeendoneonthesesets bymathematiciansinmanyareasofmathematics[2–6,8,12,13].Inaddition,manystudies

ondifferentcombinationofthesesettheorieshavebeenpresented[1,9,10].Oneofthemis Neurosophicsoftsets[14].NeutrosophicsofttopologicalspaceswaspresentedbyBerainhis work[3].

Inourstudy,theintersection,union,AND,ORanddifferenceoperationsarere-defined ontheneutrosophicsoftsetsincontrasttothestudies[3,9],andthepropertiesrelatedto theseoperationsarepresented.Then,consideringthesenewlydefinedprocesses,unlike[3], neutrosophicsofttopologyisreconstructed.Inaddition,relationsbetweenthespaces neutrosophicsofttopology,fuzzysofttopologyandfuzzytopologyareobserved.Finally,by defininginteriorandclosureoperations,fundamentaltheoremsforneutrosophicsofttopological spacesareprovedandsomeexamplesonthesubjectaregiven.

Inthepreliminariessection,wegivefundamentalinformationforthestudy.Inthenext section,theoperationsofunion,intersection,difference,AND,ORonneutrosophicsoftsetsare redefinedandtheirpropertiesareinvestigated.Then,thenextsectionstudiestheneutrosophic softtopologyandtheirnotionsbasedontheseredefinedoperations.Finally,weprovidea conclusionsectionaboutthesenewconceptsforourpaper.

2.Preliminaries

Inthissection,wewillgivesomepreliminaryinformationforthepresentstudy.

Definition1 ([14]). Aneutrosophicset A ontheuniverseofdiscourse X isdefinedas: A = {〈x, T A (x), I A (x), FA (x)〉 : x ∈ X }, where T, I, F : X →] 0, 1+[and 0 ≤ T A (x) + I A (x) + FA (x) ≤ 3+

Definition2 ([11]). Let X beaninitialuniverse, E beasetofallparametersand P(X )denotes thepowersetof X Apair(F, E)iscalledasoftsetover X , where F isamappinggivenby F : E → P(X ).

Inotherwords,thesoftsetisaparameterizedfamilyofsubsetsoftheset X .For e ∈ E, F(e) maybeconsideredasthesetof e-elementsofthesoftset(F, E), orasthesetof e-approximate elementsofthesoftset,i.e., (F, E) = {(e, F(e)): e ∈ E, F : E → P(X )}.

Firstly,neutrosophicsoftsetdefinedbyMaji[9]andlaterthisconcepthasbeenmodifiedby DeliandBromi[8]asgivenbelow:

Definition3. Let X beaninitialuniversesetand E beasetofparameters.Let P(X )denote thesetofallneutrosophicsetsof X .Then,aneutrosophicsoftset(F, E)over X isasetdefined byasetvaluedfunction F representingamapping F : E → P(X )where F iscalledapproximate functionoftheneutrosophicsoftset(F, E).Inotherwords,theneutrosophicsoftsetisa parameterizedfamilyofsomeelementsoftheset P(X )andthereforeitcanbewrittenasaset oforderedpairs, (F, E) = {(e, 〈x, TF(e)(x), IF(e)(x), FF(e)(x)〉 : x ∈ X ): e ∈ E},

CommunicationsinMathematicsandApplications,Vol.10,No.3,pp.481–493,2019

: T.Y.Ozturketal.

where TF(e)(x), IF(e)(x), FF(e)(x) ∈ [0, 1],respectivelycalledthetruth-membership, indeterminacy-membership,falsity-membershipfunctionof F(e).Sincesupremumofeach T, I, F is1sotheinequality0 ≤ TF(e)(x) + IF(e)(x) + FF(e)(x) ≤ 3isobvious.

Definition4 ([3]). Let(F, E)beneutrosophicsoftsetovertheuniverseset X Thecomplement of(F, E)isdenotedby(F, E)c andisdefinedby: (F, E)c = {(e, 〈x, FF(e)(x), 1 IF(e)(x), TF(e)(x)〉 : x ∈ X ): e ∈ E}. Obviousthat,((F, E)c)c = (F, E).

Definition5 ([9]). Let(F, E)and(G, E)betwoneutrosophicsoftsetsovertheuniverseset X .(F, E)issaidtobeneutrosophicsoftsubsetof(G, E)if TF(e)(x) ≤ TG(e)(x), IF(e)(x) ≤ IG(e)(x), FF(e)(x) ≥ FG(e)(x),forall e ∈ E, forall x ∈ X .Itisdenotedby(F, E) ⊆ (G, E) (F, E)issaidtobeneutrosophicsoftequalto(G, E)if(F, E)isneutrosophicsoftsubsetof(G, E) and(G, E)isneutrosophicsoftsubsetof(F, E). Itisdenotedby(F, E) = (G, E).

3.ANewApproachtoOperationsonNeutrosophicSoftSets

Inthissection,theoperationsofunion,intersection,difference,AND,ORonneutrosophic softsetsaredefineddifferentlyfromthestudies[3,9].Inaddition,basicpropertiesofthese operationswillbepresented.

