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Evolution of the Topological Concept of "Connected"Author(s): R. L. WilderSource: The American Mathematical Monthly, Vol. 85, No. 9 (Nov., 1978), pp. 720-726Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2321676 .Accessed: 12/08/2011 19:24

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EVOLUTION OF THE TOPOLOGICAL CONCEPI OF "CONNECTED"R. L. WILDERIn memoryfEdwinW. MillerndPaulM.Swingle

Introduction.he purpose f this aper s to trace he volutionfoneof themost asicconceptsinTopology, iz., hat f connectednotto be confused ith simply onnected"). ike many thermathematicaloncepts f a fundamentalaturee.g., ontinuousunction),t had only n intuitivemeaningsuch s "connected igure"n geometry)ntil he ncreasinglyubtle emandsfAnalysisand Topology orced ormulationf a satisfactoryefinition.he latterwasnot achieved,s onemightxpect,ntil number fdefinitionsad beenproposed-eachufficientithintsmathemati-cal context utquite nsufficients the onfigurationstudied ecamemore eneral ndabstract.We try o clearup, ncidentally,heexistingonfusionegardinghe actual authorshipfthedefinitionltimatelydopted. otsurprisingly,e uncover "multiple."or several ears, uropeantopologistsonsidered. Hausdorffo be theprime riginatorf thedefinition,pparentlyecausetheir nowledge f set theory nd fundamentalopological otionswas usually erived rom isclassic GrundziigeerMengenlehre"ublishedn 1914 5].However,y the ime fpublicationfhis1944 Mengenlehre"61,which as a third ditionfthe Grundziige,"ausdorffaddiscoveredLennes's arlier ersion f the amedefinitionseebelow), nd called ttentionheretona note tthe nd of hisbook.Thereafterhedefinitionascommonlyalled he Lennes-Hausdorffefinition"fconnected.Manymodem extbooksn Topology eem o have dopted he erm Hausdorff-LenneseparationCondition,"r "Hausdorff-Lennesondition" or he ype f separationnvolvednthedefinition.Possibly his eceivedtimulusrom he se ofthe erm yS. LefschetznhisAmerican athematicalSociety olloquium olume ntitled lgebraic opology10].On page 15,Lefschetzpeaks f the"so-called ausdorff-Lenneseparationondition."In hisclassicwork n Topology9],Kuratowskitatesna footnotep. 127) hat hedefinitionfconnected originatesrom" . Jordan's ours Analyse f 1893, nd also citesLennes'swork.AjustificationorKuratowski'statements offeredelow.W.Sierpinski,n the 1952Englishdition fhiswork n general opology19], ttributeshedefinitiono Hausdorff. owever,n his FoundationsfPoint etTheory13,p. 378],R. L. Mooreattributeshedefinitiono Lennes. n myownbook,TopologyfManifolds20], cited choenflies,Lennes,nd Hausdorff,he choenfliesefinitioneing he ame, lthoughndependentlyrrivedt,as thedefinitionf Jordan hichwas cited y Kuratowski.Without urtheriting f literature,t seemsfair o conclude hat ittle ttention as paid toUnited tates ournals uring he early artof thepresententury,ince Lennes'definition aspublished n both the Bulletin f the AmericanMathematical ociety nd the AmericanJournal fMathematicsn 1906 and 1911,respectively.erhaps, oo,the same shouldbe said about theHungarianournals, or owheren the opologicaliteratureited bove nor n any ther,o far s Ihaveobserved)s thenameof F. Riesz mentionedn connection ith hedefinitionfconnected,althoughhe amedefinitions that iven y Lenneswaspublished y him n 1906 in Hungarian)and n 1907 in German).'Professor ilder eceivedisPh.D. under . L. Moore t theUniversityfTexas.He held ositionstBrown,Texas, nd OhioStatebeforeettlingown t Michigan or longcareer p tohis retirement.e is now aResearch ssociate t Santa Barbara.He has heldvisitingppointmentst the nstituteorAdvanced tudy,Southernalifornia,alifornianstitutefTechnology,olorado, CLA,andFlorida tate.He has been Guggenheimellow nd theHenry ussel ecturert theUniversityfMichigan;nd he samemberf theNationalAcademyfSciences. e has served s Presidentfboth heAMS andtheMAAXHis main nterestsretopology,oundationsfmathematics,nd the ultural istoryf mathematics.isbooks ncludeLecturesn TopologyeditedwithW. L. Ayres); n AMS Colloquium olume,TopologyfManifolds;ntroductionothe oundationsfMathematics;ndEvolutionfMathematicaloncepts.-Editors

720

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1978] EVOLUTION OF THE TOPOLOGICAL CONCEPT OF "CONNECTED" 721The evolution.Unquestionably he roots of the concept of connected ie in the notionof thecontinuous, ut more specifically helinearcontinuum,whichgoes back as faras theGreeks,whostruggled o clarify henotion n the lightof Zeno's paradoxes.The history f this, o far as it isknown, s already adequately covered n the iterature.imilarremarks old forthecontributionsfthemedievalmathematiciansnd philosophers,specially f thescholastic radition, hose nfluence

