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1、ELEMENTARYCONCEPTSOFTOPOLOGYByPAULALEXANDROFFwithaPrefacebyDAVIDHILBERTtranslatedbyALANE.PARLEYDOVERPUBLICATIONS,INC.NewYorkTRANSLATOR'SPREFACEINTRANSLATINGthiswork,Ihavemadenoattempttoreviseit,buthavemerelytriedtopreserveitinitsproperhistoricalperspective,andhavemadeonlyafewminormodificationsa
2、ndcorrectionsintheprocess.Ihaveinsertedafootnoteandseveralparentheticalnotesinanefforttoclarifythematerialortoindicatethecurrentterminology,andhaveactuallychangedthenotationinseveralplaces(notablythatforthehomologygroups)toavoidconfusionandtobringitintoconsonancewithmodernusage.Iwishtoexpressmy
3、gratitudetoJonBeck,BasilGordonandRobertJohnstonefortheirinvaluableaidinthepreparationofthistranslation.ALANE.PARLEYAnnArbor,MichiganJanuary,1960.PREFACEFEWBRANCHESofgeometryhavedevelopedsorapidlyandsuccessfullyinrecenttimesastopology,andrarelyhasaninitiallyunpromisingbranchofatheoryturnedouttob
4、eofsuchfundamentalimportanceforsuchagreatrangeofcompletelydifferentfieldsastopology.Indeed,todayinnearlyallbranchesofanalysisandinitsfar-reachingapplications,topo-logicalmethodsareusedandtopologicalquestionsasked.Suchawiderangeofapplicationsnaturallyrequiresthattheconceptualstructurebeofsuchpre
5、cisionthatthecommoncoreofthesuperficiallydifferentquestionsmayberecognized.Itisnotsurprisingthatsuchananalysisoffundamentalgeometricalconceptsmustrobthemtoalargeextentoftheirimmediateintuitiveness—somuchthemore,whenintheapplicationtootherfields,asinthegeometryofoursurroundingspace,anextensionto
6、arbitrarydimensionsbecomesnecessary.WhileIhaveattemptedinmyAnschaulicheGeometrictoconsiderspatialperception,hereitwillbeshownhowmanyoftheseconceptsmaybeextendedandsharpenedandthus,howthefoundationmaybegivenforanew,self-containedtheoryofamuchextendedconceptofspace.Never-theless,thefactthatagaina
7、ndagainvitalintuitionhasbeenthedrivingforce,eveninthecaseofallofthesetheories,formsaglowingexampleoftheharmonybetweenintuitionandthought.ThusthefollowingbookistobegreetedasawelcomecomplementtomyAnschaulicheGeometriconthesideoftopo