Definition6. Let(F1, E)and(F2, E)betwoneutrosophicsoftsetsovertheuniverseset X . Thentheirunionisdenotedby(F1, E) ∪ (F2, E) = (F3, E)andisdefinedby: (F3, E) = {(e, 〈x, TF3(e)(x), IF3(e)(x), FF3(e)(x)〉 : x ∈ X ): e ∈ E}, where TF3(e)(x) = max{TF1(e)(x), TF2(e)(x)}, IF3(e)(x) = max{IF1(e)(x), IF2(e)(x)}, FF3(e)(x) = min{FF1(e)(x), FF2(e)(x)}.

Definition7. Let(F1, E)and(F2, E)betwoneutrosophicsoftsetsovertheuniverseset X . Thentheirintersectionisdenotedby(F1, E) ∩ (F2, E) = (F3, E)andisdefinedby: (F3, E) = {(e, 〈x, TF3(e)(x), IF3(e)(x), FF3(e)(x)〉 : x ∈ X ): e ∈ E}, where TF3(e)(x) = min{TF1(e)(x), TF2(e)(x)}, IF3(e)(x) = min{IF1(e)(x), IF2(e)(x)}, FF3(e)(x) = max{FF1(e)(x), FF2(e)(x)}.

Definition8. Let(F1, E)and(F2, E)betwoneutrosophicsoftsetsovertheuniverseset X . Then“(F1, E)difference(F2, E)”operationonthemisdenotedby(F1, E)\(F2, E) = (F3, E)andis

CommunicationsinMathematicsandApplications,Vol.10,No.3,pp.481–493,2019

definedby(F3, E) = (F1, E) ∩ (F2, E)c asfollows: (F3, E) = {(e, 〈x, TF3(e)(x), IF3(e)(x), FF3(e)(x)〉 : x ∈ X ): e ∈ E}, where TF3(e)(x) = min{TF1(e)(x), FF2(e)(x)}, IF3(e)(x) = min{IF1(e)(x), 1 IF2(e)(x)}, FF3(e)(x) = max{FF1(e)(x), TF2(e)(x)}.

Definition9. Let {(Fi, E)|i ∈ I} beafamilyofneutrosophicsoftsetsovertheuniverseset X . Then ∪ i∈I (Fi, E) = {(e, 〈x, sup[TFi (e)(x)]i∈I , sup[IFi (e)(x)]i∈I , inf[FFi (e)(x)]i∈I 〉 : x ∈ X ): e ∈ E}, ∩ i∈I (Fi, E) = {(e, 〈x, inf[TFi (e)(x)]i∈I , inf[IFi (e)(x)]i∈I , sup[FFi (e)(x)]i∈I 〉 : x ∈ X ): e ∈ E}

Definition10. Let(F1, E)and(F2, E)betwoneutrosophicsoftsetsovertheuniverseset X . Then“AND”operationonthemisdenotedby(F1, E) ∧ (F2, E) = (F3, E × E)andisdefinedby: (F3, E × E) = {((e1, e2), 〈x, TF3(e1,e2)(x), IF3(e1,e2)(x), FF3(e1,e2)(x)〉 : x ∈ X ):(e1, e2) ∈ E × E}, where TF3(e1,e2)(x) = min{TF1(e1)(x), TF2(e2)(x)}, IF3(e1,e2)(x) = min{IF1(e1)(x), IF2(e2)(x)}, FF3(e1,e2)(x) = max{FF1(e1)(x), FF2(e2)(x)}.

Definition11. Let(F1, E)and(F2, E)betwoneutrosophicsoftsetsovertheuniverseset X . Then“OR”operationonthemisdenotedby(F1, E) ∨ (F2, E) = (F3, E × E)andisdefinedby: (F3, E × E) = {((e1, e2), 〈x, TF3(e1,e2)(x), IF3(e1,e2)(x), FF3(e1,e2)(x)〉 : x ∈ X ):(e1, e2) ∈ E × E}, where TF3(e1,e2)(x) = max{TF1(e1)(x), TF2(e2)(x)}, IF3(e1,e2)(x) = max{IF1(e1)(x), IF2(e2)(x)}, FF3(e1,e2)(x) = min{FF1(e1)(x), FF2(e2)(x)}.

issaidtobenull neutrosophicsoftsetif

F(e)(x) = 0,

F, E)overtheuniverseset

1. (F1, E) ∪ [(F2, E) ∪ (F3, E)] = [(F1, E) ∪ (F2, E)] ∪ (F3, E) and (F1, E) ∩ [(F2, E) ∩ (F3, E)] = [(F1, E) ∩ (F2, E)] ∩ (F3, E);

