Wevealreadyvisitedbrie ID: 483280
Download Pdf The PPT/PDF document "p.1Math490Notes11InitialTopologies:Subsp..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
p.1Math490Notes11InitialTopologies:SubspacesandProducts We'vealreadyvisited(brie°y)subspaceand¯niteproductspacetopologies,aspresentedinMunkres.Here,we'llre-visitthesetopicsfromamoregeneralpointofview.Weconcernour-selveswithsuchtopics,inpart,becausemanyinterestingandimportanttopologicalspacesareconstructedfromotherwell-knownspaces,suchas(R;¿u),byformingsubspaces,prod-uctspaces,quotientspaces,anddisjointsums,orvariouscombinationsofthese.Subspaceandproducttopologiesarespecialcasesofinitialtopologies(de¯nedbelow);quotientanddisjointsumtopologiesarespecialcasesof¯naltopologies,whichwemaydiscusslater.Def.N11.1 LetXbeaset,letf(Xi;¿i)i2Igbeacollectionoftopologicalspaces,andletffii2Igbeacorrespondingcollectionoffunctions,wherefi:X!Xi.ThetopologyonXwithsubbasisS=ff1i(Ui)Ui2i;i2IgiscalledtheinitialtopologyonXgeneratedbyffii2Igandf(Xi;¿i)i2Ig.Notethathasabasisofsetsoftheformn\j=1f1ijUij,wherefi1;i2;:::;ingµIandUij2ijfori=1;2;:::;n;we'llrefertotheseasbasicopensetsfor.TheoremN11.1 Inthenotationofthepreviousde¯nition,isthecoarsesttopologyonXrelativetowhichfi:(X;¿)!(Xi;¿i)iscontinuousforalli2I.Proof:Toshowthatfiiscontinuousforeveryi,notethatUi2i)f1(Ui)2byde¯nition.Toshowthatisthecoarsestsuchtopology,assumeisatopologyonXsuchthatfi:(X;¹)!(Xi;¿i)iscontinuousforalli2I.Wemustshowthat.Butthisfollowsimmediatelyfromthefactthateachsubbasicsetf1i(Ui)inisalsoin,bytheassumedcontinuityoffi:(X;¹)!(Xi;¿i). p.2TopologicalSubspaces Def.N11.2 Let(X1;¿1)beatopologicalspace,andlet=XX1.LetiX:X!X1betheidentityinjection.TheinitialtopologyonXinducedbythesinglefunctioniXand(X1;¿1)iscalledthesubspacetopologyonX,and(X;¿)iscalledatopologicalsubspaceof(X1;¿1).Notethatsinceinverseimagespreserveunionsandintersections,wehave=fi1X(V)V21g,andi1X(V)=V\X,sowehave=fV\XV21g,whichagreeswiththede¯nitionofthesubspacetopologyinMunkres,giveninNotes7.If(X;¿)isatopologicalspaceandX,thenthesubspacetopologyoniscommonlydenotedbyA.Example1 ConsiderX=[0;1]asasubspaceof(R;¿u).Notethatsetsoftheform[0;²)and(1²;1](for0²1)areopeninthesubspacetopology,butarenotu-open.Aswe'vementionedbefore,inthisparticularcase,thesubspacetopologyagreeswiththeordertopologyon[0;1].Example2 ConsiderX=[0;1)[f2gasasubspaceof(R;¿u).Notethatf2g=(3 2;5 2)\isanisolatedpointinthesubspacetopology,butnotintheordertopologyon.Prop.N11.1 (a)If(A;¿A)isasubspaceof(X;¿)andB,thenBisA-closedi®a-closedsetCsuchthatB=\C.(b)If,thenClA=(Cl)\.Proof:(a)ThisisbasicallyaconsequenceofDeMorgan'slaws.BisA-closedi®BisA-open,whichistruei®thereisa-openset^CsuchthatB=^C\.Butinthiscase,X^Cis-closed,and(X^C)\=B. p.3(b)By(a),Cl\isaA-closedoversetof,andsoClA(Cl)\.Toprovetheoppositeinclusion,supposex2(Cl)\.IfUisanyA-nbhdofx,thenU=V\,whereVisa-nbhdofx.Byde¯nitionofclosure,wemusthaveV\=,whichimplies(since)that(V\)\=U\=.Thusx2ClA.TopologicalProducts