NONLINEARSTUDIES-www.nonlinearstudies.com
Vol.30,No.2,pp.437-443,2023
©CSP-Cambridge,UK;I&S-Florida,USA,2023
Generalizedsemipre-homeomorphismsinneutrosophic topologicalspaces
N.Chandramathi1 andN.Rajeshwaran2⋆
1,2 DepartmentofMathematics,GovernmentArtsCollege,Udumalpet-642126,TamilNadu,India.
E.Mails:drmathimaths@gmail.com;rajeshw851@gmail.com.
⋆ CorrespondingAuthor.E-mail:rajeshw851@gmail.com
Abstract. Asageneralizationoffuzzysetsandintuitionisticfuzzysets,neutrosophicsetshave beendevelopedbySmarandache[5]torepresentimprecise,incompleteandinconsistentinformation existingintherealworld.Aneutrosophicsetischaracterizedbyatruthvalue,anindeterminacyvalue andafalsityvalue.Inthispaper,weintroduceneutrosophicgeneralizedsemipre-Homeomorphism. Weinvestigatesomeoftheirproperties.Also,weprovidesomecharacterizationofneutrosophic generalizedsemipre-Homeomorphism.
1Introduction
In1970,Levine[9]introducedtheconceptofgeneralizedclosedsetsasaweakerformof closedsetsintopologicalspaces.Zadeh[17]introducedthenotionoffuzzysetsintheyear 1965.Infuzzysettheory,themembershipofanelementtoafuzzysetisasinglevalue between0and1.Theconceptoffuzzytopologicalspaceshavebeenintroducedanddeveloped byChang[2].In1983,Atanassov[1]introducedtheconceptofintuitionisticfuzzysetwhich wasgeneralizationoffuzzyset.Inintuitionisticfuzzysettheory,theelementshavethedegree membershipandnon-membershipvaluebetween0and1.Later,in1997Coker[4]introduced theconceptofintuitioniticfuzzytopologicalspaces,byusingthenotionoftheintuitionitic fuzzyset.FloretinSmarandache[5]introducedtheconceptofNeutrosophicset.Neutrosophic setisclassifiedintothreeindependentfunctionsnamely,membershipfunction,indeterminancy functionandnon-membershipfunctionthatareindependentlyrelated.In2012,Salama,Alblowi [12]introducedtheconceptofNeutrosophictopology.Neutrosophictopologicalspacesarevery naturalgeneralizationsoffuzzytopologicalspaces,allowmoregeneralfunctionstobemembers
2010 MathematicsSubjectClassification:18B30.
Keywords:Neutrosophictopology,Neutrosophicgeneralizedsemipreclosedsets,NeutrosophicGeneralizedsemipreHomeomorphisms.
2N.ChandramathiandN.Rajeshwaran
offuzzytopology.In2014,Salama,SmarandacheandValeri[11]introducedtheconceptof NeutrosophicclosedsetsandNeutrosophiccontinuousfunctions.Salama,Alblowi[12]introduced theconceptofgeneralizedNeutrosophicsetandgeneralizedNeutrosophictopologicalspaces.A generalizedNeutrosophicset A = {⟨
} canbeidentifiedasanordered triple ⟨µA, σA, γA⟩ ,wherethetriplefunctionsatisfiesthecondition
A
) ≤ 0.5 WadelandSmarandacheintroducedtheNeutrosophicopensetsviaNeutrosophictopologicalspaces. IshwaryaandBageerathi[8]introducedtheconceptofNeutrosophicsemiopensetsinNeutrosophic topologicalspaces.Dhavaseelan,SaiedJafari[3]introducedgeneralizedNeutrosophicclosedsets.In 2018,Shanthi,ChandrasekarandSafina[13]introducedtheNeutrosophicgeneralizedsemiclosed setsinNeutrosophictopologicalspaces.In2021,SujithraandChandramathiintroducedneutrosophic ˆ βG closedandopensetsinNeutrosophictopology.
InthispaperweintroducethenotionofNeutrosophicgeneralizedsemiprehomeomorphism.We studysomeoftheirpropertiesandprovidesomecharacterizationsofNeutrosophicgeneralizedsemi prehomeomorphism.