2. (F1, E) ∪ [(F2, E) ∩ (F3, E)] = [(F1, E) ∪ (F2, E)] ∩ [(F1, E) ∪ (F3, E)] and (F1, E) ∩ [(F2, E) ∪ (F3, E)] = [(F1, E) ∩ (F2, E)] ∪ [(F1, E) ∩ (F3, E)]; 3. (F1, E) ∪ 0(X ,E) = (F1, E) and (F1, E) ∩ 0(X ,E) = 0(X ,E); 4. (F1, E) ∪ 1(X ,E) = 1(X ,E) and (F1, E) ∩ 1(X ,E) = (F1, E) Proof. Straightforward. Proposition2. Let (F1, E) and (F2, E) betwoneutrosophicsoftsetsovertheuniverseset X Then, 1. [(F1, E) ∪ (F2, E)]c = (F1, E)c ∩ (F2, E)c; 2. [(F1, E) ∩ (F2, E)]c = (F1, E)c ∪ (F2, E)c Proof. 1.Forall e ∈ E and x ∈ X , (F1, E) ∪ (F2, E) = {〈x, max{TF1(e)(x), TF2(e)(x)}, max{IF1(e)(x), IF2(e)(x)}, min{FF1(e)(x), FF2(e)(x)}〉} [(F1, E) ∪ (F2, E)]c = {〈x, min{FF1(e)(x), FF2(e)(x)}, 1 max{IF1(e)(x), IF2(e)(x)}, max{TF1(e)(x), TF2(e)(x)}〉}

Now, (F1, E)c = {〈x, FF1(e)(x), 1 IF1(e)(x), TF1(e)(x)〉}, (F2, E)c = {〈x, FF2(e)(x), 1 IF2(e)(x), TF2(e)(x)〉}.

Then, (F1, E)c ∩ (F2, E)c = {〈x, min{FF1(e)(x), FF2(e)(x)}, min{(1 IF1(e)(x)), (1 IF2(e)(x))}, max{TF1(e)(x), TF2(e)(x)}〉} = {〈x, min{FF1(e)(x), FF2(e)(x)}, 1 max{IF1(e)(x), IF2(e)(x)}, max{TF1(e)(x), TF2(e)(x)}〉} Therefore,[(F1, E) ∪ (F2, E)]c = (F1, E)c ∩ (F2, E)c 2.Itisobtainedinasimilarway. Proposition3. Let (F1, E) and (F2, E) betwoneutrosophicsoftsetsovertheuniverseset X . Then,

(F2, E)c = {〈x, FF2(e2)(x), 1 IF2(e2)(x), TF2(e2)(x)〉 : e2 ∈ E}.

Then, (F1, E)c ∧ (F2, E)c = {〈x, min{FF1(e1)(x), FF2(e2)(x)}, min{(1 IF1(e1)(x)), (1 IF2(e2)(x))}, max{TF1(e1)(x), TF2(e2)(x)}〉} = {〈x, min{FF1(e1)(x), FF2(e2)(x)}, 1 max{IF1(e1)(x), IF2(e2)(x)}, max{TF1(e1)(x), TF2(e2)(x)}〉} Hence,[(F1, E) ∨ (F2, E)]c = (F1, E)c ∧ (F2, E)c

Example1. Supposethat,theuniverseset X givenby X = {x1, x2, x3, x4} andthesetof parameters E = {e1, e2}.Letusconsiderneutrosophicsoftsets(F1, E)and(F2, E)overthe universeset X asfollows: (F1, E) = e1 = {〈x1, 0 3, 0 7, 0 6〉, 〈x2, 0 4, 0 3, 0 8, 〈x3, 0 6, 0 4, 0 5〉, 〈x4, 0 2, 0 5, 0 4〉}, e2 = {〈x1, 0 4, 0 6, 0 8〉, 〈x2, 0 3, 0 7, 0 2〉, 〈x3, 0 3, 0 3, 0 7〉, 〈x4, 0 1, 0 4, 0 9〉} , (F2, E) = e1 = {〈x1, 0 6, 0 6, 0 8〉, 〈x2, 0 2, 0 9, 0 3〉, 〈x3, 0 1, 0 2, 0 4〉, 〈x4, 0 5, 0 4, 0 3〉}, e2 = {〈x1, 0 7, 0 9, 0 5〉, 〈x2, 0 4, 0 2, 0 3〉, 〈x3, 0 5, 0 5, 0 4〉, 〈x4, 0 4, 0 3, 0 6〉} .



 



      , (F

 



Then(X , NSS τ , E)issaidtobeaneutrosophicsofttopologicalspaceover X .Eachmembersof NSS τ issaidtobeneutrosophicsoftopenset.

Definition14. Let(X , NSS τ , E)beaneutrosophicsofttopologicalspaceover X and(F, E)be aneutrosophicsoftsetover X .Then(F, E)issaidtobeneutrosophicsoftclosedsetiffits complementisaneutrosophicsoftopenset

Proposition4. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X .Then

1. 0(X ,E) and 1(X ,E) areneutrosophicsoftclosedsetsover X

2. theintersectionofanynumberofneutrosophicsoftclosedsetsisaneutrosophicsoftclosed setover X

3. theunionoffinitenumberofneutrosophicsoftclosedsetsisaneutrosophicsoftclosedset over X .

