topology lecture notes - ward, thomas

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

2、opology52.Theproducttopology63.TheproducttopologyonRn74.Thequotienttopology95.Threeimportantexamplesofquotienttopologies9Chapter2.PropertiesofTopologicalSpaces121.Examples122.HausdorSpaces133.Examples144.Connectedness185.Pathconnectedness18Chapter3.Homotopyequivalence2

3、0Chapter4.TheFundamentalGroup261.BasedMaps272.Movingthebasepoint28Chapter5.Coveringspaces311.Liftingmaps332.Theactiononthebre35Chapter6.Classicationofsurfaces391.Orientation422.Polygonalrepresentation443.Transformationtostandardform454.Juxtapositionofsymbols495.Eulerc

4、haracteristic516.Invarianceofthecharacteristic52Chapter7.SimplicialcomplexesandHomologygroups541.Chains,cyclesandboundaries552.Theequation@2=058Chapter8.Morehomologycalculations591.Geometricalinterpretationofhomology602.Eulercharacteristic651CONTENTS2Chapter9.Simplicial

5、approximationandanapplication66Chapter10.Homologicalalgebraandtheexactsequenceofapair691.Chaincomplexesandmappings692.Relativehomology703.Theexacthomologysequenceofapair72AppendixA.Finitelygeneratedabeliangroups771.TheFundamentalTheorem782.Exactsequences79AppendixB.Revi

6、ewproblems81Index85CHAPTER1TopologicalSpacesAmetricspaceisapair(X;d)whereXisaset,anddisametriconX,thatisafunctionfromXXtoRthatsatisesthefollowingpropertiesforallx;y;z2X1.d(x;y)0,andd(x;y)=0ifandonlyifx=y,2.d(x;y)=d(y;x)(symmetry),and3.d(x;y)d(x;z)+d(z;y)(triangleine

7、quality).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=1gwiththemetricd(z;w)=jarg(z)arg(w)j,whereargischosentoliein[0;2).5

8、.S1=fz2Cjjzj=1gwiththemetricd(z;w)=jzwj:6.AnysetXwiththemetricd(x;y)=1ifx6=yand0ifx=y.Suchaspaceiscalledadisc