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1、ELEMENTARYCONCEPTSOFTOPOLOGYByPAULALEXANDROFFwithaPrefacebyDAVIDHILBERTtranslatedbyALANE.PARLEYDOVERPUBLICATIONS,INC.NewYorkTRANSLATOR'SPREFACEINTRANSLATINGthiswork,Ihavemadenoattempttoreviseit,buthavemerelytriedtopreserveitinitsproperhistoricalperspective,andhavemadeonlyafewminormodificationsandc
2、orrectionsintheprocess.Ihaveinsertedafootnoteandseveralparentheticalnotesinanefforttoclarifythematerialortoindicatethecurrentterminology,andhaveactuallychangedthenotationinseveralplaces(notablythatforthehomologygroups)toavoidconfusionandtobringitintoconsonancewithmodernusage.Iwishtoexpressmygratit
3、udetoJonBeck,BasilGordonandRobertJohnstonefortheirinvaluableaidinthepreparationofthistranslation.ALANE.PARLEYAnnArbor,MichiganJanuary,1960.PREFACEFEWBRANCHESofgeometryhavedevelopedsorapidlyandsuccessfullyinrecenttimesastopology,andrarelyhasaninitiallyunpromisingbranchofatheoryturnedouttobeofsuchfu
4、ndamentalimportanceforsuchagreatrangeofcompletelydifferentfieldsastopology.Indeed,todayinnearlyallbranchesofanalysisandinitsfar-reachingapplications,topo-logicalmethodsareusedandtopologicalquestionsasked.Suchawiderangeofapplicationsnaturallyrequiresthattheconceptualstructurebeofsuchprecisionthatth
5、ecommoncoreofthesuperficiallydifferentquestionsmayberecognized.Itisnotsurprisingthatsuchananalysisoffundamentalgeometricalconceptsmustrobthemtoalargeextentoftheirimmediateintuitiveness—somuchthemore,whenintheapplicationtootherfields,asinthegeometryofoursurroundingspace,anextensiontoarbitrarydimens
6、ionsbecomesnecessary.WhileIhaveattemptedinmyAnschaulicheGeometrictoconsiderspatialperception,hereitwillbeshownhowmanyoftheseconceptsmaybeextendedandsharpenedandthus,howthefoundationmaybegivenforanew,self-containedtheoryofamuchextendedconceptofspace.Never-theless,thefactthatagainandagainvitalintuit
7、ionhasbeenthedrivingforce,eveninthecaseofallofthesetheories,formsaglowingexampleoftheharmonybetweenintuitionandthought.ThusthefollowingbookistobegreetedasawelcomecomplementtomyAnschaulicheGeometriconthesideoftopo