Proof. Itiseasilyobtainedfromthedefinitionneutrosophicsofttopologicalspaceand Proposition3.

Definition15. Let NSS(X , E)bethefamilyofallneutrosophicsoftsetsovertheuniverse set X

1. If NSS τ = {0(X ,E), 1(X ,E)}, then NSS τ issaidtobetheneutrosophicsoftindiscretetopology and(X , NSS τ , E)issaidtobeaneutrosophicsoftindiscretetopologicalspaceover X .

2. If NSS τ = NSS(X , E), then NSS τ issaidtobetheneutrosophicsoftdiscretetopologyand (X , NSS τ , E)issaidtobeaneutrosophicsoftdiscretetopologicalspaceover X

Proposition5. Let (X , NSS τ1 , E) and (X , NSS τ2 , E) betwoneutrosophicsofttopologicalspacesover thesameuniverseset X .Then (X , NSS τ1 ∩ NSS τ2 , E) isneutrosophicsofttopologicalspaceover X . Proof. 1.Since0(X ,E), 1(X ,E) ∈ NSS τ1 and0(X ,E), 1(X ,E) ∈ NSS τ2 ,then0(X ,E), 1(X ,E) ∈ NSS τ1 ∩ NSS τ2 2.Supposethat {(Fi, E)|i ∈ I} beafamilyofneutrosophicsoftsetsin NSS τ1 ∩ NSS τ2 . Then(Fi, E) ∈ NSS τ1 and(Fi, E) ∈ NSS τ2 forall i ∈ I, so ∪ i∈I (Fi, E) ∈ NSS τ1 and ∪ i∈I (Fi, E) ∈ NSS τ2 . Thus ∪ i∈I (Fi, E) ∈ NSS τ1 ∩ NSS τ2 3.Let {(Fi, E)|i = 1, n} beafamilyofthefinitenumberofneutrosophicsoftsetsin NSS τ1 ∩ NSS τ2 Then(Fi, E) ∈ NSS τ1 and(Fi, E) ∈ NSS τ2 for i = 1, n, so n ∩ i=1 (Fi, E) ∈ NSS τ1 and n ∩ i=1 (Fi, E) ∈ NSS τ2 . Thus n ∩ i=1 (Fi, E) ∈ NSS τ1 ∩ NSS τ2 . Remark1. Theunionoftwoneutrosophicsofttopologiesover X maynotbeaneutrosophic softtopologyon X . CommunicationsinMathematicsandApplications,Vol.10,No.3,pp.481–493,2019

Example2. Let X = {x1, x2, x3} beaninitialuniverseset, E = {e1, e2} beasetofparametersand NSS

τ1 = {0(X ,E), 1(X ,E), (F1, E), (F2, E), (F3, E)} and NSS

τ2 = {0(X ,E), 1(X ,E), (F2, E), (F4, E)}

betwoneutrosophicsofttopologiesover X .Here,theneutrosophicsoftsets(F1, E), (F2, E), (F3, E)and(F4, E)over X aredefinedasfollowing:

(F1, E) = e1 = {〈x1, 0.9, 0.4, 0.3〉, 〈x2, 0.5, 0.6, 0.5〉, 〈x3, 0.4, 0.5, 0.3〉}, e2 = {〈x1, 0.7, 0.3, 0.4〉, 〈x2, 0.6, 0.6, 0.2〉, 〈x3, 0.6, 0.4, 0.5〉} ,

(F2, E) =

e1 = {〈x1, 0.7, 0.4, 0.5〉, 〈x2, 0.4, 0.5, 0.5〉, 〈x3, 0.3, 0.3, 0.4〉}, e2 = {〈x1, 0.6, 0.2, 0.4〉, 〈x2, 0.5, 0.4, 0.3〉, 〈x3, 0.4, 0.1, 0.5〉} ,

(F3, E) = e1 = {〈x1, 0.5, 0.3, 0.6〉, 〈x2, 0.3, 0.4, 0.7〉, 〈x3, 0.2, 0.2, 0.5〉}, e2 = {〈x1, 0 4, 0 1, 0 5〉, 〈x2, 0 4, 0 3, 0 4〉, 〈x3, 0 1, 0 1, 0 6〉} , (F4, E) = e1 = {〈x1, 0 8, 0 5, 0 4〉, 〈x2, 0 5, 0 6, 0 3〉, 〈x3, 0 7, 0 6, 0 2〉}, e2 = {〈x1, 0 7, 0 3, 0 3〉, 〈x2, 0 6, 0 5, 0 1〉, 〈x3, 0 7, 0 4, 0 3〉} .

Since(F1, E) ∪ (F4, E) ∉ NSS τ1 ∪ NSS τ2 ,then NSS τ1 ∪ NSS τ2 isnotaneutrosophicsofttopologyover X .