Recallourde¯nitionforsetproduct:IffXii2Igisanarbitrarycollectionofsets,thentheirproductsetisX=Yi2Xi=ff:I![i2Xif(i)2Xiforalli2Ig=f(xi)xi2Xiforalli2Ig.Inthelatterrepresentation,wewritef=(xi)i®f(i)=xiforalli2I.Thisisconsistentwiththeusual"n-tuple"notationfor¯niteproducts:nYi=1Xi=X1X2:::Xn=f(x1;x2;:::;xn)xi2Xifori=1;2;:::;ng.ForanyproductsetX=Qi2Xi,thej-thprojectionmappj:X!Xjisde¯nedbypj(f)=f(j),orinthemoreinformalnotation,pj((xi))=xj.Thatis,theprojectionmappjmapseachelementintheproductsettoits"j-thcomponent".Munkresdenotesthejthprojectionmappjbyj.Def.N11.3 LetfXii2Igbeanarbitrarycollectionoftopologicalspaces.Wede¯netheproducttopologyonX=Yi2Xitobetheinitialtopologyinducedbythegivencollectionoftopologicalspaces,togetherwiththeprojectionmapsfpji2Ig.Inthiscase,wewrite(X;¿)=Qi2(Xi;¿i).NotethathasasasubbasisS=fp1i(Ui)Ui2i;i2Ig,andabasicsetforisany¯niteintersectionofmembersofS.ByThmN11.1,theproducttopologyisthecoarsesttopologyontheproductsetXwhichmakesalltheprojectionmapspi:X!Xicontinuous. p.4IfI=f1;2;:::;ngis¯nite,thebasicopensetsforarerelativelyeasytodescribe,asfollows.IfUj2jforj=1;2;:::;n,thenU=n\j=1p1j(Uj)=U1U2:::Unisabasic-openset.Forexample,theusualtopologyforR2istheproducttopologyfor(R;¿u)(R;¿u),whichhasbasicopensetsoftheformU1U2,whereU1andU2areopenintervalsinR.IfX=Qi2Xiisanin¯niteproduct(meaningIisanin¯niteset),thenabasicopensetUfortheproducttopologyhastheformU=n\j=1p1ij(Uij),wherefi1;i2;:::;ingµIandUij2ijforj=1;2;:::;n.NotethatUis"restricted"inonly¯nitelymanycomponents.AnotherrepresentationforUisU=(U1:::Un)Yi2¡f1;:::;ngXi.AsisdoneinMunkres,theaboveexpressionisoftentakenasthede¯nitionofthebasiselementsfortheproducttopologyonanarbitraryproductspace.Itmightseemmorenaturaltode¯nebasicopensetstobeoftheformQi2Ui,whereUi2iforalli2I.Indeed,suchsetsVdoformabasisforatopologyonX=Qi2Xi,andthetopologyobtainedinthiswayiscalledtheboxtopology,denoted.Clearyeveryopensetintheproducttopologyisalso-openso.Theproductandboxtopologiesarethesamefor¯niteproducts,butnotforin¯niteproducts(unlessallbut¯nitelymanyoftheXi'sareindiscrete).Whenreferingto"productspaces",we'llalwaysassumetheproducttopolgy,unlessotherwiseindicated.Prop.N11.2 Inaproductspace,anyproductofclosedsetsisclosed.Proof:Let(X;¿)=Qi2(Xi;¿i),andletiXibei-closedforalli2I.Let=Qi2i.Toshowisclosed,letx=(xi)62.Thenthereexistsj2Isuchthatxj62j.LetUj=Xjj.Thenbyassumption,Ujisj-open,soV=p1j(Uj)is-open(bycontinuityofpj:(X;¿)!(Xj;¿j)).Finally,sincex2V2andV\=,we'veshownthatXis-open,andthusis-closed. p.5Example3 FortheEuclideanspaceRn,abasisconsistsofallproductsoftheform(a1;b1)(a2;b2)£¢¢¢£(an;bn).BecauseRnisa¯niteproduct,itsboxtopologyandproducttopologyareequivalent.Example4 Considerthein¯niteproductR!.Thissetcanbethoughtofasconsistingofallcountablyin¯nitesequencesofrealnumbersoftheform(r1;r2;r3;:::).De¯nethefunctionf:R!R!byf(t)=(t;t;t;:::).Sincethenthcomponentfunctionhereisfn(t)=t,wehave(foranycollectionfUigwithUiR)f1Yi2Z+Ui=\i2Z+Ui.AbasiselementfortheproducttopologyinR!isaproductof¯nitelymanyintervals(a;b)within¯nitelymanycopiesofR.Theinverseunderfofsuchasetwillbetheintersectionof¯nitelymanyintervalsinR,whichisopenin(R;¿u).Sofiscontinuousfrom(R;¿u)toR!withtheproducttopology.Ontheotherhand,thesetV=Yi2Z+1 i;1 i=(1;1)1 2;1 21 3;1 3£¢¢¢isopenintheboxtopologyonR!,butnotintheproducttopology.Itsinverse,f1Yi2Z+1 i;1 i=\i2Z+1 i;1 i=f0gisnotopenin(R;¿u).Sofisnotcontinuousfrom(R;¿u)toR!withtheboxtopology.