基础拓扑学习题集elementary topology problem textbook by o. ya. viro, o. a. ivanov, n. yu. netsvetaev, v. m. kharlamov

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1、ElementaryTopologyProblemTextbookO.Ya.Viro,O.A.Ivanov,N.Yu.Netsvetaev,V.M.KharlamovIntroductioniiiDedicatedtothememoryofVladimirAbramovichRokhlin(1919–1984)–ourteacherIntroductionThesubjectofthebook,ElementaryTopologyElementarymeansclosetoelements,basics.It

2、isimpossibletodeter-mineprecisely,onceandforall,whichtopologyiselementary,andwhichisnot.Theelementarypartofasubjectisthepartwithwhichanexpertstartstoteachanovice.Wesupposethatourstudentisreadytostudytopology.So,wedonottrytowinherorhisattentionandbenevolence

3、byhastyandobscurestories1aboutmisteriousandattractivethingssuchastheKleinbottle.Allingoodtime:theKleinbottlewillappearinitsturn.However,westartwithwhatatopologicalspaceis.Thatis,westartwithgeneraltopology.Generaltopologybecameapartofthegeneralmathematicalla

4、nguagelongago.Itteachesonetospeakclearlyandpreciselyaboutthingsrelatedtotheideaofcontinuity.Itisneedednotonlyinordertoexplainwhat,finally,theKleinbottleis.Thisisalsoawaytointroducegeometricalimagesintoanyareaofmathematics,nomatterhowfarfromgeometrytheareamay

5、beatfirstglance.Asanactiveresearcharea,generaltopologyispracticallycompleted.Apermanentusageinthecapacityofageneralmathematicallanguagehaspolisheditssystemofdefinitionsandtheorems.Nowadays,studyofgeneraltopologyindeedresemblesratherastudyofalanguagethanastudy

6、ofmathematics:onehastolearnmanynewwords,whiletheproofsofthemajorityoftheoremsareextremelysimple.Butthequantityofthetheoremsishuge.Thiscomesasnosurprisebecausetheyplaytheroleofrulesthatregulateusageofwords.Thebookconsistsoftwoparts.Generaltopologyisthesubjec

7、tofthefirstone.Thesecondpartisanintroductiontoalgebraictopologyviaitsmostclassicalandelementarysegmentwhichemergesfromthenotionsoffundamentalgroupandcoveringspace.1Apersonwhoislookingforsuchelementarytopologywilleasilyfinditinoneofthenumerousbookswithbeautifu

8、lpicturesonvisualtopology.ivIntroductionInouropinion,elementarytopologyalsoincludesbasictopologyofman-ifolds,i.e.,spacesthatlooklocallyastheEuclideanspace.One-andtwo-dimensionalmanifolds,i.e.,curvesand