Proposition6. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X and NSS τ = {(Fi, E):(Fi, E) ∈ NSS(X , E)} = {[e, Fi(e)]e∈E :(Fi, E) ∈ NSS(X , E)} where Fi(e) = {〈x, TFi (e)(x), IFi (e)(x), FFi (e)(x)〉 : x ∈ X }.Then τ1 = {[TFi (e)(X )]e∈E } , τ2 = {[IFi (e)(X )]e∈E } , τ3 = {[FFi (e)(X )]c e∈E } definefuzzysofttopologieson X .

Proof. 1.0(X ,E), 1(X ,E) ∈ NSS τ ⇒ 0, 1 ∈ τ1,0, 1 ∈ τ2 and0, 1 ∈ τ3

2.Supposethat {(Fi, E)|i ∈ I} beafamilyofneutrosophicsoftsetsin NSS τ Then{[TFi (e)(X )]e∈E }i∈I isafamilyoffuzzysoftsetsin τ1, {[IFi (e)(X )]e∈E }i∈I isafamilyof fuzzysoftsetsin τ2 and {[FFi (e)(X )]c e∈E }i∈I isafamilyoffuzzysoftsetsin τ3.Since NSS τ isa neutrosophicsofttopology,then ∪ i∈I (Fi, E) ∈ NSS τ .Thatis, ∪ i∈I (Fi, E) = {〈sup[TFi (e)(X )]e∈E , sup[IFi (e)(X )]e∈E , inf[FFi (e)(X )]e∈E 〉}i∈I ∈ NSS τ . Therefore, {sup[TFi (e)(X

offuzzysoftsetsin τ2 and{[FFi (e)(X )]c e∈E }i=1,n isafamilyoffuzzysoftsetsin τ3.Since NSS τ isa neutrosophicsofttopology,then n ∩ i=1 (Fi, E) ∈ NSS τ .Thatis, n ∩ i=1 (Fi, E) = {〈min[TFi (e)(X )]e∈E , min[IFi (e)(X )]e∈E , max[FFi (e)(X )]e∈E 〉}i=1,n ∈ NSS τ . Therefore, {min[TFi (e)(X )]e∈E }i∈I ∈ τ1 , {min[IFi (e)(X )]e∈E }i∈I ∈ τ2 , {min[FFi (e)(X )]c e∈E }i∈I ∈ τ3 . Thiscompletestheproof. Remark2. Generally,converseoftheabovepropositionisnottrue. Example3. Let X = {x1, x2, x3} beainitialuniverse, E = {e1, e2} beasetofparametersand NSS τ = {0(X ,E), 1(X ,E), (F1, E), (F2, E), (F3, E)} beafamilyofneutrosophicsoftsetsover X .Here,theneutrosophicsoftsets(F1, E), (F2, E)and (F3, E)over X aredefinedasfollowing: (F1, E) = e1 = {〈x1, 0.7, 0.3, 0.2〉, 〈x2, 0.5, 0.4, 0.6〉, 〈x3, 0.5, 0.3, 0.1〉}, e2 = {〈x1, 0 4, 0 5, 0 3〉, 〈x2, 0 2, 0 3, 0 5〉, 〈x3, 0, 0 4, 0 5〉} , (F2, E) = e1 = {〈x1, 0 6, 0 5, 0 3〉, 〈x2, 0 4, 0 6, 0 6〉, 〈x3, 0 3, 0 5, 0 6〉}, e2 = {〈x1, 0 3, 0 6, 0 5〉, 〈x2, 0 1, 0 7, 0 6〉, 〈x3, 0, 0 6, 0 5〉} , (F3, E) = e1 = {〈x1, 0 8, 0 4, 0 5〉, 〈x2, 0 5, 0 5, 0 8〉, 〈x3, 0 5, 0 4, 0 7〉}, e2 = {〈x1, 0.5, 0.5, 0.7〉, 〈x2, 0.4, 0.6, 0.7〉, 〈x3, 0.6, 0.5, 0.9〉}

Then,

τ1 = {〈TF0(X ,E) (e)(X ), TF1(X ,E) (e)(X ), TF1(e)(X ), TF2(e)(X ), TF3(e)(X )〉e∈E },

τ2 = {〈IF0(X ,E) (e)(X ), IF1(X ,E) (e)(X ), IF1(e)(X ), IF2(e)(X ), IF3(e)(X )〉e∈E },

τ3 = {〈FF0(X ,E) (e)(X ), FF1(X ,E) (e)(X ), FF1(e)(X ), FF2(e)(X ), FF3(e)(X )〉e∈E }, arefuzzysofttopologieson X .Forexample,

