Toplogical and Geometric KM-Single Valued Neutrosophic Metric Spaces by Florentin Smarandache

MathematicsInterdisciplinaryResearch x (202x) xx yy

ToplogicalandGeometric KM-SingleValued NeutrosophicMetricSpaces

MohammadHamidi ⋆ ,MahdiMollaei-AraniandYousefAlipour-Fakhri

Abstract

Thispaperintroducesthenovelconceptof KM-singlevaluedneutrosophicmetricspacesasanespecialgeneralizationof KM-fuzzymetricspaces, investigatesseveraltopologicalandstructuralpropertiesandpresentssome ofitsapplications.Thisstudyalsoconsidersthemetricspacesandconstructs KM-singlevaluedneutrosophicspaceswithrespecttoanygiventriangular normsandtriangularconorms.Moreover,wetrytoextendtheconcept of KM-singlevaluedneutrosophicmetricspacestoalargerclassof KMsinglevaluedneutrosophicmetricspacessuchasunionof KM-singlevalued neutrosophicmetricspacesandproductof KM-singlevaluedneutrosophic metricspaces.

Keywords:KM-singlevaluedneutrosophicmetric,left-continuoustriangular (co)norm,Cauchysequence

2010MathematicsSubjectClassiﬁcation: 31C12,26E50.

Howtocitethisarticle

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri,Toplogicaland geometric KM-singlevaluedneutrosophicmetricspaces, Math.Interdisc. Res. x (202x)xx-yy.

1.Introduction

Classicalsettheoryisapureconceptandwithoutqualityorcriteria,soitisnot attractivetouseinourworld,that’swhyweusetheneutrosophicsetstheoryasone ofageneralizationsofsettheoryinordertodealwithuncertainties,whichisakey actioninthecontemporaryworldintroducedbySmarandachefortheﬁrsttimein ⋆

Correspondingauthor(m.hamidi@pnu.ac.ir)

AcademicEditor:BehrozBidabad

Received15April2020,Accepted27July2020 DOI:10.22052/mir.2020.227202.1209

c ⃝2020UniversityofKashan

ThisworkislicensedundertheCreativeCommonsAttribution4.0InternationalLicense.

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

1998and2005[11].Thisconceptisanewmathematicaltoolforhandlingproblems involvingimprecise,indeterminacy,andinconsistentdata.Thistheorydescribes animportantroleinmodelingandcontrollingunsurehypersystemsinnature, societyandindustry.Inaddition,fuzzytopologicalspacesasageneralizationof topologicalspaces,haveafundamentalroleinconstructionoffuzzymetricspaces asanextensionoftheconceptofmetricspaces.Thetheoryoffuzzymetricspaces worksonfindingthedistancebetweentwopointsasnon-negativefuzzynumbers, whichhavevariousapplications.Thestructureoffuzzymetricspacesisequipped withmathematicaltoolssuchastriangularnormsandfuzzysubsetsdependingon timeparameterandonothervariables.Thistheoryhasbeenproposedbydifferent researcherswithdifferentdefinitionsfromseveralpointsofviews([1,2,3,7]),and thatthisstudywasappliedtothenotionofKM-fuzzymetricspaceintroducedin 1975[2]byKramosilandMichalek.Furthermaterialsregardingthesinglevalued neutrosophicmetricsetsandtheirapplicationsin,graphs,hypergraphsandneutro algebrasareavailableintheliteraturetoo[4,5,6].

Regardingthesepoints,weintroducetheconceptof KM -singlevaluedneutrosophicmetricspacesasanapplicationofneutrosophicsets.Althoughweapply threefuzzysubsetsinourdefinitionbutithaslimitedtheirsumofthreefuzzy subsetsintoafuzzysubset.Alsoweprovedthat KM -singlevaluedneutrosophic metricspaceshavebothnon-increasingfuzzysubsetsandnon-decreasingfuzzy subsets.Ithastriedtoconstructalargerclassof KM -singlevaluedneutrosophic metricspaceswithrespecttounionandproductoperations.Metricspaceshave animportantroleingeneratingthe KM -singlevaluedneutrosophicmetricspaces viaanytriangularnormsandtriangularconorms;therefore,weanalyzedtherelationbetweentheclassofmetricspacesand KM -singlevaluedneutrosophicmetric spaces.Moreover,wepresentedtheballsubsetsin KM -singlevaluedneutrosophicmetricspacesandprovedthatballsubsetsareopensubsets.Furthermore, thepresentstudyaimedtogeneratesometopologiesonthebasesetof KMsinglevaluedneutrosophicmetricspaceswithrespecttoopenballsanditisone ofthemainmotivationsofintroducingthe KM -singlevaluedneutrosophicmetric spaces.The KM -singlevaluedneutrosophicmetricspacesarenotnecessarilyinfinitespaces.Thus,anotherimportantmotivationofthisstudyistheconstructionof finite KM -singlevaluedneutrosophicmetricspaces.Thisstudyalsopresentedan inducedequivalenceinrelationto KM -singlevaluedneutrosophicmetricspaces suchthataquotientofgiven KM -singlevaluedneutrosophicmetricspaceisa KM -singlevaluedneutrosophicmetricspace.Thisstudygeneratedsomemetrics onanynonemptysetsusingtheconceptof KM -singlevaluedneutrosophicmetric spacesandextended KM -singlevaluedneutrosophicmetricspacestoafamilyof metricspaceswithleft-continuousmetrics.Forthesignificanceoftheapplicabilityofthisargument,wepresentedanexampleofapplicationof KM -singlevalued neutrosophicmetricspaceoneconomicanditencouragedustodevelopthisstudy.

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 3

2.Preliminaries

Thissectionpresentedsomedeﬁnitionsandresultswhichareusedinfollowing sections.

Deﬁnition2.1. [11]Let V beauniversalset.Aneutrosophicset (NS) X in V isanobjecthasthefollowingform X = {(x,TX (x),IX (x),FX (x)) | x ∈ V }, or X : V → [0, 1]×[0, 1]×[0, 1] whichischaracterizedbyatruth-membershipfunction TX ,anindeterminacy-membershipfunction IX andafalsity-membershipfunction FX .Thereisnorestrictiononthesumof TX (x),IX (x) and FX (x),therefore 0 ≤ sup TX (x)+sup IX (x)+sup FX (x) ≤ 3+

Deﬁnition2.2. [10]Abinaryoperation T :[0, 1] × [0, 1] → [0, 1] isat-normifit forall x,y,z,w ∈ [0, 1] satisﬁesthefollowing:

(i) T (1,x)= x;

(ii) T (x,y)= T (y,x);

(iii) T (T (x,y),z)= T (x,T (y,z));

(iii) If w ≤ x and y ≤ z then T (w,y) ≤ T (x,z).

Deﬁnition2.3. [8]Atriplet (X,ρ,T ) iscalleda KM -fuzzymetricspace,if X is anarbitrarynon–emptyset, T isaleft-continuoust-normand ρ : X2 ×R≥0 → [0, 1] isafuzzyset,suchthatforeach x,y,z, ∈ X and t,s ≥ 0,wehave:

(i) ρ(x,y, 0)=0,

(ii) ρ(x,x,t)=1 forall t> 0,

(iii) ρ(x,y,t)= ρ(y,x,t)(commutativeproperty),

(iv) T (ρ(x,y,t),ρ(y,z,s)) ≤ ρ(x,z,t + s)(triangularinequality),

(vi) ρ(x,y, ): R≥0 → [0, 1] isaleft-continuousmap,

(vii) ρ(x,y,t) → 1,when t →∞.

(viii) ρ(x,y,t)=1, ∀ t> 0 impliesthat x = y.

If (X,ρ,T ) issatisﬁedinconditions (i)–(vii),thenitiscalleda KM -fuzzy pseudometricspaceand ρ iscalleda KM -fuzzypseudometric.

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

3. KM-SingleValuedNeutrosophicMetricSpace

Inthissection,weintroducetheconceptof KM -singlevaluedneutrosophicmetric spacesandinvestigatetheirproperties.Inaddition,wegenerate KM -singlevalued neutrosophicmetricspaceswithrespecttometricspaces.

Deﬁnition3.1. Atriplet (X,ρ1,ρ2,ρ3,T,S) iscalleda KM -singlevaluedneutrosophicmetricspace,if X isanarbitrarynonemptyset, T isaleft-continuous t-norm, S isaleft-continuoust-conorm,and ρ1,ρ2,ρ3 : X2 × R≥0 → [0, 1] are fuzzysubsets,suchthatforeach x,y,z, ∈ X and t,s ≥ 0,wehave:

(i) ρ1(x,y, 0)=0 andforall i ∈{2, 3}, ρi(x,y, 0)=1,

(ii) ρ1(x,x,t)=1 andforall i ∈{2, 3}, ρi(x,x,t)=0,where t> 0,

(iii) forall i ∈{1, 2, 3}, ρi(x,y,t)= ρi(y,x,t)(commutativeproperty),

(iv) T (ρ1(x,y,t),ρ1(y,z,s)) ≤ ρ1(x,z,t + s) andforall i ∈{2, 3},S(ρi(x,y,t), ρi(y,z,s)) ≥ ρi(x,z,t + s)(triangularinequality),

(vi) forall i ∈{1, 2, 3}, ρi(x,y, ): R≥0 → [0, 1] areleft-continuousmaps,

(vii) lim t→∞ ρ1((x,y,t))=1 andforall i ∈{2, 3}, lim t→∞ ρi((x,y,t))=0,

(viii) forall t ∈ R+ andforall x,y ∈ X,wehave 0 ≤ 3 ∑ i=1 ρi(x,y,t) ≤ 1,

(ix) ∀ t> 0,ρ1(x,y,t)=1 impliesthat x = y andforall i ∈{2, 3}, ∀ t> 0,ρi(x,y,t)=0 impliesthat x = y

If (X,ρ1,ρ2,ρ3,T,S) satisﬁesinconditions (i)–(viii),thenitiscalleda KMsinglevaluedneutrosophicpseudometricspaceandtriple (ρ1,ρ2,ρ3) iscalleda KM -singlevaluedneutrosophicpseudometric.

Thefollowingpropositionshowsthat KM -singlevaluedneutrosophicmetrics arediﬀerentfrom KM -fuzzymetrics.

