General Topology.pdf

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1、GeneralTopologyCourseatParisVIUniversity,2010/2011(v.3)PierreSchapirahttp://www.math.jussieu.fr/˜schapira/lectnotesschapira@math.jussieu.fr2Contents1Topologicalspaces71.1AshortreviewofSetTheory..................71.2Review:normsanddistances..................91.3Topologies.................

2、............111.4Continuousmaps.........................161.5Inverseimagetopology......................191.6Producttopology.........................191.7Directimagetopology......................22Exercises.................................242Metricspaces272.1Basicpropertiesofmetricspaces.

3、...............272.2Completemetricspace......................312.3TheBaireTheorem........................362.4Contractingmaps.........................37Exercises.................................393Compactspaces413.1Basicpropertiesofcompactspaces...............413.2Compactspacesandrealnum

4、bers................443.3Compactmetricspaces......................463.4Locallycompactspaces......................48Exercises.................................504Banachspaces534.1Normedspaces..........................534.2Banachspaces...........................584.3StudyofthespaceC0(K;R)..

5、.................60Exercises.................................6434CONTENTS5Connectnessandhomotopy675.1Connectness............................675.2Homotopy.............................705.3Fundamentalgroup........................745.4C0-manifolds...........................76Exercises......

6、...........................78IntroductionTheaimoftheseNotesistoprovideashortandself-containedpresentationofthemainconceptsofgeneraltopology.Ofcourse,wecertainlydonotclaimforanyoriginalityhere.Indeed,whenwritingtheseNotes,wehavebeendeeplyinfluencedbytheexcellentbook[3]ofJacquesDixmier.Weha

7、veincludedafewexercisesattheendofthechapters.Hereagainnooriginalityshouldbeexpected.BesidesDixmier’sbook,andamongavastliteratureonthesubject,letusonlymentionthefewbooksbelow.FortheFrenchstudentswhowouldlearnMathematicalEnglish,wereco-mmandtheNotes[5]byJanNekovar.Bibliogra