τ1 = 〈(0, 0, 0), (1, 1, 1), (0.7, 0.5, 0.5), (0.6, 0.4, 0.3), (0.8, 0.5, 0.5)〉e1 , 〈(0, 0, 0), (1, 1, 1), (0.4, 0.2, 0), (0.3, 0.1, 0), (0.5, 0.4, 0.6)〉e2 andsoon. Ontheotherhand,since(F2, E) ∩ (F3, E) ∉ NSS τ , NSS τ isnotaneutrosophicsofttopologyon X . Proposition7. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X .Then τ1e = {[TF(e)(X )]:(F, E) ∈ NSS τ } , τ2e = {[IF(e)(X )]:(F, E) ∈ NSS τ } , τ3e = {[FF(e)(X )]c :(F, E) ∈ NSS τ } foreach e ∈ E,definefuzzytopologieson X Proof. Straightforward. CommunicationsinMathematicsandApplications,Vol.10,No.3,pp.481–493,2019

Remark3. Generally,converseoftheabovepropositionisnottrue. Example4. LetusconsidertheExample3.Then,

τ1e1 = {TF0(X ,E) (e1)(X ), TF1(X ,E) (e1)(X ), TF1(e1)(X ), TF2(e1)(X ), TF3(e1)(X )},

τ2e1 = {IF0(X ,E) (e1)(X ), IF1(X ,E) (e1)(X ), IF1(e1)(X ), IF2(e1)(X ), IF3(e1)(X )},

τ3e1 = {FF0(X ,E) (e1)(X ), FF1(X ,E) (e1)(X ), FF1(e1)(X ), FF2(e1)(X ), FF3(e1)(X )}, arefuzzytopologieson X .Forexample,

τ1e1 = {(0, 0, 0), (1, 1, 1), (0.7, 0.5, 0.5), (0.6, 0.4, 0.3), (0.8, 0.5, 0.5)} andsoon.Here, {τ1e1 , τ2e1 , τ3e1 } and {τ1e2 , τ2e2 , τ3e2 } arefuzzytritopologyon X .But NSS τ isnota neutrosophicsofttopologyon X

Definition16. Let(X , NSS τ , E)beaneutrosophicsofttopologicalspaceover X and(F, E) ∈ NSS(X , E)beaneutrosophicsoftset.Then,theneutrosophicsoftinteriorof(F, E),denoted (F, E)◦,isdefinedastheneutrosophicsoftunionofallneutrosophicsoftopensubsetsof(F, E). Clearly,(F, E)◦ isthebiggestneutrosophicsoftopensetthatiscontainedby(F, E) Example5. Letusconsidertheneutrosophicsofttopology NSS τ1 giveninExample2.Suppose thatanany(F, E) ∈ NSS(X , E)isdefinedasfollowing: (F, E) = e1 = {〈x1, 0.8, 0.5, 0.2〉, 〈x2, 0.5, 0.6, 0.3〉, 〈x3, 0.4, 0.4, 0.3〉}, e2 = {〈x1, 0.8, 0.4, 0.1〉, 〈x2, 0.7, 0.6, 0.2〉, 〈x3, 0.8, 0.4, 0.4〉}

Then0(X ,E), (F2, E), (F3, E) ⊆ (F, E).Therefore,(F, E)◦ = 0(X ,E) ∪ (F2, E) ∪ (F3, E) = (F2, E).

Theorem1. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X and (F, E) ∈ NSS(X , E). (F, E) isaneutrosophicsoftopensetiff (F, E) = (F, E)◦ .

Proof. Let(F, E)beaneutrosophicsoftopenset.Thenthebiggestneutrosophicsoftopenset thatiscontainedby(F, E)isequalto(F, E).Hence,(F, E) = (F, E)◦ Conversely,itisknownthat(F, E)◦ isaneutrosophicsoftopensetandif(F, E) = (F, E)◦ , then(F, E)isaneutrosophicsoftopenset.

Theorem2. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X and (F1, E), (F2, E) ∈ NSS(X , E).Then,

Proof. 1.Let(F1, E)◦ = (F2, E).Then(F2, E) ∈ NSS τ iff(F2, E) = (F2, E)◦.So,[(F1, E)◦]◦ = (F1, E)◦ 2.Straighforward. 3.Itisknownthat(F1, E)◦ ⊆ (F1, E) ⊆ (F2, E)and(F2, E)◦ ⊆ (F2, E).Since(F2, E)◦ isthebiggest neutrosophicsoftopensetcontainedin(F2, E)andso,(F1, E)◦ ⊆ (F2, E)◦ 4.Since(F1, E) ∩ (F2, E) ⊆ (F1, E)and(F1, E) ∩ (F2, E) ⊆ (F2, E), then[(F1, E) ∩ (F2, E)]◦ ⊆ (F1, E)◦ and[(F1, E) ∩ (F2, E)]◦ ⊆ (F2, E)◦ andso, [(F1, E) ∩ (F2, E)]◦ ⊆ (F1, E)◦ ∩ (F2, E)◦ . Ontheotherhand,since(F1, E)◦ ⊆ (F1, E)and(F2, E)◦ ⊆ (F2, E),then(F1, E)◦ ∩ (F2, E)◦ ⊆ (F1, E) ∩ (F2, E).Besides,[(F1, E) ∩ (F2, E)]◦ ⊆ (F1, E) ∩ (F2, E)anditisthebiggestneutrosophic softopenset.Therefore,(F1, E)◦ ∩ (F2, E)◦ ⊆ [(F1, E) ∩ (F2, E)]◦.Thus,[(F1, E) ∩ (F2, E)]◦ = (F1, E)◦ ∩ (F2, E)◦ . 5.Since(F1, E) ⊆ (F1, E) ∪ (F2, E)and(F2, E) ⊆ (F1, E) ∪ (F2, E),then(F1, E)◦ ⊆ [(F1, E) ∪ (F2, E)]◦ and(F2, E)◦ ⊆ [(F1, E) ∪ (F2, E)]◦.Therefore,(F1, E)◦ ∪ (F2, E)◦ ⊆ [(F1, E) ∪ (F2, E)]◦