Proposition3.2. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophic metricspace.Thenforall x,y ∈ X and t ∈ R+

(i) ρ1(x,y,t)+ ρ2(x,y,t)+ ρ3(x,y,t) =0;

(ii) ρ2(x,y,t)+ ρ3(x,y,t) =1;

(iii) if ρ2(x,y,t)= ρ3(x,y,t),then ρ2(x,y,t) < 1 2 ,where t> 0;

(iv) ρ1(x,y,t)=1,ifandonlyif ρ2(x,y,t)+ ρ3(x,y,t)=0.

Proof. Itisimmediatebydeﬁnition.

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 5

Fromnowon,forall x,y ∈ [0, 1] weconsider Tmin(x,y)=min{x,y}, Tpr(x,y)= xy,Tlu(x,y)=max(0,x + y 1),Tdo(x,y)=    xy x + y xy if (x,y) =(0, 0) 0 if (x,y)=(0, 0) , CT = {T :[0, 1] × [0, 1] → [0, 1] | T isaleft-continuoust-norm}, Smax(x,y)= max{x,y}, Spr(x,y)= x + y xy,Slu(x,y)=min(1,x + y) and CS = {S : [0, 1] × [0, 1] → [0, 1] | S isaleft-continuoust-conorm} Inwhatfollows,weinvestigatesomepropertiesofthe KM -singlevaluedneutrosophicmetricspaces.

Deﬁnition3.3. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetricspace,let (xn)n beasequencein X and x ∈ X.Wesaythat

(i) (xn)n convergesto x,ifforall t ∈ R+ wehave lim n→∞ ρ1(xn,x,t)=1 and lim n→∞ ρ2(xn,x,t)=lim n→∞ ρ3(xn,x,t)=0.Itmeansthatforall t ∈ R+ and forall 0 <ϵ< 1,thereexists N ∈ N,suchthatforall n ≥ N , 1 ϵ< ρ1(xn,x,t),ρ2(xn,x,t) <ϵ and ρ3(xn,x,t) <ϵ. (ii) (xn)n isaCauchysequenceifandonlyifforeach t ∈ R+ and p> 0 lim n→∞ ρ1(xn,xn+p,t)=1 and lim n→∞ ρ2(xn,xn+p,t)=lim n→∞ ρ3(xn,xn+p,t)=0

In[3],GeorgeandVeeramaniprovedthatevery GV -fuzzymetricisanondecreasingmap.InasimilarwaywehavethefollowingTheoremonthe KM -single valuedneutrosophicmetricspaces.

Theorem3.4. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetric space.Then ρ1(x,y, ): R≥0 → [0, 1] isanon-decreasingmapandforall i ∈ {2, 3}, ρi(x,y, ): R≥0 → [0, 1] arenon-increasingmaps.

Proof. Let 0 ≤ t<s.If t =0, thenforall t> 0,ρ2(x,y, 0)=1 ≥ ρ2(x,y,s).But forall t = s if ρ2(x,y,t) <ρ2(x,y,s),then S(ρ2(x,y,t),ρ2(y,y,s t)) ≥ ρ2(x,y,s) Bydeﬁnition, ρ2(y,y,s t)=0,andthusobtainthat ρ2(x,y,t) ≥ ρ2(x,y,s),which isacontradictionanditimpliesthat ρ2(x,y, ): R≥0 → [0, 1] isanon-increasing map.Inasimilarway, ρ3(x,y, ): R≥0 → [0, 1] isanon-increasingmapand ρ1(x,y, ): R≥0 → [0, 1] isanon-decreasingmap.

In[12],Rodrguez-LopezandRomaguera,provedthatevery GV -fuzzymetricisacontinuousmap.Inasimilarway,onecanseethat KM -singlevalued neutrosophicmetricsareleft-continuousmaps.

Theorem3.5. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetric space.Thenforall i ∈{1, 2, 3},ρi’sareleft-continuousmapson X2 × R≥0

Proof. Let x,y ∈ X,t ∈ R+ and (x′ n,y′ n,t′ n)n beasequencein X2 × R≥0 that lim n→∞ (x ′ n,y ′ n,t′ n)=(x,y,t).Supposethat t/2 >ϵ> 0 bearbitrary.Since (ρ1(x′ n,y′ n,t′ n))n isasequencein [0, 1],thereisasubsequence (xn,yn,tn)n of

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

(x′ n,y′ n,t′ n)n suchthat ρ2((xn,yn,tn)) convergestosomepointsof [0, 1].Now, lim n→∞ tn = t impliesthatthereis N ∈ N suchthatforall n ≥ N wehave t ϵ<tn.Hence,forall n ≥ N wehave ρ2(xn,yn,tn) ≤ ρ2(xn,yn,t ϵ) ≤ S(ρ2(xn,x,ϵ/2),ρ2(x,y,t 2ϵ),ρ2(y,yn,ϵ/2)) and ρ2(x,y,t+2ϵ) ≤ ρ2(x,y,tn+ϵ) ≤ S(ρ2(xn,x,ϵ/2),ρ2(xn,yn,tn),ρ2(y,yn,ϵ/2)).Thus lim n→∞ ρ2(xn,yn,tn) ≤ S(0,ρ2(x, y,t 2ϵ), 0)= ρ2(x,y,t 2ϵ),sobyleft-continuityofthefunction ρ2 on R≥0 , getthat lim ϵ→0(lim n→∞ ρ2(xn,yn,tn)) ≤ lim ϵ→0(ρ2(x,y,t 2ϵ))= ρ2(x,y,t).Inaddition, ρ2(x,y,t +2ϵ) ≥ S(0, lim n→∞ ρ2(xn,yn,tn), 0)=lim n→∞ ρ2(xn,yn,tn).Thus ρ2(x,y,t +2ϵ) ≤ lim n→∞ ρ2(xn,yn,tn) andbyleft-continuityofthefunction ρ2 on R≥0,getthat lim ϵ→0(lim n→∞ ρ2(xn,yn,tn)) ≥ lim ϵ→0(ρ2(x,y,t +2ϵ))= ρ2(x,y,t).Itfollowsthat ρ2 isaleft-continuousmapon X2 × R≥0.Inasimilarwayonecansee that ρ1 and ρ3 areleft-continuousmapson X2 × R≥0

Let x1,x2,...,xn ∈ [0, 1].Thenforall T ∈CT and S ∈CS ,wehave T (x1,x2, x3)= T (T (x1,x2),x3) and T (x1,x2,...,xn 1,xn)= T (T (x1,x2,...,xn 1),xn) Inasimilarway S(x1,x2,x3)= S(S(x1,x2),x3) and S(x1,x2,...,xn 1,xn) = S(S(x1,x2,...,xn 1),xn)

Lemma3.6. Let x1,x2,...,xn ∈ [0, 1], T ∈CT and S ∈CS .Then

(i) if 0 ∈{x1,x2,...,xn},then T (x1,x2,...,xn 1,xn)=0;

(ii) if 1 ∈{x1,x2,...,xn},then S(x1,x2,...,xn 1,xn)=1;

(iii) Smax(x1,x2,...,xn 1,xn) ≤ S(x1,x2,...,xn 1,xn);

(iv) T (x1,x2,...,xn 1,xn) ≤ Tmin(x1,x2,...,xn 1,xn); (v) T (x1,x2,...,xn 1,xn)=1 ifandonlyif x1 = x2 = ... = xn 1 = xn =1;

(vi) S(x1,x2,...,xn 1,xn)=0 ifandonlyif x1 = x2 = ... = xn 1 = xn =0;

(vii) forall p ∈ R+ , Smax( x1 p , x2 p ,..., xn 1 p , xn p )= 1 p Smax(x1,x2,...,xn 1,xn) and Tmin( x1 p , x2 p ,..., xn 1 p , xn p )= 1 p Tmin(x1,x2,...,xn 1,xn)

Theorem3.7. If (X,ρ1,ρ2,ρ3,Tmin,Smax) isa KM -fuzzymetricspace, T ∈CT and S ∈CS .Then (X,ρ1,ρ2,ρ3,T,S) isa KM -singlevaluedneutrosophicmetric space.

Proof. Itisclear.

Inthefollowing,wegeneratethe KM -singlevaluedneutrosophicmetricspaces withrespecttothemetricspaces.

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 7

Forall x,y ∈ X andforall 0 >p,p′,m, (p,p′ ≥ 3m),t,s ∈ R≥0,deﬁne ρ1(x,y,t)=    0 t =0 φ(t) φ(t)+ md(x,y) t =0 , ρ2(x,y,t)=    md(x,y) p(md(x,y)+ φ(t)) if t> 0 1 if t =0 and ρ3(x,y,t)=    md(x,y) p′(md(x,y)+ φ(t)) if t> 0 1 if t =0 ,where φ : R≥0 → R≥0 isincreasingthe left-continuousmapand φ(t)+ md(x,y) =0 and φ(t + s) ≥ φ(t)+ φ(s)

Theorem3.8. Let (X,d) beametricspace.If p′,p> 2,then (X,ρ1,ρ2,ρ3,Tmin, Smax) isa KM -singlevaluedneutrosophicmetricspace.

Proof. If p,p′ > 2,thenforall x,y ∈ X,t ∈ R+,wehave 0 ≤ 3 ∑ i=1 ρi(x,y,t) ≤ 1. Now,weonlyprovethetriangularinequalityproperty.Let x,y,z ∈ X.For 0 ∈{t,s} isclear,nowfor 0 ̸∈{t,s} weinvestigateit.Withoutlossofgenerality, if φ(t) p(φ(t)+ md(x,y)) ≤ φ(s) p(φ(s)+ md(z,y)) ,wegetthat φ(t)d(z,y) ≤ φ(s)d(x,y) Sinceforall s,t,m ∈ R+,φ(t + s) ≥ φ(t)+ φ(s),wegetthat φ(t)d(x,z) ≤ φ(t + s)d(x,y) andso Tmin( φ(t) p(φ(t)+ md(x,y)) , φ(s) p(φ(s)+ md(y,z)) ) ≤ φ(t + s) p(φ(t + s)+ md(x,z)) .