2Preliminaries
Werecallsomebasicdefinitionsthatareusedinthesequel
Definition2.1. [11]Aneutrosophictopology(NTforshort)anon-emptysetXisafamily τN of neutrosophicsubsetsinXsatisfyingthefollowingaxioms
(NT1) 0N , 1N ∈ τN (NT2) G1 ∩ G2 ∈ τN (NT3) ∪Gi ∈ τN , ∀Gi : i ∈ J ⊆ τN inthiscasepair (X , τN ) iscalledaneutrosophictopologicalspace(NTSforshort)andany neutrosophicsetin τN isknownasneutrosophicopenset(NOSforshort)in X .Aneutrosophicset F isclosedifandonlyifitscompliment (FC ) isNeutrosophicopenset.
Definition2.2. [11]Let A and B betwoNeutrosophicSets(NSforShort)oftheform A = {⟨X , µA(X ), σA(X ), γA(X )⟩ : x ∈ X } and B = {⟨X , µB(X ), σB(X ), γB(X )⟩ : x ∈ X }.Then
(a) A ⊆ B iff µA(X ) ≤ µB(X ), σA(X ) ≤ σB(X )andγ
(b) AC = {⟨X , γA(X ), 1 σA(X ), µA(X )⟩ : x ∈ X }
(c) A ∩ B = {⟨X , µA(X ) ∧ µB(X ), σ
(
)⟩ : x ∈ X }
(d) A ∪ B = {⟨X , µA(X ) ∨ µB(X ), σA(X ) ∨ σB(X ), γA(X ) ∧ γB(X )⟩ : x ∈ X } fortheSakeofSimplicity, WeShallusethenotation A = {⟨X , µA(X ), σA(X ), γA(X )⟩ : x ∈ X } TheNeutrosophicSets0N = {⟨X , 0, 0, 1⟩ : x ∈ X } and1N = {⟨X , 1, 1, 0⟩ : x ∈ X } arerespectively theemptysetandthewholesetof (X , τu).
Definition2.3. [11]Let (X , τN ) and A = {⟨X , µA, σA, γA⟩} beNSin (X , τN ).ThentheNeutrosophic ClosureandNeutrosophicinteriorofAaredefinedby.
Ncl(A)= ∩{K : KisaNCSin(X , τN )andA ⊆ K}
Nint(A)= ∪{G : GisaNOSin(X , τN )andG ⊆ A}
Definition2.4. [10]Let (X , τN ) beaneutrosophictopologicalspace.AsubsetAof (X , τN ) iscalled Neutrosophicgeneralizedclosedset(NGCS)if Ncl(A) ⊆ G whenever (A) ⊆ G
Definition2.5. [13]LetAbeaNeutrosophicsetinNeutrosophictopology (X , τN ).ThenNissemi preinteriorof A [Nspint(A)]andNeutrosophicsemipreclosureof A [Nspcl(A)]aredefinedtobe
3.NEUTROSOPHICGENERALIZEDSEMIPREHOMEOMORPHISM3 (i). [Nspint(A)]= ∪{G :GisaNSPOSin(X , τN )andG ⊆ A} (ii).[Nspcl(A)]= ∩{K :KisaNSPCSin(X , τN )andA ⊆ K}
Definition2.6. [10]Let (X , τN ) beaneutrosophictopologicalspace.AsubsetAof (X , τN ) iscalled Neutrosophicgeneralizedsemipreclosedset[Neutrosophicgsp-closed]ifNspcl(A) ⊆ G,whenever A ⊆ G andGisNeutrosophicopenset.ComplementofNeutrosophicgspclosedset[NGSPCSfor short]iscalledtheNeutrosophicgsp-openset.
Definition2.7. [13]Let (X , τN ) beaneutrosophictopologicalspace.AsubsetAof (X , τN ) iscalled Neutrosophicgeneralizedclosedset(NGCS)if Ncl(A) ⊆ G whenever (A) ⊆ G.