Definition17. Let(X , NSS τ , E)beaneutrosophicsofttopologicalspaceover X and(F, E) ∈ NSS(X , E)beaneutrosophicsoftset.Then,theneutrosophicsoftclosureof(F, E),denoted (F, E),isdefinedastheneutrosophicsoftintersectionofallneutrosophicsoftclosedsupersets of(F, E). Clearly, (F, E)isthesmallestneutrosophicsoftclosedsetthatcontaining(F, E).

Example6. Letusconsidertheneutrosophicsofttopology NSS τ1 giveninExample2.Suppose thatanany(F, E) ∈ NSS(X , E)isdefinedasfollowing: (F, E) = e1 = {〈x1, 0 2, 0 5, 0 9〉, 〈x2, 0 5, 0 3, 0 7〉, 〈x3, 0 2, 0 4, 0 6〉}, e2 = {〈x1, 0.1, 0.4, 0.8〉, 〈x2, 0.1, 0.3, 0.7〉, 〈x3, 0.3, 0.4, 0.8〉} Obviously,0c (X ,E), 1c (X ,E), (F1, E)c , (F2, E)c and(F3, E)c areallneutrosophicsoftclosedsetsover (X , NSS τ1 , E).Theyaregivenasfollowing:

0c (X ,E) = 1(X ,E), 1c (X ,E) = 0(X ,E) (F1, E)c = e1 = {〈x1, 0 3, 0 6, 0 9〉, 〈x2, 0 5, 0 4, 0 5〉, 〈x3, 0 3, 0 5, 0 4〉}, e2 = {〈x1, 0 4, 0 7, 0 7〉, 〈x2, 0 2, 0 4, 0 6〉, 〈x3, 0 5, 0 6, 0 6〉} , (F2, E)c = e1 = {〈x1, 0 5, 0 6, 0 7〉, 〈x2, 0 5, 0 5, 0 4〉, 〈x3, 0 4, 0 7, 0 3〉}, e2 = {〈x1, 0.4, 0.8, 0.6〉, 〈x2, 0.3, 0.6, 0.5〉, 〈x3, 0.5, 0.9, 0.4〉} , (F3, E)c = e1 = {〈x1, 0.6, 0.7, 0.5〉, 〈x2, 0.7, 0.6, 0.3〉, 〈x3, 0.5, 0.8, 0.2〉}, e2 = {〈x1, 0.5, 0.9, 0.4〉, 〈x2, 0.4, 0.7, 0.4〉, 〈x3, 0.6, 0.9, 0.1〉} . Then1c (X ,E), (F1, E)c , (F2, E)c , (F3, E)c ⊇ (F, E).Therefore, (F, E) = 1c (X ,E) ∩ (F1, E)c ∩ (F2, E)c ∩ (F3, E)c = (F1, E)c .

Theorem3. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X and (F, E) ∈ NSS(X , E). (F, E) isneutrosophicsoftclosedsetiff (F, E) = (F, E).

CommunicationsinMathematicsandApplications,Vol.10,No.3,pp.481–493,2019

Proof. Straightforward.

Theorem4. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X and (F1, E), (F2, E) ∈ NSS(X , E).Then, 1. [(F1, E)] = (F1, E), 2. (0(X ,E)) = 0(X ,E) and (1(X ,E)) = 1(X ,E) 3. (F1, E) ⊆ (F2, E) ⇒ (F1, E) ⊆ (F2, E), 4. [(F1, E) ∪ (F2, E)] = (F1, E) ∪ (F2, E), 5. [(F1, E) ∩ (F2, E)] ⊆ (F1, E) ∩ (F2, E)

Proof. 1.Let (F1, E) = (F2, E).Then,(F2, E)isaneutrosophicsoftclosedset.Hence,(F2, E)and (F2, E)areequal.Therefore, [(F1, E)] = (F1, E).

2.Straightforward.

3.Itisknownthat(F1, E) ⊆ (F1, E) and(F2, E) ⊆ (F2, E) andso,(F1, E) ⊆ (F2, E) ⊆ (F2, E).Since (F1, E)isthesmallestneutrosophicsoftclosedsetcontaining(F1, E), then (F1, E) ⊆ (F2, E).