Inasimilarway,onecanseethat Smax( md(x,y) p(md(x,y)+ φ(t)) , md(z,y) p(md(z,y)+ φ(s)) ) ≥ md(x,z) p(md(x,z)+ φ(t + s)) ,where i ∈{2, 3}.Itfollowsthat Tmin(ρ(x,y,t),ρ1(y,z,s)) ≤ ρ1(x,z,t+s),Smax(ρi(x,y,t),ρ(y,z,s)) ≥ ρ(x,z,t+s) andso (X,ρ1,ρ2,ρ3,Tmin, Smax) isa KM -singlevaluedneutrosophicmetricspace.

ByLemma3.6andTheorem3.8,onecanconstructa KM -singlevaluedneutrosophicmetricspacewithanytriangularnormsandtriangularconormsonany givenmetricspaceasfollows.

Corollary3.9. Let (X,d) beametricspace.Thenthereexistfuzzysubsets ρ1,ρ2,ρ3 on X2 × R≥0 suchthatforeach T ∈CT and S ∈CS , (X,ρ1,ρ2,ρ3,T,S) isa KM -singlevaluedneutrosophicmetricspace.

Example3.10. Consider X = N andthemetricspace (X,d),where d(x,y)= |x y| Deﬁneforall x,y ∈ X,ρ1(x,y,t)=    38t (38t +7d(x,y)) if t> 0 0 if t =0 ,ρ2(x,y,t)=    7d(x,y) 21(7d(x,y)+38t) if t> 0 1 if t =0 , and ρ3(x,y,t)=    7d(x,y) 37(7d(x,y)+38t) if t> 0 1 if t =0 .

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

Then (X,ρ1,ρ2,ρ3,T,S) isa KM -singlevaluedneutrosophicmetricspace,where T ∈CT and S ∈CS .

3.1.Operationson KM-SingleValuedNeutrosophicMetric Spaces

Inthissubsection,weextend KM -singlevaluedneutrosophicmetricspacesto unionof KM -singlevaluedneutrosophicmetricspacesandproductof KM -single valuedneutrosophicmetricspaces.Let (X,ρ1,ρ2,ρ3,T,S) and (Y,ρ′ 1,ρ′ 2,ρ′ 3,T,S) be KM -singlevaluedneutrosophicmetricspaces, (x1,y1), (x2,y2) ∈ X × Y and t ∈ R≥0.Foranarbitrary T ∈CT and S ∈CS deﬁne T (ρ1,ρ′ 1),S(ρ2,ρ′ 2),S(ρ3,ρ′ 3): (X×Y )2×R≥0 → [0, 1] by T (ρ1,ρ′ 1)((x1,y1), (x2,y2),t) = T (ρ1(x1,x2,t),ρ′ 1(y1,y2, t)),S(ρ2,ρ′ 2)((x1,y1), (x2,y2),t) = S(ρ2(x1,x2,t),ρ′ 2(y1,y2,t)) and S(ρ3,ρ ′ 3)((x1,y1), (x2,y2),t) = S(ρ3(x1,x2,t),ρ ′ 3(y1,y2,t))

Sowehavethefollowingtheorem.

Theorem3.1. Let (X,ρ1,ρ2,ρ3,T,S) and (Y,ρ′ 1,ρ′ 2,ρ′ 3,T,S) be KM -singlevaluedneutrosophicmetricspaces.Then (X ×Y,Tmin(ρ1,ρ′ 1),Smax(ρ2,ρ′ 2),Smax(ρ3, ρ′ 3),T,S) isa KM -singlevaluedneutrosophicmetricspace.

Proof. Let (x1,y1), (x2,y2), (x3,y3) ∈ X × Y and t,s ∈ R≥0 (i) Sinceforall x1,x2 ∈ X,y1,y2 ∈ Y,ρ1(x1,x2, 0)=0 and ρ2(x1,x2, 0)= ρ3(x1,x2, 0)= ρ′ 2(y1,y2, 0)= ρ′ 3(y1,y2, 0)=1,wehave Tmin(ρ1,ρ′ 1)((x1,y1), (x2,y2 ), 0) =0, Smax(ρ2,ρ′ 2)((x1,y1), (x2,y2), 0) =1 and Smax(ρ3,ρ′ 3)((x1,y1), (x2,y2), 0) =1

(ii) Tmin(ρ1,ρ′ 1)((x1,y1), (x2,y2),t) =1 ifandonlyif ρ1(x1,x2,t)= ρ′ 1(y1,y2,t) =1 ifandonlyif (x1,y1)=(x2,y2).Inaddition, Smax(ρ2,ρ′ 2)((x1,y1), (x2,y2),t) = 0 ifandonlyif ρ2(x1,x2,t)= ρ′ 2(y1,y2,t)=0 ifandonlyif (x1,y1)=(x2,y2).Ina similarway, Smax(ρ3,ρ′ 3)((x1,y1), (x2,y2),t) =0 ifandonlyif (x1,y1)=(x2,y2)

(iii) Itisclearthat Tmin(ρ1,ρ′ 1),Smax(ρ2,ρ′ 2) and Smax(ρ3,ρ′ 3) arecommutativemaps.

(iv) ByLemma3.6,

T (Tmin(ρ1,ρ ′ 1)((x1,y1), (x2,y2),t),Tmin(ρ1,ρ ′ 1)((x2,y2), (x3,y3),s))

= T (Tmin(ρ1(x1,x2,t),ρ ′ 1(y1,y2,t)),Tmin(ρ1(x2,x3,s),ρ ′ 1(y2,y3,s)))

≤ Tmin(T (ρ1(x1,x2,t),ρ1(x2,x3,s)),T (ρ ′ 1(y1,y2,t),ρ ′ 1(y2,y3,s))))

≤ Tmin(ρ1(x1,x3,t + s),ρ ′ 1(y1,y3,t + s))

= Tmin(ρ1,ρ ′ 1)((x1,y1), (x3,y3),t + s)

Also,

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 9

S(Smax(ρ2,ρ ′ 2)((x1,y1), (x2,y2),t),Smax(ρ2,ρ ′ 2)((x2,y2), (x3,y3),s))

= S(Smax(ρ2(x1,x2,t),ρ ′ 2(y1,y2,t)),Smax(ρ2(x2,x3,s),ρ ′ 2(y2,y3,s)))

≥ Smax(S(ρ2(x1,x2,t),ρ2(x2,x3,s)),S(ρ ′ 2(y1,y2,t),ρ ′ 2(y2,y3,s))))

≥ Smax(ρ2(x1,x3,t + s),ρ ′ 2(y1,y3,t + s))

= Smax(ρ2,ρ ′ 2)((x1,y1), (x3,y3),t + s).

Inasimilarway,onecanseethat

S(Smax(ρ3,ρ ′ 3)((x1,y1), (x2,y2),t),Smax(ρ3,ρ ′ 3)((x2,y2), (x3,y3),s))

≥ Smax(ρ3,ρ ′ 3)((x1,y1), (x3,y3),t + s)

(v) Since ρ1,ρ2,ρ3,ρ′ 1,ρ′ 2,ρ′ 3 areleft-continuousmaps,wegetthat Tmin(ρ1,ρ′ 1),Smax(ρ2,ρ′ 2),Smax(ρ3,ρ′ 3) areleft-continuousmap.

(vi) Since Tmin and Smax areleft-continuousmaps,wegetthat lim t→∞ Tmin(ρ1(x1,x2,t),ρ ′ 1(y1,y2,t)) = Tmin(lim t→∞ ρ1(x1,x2,t), lim t→∞ ρ ′ 1(y1,y2,t))= Tmin(1, 1)=1 Inasimilarway, lim t→∞ Smax(ρ2(x1,x2,t),ρ ′ 2(y1,y2,t))=1, and lim t→∞ Smax(ρ3(x1,x2,t),ρ ′ 3(y1,y2,t))=1. Theothercases,clearlyobtained,so (X × Y,Tmin(ρ1,ρ ′ 1),Smax(ρ2,ρ ′ 2),Smax(ρ3,ρ ′ 3),T,S) isa KM -singlevaluedneutrosophicmetricspace. Let X ∩ Y = ∅, (X,ρ1,ρ2,ρ3,T,S) and (Y,ρ′ 1,ρ′ 2,ρ′ 3,T,S) be KM -singlevaluedneutrosophicmetricspaces, x,y ∈ X ∪ Y and t ∈ R≥0.Consider ϵ(x,y,t)= ∧ x,u

∈X y,v∈Y (ρ ′ 1(x,u,t)∧ρ ′ 1(y,v,t))), σ(x,y,t)= ∨ x,u∈X y,v∈Y (ρ2(x,u,t)∨ρ ′ 2(y,v,t))) and δ(x,y,t) = ∨ x,u∈X y,v∈Y (ρ3(x,u,t) ∨ ρ ′ 3(y,v,t))).Deﬁne ρ1 ∪ ρ′ 1,ρ2 ∪ ρ′ 2,ρ3 ∪ ρ′ 3 :(X ∪ Y )2 ×R≥0 →

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

[0, 1] by (ρ1 ∪ ρ ′ 1)(x,y,t)=      ρ1(x,y,t) if x,y ∈ X, ρ′ 1(x,y,t) if x,y ∈ Y, ϵ(x,y,t) if x ∈ X,y ∈ Y, , (ρ2 ∪ ρ ′ 2)(x,y,t)=      ρ2(x,y,t) if x,y ∈ X, ρ′ 2(x,y,t) if x,y ∈ Y, σ(x,y,t) if x ∈ X,y ∈ Y, and (ρ3 ∪ ρ′ 3)(x,y,t)=      ρ3(x,y,t) if x,y ∈ X, ρ′ 3(x,y,t) if x,y ∈ Y, δ(x,y,t) if x ∈ X,y ∈ Y,

Sowehavethefollowing theorem.