Definition2.8. [8]Let (X , τN ) beaneutrosophictopologicalspace.AneutrosophicsubsetAofthe neutrosophictopologicalspaceXissaidtobeneutrosophicsemi-closedset(NSCS)if Nint(Ncl(A)) ⊆ A.
Definition2.9. [8]Let (X , τN ) beaneutrosophictopologicalspace.AsubsetAof (X , τN ) iscalled Neutrosophicsemipreopenset(NSPOS)if A ⊆ Ncl(Nint(Ncl(A))).
Definition2.10. [14]Let (X , τN ) and (Y, σN ) betwoneutrosophictopologicalspaces,thenAmap f : (X , τN ) → (Y, σN ) iscalledN-continuous(inshortN-continuous)iftheinverseimageofevery closedsetin (Y, σN ) isclosedin (X , τN ).
Preposition2.1. [14]Let (X , τN ) and (Y, σN ) betwoneutrosophictopologicalspaces,if f : (X , τN ) → (Y, σN ) neutrosophiccontinuousthenitisN-continuous
Definition2.11. [14]Aneutrosophictopologicalspace (X , τN ) issaidtobeneutrosophic T1/2 ifevery Neutrosophicclosedsetin (X , τN ) isneutrosophicclosedin (X , τN )
Definition2.12. [24]Amapping f : (X , τN ) → (Y, σN ) iscalledsemicontinuous(NScontinuousfor short)mappingif f 1(v) isNSOSin (X , τN ) foreveryNOSVof (Y, σN ).
Definition2.13. [24]Amapping f : (X , τN ) → (Y, σN ) iscalledsemicontinuous(NScontinuousfor short)mappingif f 1(v) isNGOSin (X , τN ) foreveryNOSVof (Y, σN ).
Definition2.14. [26]Amapping f : (X , τN ) → (Y, σN ) Neutrosophicgeneralizedsemiprecontinuous (NGSPcontinuousforshort)mappingif f 1(v) isNGSPCSin (X , τN ) foreveryNOSVof (Y, σN ).
Definition2.15. [26]]Afunction f : (X , τN ) → (Y, σN ) issaidtobehomeomorphism(topological mapping)ifandonlyifthefollowingconditionsaresatisfied:
(i) f isbijective
(ii) f iscontinnuous
(iii)f 1 iscontinnuous
3NeutrosophicGeneralizedSemiPreHomeomorphism
InThisSectionWeIntroducetheconceptsofNeutrosophicgeneralizedsemipreopenmap (briefly N gsp openmap),NeutrosophicgeneralizedsemipreHomeomorphism(briefly N gsp Homeomorphism)andinvestigatesomeofitsproperties.
Definition3.1. Afunction f : (X , τN ) → (Y, σN ) iscalledNeutrosophicgspopenif f (V ) is N gsp openin (Y, σN ) foreveryneutrosophicopenset V of (X , τN ).
Definition3.2. Afunction f : (X , τN ) → (Y, σN ) iscalledNeutrosophicgspirresoluteif f 1(V ) is N gsp closedin (X , τ) forevery N gsp closedset V of (Y, σN ).
4N.ChandramathiandN.Rajeshwaran
Definition3.3. Abijection f : (X , τN ) → (Y, σN ) iscalled N gsp homeomorphismif f isboth N gsp continuousand N gsp openmap.
Theorem3.1. EveryNeutrosophicsemihomeomorphismis N gsp homeomorphismbutnot conversly.
Proof. Let f beaNeutrosophicsemihomeomorphismfromatopologicalspace (X , τN ) to (Y, σN ).SinceeveryNeutrosophicsemicontinuousmapis N gsp continuousandeveryNeutrosophic semiopenmapis N gsp open,weconcludethatfis N gsp homeomorphism.
Theconverseoftheabovetheoremisnottrueasitcanbeseeninthefollowingexample.