4.Since(F1, E) ⊆ (F1, E) ∪ (F2, E)and(F2, E) ⊆ (F1, E) ∪ (F2, E),then (F1, E) ⊆ [(F1, E) ∪ (F2, E)] and (F2, E) ⊆ [(F1, E) ∪ (F2, E)]andso, (F1, E) ∪ (F2, E) ⊆ [(F1, E) ∪ (F2, E)].

Conversely,since(F1, E) ⊆ (F1, E) and(F2, E) ⊆ (F2, E),then(F1, E)∪(F2, E) ⊆ (F1, E)∪(F2, E) Besides, [(F1, E) ∪ (F2, E)] isthesmallestneutrosophicsoftclosedsetthatcontaining(F1, E) ∪ (F2, E).Therefore, [(F1, E) ∪ (F2, E)] ⊆ (F1, E) ∪ (F2, E).Thus, [(F1, E) ∪ (F2, E)] = (F1, E) ∪ (F2, E).

5.Since(F1, E) ∩ (F2, E) ⊆ (F1, E) ∩ (F2, E) and [(F1, E) ∩ (F2, E)] isthesmallestneutrosophic softclosedsetthatcontaining(F1, E) ∩ (F2, E),then [(F1, E) ∩ (F2, E)] ⊆ (F1, E) ∩ (F2, E). Theorem5. Let (X , NSS τ , E) beaneutrosophicsofttopologicalspaceover X and (F, E) ∈ NSS(X , E).Then, 1. [(F, E)]c = [(F, E)c]◦ ,

fuzzysofttopology.Wehopethat,theresultsofthisstudymayhelpintheinvestigationof neutrosophicsoftcontinuousfunctionspacesandinmanyresearches.

CompetingInterests

Theauthorsdeclarethattheyhavenocompetinginterests.

Authors’Contributions

Alltheauthorscontributedsignificantlyinwritingthisarticle.Theauthorsreadandapproved thefinalmanuscript.

References

[1] K.Atanassov,Intuitionisticfuzzysets, FuzzySetsandSystems 20(1)(1986),87–96, DOI:10.1007/978-3-7908-1870-3-1.

[2] S.BayramovandC.Gunduz(Aras),Onintuitionisticfuzzysofttopologicalspaces, TWMSJ.Pure Appl.Math. 5(1)(2014),66–79.

[3] T.BeraandN.K.Mahapatra,Introductiontoneutrosophicsofttopologicalspace, Opsearch 54(4) (2017),841–867,DOI:10.1007/s12597-017-0308-7.

[4] T.BeraandN.K.Mahapatra,Onneutrosophicsoftfunction, AnnalsofFuzzyMathematicsand Informatics 12(1)(2016),101–119.

[5] N.Ca ˘ gman,S.Karata¸sandS.Engino ˘ glu,Softtopology, Computers&Mathematicswith Applications 62(1)(2011),351–358,DOI:10.1016/j.camwa.2011.05.016.

[6] C.L.Chang,Fuzzytopologicalspaces, JournalofMathematicalAnalysisandApplications 24(1) (1968),182–190,DOI:10.1016/0022-247x(68)90057-7.

[7] D.Coker,Anoteonintuitionisticsetsandintuitionisticpoints, TurkishJournalofMathematics 20(3)(1996),343–351.

[8] I.DeliandS.Broumi,Neutrosophicsoftrelationsandsomeproperties, AnnalsofFuzzy MathematicsandInformatics 9(1)(2015),169–182,DOI:10.5281/zenodo.23153.

[9] P.K.Maji,Neutrosophicsoftset, AnnalsofFuzzyMathematicsandInformatics 5(1)(2013),157–168.

[10] P.K.Maji,R.BiswasandA.R.Roy,Fuzzysoftsets, TheJournaloffuzzymathematics 9(3)(2001), 589–602.

[11] D.Molodtsov,Softsettheory-firstresults, Computers&MathematicswithApplications 37(4/5) (1999),19–31,DOI:10.1016/s0898-1221(99)00056-5.

[12] A.A.SalmaandS.A.Alblowi,Neutrosophicsetandneutrosophictopologicalspaces, IOSRJournal ofMathematics 3(4)(2012),31–35,DOI:10.9790/5728-0343135.

[13] M.ShabirandM.Naz,Onsofttopologicalspaces, Computers&MathematicswithApplications 61(7)(2011),1786–1799,DOI:10.1016/j.camwa.2011.02.006.

[14] F.Smarandache,Neutrosophicset-ageneralisationoftheintuitionisticfuzzysets, International JournalofPureandAppliedMathematics 24(3)(2005),287–297,DOI:10.1109/grc.2006.1635754.

[15] L.A.Zadeh,Fuzzysets, InformationandControl 8(3)(1965),338–353,DOI:10.1016/s00199958(65)90241-x.

CommunicationsinMathematicsandApplications,Vol.10,No.3,pp.481–493,2019