Theorem3.2. Let (X,ρ1,ρ2,ρ3,T,S) and (Y,ρ′ 1,ρ′ 2,ρ′ 3,T,S) be KM -singlevaluedneutrosophicmetricspaces.Then (X ∪ Y,ρ1 ∪ ρ′ 1,ρ2 ∪ ρ′ 2,ρ3 ∪ ρ′ 3,T,S) isa KM -singlevaluedneutrosophicmetricspace,where X ∩ Y = ∅

Proof. Let x,y,z ∈ X ∪ Y and t,s ∈ R≥0.Weonlyprovethetriangularinequality propertyandothercasesareimmediate.Let x,y ∈ X(for x,y ∈ Y issimilar), then T ((ρ1 ∪ ρ′ 1)(x,y,t), (ρ1 ∪ ρ′ 1)(y,z,s)) = T (ρ1(x,y,t), (ρ1 ∪ ρ′ 1)(y,z,s)). If z ∈ X,then T ((ρ1 ∪ ρ′ 1)(x,y,t), (ρ1 ∪ ρ′ 1)(y,z,s)) = T (ρ1(x,y,t),ρ1(y,z,s)) ≤ ρ1(x,z,t + s)=(ρ1 ∪ ρ′ 1)(x,z,t + s) If z ∈ Y ,then T ((ρ1 ∪ ρ′ 1)(x,y,t), (ρ1 ∪ ρ′ 1)(y,z,s)) = T (ρ1(x,y,t),ϵ) ≤ ϵ =(ρ1 ∪ ρ′ 1)(x,z,t + s). Let x ∈ X,y ∈ Y .Then T ((ρ1 ∪ρ′ 1)(x,y,t), (ρ1 ∪ρ′ 1)(y,z,s)) = T (ϵ, (ρ1 ∪ρ′ 1)(y,z,s)) If z ∈ Y ,since x ∈ X and y ∈ Y ,wegetthat (ρ1 ∪ ρ′ 1)(x,z,t + s)= ϵ andso T (ϵ, (ρ1 ∪ ρ′ 1)(y,z,s)) = T (ϵ,ρ2(y,z,s)) ≤ ϵ =(ρ1 ∪ ρ′ 1)(x,z,t + s) If z ∈ X,since x ∈ X and y ∈ Y ,we getthat (ρ1 ∪ ρ′ 1)(x,z,t + s) = ϵ andso T (ϵ, (ρ1 ∪ ρ′ 1)(y,z,s)) = T (ϵ,ϵ) ≤ ϵ ≤ ρ1(x,z,t + s)) =(ρ1 ∪ ρ′ 1)(x,z,t + s). Supposethat x,y ∈ X(for x,y ∈ Y issimilar),then S((ρ2 ∪ ρ′ 2)(x,y,t), (ρ2 ∪ ρ′ 2)(y,z,s)) = S(ρ2(x,y,t), (ρ2∪ρ′ 2)(y,z,s)) If z ∈ X,then S((ρ2∪ρ′ 2)(x,y,t), (ρ2∪ ρ′ 2)(y,z,s)) = S(ρ2(x,y,t),ρ2(y,z,s)) ≥ ρ2(x,z,t + s)=(ρ2 ∪ ρ′ 2)(x,z,t + s) If z ∈ Y ,then S((ρ2 ∪ ρ′ 2)(x,y,t), (ρ2 ∪ ρ′ 2)(y,z,s)) = S(ρ2(x,y,t),σ) ≥ σ = (ρ2 ∪ρ′ 2)(x,z,t+s) Let x ∈ X,y ∈ Y .Then S((ρ2 ∪ρ′ 2)(x,y,t), (ρ2 ∪ρ′ 2)(y,z,s)) = S(σ, (ρ2 ∪ ρ′ 2)(y,z,s)) If z ∈ Y ,since x ∈ X and y ∈ Y ,wegetthat (ρ2 ∪ ρ′ 2)(x,z,t + s)= σ andso S(σ, (ρ2 ∪ ρ′ 2)(y,z,s)) = S(σ,ρ2(y,z,s)) ≥ σ =(ρ2 ∪ ρ′ 2)(x,z,t+s) If z ∈ X,since x ∈ X and y ∈ Y ,wegetthat (ρ2 ∪ρ′ 2)(x,z,t+s) = σ andso S(σ, (ρ2 ∪ρ′ 2)(y,z,s)) = S(σ,σ) ≥ σ ≥ ρ2(x,z,t+s)) =(ρ2 ∪ρ′ 2)(x,z,t+s) Incommonaway,wecanprovethatforall x,y ∈ X ∪ Y , S((ρ3 ∪ ρ′ 3)(x,y,t), (ρ3 ∪ ρ′ 3)(y,z,s)) ≥ (ρ3 ∪ ρ′ 3)(x,z,t + s) (X ∪ Y,ρ1 ∪ ρ′ 1,ρ2 ∪ ρ′ 2,ρ3 ∪ ρ′ 3,T,S) isa KM -singlevaluedneutrosophicmetricspace,where X ∩ Y = ∅

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 11

4.InducedTopologyfrom KM-SingleValuedNeutrosophicMetricSpace

Inthissection,in KM -singlevaluedneutrosophicmetricspace (X,ρ1,ρ2,ρ3,T,S), weintroducethesubsetsasballsandshowthattheyareopensubsets.Alsowe provethatevery KM -singlevaluedneutrosophicmetrics ρ1,ρ2,ρ3 on X whichhas asabasethefamilyofopensetsoftheform O = {O(x,ϵ,t) | x ∈ X, 0 <ϵ< 1,t ∈ R+}

Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetricspace, t ∈ R≥0,x ∈ X and 0 <ϵ< 1.Deﬁne Oρ1 (x,ϵ,t)= {y ∈ X | ρ1(x,y,t) > 1 ϵ},Oρ2 (x,ϵ,t)= {y ∈ X | ρ2(x,y,t) <ϵ},Oρ3 (x,ϵ,t)= {y ∈ X | ρ3(x,y,t) <ϵ} and O(x,ϵ,t)= {y ∈ X | ρ1(x,y,t) > 1 ϵ,ρ2(x,y,ϵ) <ϵ,ρ3(x,y,t) <ϵ} asaball withcenter x,radius ϵ,andatthetime t.Clearly Oρ1 (x,ϵ, 0)= Oρ2 (x,ϵ, 0)= Oρ3 (x,ϵ, 0)= O(x,ϵ, 0)= ∅

Theorem4.1. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetric space, t,t1,t2 ∈ R≥0,x,y ∈ X and 0 <ϵ,ϵ1,ϵ2 < 1.Then

(i) if t1 <t2,then Oρ1 (x,ϵ,t1) ⊆ Oρ1 (x,ϵ,t2);

(ii) if ϵ1 <ϵ2,then Oρ1 (x,ϵ1,t) ⊆ Oρ1 (x,ϵ2,t);

(iii) if ϵ1 <ϵ2 and t1 <t2,then Oρ1 (x,ϵ1,t1) ⊆ Oρ1 (x,ϵ2,t2);

(iv) forall t ∈ R+ , {x}⊆ Oρ1 (x,ϵ,t) = ∅;

(v) if X isﬁnite, t ∈ R+ and ϵt =(1 δ) ∨ x,y∈X ρ1(x,y,t),then Oρ1 (x,ϵt,t)= {x},where 0 <δ< 1;

(vi) if X isﬁniteand t ∈ R+,thereexists ϵmin suchthat Oρ1 (x,ϵmin,t)= X

Proof. (i) Let y ∈ Oρ1 (x,ϵ,t1).Then ρ1(x,y,t1) > 1 ϵ,sobyTheorem3.4, t1 <t2 impliesthat ρ1(x,y,t2) >ρ1(x,y,t1) > 1 ϵ.Thus Oρ1 (x,ϵ,t1) ⊆ Oρ1 (x,ϵ,t2)

(ii) Let y ∈ Oρ1 (x,ϵ1,t).Then ρ1(x,y,t) > 1 ϵ1,so ϵ1 <ϵ2 impliesthat ρ1(x,y,t) > 1 ϵ1 > 1 ϵ2.Thus Oρ1 (x,ϵ,t1) ⊆ Oρ1 (x,ϵ,t2) (iii), (iv) Itisclear.

(v) Let y ∈ Oρ1 (x,ϵt,t).Since ρ1(x,y,t) ≤ ∨ x,y∈X ρ1(x,y,t),if ρ1(x,y,t) > 1 ϵt,wegetthat ρ1(x,y,t)=1 andso x = y.

(vi) Consider ϵmin =1+ δ ∧ x,y∈X ρ1(x,y,t),where ∧ x,y∈X ρ1(x,y,t) δ> 0 Sinceforall y ∈ X,ρ1(x,y,t) >ϵmin,wegetthat X ⊆ Oρ1 (x,ϵmin,t) InasimilarwaytoTheorem4.1,wehavethefollowingTheorem.

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

Theorem4.2. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetric space, t,t1,t2 ∈ R≥0,x ∈ X and 0 <ϵ,ϵ1,ϵ2 < 1.Thenforall i ∈{2, 3}

(i) if t1 <t2,then Oρi (x,ϵ,t1) ⊆ Oρi (x,ϵ,t2); (ii) if ϵ1 <ϵ2,then Oρi (x,ϵ1,t) ⊆ Oρi (x,ϵ2,t);

(iii) if ϵ1 <ϵ2 and t1 <t2,then Oρi (x,ϵ1,t1) ⊆ Oρi (x,ϵ2,t2);

(iv) forall t ∈ R+ , {x}⊆ Oρi (x,ϵ,t) = ∅;

(v) if X isﬁnite, t ∈ R+ and ϵ′ t = δ + ∧ x,y∈X ρ2(x,y,t),then Oρi (x,ϵ′ t,t)= {x}, where 0 <δ< 1;

(vi) if X isﬁnite, t ∈ R+ and ϵ′ max = δ + ∨ x=y∈X ρ2(x,y,t),then Oρi (x,ϵ′ max,t)= X,where 0 <δ< 1

Corollary4.3. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetricspace, t,t1,t2 ∈ R≥0,x ∈ X and 0 <ϵ,ϵ1,ϵ2 < 1.Then

(i) if t1 <t2,then O(x,ϵ,t1) ⊆ O(x,ϵ,t2);

(ii) if ϵ1 <ϵ2,then O(x,ϵ1,t) ⊆ O(x,ϵ2,t);

(iii) if ϵ1 <ϵ2 and t1 <t2,then O(x,ϵ1,t1) ⊆ O(x,ϵ2,t2); (iv) forall t ∈ R+ , {x}⊆ O(x,ϵ,t) = ∅; (v) if X isﬁniteand t ∈ R+,then O(x,ϵt,t)= O(x,ϵ′ t,t)= {x}; (vi) if X isﬁniteand t ∈ R+,then O(x,ϵmin,t)= O(x,ϵ′ max,t)= X

Theorem4.4. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetric space, t ∈ R≥0,x,y ∈ X and 0 <r< 1.Then

(i) x ∈ O(y,ϵ,t) ifandonlyif y ∈ O(x,ϵ,t);

(ii) if O(x,ϵ,t) ∩ O(y,ϵ,t) = ∅ thenforall 2 ≤ k ∈ N wehave O(x,ϵ,kt) ∩ O(y,ϵ,kt) = ∅;

Proof. Itisobvious,bydeﬁnition.

