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首页 A Guide to Topology (Steven G. Krantz).pdf

A Guide to Topology (Steven G. Krantz).pdf

A Guide to Topology (Steven G. …

上传者: mathworker 2014-01-08 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《A Guide to Topology (Steven G. Krantz)pdf》,可适用于高等教育领域,主题内容包含“topguide”:pagei#iiiiiiiiAGuidetoTopology“topguide”:pageii#iiiiiiiicbyTheM符等。

“topguide”:pagei#iiiiiiiiAGuidetoTopology“topguide”:pageii#iiiiiiiicbyTheMathematicalAssociationofAmerica(Incorporated)LibraryofCongressCatalogCardNumberPrintEditionISBNElectronicEditionISBNPrintedintheUnitedStatesofAmericaCurrentPrinting(lastdigit):“topguide”:pageiii#iiiiiiiiTheDolcianiMathematicalExpositionsNUMBERFORTYMAAGuides#AGuidetoTopologyStevenGKrantzWashingtonUniversity,StLouisPublishedandDistributedbyTheMathematicalAssociationofAmerica“topguide”:pageiv#iiiiiiiiDOLCIANIMATHEMATICALEXPOSITIONSCommitteeonBooksPaulZorn,ChairDolcianiMathematicalExpositionsEditorialBoardUnderwoodDudley,EditorJeremySCaseRosalieADanceTevianDrayPatriciaBHumphreyVirginiaEKnightMarkAPetersonJonathanRognessThomasQSibleyJoeAlynStickles“topguide”:pagev#iiiiiiiiTheDOLCIANIMATHEMATICALEXPOSITIONSseriesoftheMathematicalAssociationofAmericawasestablishedthroughagenerousgifttotheAssociationfromMaryPDolciani,ProfessorofMathematicsatHunterCollegeoftheCityUniversityofNewYorkInmakingthegift,ProfessorDolciani,herselfanexceptionallytalentedandsuccessfulexpositorofmathematics,hadthepurposeoffurtheringtheidealofexcellenceinmathematicalexpositionTheAssociation,foritspart,wasdelightedtoacceptthegraciousgestureinitiatingtherevolvingfundforthisseriesfromonewhohasservedtheAssociationwithdistinction,bothasamemberoftheCommitteeonPublicationsandasamemberoftheBoardofGovernorsItwaswithgenuinepleasurethattheBoardchosetonametheseriesinherhonorThebooksintheseriesareselectedfortheirlucidexpositorystyleandstimulatingmathematicalcontentTypically,theycontainanamplesupplyofexercises,manywithaccompanyingsolutionsTheyareintendedtobesufficientlyelementaryfortheundergraduateandeventhemathematicallyinclinedhighschoolstudenttounderstandandenjoy,butalsotobeinterestingandsometimeschallengingtothemoreadvancedmathematicianMathematicalGems,RossHonsbergerMathematicalGemsII,RossHonsbergerMathematicalMorsels,RossHonsbergerMathematicalPlums,RossHonsberger(ed)GreatMomentsinMathematics(Before),HowardEvesMaximaandMinimawithoutCalculus,IvanNivenGreatMomentsinMathematics(After),HowardEvesMapColoring,Polyhedra,andtheFourColorProblem,DavidBarnetteMathematicalGemsIII,RossHonsbergerMoreMathematicalMorsels,RossHonsbergerOldandNewUnsolvedProblemsinPlaneGeometryandNumberTheory,VictorKleeandStanWagonProblemsforMathematicians,YoungandOld,PaulRHalmosExcursionsinCalculus:AnInterplayoftheContinuousandtheDiscrete,RobertMYoungTheWohascumCountyProblemBook,GeorgeTGilbert,MarkKrusemeyer,andLorenCLarsonLionHuntingandOtherMathematicalPursuits:ACollectionofMathematics,Verse,andStoriesbyRalphPBoas,Jr,editedbyGeraldLAlexandersonandDaleHMuglerLinearAlgebraProblemBook,PaulRHalmosFromErdostoKiev:ProblemsofOlympiadCaliber,RossHonsberger“topguide”:pagevi#iiiiiiiiWhichWayDidtheBicycleGoandOtherIntriguingMathematicalMysteries,JosephDEKonhauser,DanVelleman,andStanWagonInPolya’sFootsteps:MiscellaneousProblemsandEssays,RossHonsbergerDiophantusandDiophantineEquations,IGBashmakova(UpdatedbyJosephSilvermanandtranslatedbyAbeShenitzer)LogicasAlgebra,PaulHalmosandStevenGivantEuler:TheMasterofUsAll,WilliamDunhamTheBeginningsandEvolutionofAlgebra,IGBashmakovaandGSSmirnova(TranslatedbyAbeShenitzer)MathematicalChestnutsfromAroundtheWorld,RossHonsbergerCountingonFrameworks:MathematicstoAidtheDesignofRigidStructures,JackEGraverMathematicalDiamonds,RossHonsbergerProofsthatReallyCount:TheArtofCombinatorialProof,ArthurTBenjaminandJenniferJQuinnMathematicalDelights,RossHonsbergerConics,KeithKendigHesiod’sAnvil:fallingandspinningthroughheavenandearth,AndrewJSimosonAGardenofIntegrals,FrankEBurkAGuidetoComplexVariables(MAAGuides#),StevenGKrantzSinkorFloatThoughtProblemsinMathandPhysics,KeithKendigBiscuitsofNumberTheory,ArthurTBenjaminandEzraBrownUncommonMathematicalExcursions:PolynomiaandRelatedRealms,DanKalmanWhenLessisMore:VisualizingBasicInequalities,ClaudiAlsinaandRogerBNelsenAGuidetoAdvancedRealAnalysis(MAAGuides#),GeraldBFollandAGuidetoRealVariables(MAAGuides#),StevenGKrantzVoltaire’sRiddle:Micromegasandthemeasureofallthings,AndrewJSimosonAGuidetoTopology,(MAAGuides#),StevenGKrantzMAAServiceCenterPOBoxWashington,DCMAAFAX:“topguide”:pagevii#iiiiiiiiPrefaceTopologyistrulyatwentiethcenturychapterinthehistoryofmathematicsAlthoughitsawitsrootsinworkofEuler,Mobius,andothers,thesubjectdidnotseeitsfullfloweruntiltheseminalworkofPoincareandothertwentiethcenturymathematicianssuchasLefschetz,MacLane,andSteenrodTopologytakestheideaofnonEuclideangeometrytoanewplateau,andgivesustherebynewpowerandinsightTodaytopology(alongsidedifferentialgeometry)isasignificanttoolintheoreticalphysics,itisoneofthekeyideasindevelopingatheoreticalstructurefordatamining,anditplaysaroleinmicrochipdesignMostimportantly,itmustbesaidthattopologyhaspermeatedeveryfieldofmathematics,andhastherebyhadaprofoundandlastingeffectEverymathematicsstudentmustlearntopologyAndphysics,engineeringandotherstudentsarenowlearningthesubjectaswellNotonlythecontent,butalsothestyleandmethodology,oftopologyhaveprovedtobeofseminalimportanceBasictopologyiscertainlywellsuitedfortheaxiomaticmethodHencethefloweringoftheschoolsofHilbertandBourbakicameforthhandinhandwiththedevelopmentoftopologyTheRLMooremethodofmathematicalteachingwasfashionedinthecontextoftopology,andtheinteractionwasjustperfectModerngaugetheoryandstringtheory,andmuchofcosmology,arebestformulatedinthelanguageoftopologyThepurposeofthisbookistogiveabriefcourseintheessentialideasofpointsettopologyAfterreadingthisbook,thestudentwillbewellversedinallthebasicideasandtechniques,andcanmoveontostudyfieldsinwhichtopologyisusedwithconfidenceandskillWeleavemanifoldtheoryandalgebraictopologyforanothervenuesothatwecanconcentratehereonthemostcentralandbasictechniquesinthesubjectStudentsstudyingforqualifyingexamswillfindthisbooktobeausefulresourcePracticingmathematicianswhoneedaplacetolookupakeyideacanlookherevii“topguide”:pageviii#iiiiiiiiviiiThebookiswrittensoastobeaccessibleandselfcontainedTherearemanyexamplesandpictures,togetherwithtimelydiscussiontoputthekeyideasinperspectiveInjustpagesthereadercancomeawayknowingwhattopologyisandwhatitisgoodforWemakeanefforttohooktopologicalideastomorefamiliarconceptsfromcalculus,analysis,andalgebrasothatthereaderwillalwaysfeelfirmlygroundedEveryconceptinthisbookhasacontextThebookconcludeswithatreatmentoffunctionspaces,includingtheAscoliArzelatheoremandtheWeierstrassapproximationtheoremThisseemstoprovideafittingclimax,andanicesetofapplications,forwhathasgonebeforeSincethisisnotaformaltextbook,itdoesnothaveexercisesetsMostresultsinthisbookareprovedinthetraditionalmathematicalmannerWealsoincludeaTableofNotationandaGlossaryinordertofacilitatethereader’srapidacclimatizationtothesubjectBecausethisbookisinthenatureofahandbookratherthanaformaltext,wehaveindulgedincertaininformalitiesWesometimeswilluseatermoraconceptbeforewehavegivenitsformaldefinitionWedosotoavoidthesometimescumbersomebaggageofformalmathematics,andtokeeptheexpositionlightandaccessibleInallsuchcaseswemakeitclearfromcontextwhattheconceptmeans,andalsomakeitclearwherethemorecarefultreatmentofthetermwillappearThereadershouldnotexperienceanydiscomfortfromthisfeature,andwehopewillfindthatitmakesthebookeasiertoreadWeassumethatthereaderofthisbookhasasolidbackgroundinundergraduatemathematicsThiswouldofcourseincludecalculusandlinearalgebra,andasmatteringofrealanalysisespeciallythetopologyofthereallinewouldbehelpfulTotheextentpossible,wehaveendeavoredtofillingapssothatminimalprerequisitesarenecessaryThetypical(thoughnotexclusive)readerofthisbookwillbeagraduatestudentstudyingforqualifyingexamsThisismeanttobeabookthatcanbereadinafewsittings,justtogetthesenseofwhatthissubjectisaboutandhowitfitstogetherItisdifferentfromatypicalmathematicstextormonographAfterreadingthisbook(orevenwhilereadingthisbook),thereadermaywanttopickupamoretraditionalandcomprehensivetomeandworkhisherwaythroughitCertainly,ifonereallywantstolearnthesubject,itisnecessarytodoplentyofexercisesfromthoseancillarytextsThepresentbookwillserveasagoodstartonthatjourneyThisvolumeispartofaseriesbytheMathematicalAssociationofAmericathatisintendedtoaugmentgraduateeducationinthiscountryAs“topguide”:pageix#iiiiiiiiixalways,ithasbeenapleasuretoworkwiththeMAAandwithDonAlberstodevelopthebookUnderwoodDudleyservedaseditorofthisbookseries,andcontributedmanyusefulideasandeditstohelpsharpenmyexpositionTheeditorialboardforthisseriesalsoreadthebookextremelycarefullyandofferedmuchwisdomandadviceManyofthefiguresthroughoutthebookwereproducedbyEllenKleinandJuliaNeidert,highschoolstudentsintheUniversityofMinnesotaTalentedYouthMathematicsProgramunderthedirectionofJohnRognessToallIamhumblygratefulWehopethatthepresentvolumeisapositivecontributiontothisnewMAAbookseriesStLouis,MissouriStevenGKrantz“topguide”:pagex#iiiiiiiiToBobBonic“topguide”:pagexi#iiiiiiiiContentsPrefaceviiFundamentalsWhatisTopologyFirstDefinitionsMappingsTheSeparationAxiomsCompactnessHomeomorphismsConnectednessPathConnectednessContinuaTotallyDisconnectedSpacesTheCantorSetMetricSpacesMetrizabilityBaire’sTheoremLebesgue’sLemmaandLebesgueNumbersAdvancedPropertiesofTopologicalSpacesBasisandSubbasisProductSpacesRelativeTopologyFirstCountableandSecondCountableCompactificationsQuotientTopologiesUniformitiesMorseTheoryProperMappingsParacompactnessxi“topguide”:pagexii#iiiiiiiixiiContentsMooreSmithConvergenceandNetsIntroductoryRemarksNetsFunctionSpacesPreliminaryIdeasTheTopologyofPointwiseConvergenceTheCompactOpenTopologyUniformConvergenceEquicontinuityandtheAscoliArzelaTheoremTheWeierstrassApproximationTheoremTableofNotationGlossaryBibliographyIndexAbouttheAuthor“topguide”:page#iiiiiiiiCHAPTERFundamentalsWhatisTopologyInmathematicsandthephysicalsciencesitisimportanttobeabletocomparetheshapesorformsofobjectsJustwhatdowemeanby“shape”Whatdoesitmeantosaythatanobjecthasa“hole”initIstheholeinthecenterofabasketballthesameastheholeinthecenterofadonutIsitcorrecttosaythatarulerandasheetofpaperhavethesameshapebothare,afterall,rectanglesWhatisarigorousandmathematicalmeansofestablishingthattwoobjectsareequivalentfromthepointofviewofshapeorformFirstDefinitionsAtopologicalspaceisasetXtogetherwithacollectionofsubsetsUDfUgAthatwecalltheopensetsWeassumethat(a)TheentirespaceXisopen(b)Theemptysetisopen(c)IfUˇaretheopensetsthenSˇUˇisanotheropenset(Uisclosedundertheunionoperation)(d)IfUandVareopensetsthenUVisanopenset(Uisclosedunderpairwise,indeedfinite,intersection)TheentiresubjectoftopologyisbasedontheideathatifyouknowtheopensubsetsofaspacethenyouknowaboutitsformWesometimeswriteourtopologicalspaceasXU“topguide”:page#iiiiiiiiFundamentalsEXAMPLELetXDR,thefamiliarrealnumbersystemLetusdeclareeveryopenintervalabtobeanopenset(hereweallowatobeorbtobeC)WealsodeclareanyunionofsuchintervalsintervalstobeanopensetItiseasytoseethatthecollectionUofalltheopensetsthatwehavedescribedformsatopology(ofcoursetheemptysetistheunionofnoopenintervals)TheentirereallineisanopensetaccordingtoourdefinitionTypicalopensetsareUDandUDandC:WecallthistopologythestandardtopologyortheusualtopologyorsometimestheEuclideantopologyonthereallineEXAMPLELetXDRN,thestandardEuclideanspaceofNdimensionsWeusecoordinatestodenoteapointxXbyxDxx:::xNLetussaythatUisanopensetif,wheneverxU,thenthereisan">suchthattheballBx"ftRNWjtxj<"gliesinUWecommonlyrefertoBx"astheopenballwithcenterxandradius"TheentirespaceUDRNisanopensetaccordingtothisdefinition,andsoistheemptysetButtherearelotsofotheropensetsaswellThesetTDftDtt:::tNRNWjtjj<forjD:::NgiseasilycheckedtobeanopensetForifxDx:::xNT,thenlet"DminfjxjjWjD:::NgThenBx"TasrequiredSeeFigurexe––Bx,()eFIGUREThesetTisopen“topguide”:page#iiiiiiiiFirstDefinitionsTheopensetsareclosedundertheunionoperationandunderfiniteintersectionSoRN,equippedwiththenotionofopennessspecifiedhere,isatopologicalspaceWecallthistopologythestandardtopologyortheusualtopologyorsometimestheEuclideantopologyonRNItisworthnotingthatthetopologydescribedhereis“generated”byalltheballsBx"Eachsuchballisopen,justastheexamplesetTabovewasopenSecondly,aunionofsuchballsisclearlyopenThird,ifUisanyopensetaccordingtoourdefinitioninthefirstparagraph,theneachpointinUhasaballaboutit(thatliesinU)SoUisaunionofopenballsThusthecollectionofunionsofopenballsisjustthesameasthecollectionofopensetsItisalsoworthobservingexplicitlythatthetopologydescribedhereincasethedimensionNisisjustthesameasthetopologyconsideredinthelastexampleRemarkFortherecord,wenotethattheclosedballwithcenterxandradius"inEuclideanspaceisgivenbyBx"DftRNWjtxj"g:EXAMPLELetXbetheunitintervalŒandlettheonlyopensetsbetheemptysetandtheentireintervalŒOnemaycheckdirectlythatalltheaxiomsforatopologicalspacearesatisfiedThelastexamplewhichwesometimescallthetrivialtopologywillworkonanynonemptysetXEXAMPLELetXbethesetofintegersZCallasetopenifitiseitheremptyorallofZoristhecomplementofafinitesetThenitisstraightforwardtoconfirmthatthisisatopologicalspaceWemaynotethatthelastexampleworksforanyinfinitesetX(notjusttheintegersZ)EXAMPLEConsiderthetopologyonthereallinegeneratedbyintervalsoftheformŒaborŒaC(herewemean“generated”inthesenseoftakingfiniteintersectionandarbitraryunion)ThisiscalledtheSorgenfreyline,namedafterRobertSorgenfrey(–)TheSorgenfreyl

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