湖南师范大学-拓扑学top.2

2.Homeomorphisms,Def.Iff:XY，f,Thenfissaidtobeahomeomorphism,andXissaidtobehomeomorphictoY,denotedby.,Examples(1)a,b0,1,.,(2)(a,b)(0,1),(3)(1,1)(,),f(x)=tan.,(4)S2NR2,RemarkIff:XYbeahomeomorphism,Thentheimageofanopen/closedisstillopen/closed.,Def.Supposethatf:XY.Iff:Xf(X)isahomeomorphism,thenfissaidtobeatopologyimbedding.,Example.Let,bijective.Butnotahomeomorphism.,3.ConstructingcontinuousFunctions,TheoremLetX,YandZbetop.sp.s.,(a)(Constantfunction)If,(b)Inclusion.AX,(c)Composities:f:X,g:YZ,(d)Restrictingthedomain:f:X,A.,(e)Restrictingorexpandingtherange:f:XY.f(X),(whereg(x)=f(x).IfWY,h:XW,here.,(f)Localformulationofcontinuity:Themapf:XYifX=s.t.,.,Theorem(Thepastinglemma)LetX=AB,A,BLet,g:Band.LetThen.(,).,Theorem(Mapsintoproduct)LetThen.Heref1,f2arecalledthecoordinatefunctionoff.,19Theproducttop.20-21TheMetricTop,1.Theboxtop.andtheproducttop.,Def.Letbeanindexedfamilyoftopologyspaces.Let,isabasisforatop.onthetop.generatedbyiscalledtheboxtop.,Themap,projectionmappingassociatedwiththeindex.,.,iscalledthe,Def.,Thetop.generatedbythesubbasisCiscalledtheproducttop.()iscalledaproductsp.,Theorem(Comparisonoftheboxandproducttop.)Theboxtop.onhasasbasisallsetsoftheform,whereforeach.Theproducttop.onhasasbasisallsetsoftheform,whereforeachandexceptforfinitelymanyvaluesof.,TheoremSupposethatthetop.onXisgivenbyabasis.Thenthecollectionofallsetsoftheform,isabasisfortheboxtop.on.,TheoremIfeachsp.isHausdorff.ThenisHausdorffinboththeboxtop.andproducttop.,TheoremIfAX,=,undertheproducttoportheboxtop.,TheoremLet:AX,theproducttop.onthen.,Example.，f:.foreachn.butundertheboxtopology.,2.TheMetricTop.,Def.AmetriconasetXisamapdhavingthefollowingproperties:,(1),equalityholdsifandonlyifx=y.,(2),(3).,Examples.(1)GivenasetX，define,thendisametriconX,andiscalledthediscretemetric.Itinducesthediscretetop.onX.,Def.IfXisatopologyspaceXissaidtobemetrizableifthereexistsametricdonthesetXthatinducesthetopologyofX.Ametricspacesisametrizablesp.Xtogetherwithaspecificmetricdthatgivesthetop.ofX.,Def.Let(X,d)beametricsp.AissaidtobeboundedifM0s.t.,IfAisboundedandnonempty,thediameterofAisdefinedtobethenumber.,TheoremLet(X,d)beametricsp.Define,Thenisametricthatinducesthesametop.asd.,LemmaLetdanddbetwometricsonthesetX,letandbethetopstheyinduce,respectively.The,Examples.(1),theEuclideanmetric,thesquaremetricthetop.inducedbyd=thetop.inducedby=theproducttop.,Def.GivenanindexsetJandgivenpointandDefineonbytheequationwhereisthestandardboundedmetricon.,TheoremLet,isametriconandthetop.inducedbyDisjusttheproducttop.on.,3.continuityoffunctionsonmetricspaces,TheoremLetXandYbemetrizablewithmetricsand,respectively.Thens.t.,ThesequencelemmaLetXbeatop.sp.Ifasequences.t.TheconverseholdsifXismetrizable.,TheoremLetIfaconvergentsequencein,thesequenceTheconverseholdsifXismetrizable.,22TheQuotientTopology,Def.LetXandYbetop.spaces.Letp:XYbeasurjectivemap.If,thenpissaidtobeaquotientmap.,Def.Letp:XYbesurjective,C.IfCcontainseverysetp(y)thatitintersects,thenCissaidtobesaturated.(1)CissaturatedifitequalsthatthecompleteinverseimageofasubsetofY.(2)pisaquotientmapandpmapssaturatedopensetsofXtoopensetsifY.,Prop.Letf:XYbesurjectiveandcontinuous.Iffisopen(orclosed).Thenfisaquotientmap.,Remark.Aquotientmapneednotbeopen(orclosed.),Examples.