In[3],GeorgeandVeeramaniprovedthatinanymetricspace (X,d) (Remark 2.8)theopenball Oρ1 isanopenset.Inthefollowingtheoremweshowthatthe openball O isanopenset.

Theorem4.5. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetric space, t ∈ R≥0,x ∈ X and 0 <r< 1.Thenforall i ∈{2, 3}

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 13

(i) O(x,ϵ,t)= Oρ1 (x,ϵ,t) ∩ Oρ2 (x,ϵ,t) ∩ Oρ3 (x,ϵ,t);

(ii) Oρi(x,ϵ,t) isanopenset;

(iii) O(x,ϵ,t) isanopenset.

Proof. (i) Itisobvious.

(ii) Let y ∈ Oρi(x,ϵ,t).Thenthereexists t>t0 ∈ R+ and ϵ′ suchthat ρi(x,y,t0) <ϵ and S(ϵ′,ϵ) <ϵ.Consider B = Oρi(y,ϵ′,t t0).Forall z ∈ Oρi(y,ϵ′,t t0),wehave ϵ>S(ϵ,ϵ′) ≥ S(ρi(x,y,t0),ρi(y,z,t t0)) ≥ ρi(x,z,t) Itfollowsthat y ∈ B ⊆ Oρi(x,ϵ,t) andso Oρi(x,ϵ,t) isanopenset. (iii) Itisimmediateby (i), (ii)

Theorem4.6. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetric space, t ∈ R≥0,x ∈ X and 0 <r< 1.Thenforall i ∈{1, 2, 3} (i) if X isaﬁnitesetand τ = {O(x,ϵ, 0),O(x,ϵmax,t) | x ∈ X, 0 <ϵ< 1,t ∈ R+},then (X,τ ) isatopologicalspace;

(ii) O = {Oρ2 (x,ϵ,t) | x ∈ X, 0 <ϵ< 1,t ∈ R+} formsabaseofatopology τρ2 in X;

(iii) O = {Oρ3 (x,ϵ,t) | x ∈ X, 0 <ϵ< 1,t ∈ R+} formsabaseofatopology τρ3 in X;

(iv) O = {Oρ1 (x,ϵ,t) | x ∈ X, 0 <ϵ< 1,t ∈ R+} formsabaseofatopology τρ1 in X.

Proof. (i) Itisobvious. (ii), (iii) Let x ∈ X and i ∈{2, 3}.ThenbyTheorem4.2,forall 0 <ϵ< 1,t ∈ R+,x ∈ Oρi (x,ϵ,t) andso X ⊆ ∪ ϵ,t Oρi (x,ϵ,t).Letfor x,y,z ∈ X, 0 < ϵ,ϵ′ < 1,t,t′ ∈ R+,wehave z ∈ Oρi (x,ϵ,t) ∩ Oρi (y,ϵ′,t′).Then ρ2(x,z,t) < ϵ,ρ2(y,z,t′) <ϵ′,ρ3(x,z,t) <ϵ and ρ3(y,z,t′) <ϵ′.Thusthereexists t0 ∈ R+ suchthat t0 <t,t0 <t′,which ρi(x,z,t0) <ϵ and ρi(y,z,t0) <ϵ′.Nowconsider 0 <ϵ′′ < 1, t′′ ∈ R+ suchthat Tmin(ϵ,ϵ′) >ϵ′′ and t′′ <Tmin{t t0,t′ t0} If m ∈ Oρi (z,ϵ′′,t′′),then ρi(m,y,t′) ≤ S(ρi(m,z,t′′),ρi(z,y,t0)) <S(ϵ′′,ϵ′) <ϵ′ andso m ∈ Oρi (y,ϵ′,t′).Analogously,onecanseethat Oρi (z,ϵ′′,t′′) ⊆ Oρi (x,ϵ,t). (iv) Let x ∈ X.ThenbyTheorem4.1,forall 0 <ϵ< 1,t ∈ R+,x ∈ Oρ1 (x,ϵ,t) andso X ⊆ ∪ ϵ,t Oρ1 (x,ϵ,t).Letfor x,y,z ∈ X, 0 <ϵ,ϵ′ < 1,t,t′ ∈ R+,wehave z ∈ Oρ1 (x,ϵ,t) ∩ Oρ1 (y,ϵ′,t′).Then ρ1(x,z,t) > 1 ϵ and ρ1(y,z,t′) > 1 ϵ′ Thusthereexists t0 ∈ R+ suchthat t0 <t,t0 <t′,which ρ1(x,z,t0) > 1 ϵ and ρ1(y,z,t0) > 1 ϵ′.Nowconsider 0 <ϵ′′ < 1, t′′ ∈ R+ suchthat Tmin(ϵ,ϵ′) > ϵ′′ and t′′ <Tmin{t t0,t′ t0}.If m ∈ Oρ1 (z,ϵ′′,t′′),then ρ1(m,y,t′) ≥ T (ρ1(m,z,t′′),ρ1(z,y,t0)) >T (1 ϵ′′ , 1 ϵ′) > 1 ϵ′ andso m ∈ Oρ1 (y,ϵ′,t′) Analogously,onecanseethat Oρ1 (z,ϵ′′,t′′) ⊆ Oρ1 (x,ϵ,t)

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

Corollary4.7. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetricspace, t ∈ R≥0,x ∈ X and 0 <r< 1.Then O = {O(x,ϵ,t) | x ∈ X, 0 <ϵ< 1,t ∈ R+} formsabaseofatopology τρ1,ρ2,ρ3 in X.

5. KM-SingleValuedNeutrosophicMetricSpaces andMetricSpaces

Inthissection,wepresenttheconnectionbetween KM -singlevaluedneutrosophic metricspacesandmetricspaceswithrespecttoinducedequivalencerelationbased onuniteintervalvalues.

Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetricspaceand α,β,γ ∈ [0, 1].Foreverytime t,on X2 , deﬁne ρ(α,t) 1 = {(x,y) | ρ1(x,y,t) ≥ α},ρ(β,t) 2 = {(x,y) | ρ2(x,y,t) ≤ β} and ρ(γ,t) 3 = {(x,y) | ρ3(x,y,t) ≤ γ} as α-part, β-partand γ-part. Theorem5.1. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspaceand α,β,γ ∈ [0, 1].Then

(i) if t =0,then ρ(α,t) 1 = ρ(β,t) 2 = ρ(γ,t) 3 = X2 ifandonlyif α =0 and β = γ =1;

(ii) if α = β = γ =0,then ρ(α,t) 1 = X2,ρ(β,t) 2 = ρ(γ,t) 3 = △; (iii) if α = β = γ =1,then ρ(α,t) 1 = △,ρ(β,t) 2 = ρ(γ,t) 3 = X2; (iv) ρ(α,t) 1 ,ρ(β,t) 2 and ρ(γ,t) 3 arereﬂexiveandsymmetricrelationson X(t> 0); (v) ρ(α,β,γ,t) = ρ(α,t) 1 ∩ ρ(β,t) 2 ∩ ρ(γ,t) 3 isareﬂexiveandsymmetricrelationon X(t> 0)

Proof. (i) Let t =0 and α,β,γ ∈ [0, 1].Then ρ(α,t) 1 = {(x,y) | ρ1(x,y, 0) ≥ α} = {(x,y) | 0 ≥ α}.Hence ρ(α,t) 1 = X2 ifandonlyif α =0 Inasimilarway, ρ(β,t) 2 = ρ(γ,t) 3 = X2 ifandonlyif β = γ =1

(ii), (iii) Obviouslyareproved.

(iv) Let α,β,γ ∈ [0, 1], x,y,z ∈ X and t,s ∈ R≥0.Since ρ1(x,x,t)=1 ≥ α,ρ2(x,x,t)=0 ≤ β and ρ3(x,x,t)=0 ≤ γ,wegetthat ρ(α,t) 1 ,ρ(β,t) 2 ,ρ(γ,t) 3 are reﬂexiverelations.Clearly ρ(α,t) 1 ,ρ(β,t) 2 ,ρ(γ,t) 3 aresymmetricrelations.

(v) Itissimilartoitem (iv)

InthefollowingExample,wedescribesomeapplicationsof KM -singlevalued neutrosophicmetricspace.Wediscussapplicationsof KM -singlevaluedneutrosophicmetricspaceforstudyingthecompetitionalongwithalgorithms.The KM -singlevaluedneutrosophicmetricspacehasmanyutilizationsindiﬀerent areas,whereweconnect KM -singlevaluedneutrosophicmetricspacetoother

sciencessuchaseconomics,computersciencesandotherengineeringsciences.We presentanexampleofapplicationof KM -singlevaluedneutrosophicmetricspace inoptimizationineconomic.

Example5.2. (DecisioninEconomic) Let X = {x1,x2,x3,x4} beasetof allfactoriesofatownandbyTable1bemetrics.Wewanttocombinethe performancesofthesefactoriesinaoneyearcourse (t =1) todecideonhow theyworktogether.InTable2,weisextractedalistofenergyconsumption offactoriesby ρ1,alistoftheproﬁtsofproductionoffactoriesby ρ2 anda listoflossesofproducingoffactoriesby ρ3.ForExample,ifenergyconsumptionoffactories x1,x2 isequalto 60/100,itisdenotedby ρ1(x1,x2, 1)=0 6,if lossesofproducingareequalto 30/100,itisdenotedby ρ1(x1,x2, 1)=0 3 and ifproﬁtsofproducingareequalto 20/100,itisdenotedby ρ1(x1,x2, 1)=0.2.

