topology lecture notes - ward, thomas

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1、SeetheLaTeXmanualorLaTeXCompanionforexplanation.TypeHforimmediatehelpRnchapter.1SeetheLaTeXmanualorLaTeXCompanionforexplanation.TypeHforimmediatehelpRnchapter.11TopologyLectureNotesThomasWard,UEAJune4,2001ContentsChapter1.TopologicalSpaces31.The

2、subspacetopology52.Theproducttopology63.TheproducttopologyonRn74.Thequotienttopology95.Threeimportantexamplesofquotienttopologies9Chapter2.PropertiesofTopologicalSpaces121.Examples122.HausdorSpaces133.Examples144.Connectedness185.Pathconnectedness18Chapter3.Ho

3、motopyequivalence20Chapter4.TheFundamentalGroup261.BasedMaps272.Movingthebasepoint28Chapter5.Coveringspaces311.Liftingmaps332.Theactiononthebre35Chapter6.Classicationofsurfaces391.Orientation422.Polygonalrepresentation443.Transformationtostandardform454.Juxta

4、positionofsymbols495.Eulercharacteristic516.Invarianceofthecharacteristic52Chapter7.SimplicialcomplexesandHomologygroups541.Chains,cyclesandboundaries552.Theequation@2=058Chapter8.Morehomologycalculations591.Geometricalinterpretationofhomology602.Eulercharacter

5、istic651CONTENTS2Chapter9.Simplicialapproximationandanapplication66Chapter10.Homologicalalgebraandtheexactsequenceofapair691.Chaincomplexesandmappings692.Relativehomology703.Theexacthomologysequenceofapair72AppendixA.Finitelygeneratedabeliangroups771.TheFundame

6、ntalTheorem782.Exactsequences79AppendixB.Reviewproblems81Index85CHAPTER1TopologicalSpacesAmetricspaceisapair(X;d)whereXisaset,anddisametriconX,thatisafunctionfromXXtoRthatsatisesthefollowingpropertiesforallx;y;z2X1.d(x;y)0,andd(x;y)=0ifandonlyifx=y,2.d(x;y)=

7、d(y;x)(symmetry),and3.d(x;y)d(x;z)+d(z;y)(triangleinequality).Example1.1.Thefollowingareallmetricspaces(checkthis).1.Rwiththemetricd(x;y)=jxyj.2.Rdwiththemetricd(x;y)=((xy)p+


+(xy)p)1=p=11ddjxyjpforanyp1.3.Cwiththemetricd(z;w)=jxwj.4.S1=fz2Cjjzj=1gwith

8、themetricd(z;w)=jarg(z)arg(w)j,whereargischosentoliein[0;2).5.S1=fz2Cjjzj=1gwiththemetricd(z;w)=jzwj:6.AnysetXwiththemetricd(x;y)=1ifx6=yand0ifx=y.Suchaspaceiscalledadisc