Let X = {a, b, c}, Y = {
and A = {⟨
⟩} B = {⟨0 1, 0 2, 0 1⟩, ⟨0 4, 0 7, 0 6⟩, ⟨0 3, 0 7, 0 9⟩}.Then τN = {0N , 1N , A} and σN = {0N , 1N , B} are NTSon (X , τN ) and (Y, σN ) respectively.Defineamapping f : (X , τN ) → (Y, σN ) by f (a)= v, f (b)= u, f (c)= w,SincefisnotaNeutrosophicsemicontinuousmap, f isnotsemihomeomorphism. However f is N gsp homeomorphism.
Theorem3.2. EveryNeutrosophicprehomeomorphismis N gsp homeomorphismbutnot conversely.
Proof. Let f beaNeutrosophicprehomeomorphismfromatopologicalspace (X , τN ) to (Y, σN ) SinceeveryNeutrosophicprecontinuousmapis N gsp continuousandeveryNeutrosophicpreopen mapis N gsp open,weconcludethat f is N gsp homeomorphism.
Theconverseoftheabovetheoremisnottrueasitcanbeseeninthefollowingexample. X = {a, b}, Y = {u, v} and A = {⟨
.
,
⟩}.Then τN = {0N , 1N , A} and σN = {0N , 1N , B} areNTSon (X , τN ) and (Y, σN ) respectively.Defineamapping f : (X , τN ) → (Y, σN ) by f (a)= v, f (b)= u.Since f isnotaNeutrosophicpreopenmap, f isnot prehomeomorphism.However f is N gsp homeomorphism.
Theorem3.3. EveryNeutrosophicsemiprehomeomorphismis N gsp homeomorphismbutnot conversely.
Proof. Let f beaNeutrosophicsemiprehomeomorphismfromatopologicalspace (X , τN ) to (Y, σN ) .SinceeveryNeutrosophicsemiprecontinuousmapis N gsp continuousandeveryNeutrosophic semipreopenmapis N gsp open,then f is N gsp homeomorphism.
Theconverseoftheabovetheoremisnottrueasitcanbeseeninthefollowingexample
Let X = {a, b, c}, Y = {u, v, w} and A = {⟨0 2, 0 4, 0 6⟩, ⟨0 1, 0 7, 0 9⟩, ⟨0 3, 0 6, 0 9⟩} B = {⟨0 1, 0 2, 0 1⟩, ⟨0 4, 0 7, 0 6⟩, ⟨0 3, 0 7, 0 9⟩}.Then τN = {0N , 1N , A} and σN = {0N , 1N , B} are NTSon (X , τN ) and (Y, σN ) respectively.Defineamapping f : (X , τN ) → (Y, σN ) by f (a)= v, f (b)= u, f (c)= w,SincefisnotaNeutrosophicsemipreopen, f isnotsemihomeomorphism. However f is N gsp homeomorphism.
Theorem3.4. EveryNeutrosophicgeneralizedhomeomorphismis N gsp homeomorphismbutnot conversely.
Proof. Let f beaNeutrosophicgeneralizedhomeomorphismfromatopologicalspace (X , τN ) to (Y, σN ).SinceeveryNeutrosophicgeneralizedcontinuousmapis N gsp continuousandevery Neutrosophicgeneralizedopenmapis N gsp open,then f is N gsp homeomorphism.
3.NEUTROSOPHICGENERALIZEDSEMIPREHOMEOMORPHISM5
Theconverseoftheabovetheoremisnottrueasitcanbeseeninthefollowingexample
Let X = {a, b, c}, Y = {u, v, w} and A = {⟨0
,
B = {⟨0.9, 0.8, 0.7⟩, ⟨0.2, 0.4, 0.6⟩, ⟨0.1, 0.7, 0.3⟩}.Then τN = {0N , 1N , A} and σN = {0N , 1N , B} are NTSon (X , τN ) and (Y, σN ) respectively.Defineamapping f : (X , τN ) → (Y, σN ) by f (a)= v, f (b)= u, f (c)= w,Since f isnotaNeutrosophicgeneralizedcontinuous, f isnotgeneralized homeomorphism.However f is N gsp homeomorphism.
Theorem3.5. EveryNeutrosophic αg homeomorphismis N gsp homeomorphismbutnot conversely.