Clearly (X,ρ1,ρ2,ρ3,Tpr,Smax) isa KM -singlevaluedneutrosophicmetricspace. Soconsideracontrolforenergyconsumption,proﬁtsofproducingandlosses ofproducingwithfactorycooperationby α-part, β-partand γ-part.For α = 0 3,β =0 2,γ =0 12,t =1, weobtain ρ(α,t) 1 = X2 \{(x1,x4), (x4,x1)},ρ(β,t) 2 = △∪{(x1,x2), (x2,x1), (x2,x3), (x3,x2), (x3,x4), (x4,x3)} = ρ(γ,t) 3 .

Table1:Metricspace (X,d) d x1 x2 x3 x4 x1 0123 x2 1012 x3 2101 x4 3210

Table2:Fuzzymetricsubsets ρ1,ρ2,ρ3 on X2 × R≥0 . ρ1 x1 x2 x3 x4 x1 1 1 2 1 3 1 4 x2 1 2 1 1 2 1 3 x3 1 3 1 2 1 1 2 x4 1 4 1 3 1 2 1

ρ2 x1 x2 x3 x4 x1 0 1 6 2 9 1 4 x2 1 6 0 1 6 2 9 x3 1 9 1 6 0 1 6 x4 1 4 2 9 1 6 0

and ρ3 x1 x2 x3 x4 x1 0 1 10 2 15 3 20 x2 1 10 0 1 10 2 15 x3 2 15 1 10 0 1 10 x4 3 20 2 15 1 10 0

Clearly ρ(α,t) 1 ,ρ(β,t) 2 and ρ(γ,t) 3 arenottransitiverelationsandso ρ(α,β,γ,t) isnota transitiverelation.

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 15 M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

Let ρ(α,t,∗) 1 ,ρ(β,t,∗)) 2 , ρ(γ,t,∗)) 3 and ρ(α,β,γ,t,∗) bethe transitiveclosure of ρ(α,t) 1 , ρ(β,t)) 2 , ρ(γ,t)) 3 and ρ(α,β,γ,t),respectively(thesmallesttransitiverelationinsuch awaythatcontains ρ(α,t) 1 ,ρ(β,t)) 2 , ρ(γ,t)) 3 and ρ(α,β,γ,t),respectively).Thenin thefollowingtheoremweshowthat ρ(α,t,∗) 1 ,ρ(β,t,∗)) 2 , ρ(γ,t,∗)) 3 and ρ(α,β,γ,t,∗) are regularrelations.Deﬁne X/ρ(α,β,γ,t,∗) = {ρ(α,β,γ,t,∗)(x,y,t) | x,y ∈ X)} assetof allequivalenceclassof X on ρ(α,β,γ,t,∗)

Theorem5.3. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspaceand α,β,γ ∈ [0, 1].Then

(i) if t =0,then ρ(α,t,∗) 1 = ρ(α,t) 1 ,ρ(β,t,∗) 2 = ρ(β,t) 2 and ρ(γ,t,∗) 3 = ρ(γ,t) 3 ;

(ii) if α = β = γ =0,then ρ(α,t,∗) 1 = ρ(α,t) 1 ,ρ(β,t,∗) 2 = ρ(β,t) 2 and ρ(γ,t,∗) 3 = ρ(γ,t) 3 ;

(iii) if α = β = γ =1,then ρ(α,t,∗) 1 = ρ(α,t) 1 ,ρ(β,t,∗) 2 = ρ(β,t) 2 and ρ(γ,t,∗) 3 = ρ(γ,t) 3

Proof. (i) Let α,β,γ ∈ [0, 1] and x,y ∈ X.If t =0,then ρ(α,t) 1 ∈{∅,X2},ρ(β,t) 2 ∈ {∅,X2} and ρ(γ,t) 3 ∈{∅,X 2}.Thusinanycase, ρ(α,t,∗) 1 = ρ(α,t) 1 ,ρ(β,t,∗) 2 = ρ(β,t) 2 and ρ(γ,t,∗) 3 = ρ(γ,t) 3 . (ii) Let α = β = γ =0.ThenbyTheorem5.1, ρ(α,t) 1 = X2,ρ(β,t) 2 = ρ(γ,t) 3 = △ andso ρ(α,t,∗) 1 = ρ(α,t) 1 ,ρ(β,t,∗) 2 = ρ(β,t) 2 and ρ(γ,t,∗) 3 = ρ(γ,t) 3 (iii) Itissimilartoitem (ii)

Theorem5.4. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspaceand α,β,γ ∈ [0, 1].Thenforall t ∈ R≥0 and n ∈ N,

(i) ρ(α,t) 1 ⊆ ρ(α,nt) 1 andforall k ≥ n,ρ(α,nt) 1 ⊆ ρ(α,kt) 1 ;

(ii) ρ(β,t) 2 ⊆ ρ(β,nt) 2 andforall k ≥ n,ρ(β,nt) 2 ⊆ ρ(β,kt) 2 ;

(iii) ρ(γ,t) 3 ⊆ ρ(γ,nt) 3 andforall k ≥ n,ρ(γ,nt) 3 ⊆ ρ(γ,kt) 3

Proof. (i) Let x,y ∈ X,t ∈ R≥0 and n ∈ N.If (x,y) ∈ ρ(α,t) 1 ,thensimpleinduction concludesthat α ≤ Tmin(ρ1(x,y,t),ρ1(y,y,t),...,ρ1(y,y,t) (n 1)-times

) ≤ ρ1(x,y,nt),so ρ(α,t) 1 ⊆ ρ(α,nt) 1 andforall k ≥ n,ρ(α,nt) 1 ⊆ ρ(α,kt) 1

(ii,iii) Let x,y ∈ X,t ∈ R≥0 and n ∈ N.Since nt>t,byTheorem3.4,weget that ρ2(x,y,t) ≥ ρ2(x,y,nt),ρ3(x,y,t) ≥ ρ3(y,y,nt) anditconcludesthat ρ(β,t) 2 ⊆ ρ(β,nt) 2 , ρ(γ,t) 3 ⊆ ρ(γ,nt) 3 andforall k ≥ n,ρ(β,nt) 2 ⊆ ρ(β,kt) 2 and ρ(γ,nt) 3 ⊆ ρ(γ,kt) 3 .

Theorem5.5. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspaceand α,β,γ ∈ (0, 1).Then,forall t> 0,thereexiststhesmallest n ∈ N, suchthat

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 17

(i) ρ(α,t,∗) 1 = n ∪ k=1 ρ(α,kt) 1 ,whereforall k ≥ n, wehave ρ(α,kt) 1 = ρ(α,nt) 1 ;

(ii) ρ(β,t,∗) 2 = n ∪ k=1 ρ(β,kt) 2 ,whereforall k ≥ n, wehave ρ(β,kt) 2 = ρ(β,nt) 1 ;

(iii) ρ(γ,t,∗) 3 = n ∪ k=1 ρ(γ,kt) 3 ,whereforall k ≥ n, wehave ρ(γ,kt) 3 = ρ(γ,nt) 3

Proof. Let n ∈ N,α,β,γ ∈ (0, 1) and x,y ∈ X. (i) ByTheorem5.4, ρ1(x,y,t) ≤ ρ1(x,y,nt).Since lim t→∞ ρ1(x,y,t)=1,weget that lim t→∞ ρ1(x,y,nt)=1.Thusthereexists N ∈ N suchthatforall t> ⌊N/n⌋, wehave 1 α<ρ1(x,y,t).Hence Aα = {m ∈ N | ρ1(x,y,m) ≥ α}̸= ∅ andusing well-orderingprinciple,thereexiststhesmallest n ∈ N suchthat ρ1(x,y,nt) ≥ α. ByTheorem5.4,forall k ≥ n,ρ(α,nt) 1 ⊆ ρ(α,kt) 1 .Inaddition,since n isthesmallest which ρ1(x,y,m) ≥ α,wegetthat ρ(α,kt) 1 ⊆ ρ(α,nt) 1 andsoforall k ≥ n,ρ(α,nt) 1 = ρ(α,kt) 1 .Now,if (x,y), (y,z) ∈ ρ(α,t,∗) 1 ,then,thereexists k,k′ ∈ N insucha waythat (x,y) ∈ ρ(α,kt) 1 and (x,y) ∈ ρ(α,k′t) 1 .Theorem5.4,impliesthat α ≤ Tmin(ρ1(x,y,t),ρ1(y,z,t)) ≤ Tmin(ρ1(x,y,kt),ρ1(y,z,kt)) ≤ ρ1(x,y, (k + k′)t) Thus (x,z) ∈ ρ(α,(k+k′)t) 1 ⊆ n ∪ k=1 ρ(α,kt) 1 = ρ(α,t,∗) 1 andso ρ(α,t,∗) 1 isatransitive relation.Supposethat R beatransitiverelationsuchthat ρ(α,t) 1 ⊆ R,byinduction weshowthat ρ(α,t,∗) 1 ⊆ R.Itisclearfor n =1.Assumethatitissatisﬁed for n,ifmeansthat n ∪ k=1 ρ(α,kt) 1 ⊆ R.If (x,y) ∈ n+1 ∪ k=1 ρ(α,kt) 1 ,since n+1 ∪ k=1 ρ(α,kt) 1 = n ∪ k=1 ρ(α,kt) 1 ∪ ρ(α,t) 1 ,byassumptionofinductionwegetthat (x,y) ∈ n ∪ k=1 ρ(α,kt) 1 ⊆ R or (x,y) ∈ ρ(α,t) 1 ⊆ R.Itfollowsthat ρ(α,t,∗) 1 ⊆ R andsoitis transitiveclosure of ρ(α,t) 1 Inasimilarway,onecanseethat (ii), (iii) areproved.

Corollary5.6. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspaceand α,β,γ ∈ (0, 1).Then,forall t> 0,thereexiststhe smallest n ∈ N, suchthat ρ(α,β,γ,t,∗) = n ∪ k=1 ρ(α,β,γ,nt),whereforall k ≥ n, wehave ρ(α,β,γ,kt) = ρ(α,β,γ,nt) .