(1)X=0,12,3,Y=0,2,Define,(2),Def.(thequotienttopology)LetXbeatopologyspaceandAbeset.p:XA,asurjectivemap.Letisatop.onAandp:X(A,)isaquotientmap,hereiscalledthequotienttopologyinducedbyp.,Example.Let,Similarly,.,Def.LetXbeatop.space,letXbeapartitionofXintodisjointsubsetswhoseunionisX.Letp:XXbethesurjectivemapthatcarrieseachpointofXtotheelementofX*containingit.Inthequotienttop.inducedbyp,thesp.X*iscalledisquotienttop.,TheoremLetp:XYbeaquotientmap.LetZbeaspace,g:XZbeamap.If,g(p(y)=aone-pointset.,Ch.3ConnectednessandCompactness,23Connectedspace,1.Conceptsandthenecessaryandsufficientcondition,Def.LetXbeatop.sp.IfthenthepairU,ViscalledaseparationofX.IftheredoesnotexistaseparationofX,thenwesaythatXisconnected.,Example.Qisnotconnected,Randallintervalsareconnected,TheoremLetXbeatopologyspace.ThenXisconnectedtheredoesnotexistapropernonemptysubsetofXwhichisbothopenandclosed.theredoesnotexistapairofclosednonemptysetswhicharedisjointandAB=X.,Example.LetX=aninfiniteset,=thefinitecomplementtopology.Then(X,)isconnected.,Def.IfYisconnectedasasubsp.theYiscalledaconnectedsubset.,Def.LetYbeasubsp.ofX,aseparationofYisapairofnonemptysetsAandBs.t.,LemmaLetYisconnectedthereexistsnoseparationofY.,Examples.(1)X:containingtwopts,=thetrivialtop.(X,)isconnected.,(2)X=(x,y)|y=0(x,y)|x0，y=isnotconnected.,(3),(4)isnotconnected.,2.Properties,LemmaIfthesetsCandDformaseparationofX,andifYisaconnectedsubsp.ofX.ThenY,TheoremTheimageofaconnectedspaceunderacontinuousmapisconnected.,3.Sufficientconditionsofaconnectedsp.,TheoremTheunionofacollectionofconnectedsubspacesofXthathaveapt.incommonisconnected.Thatis,isconnected.isconnected.,TheoremAfinitecartesianproductofconnectedsp.sisconnected.,TheoremLetAbeaconnectedsubsp.ofX.If,thenBisalsoconnected.,ExampleLetisconnected.,24ConnectedsubspacesoftheRealLine,Def.AsimplyorderedsetLhavingmorethanoneelementiscalledaLinearcontinuumifthefollowinghold:(1)Lhastheleastupperboundproperty;(2)Ifxy,thereexistszs.t.xzy.,Theorem24.1IfLisalinearcontinuumintheordertopology,thenLisconnected,andsoareintervalsandraysinL.,Cor.ThereallineRisconnectedandsoareallintervalsandraysinR.,Theorem(Intermediatevaluetheorem)Letf:XYbeacontinuousmap,whereXisaconnectedspaceandYisanorderedsetintheordertop.Ifa,then.,4.PathConnectedSpaces,Def.Aspaceiscalledpathconnectedifeachpairof,pointshasapathconnectingthem.,25ComponentsandLocalConnectedness,1.Components,Def.LetXbeatopologyspace,x,yyaconnectedsubsp.AofXs.t.x,y“”isanequivalentrelation.TheequivalenceclassesarecalledtheconnectedcomponentsofX.,TheoremLetbethecollectionofallthecomponentsofX.,(1),(2),(3)isclosedandconnected.,(4)IfYisaconnectedsubsetofX,forsome.,Then,Examples.(1)(0,1)2,3)isnotconnectedandhastwocomponents.,(2)Everyconnectedsp.hasonlyonecomponent.,(3)Qisnotconnected,eachcomponentofQconsistsofasinglept.,Remark.Acomponentofatopologicalspaceneednotbeopen.,CounterexampleEverycomponentofQisone-ptsetwhichisnotopeninQ.,2.Pathcomponents.,Def.xyapathinXfromxtoy,“”isanequivalencerelation,eachclassunderiscalledapathcomponentofX.,Example.X=AB,Xisconnected,butnotpathconnected,Xhastwopathcomponents:AandB.,3.Locallyconnectednessandlocallypathconnectedness,Def.Asp.XissaidtobelocallyconnectedatxifforeveryneighborhoodUofx,ther