Proof. Let f beaNeutrosophicgeneralizedhomeomorphismfromatopologicalspace (X , τN ) to (Y, σN ) .SinceeveryNeutrosophic αg continuousmapis N gsp continuousandeveryNeutrosophic g αg openmapis N gsp open,then f is N gsp homeomorphism.
Theconverseoftheabovetheoremisnottrueasitcanbeseeninthefollowingexample.
Let X = {a, b, c}, Y = {u, v, w} and A = {⟨0.2, 0.5, 0.2⟩, ⟨0.1, 0.4, 0.5⟩, ⟨0.5, 0.2, 0.5⟩}
B = {⟨0.9, 0.8, 0.7⟩, ⟨0.2, 0.4, 0.6⟩, ⟨0.1, 0.7, 0.3⟩}.Then τN = {0N , 1N , A} and σN = {0N , 1N , B} are NTSon (X , τN ) and (Y, σN ) respectively.Defineamapping f : (X , τN ) → (Y, σN ) by f (a)= v, f (b)= u, f (c)= w,Since f isnotaNeutrosophic αg opencontinuous, f isnot s αg homeomorphism .However f is N gsp homeomorphism.
Preposition3.1. Foranybijection f : (X , τN ) → (Y, σN ) thefollowingstatementsareequivalent (i) Itsinversemap f ( 1) : (X , τN ) → (Y, σN ) is N gsp continuous. (ii) f is N gsp openmap. (iii)f is N gsp closedmap.
Proof. Toprove (i)⇒(ii)Let V bean N opensetof (X , τN ).Byassumption ( f 1) 1(V )= f (V is N gsp openin (Y, σN ) andso f is N gsp open.
Toprove (ii)⇒ (iii)Let V bean N closedsetof (X , τN ).Then V C bean N opensetof (X , τN ) .By assumption f (V C ) is N gsp openin (Y, σN ). ie., f (V C )=( f (V ))C is N gsp openin (Y, σN ) and therefore f (V ) is N gsp closedin (Y, σN ).Hence f is N gsp closed.
Toprove (iii)⇒ (i)Let V bean N closedsetin (X , τN ).Byassumption f (V ) is N gsp closedin (Y, σN ).But f (V )=( f 1) 1(V ) andtherefore f 1 is N gsp continuouson (Y, σN )
Preposition3.2. Let f : (X , τN ) → (Y, σN ) beabijectiveand N gsp continuous.Thenthefollowing statementsareequivalent.
(i) f isa N gsp openmap. (ii) f isa N gsp homeomorphism. (iii)f is N gsp closedmap.
Proof. Toprove (i)⇒ (ii)Bythehypothesisandassumption f isa N gsp homeomorphism.
Toprove (ii)⇒(iii)Let V bean N closedsetin (X , τN ).Then V C bean N opensetof (X , τN ).By assumption,( f isa N gsp homeomorphism,itsis N gsp open). f (V C ) is N gsp openin (Y, σN ). ie., f (V C )=( f (V ))C is N gsp openin (Y, σN ) andtherefore f (V ) is N gsp closedin (Y, σN ). Hence f is N gsp closed.
Toprove (iii)⇒(i)Let V bean N opensetin (X , τN ).Then V C bean N opensetof (X , τN ).By assumption, f (V C ) is N gspclosedin (Y, σN ) ie., f (V C )=( f (V ))C is N gsp closedin (Y, σN ) andtherefore f (V ) is N gsp closedin (Y, σN ).Hence f is N gsp closed.
6N.ChandramathiandN.Rajeshwaran
4Conclusion
Inthispaperweintroducedtheconceptofneutrosophicgeneralizedsemi-homeomorphism anddiscussedtheirpropertiesFurther,therelationbetweenneutrosophicgeneralizedsemipre homeomorphismandexistingneutrosophichomeomorphisminneutrosophictopologicalspaceswere established.Manyexamplesaregiventojustifytheresults.Wehopethat,manynewinvestigations canbedoneinthefuturebasedonthedevelopednotionsofneutrosophichomeomorphismforvarious setsviaNTSs.
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