Theorem5.7. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspace.Thenthereexists α,β,γ ∈ [0, 1] suchthatforall t ≥ 0, ρ(α,β,γ,t,∗) = ρ(α,β,γ,t)

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

Proof. Let x,y ∈ X.Thenconsider α = ∧ x,y∈X ρ1(x,y,t),β = ∨ x,y∈X ρ1(x,y,t) and γ = ∨ x,y∈X ρ1(x,y,t) Sinceforall x,y ∈ X,ρ1(x,y,t) ≥ α,ρ2(x,y,t) ≤ β and ρ2(x,y,t) ≤ γ,forall t ≥ 0 wegethat ρ(α,t,∗) 1 = ρ(α,t) 1 ,ρ(β,t,∗) 2 = ρ(β,t) 2 and ρ(γ,t,∗) 3 = ρ(γ,t) 3 .Itconcludesthatthereexists α,β,γ ∈ [0, 1] suchthatforall t ≥ 0, ρ(α,β,γ,t,∗) = ρ(α,β,γ,t)

Example5.8. Considerthe KM -singlevaluedneutrosophicmetricspace (X,ρ1,ρ2 ,ρ3,T,S) inExample5.2.For α =0.3,β =0.2,γ =0.12,byCorollary5.6, weobtain n =2(wemustconsidertheperformancesofthesefactoriesintwo yearscourse (t =2)) andgetfuzzysubsets ρ1,ρ2,ρ3 inTable3.Thus ρ(α,2t) 1 =

Table3:Fuzzymetricsubsets ρ1,ρ2,ρ3 on X2 × R≥0 ρ1 x1 x2 x3 x4 x1 1 1 3 1 4 1 5 x2 1 3 1 1 3 1 4 x3 1 4 1 3 1 1 3 x4 1 5 1 4 1 3 1

ρ2 x1 x2 x3 x4 x1 0 1 9 1 6 1 5 x2 1 9 0 1 9 1 6 x3 1 6 1 9 0 1 9 x4 1 5 1 6 1 9 0

and ρ3 x1 x2 x3 x4 x1 0 1 15 1 10 3 25 x2 1 15 0 1 15 1 10 x3 1 10 1 15 0 1 15 x4 3 25 1 10 1 15 0

△∪{(x1,x2)},ρ(β,2t) 2 = ρ(γ,2t) 3 = X2 andso ρ(α,β,γ,2t) = △∪{(x1,x2)} isa transitiverelation.

Fromnowon,forall KM -singlevaluedneutrosophicmetricspace (X,ρ1,ρ2,ρ3, T,S),weconsider A,B ∈ P ∗(X) areﬁnitesubsets.

Deﬁnition5.9. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetricspace, A,B ∈ P ∗(X),t ∈ R+,x,y ∈ X.Deﬁne (i) ρ1(x,B,t)= ∨ y∈B ρ1(x,y,t), ρ2(x,B,t)= ∧ y∈B ρ2(x,y,t) and ρ3(x,B,t)= ∧ y∈B ρ3(x,y,t)

(ii) ρ1(A,y,t)= ∨ x∈A ρ1(x,y,t), ρ2(A,y,t)= ∧ x∈A ρ2(x,y,t) and ρ3(A,y,t)= ∧ x∈A ρ3(x,y,t)

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 19

(iii) ρ1(A,B,t)=( ∧ x∈A ∨ y∈B ρ1(x,y,t)) ∧ ( ∧ y∈B ∨ x∈A ρ1(x,y,t)),

(iv) ρ2(A,B,t)=( ∨ x∈A ∧ y∈B ρ2(x,y,t)) ∨ ( ∨ y∈B ∧ x∈A ρ2(x,y,t)),

(v) ρ3(A,B,t)=( ∨ x∈A ∧ y∈B ρ3(x,y,t)) ∨ ( ∨ y∈B ∧ x∈A ρ3(x,y,t))

In[12],Rodrguez-LopezandRomaguera,provedthatinevery GV -fuzzymetric space (X,ρ1,T ),thesetofnonemptycompactsubsets (K∗(X)) of (X; τX ) constructafuzzymetricspaceasHausdorffuzzymetric.Inasimilarwaywehave thefollowingTheorem.

Corollary5.10. Let (X,ρ1,ρ2,ρ3,T,S) bea KM -singlevaluedneutrosophicmetricspace, A,B,C ∈K∗(X),t ∈ R+.Then (K∗(X), ρ1, ρ2, ρ3,T,S) isa KM -single valuedneutrosophicmetricspace.

Theorem5.11. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspaceand α,β,γ ∈ [0, 1].Then, (X/ρ(α,β,γ,t,∗) , ρ1, ρ2, ρ3,T,S) isa KM -singlevaluedneutrosophicmetricspace.

Proof. TheproofisobtainedbyCorollary5.10.

Deﬁnition5.12. Let (X,ρ1,ρ2,ρ3,T,S) and (X′,ρ′ 1,ρ′ 2,ρ′ 3,T ′,S′) be KM -single valuedneutrosophicmetricspaces.Abijection φ : X → X′ iscalledanisomorphismifforall x,y ∈ X andforall i ∈{1, 2, 3},ρi(x,y,t)= ρ′ i(φ(x),φ(y),t) and denoteditby (X,ρ1,ρ2,ρ3,T,S) ∼ = (X′,ρ′ 1,ρ′ 2,ρ′ 3,T ′,S′)

Corollary5.13. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspaceand α,β,γ ∈ [0, 1].Then

(i) if t =0,then (X/ρ(α,β,γ,t,∗) , ρ1, ρ2, ρ3,T,S) ∼ = (X,ρ1,ρ2,ρ3,T,S); (ii) if α = β = γ =0,then (X/ρ(α,β,γ,t,∗) , ρ1, ρ2, ρ3,T,S) ∼ = (X,ρ1,ρ2,ρ3,T,S); (iii) if α = β = γ =1,then (X/ρ(α,β,γ,t,∗) , ρ1, ρ2, ρ3,T,S) ∼ = (X,ρ1,ρ2,ρ3,T,S)

Deﬁnition5.14. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspaceand α,β,γ ∈ [0, 1].Forall x,y ∈ X, deﬁne

d(α,t,∗)(x,y)= {0 if x = y |x/ρ(α,t,∗) 1 | + |y/ρ(α,t,∗) 1 | if x = y , d(β,t,∗)(x,y)= {0 if x = y |x/ρ(β,t,∗) 2 | + |y/ρ(β,t,∗) 2 | if x = y , d(γ,t,∗)(x,y)= {0 if x = y |x/ρ(γ,t,∗) 3 | + |y/ρ(γ,t,∗) 3 | if x = y ,

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri and d(α,β,γ,t,∗)(x,y)= {0 if x = y |x/ρ(α,β,γ,t,∗)| + |y/ρ(α,β,γ,t,∗)| if x = y

Theorem5.15. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspaceand 0 ≤ α,β,γ ≤ 1.Then (i) (X,d(α,t,∗)), (X,d(β,t,∗)), (X,d(γ,t,∗)) and (X,d(α,β,γ,t,∗)) aremetricspaces. (ii) d(α,β,γ,t,∗) =(d(α,t,∗)) ∧ (d(β,t,∗)) ∧ (d(γ,t,∗))

Proof. Itisclearthat. Let A = {(X,ρ1,ρ2,ρ3,T,S) | ρ1,ρ2,ρ3 arefuzzysubsetson X2 × R≥0} and B = {(X,d) | d isametric }.Deﬁne φ : A→B,by φ((X,ρ1,ρ2,ρ3,T,S))= (X,d(α,β,γ,t,∗)) basedTheorem5.15and ψ : B→A by φ((X,d))=(X,ρ1,ρ2,ρ3,T, S) basedCorollary3.9.

Corollary5.16. Let α,β,γ ∈ [0, 1].ThenForsets A and B,wehaveadiagram inFigure1. A • B • φ ψ

Figure1:Diagrambetween KM -singlevaluedneutrosophicmetricspaceandmetricspace.

5.1.Extended(KM-SingleValuedNeutrosophic)Metric

Inthissubsection,weobtaincontinuousmetricsfrom KM -singlevaluedneutrosophicmetricspacesandcontinuous KM -singlevaluedneutrosophicmetricfrom metricspaces.

Deﬁnition5.1. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, α,β,γ ∈ (0, 1).Forall x,y ∈ X, deﬁne dα(x,y)= ∧{t ∈ R+ | ρ1(x,y,t) ≥ α },dβ (x,y)= ∧{t ∈ R+ | ρ2(x,y,t) ≤ β } and dγ (x,y)= ∧{t ∈ R+ | ρ3(x,y,t) ≤ γ }.

Inwhatfollows,wegenerateafamilyofmetricspacesfroma KM -singlevalued neutrosophicmetricspace.

Theorem5.2. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspace, α,β,γ ∈ (0, 1).Then (X,dα), (X,dβ ) and (X,dγ ) aremetricspaces.

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 21

Proof. Let x,y,z ∈ X and β ∈ (0, 1).Then dβ (x,x)= ∧{t ∈ R+ | ρ2(x,x,t) ≤ β} = ∧{t ∈ R+ | 0 ≤ β } =0.Itisclearthat dβ (x,y)= dβ (y,x).Let ∧{t ∈ R+ | ρ2(x,y,t) ≤ β} = t0 and ∧{t ∈ R+ | ρ2(y,z,t) ≤ β} = s0.Thus β = Smax(ρ2(x,y,t0),ρ2(y,z,s0)) ≥ ρ2(x,z,t0 + s0) impliesthat ρ2(x,z,t0 + s0) ≤ β.Itfollowsthat (t0 + s0) ∈ ∧{t ∈ R+ | ρ2(x,z,t) ≤ β} andso dβ (x,y)+ dβ (y,z)= ∧{t ∈ R+ | ρ2(x,y,t) ≤ β} + ∧{t ∈ R+ | ρ2(y,z,t) ≤ β}≥ ∧{t ∈ R+ | ρ2(x,z,t) ≤ β} = dβ (x,z) Inasimilarwayonecanseethat (X,dα) and (X,dγ ) aremetricspaces.

Theorem5.3. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspace, α,α′,β,β′,γ,γ′ ∈ (0, 1) and x,y ∈ X.Then (i) if α ≤ α′,then dα(x,y) ≤ dα′ (x,y); (ii) if β ≤ β′,then dβ′ (x,y) ≤ dβ (x,y); (iii) if γ ≤ γ′,then dγ′ (x,y) ≤ dγ (x,y)

Proof. Let x,y,z ∈ X and α,α′,β,β′,γ,γ′ ∈ (0, 1).If dβ (x,y)= t0,then ρ2(x,y,t0) ≤ β ≤ β′ andso t0 ∈ ∧{t ∈ R+ | ρ2(x,y,t) ≤ β′}.Itconcludes that dβ′ (x,y) ≤ t0 = dβ (x,y). Otheritemsareprovedinasimilarway.

Theorem5.4. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspace, α,β,γ ∈ (0, 1).Then dα,dβ ,dγ :(X,ρ1,ρ2,ρ3,Tmin,Smax) × (X,ρ1,ρ2,ρ3,Tmin,Smax) → R≥0 , arecontinuousmaps.

Proof. Let (xn,yn)n beasequencein X × X thatconvergesto (x,y) withrespect tothefuzzymetric (X,ρ1,ρ2,ρ3,Tmin,Smax).Thusforall ε> 0 and t ∈ R+,there exists N ∈ N,suchthatforall n ≥ N , 1 ϵ<ρ1((xn,x,t)),ρ2((xn,x,t)) <ϵ and ρ3((xn,x,t)) <ϵ. Henceforall α,β,γ, ∈ (0, 1), {t ∈ R+ | ρ1(xn,x,t) ≥ α}̸= ∅, {t ∈ R+ | ρ2(xn,x,t) ≤ β}̸= ∅ and {t ∈ R+ | ρ2(xn,x,t) ≤ γ}̸= ∅. Hencethereexists N ∈ N,suchthatforall n ≥ N, andall α< 1 ε,β>ε and γ>ε,dα(xn,x) < t,dβ (xn,x) <t and dγ (xn,x) <t andso lim n→∞ dα(xn,x)= dβ (xn,x)= dγ (xn,x)= 0.Inasimilarway, lim n→∞ dα(yn,y)= dβ (yn,y)= dγ (yn,y)=0.Sinceforall, n ∈ N,dα(xn,x)+ dα(xn,yn)+ dα(yn,y) ≥ dα(x,y),wegetthat | lim n→∞ dα(xn,yn) dα(x,y)|≤ (lim n→∞ dα(xn,x)+lim n→∞ dα(y,yn))=0.Inasimilarway, lim n→∞ dβ (xn,yn)= dβ (x,y) and lim n→∞ dγ (xn,yn)= dγ (x,y)

Theorem5.5. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspace, α,α′,β,

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

β′,γ,γ′ ∈ (0, 1) and x,y ∈ X.Thenasequence (xn)n isaCauchysequencein (X,ρ1,ρ2,ρ3,Tmin ,Smax) ifandonlyifitisaCauchysequencein (X,dα), (X,dβ ) and (X,dγ ).

Proof. Suppose (xn)n isaCauchysequenceinmetricspaces (X,dα), (X,dβ ) and (X,dγ ) Fixed ε> 0 Since (xn)n isaCauchysequencein (X,dβ ),thereexists N ∈ N suchthatforall n,m ≥ N,dβ (xn,xm) <ε.Itfollowsthat t0 = ∧{t ∈ R+ | ρ2(xn,xm,t) ≤ β} <ε andso ρ2(xn,xm,t) ≤ ρ2(xn,xm,t0) <ε.Ina similarway,onecanseethat ρ3(xn,xm,t) <ε and ρ1(xn,xm,t) > 1 ε andso sequence (xn)n isaCauchysequencein KM -singlevaluedneutrosophicmetric space (X,ρ1,ρ2,ρ3,Tmin,Smax) Conversely,letsequence (xn)n isaCauchysequencein (X,ρ1,ρ2,ρ3,Tmin,Smax) andFixed ε> 0 Thusforall t ∈ R+,thereexists N ∈ N suchthatforany m,n ≥ N, wehave 1 ε<ρ1(xn,xm,t),ρ2(xn,xm,t) <ε and ρ3(xn,xm,t) <ε. Henceforall α,β,γ, ∈ (0, 1), {t ∈ R+ | ρ1(xn,xm,t) ≥ α}̸= ∅, {t ∈ R+ | ρ2(xn,xm, t) ≤ β}̸= ∅ and {t ∈ R+ | ρ2(xn,xm,t) ≤ γ}̸= ∅ Sothereexists N ∈ N suchthatforall n,m ≥ N andall α< 1 ε,β>ε and γ>ε,wehave dα(xn,xm) ≤ t<ε,dβ (xn,xm) <t<ε and dγ (xn,xm) <t<ε.Itisconcluded thatthesequence (xn)n isaCauchysequenceinmetricspaces (X,dα), (X,dβ ) and (X,dγ )

Deﬁnition5.6. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, α,β,γ ∈ (0, 1).Forall x ∈ X and k ∈ N, deﬁne dα(x,y)= kdα(x,y) 1+ kdα(x,y) , dβ (x,y)= kdβ (x,y) 1+ kdβ (x,y) and dγ (x,y)= kdγ (x,y) 1+ kdγ (x,y)

Theorem5.7. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspace, α,β,γ ∈ (0, 1).Then (X, dα), (X, dβ ) and (X, dγ ) aremetricspaces.

Proof. TheproofissimilartoTheorem5.2.

Theorem5.8. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophic metricspace, α,β,γ ∈ (0, 1).Then dα, dβ , dγ :(X,ρ1,ρ2,ρ3,Tmin,Smax) × (X,ρ1,ρ2,ρ3,Tmin,Smax) → R≥0 , arecontinuousmaps.

Proof. TheproofissimilartoTheorem5.4.

Definition5.9. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, α,β,γ ∈ (0, 1) and ∅̸= A ⊆ X.Forall x ∈ X, define dα(x,A)= ∧{dα(x,y) | y ∈ A},dβ (x,A)= ∧{dβ (x,y) | y ∈ A},dγ (x,A)= ∧{dγ (x,y) | y ∈ A}.Inasimilarway dα(x,A), dβ (x,A) and dγ (x,A) aredefined.

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces

Theorem5.10. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, α,β,γ ∈ (0, 1) and ∅̸= A ⊆ X.Thenforall x,y ∈ X

(i) if x ∈ A,then dα(x,A)= dβ (x,A)= dγ (x,A)=0;

(ii) forall a ∈ A, dα(x,A) ≤ dα(x,a),dβ (x,A) ≤ dβ (x,a),dγ (x,A) ≤ dγ (x,a);

(iii) |dα(x,A) dα(y,A)|≤ dα(x,y);

(iv) |dβ (x,A) dβ (y,A)|≤ dβ (x,y);

(v) |dγ (x,A) dγ (y,A)|≤ dγ (x,y).

Corollary5.11. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, β,γ ∈ (0, 1) and ∅̸= A ⊆ X.Thenforall x,y ∈ X

Theorem5.12. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, α,β,γ ∈ (0, 1).Thenforall x ∈ X and ∅̸= A ⊆ X, dα(x,A),dα(x,A),dα(x,A):(X,ρ1,ρ2,ρ3,Tmin,Smax)×(X,ρ1,ρ2,ρ3,Tmin,Smax) → R≥0 arecontinuousmaps.

Proof. Suppose (xn)n isasequencein X andinsuchawaythatforall t ∈ R+ , lim n→∞ ρ2(xn,x,t)=1 and x ∈ X. Fixed ε> 0 Thusthereexists N ∈ N suchthatforall n ≥ N,ρ2(xn,x,t) <ε.Itfollowsthatforall ε<β, wehave ∧{t ∈ R+ | ρ2(xn,x,t) ≤ β}̸= ∅ andso dβ (xn,x) <t.Hence lim n→∞ dβ (xn,x)=0 UsingTheorem5.10, |dβ (x,A) dβ (xn,A)|≤ dβ (x,xn) Consequently, lim n→∞ dβ (xn,A)= dβ (x,A). Inasimilarway,onecanseethat dα(x,A),dα(x,A):(X,ρ1,ρ2,ρ3,Tmin,Smax) × (X,ρ1,ρ2,ρ3,Tmin,Smax) → R≥0 arecontinuousmaps.

M.Hamidi,M.Mollaei-AraniandY.Alipour-Fakhri

Corollary5.13. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, αβ,γ ∈ (0, 1).Thenforall x ∈ X and ∅̸= A ⊆ X, dα(x,A), dα(x,A), dα(x,A):(X,ρ1,ρ2,ρ3,Tmin,Smax)×(X,ρ1,ρ2,ρ3,Tmin,Smax) → R≥0 arecontinuousmaps.

Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, α,β,γ ∈ (0, 1).Thenforall x ∈ X and ∅̸= A ⊆ X,deﬁne A = {y | ρ1(y,A,t)= 1, ρ2(y,A,t)= ρ3(y,A,t)=0 forall t> 0} and A = {y | dα(y,A)= dβ (y,A)= dγ (y,A)=0}.

Corollary5.14. Let (X,ρ1,ρ2,ρ3,Tmin,Smax) bea KM -singlevaluedneutrosophicmetricspace, α,β,γ ∈ (0, 1).Thenforall x ∈ X and ∅̸= A ⊆ X, x ∈ A if andonlyif x ∈ A

6.Conclusion

Thepresentstudyhasintroducedanovelconceptfuzzyalgebraas KM -singlevaluedneutrosophicmetricspacesandhasconstructedﬁniteorinﬁnite KM -single valuedneutrosophicmetricspacesbasedoninducedunitintervalvalues.Wecan makeacorrespondencebetweenmetricspacesand KM -singlevaluedneutrosophic metricspaces,soweshowthat KM -singlevaluedneutrosophicmetricspacesgeneratesometopologicalspacesandmetricspaces.

ConﬂictsofInterest. Theauthorsdeclarethattherearenoconﬂictsofinterest regardingthepublicationofthisarticle.

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MohammadHamidi

DepartmentofMathematics, UniversityofPayameNoor, Tehran,Iran M.Hamidi@pnu.ac.ir

MahdiMollaei-Arani DepartmentofMathematics, UniversityofPayameNoor, Tehran,Iran M.Mollaei@pnu.ac.ir

YousefAlipour-Fakhri DepartmentofMathematics, UniversityofPayameNoor, Tehran,Iran y_alipour@pnu.ac.ir

ToplogicalandGeometric KM-SingleValuedNeutrosophicMetricSpaces 25