on both Bolzano and Cantor have beenwidely ommented pon.Bolzano's contribution. lthough hetheory fproportion ivenby Eudoxus (and reproduced nEuclid's Elements)has been creditedby some as the equivalentof Dedekind's definition f therealcontinuum, t seems not to have figuredn the analysisof the earlypartof the nineteenthentury.During the atter eriod, he growingtress or properbasis for stablishinghe"locationtheorem"2of algebra, using only arithmeticas opposed to geometric)means, ed BernardBolzano to offerproofof the theorem n 1817 [1]. A casual readingof Bolzano's works onvincesone thathe had aremarkable ntuitive nowledgeof the structure f the real continuum3s it is understood oday.Alongwith his,he evidently onceivedof the notionof a general ontinuum. onsiderthefollowingdefinitiongiven n hisParadoxien 2, p. 129]): "... a continuums presentwhen, nd only when,wehave an aggregate of simple entities instances or points or substances) so arrangedthat eachindividualmember fthe aggregate as,at each individual nd sufficientlymall distancefromtself,at least one othermember f theaggregate or neighbor.When thisdoes notobtain,whenso muchas a singlepoint of theaggregate s notso thickly urrounded y neighbors s to have at least one ateach individual nd sufficientlymalldistancefrom t,thenwe call sucha point solated, nd sayforthisreason thatouraggregate oes notform continuum."Curiously, he motivation or thisdefinition,ccording to Bolzano's own testimony,ay in theparadoxes that plagued thephilosophical nd mathematical onceptions f time, pace, and "sub-stance."Bolzano reasoned that,by establishing suitablecharacterizationf the abstract tructuralpattern ommon to all these oncepts, he paradoxescould be explained.The analogywith heGreekdilemma nd the effortso resolve t is striking.Now the Paradoxienwas writtenoward he end ofBolzano's life nd publishedposthumouslyn1851,while the proof fthe "locationtheorem," ited above, was published ome 34yearspreviously.But the motivation or the latterwas strictlymathematical n that t was to freeanalysis of itsnotoriousreliance on the geometric spects of continuity. here can be littledoubt, however, hatBolzano's development f his intuition f thecontinuous n the atterwork was contributoryo hisphilosophical onception f time, pace,and substance s continua.And itseems torepresent hefirstattempt t a mathematical ormulationf thetopologicalnotionofconnected. ince,as was to be thecase forover a half-centuryhereafter,he definition f"connected"was tied to thatof "continuum,"itwouldperhapsbe more proper o term t themathematical rogenitor f thenotionof continuum.However, hetimefor onsideration f point etshavingno compactness roperties ad not arrived nmathematics,nd there s little oubtthatthe ntuitive otionwhichBolzano (and after im Cantor)was trying o makeprecisewas equivalent, n itscontext, o thatwhich ed laterto the "unrestricted"notionoftopological onnectedness.

Cantor'scontribution.antor,who was familiarwithBolzano'swork, aw clearly hat he propertyused by Bolzano was insufficientomake precisethe ntuitive otionof continuum.n a paper oftencalled theGrundlagen3, ? 10],he pointed out, forexample, hat setsconsisting f several separatedcontinua satisfyBolzano's condition.Moreover, he recognized ntuitively hat the compactnesspropertiesnow associated in topologywith the notion of continuumhad not been required nBolzano's definition,nd pointed utthat hecomplement f an "isolated" point et n n-dimensionalcoordinatespace, En, n> 2, is a continuum ccording to Bolzano.4 He also rejected enlisting heconceptsof time or space as aids in exploring hemathematical otion of continuum, eeming herelationship uite thereverse.

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722 R. L. WILDER [NovemberAccordingoCantor, continuumnEn must ossess wo asicproperties,amely,hepropertyofbeing erfectnd that fbeing onnected.nmodemerms,point et nEn isperfectf t sclosedanddense-in-itselfi.e., achof tspointssa limit oint f t).5n this onnectionepointedut heinsufficiencyfrequiringpoint ettobe only erfectn order hatt be a continuumy giving is

classical xample fa totally isconnectederfectet-the"Cantor ernaryet,"now often alledsimply he Cantor et." See footnote1,p. 590, oc. cit.)He then efinedonnecteds follows:point et T isconnectedffor verywo f tspoints andt',andarbitraryiven ositive umber,therelways xistsfinitenumberfpointsl,t2,.. .,tn fT such hat hedistancestl, lt2,.. ,tnt'areall smallerhan (loc.cit., 75-576). hen nyperfectndconnectedubset fEn is a continuum,accordingoCantor, hopointed ut n a footnote# 12,p. 590, oc.cit.) hat ospecial imensionwas impliednthedefinition; line, urface,olid, tc., re all continua.ncidentally,or oundedsubsets f El, this sequivalento themodernefinitionf a continuum.The most mportantspectof Cantor's efinitionf continuums his separationf the twoconcepts erfectnd connected,hus dentifyingor the first imethe atter s an independentproperty.t the ime, owever,opologywasvirtuallyonexistents a field fstudy,nd tcouldnotbeexpected hat etshaving he olepropertyfconnectednessould eceiveny ttention.ndas already mplied bove, Cantor's efinitionf connected as quiteadequateforthestudy fcontinua.

C. Jordan'sontribution.he next oteworthytepnthe volutionfthe onceptfconnectedsfoundnC. Jordan'sours 'Analyse.6 pparentlyordan as not amiliarith antor's efinitionftenyears arlier,incehemakes omentionf t.Following discussionfclosed ets,7stablishingthenotion fdistance"ecart")between hem,nd definingets as separated hen he distancebetweenhemsgreaterhan ero, e gives definitionfwhat e calls un seul enant"-inmoderntermscomponent"8-of bounded, losed et, owit:a bounded ndclosed et ofpoints as asingle omponentf tcannot edecomposednto everal losed eparatedets. One seeseasily hatthedistinctiveharacterfsuch set s thefollowing:Forarbitrary,onecanintercalate,etweenany twoof tspoints , p', a chain f ntermediateoints f the et uch hat hedistance etweenconsecutive oints s less thane."' It is this tatementhatJordantalicized, ot theprecedingdefinition.Thiscoincides, fcourse,withCantor's efinitionf connectedndis proved necessaryndsufficientonditionor bounded nd closed et oconsistf a single omponentloc.cit., . 26). tis then imple oproveloc. cit., . 27) that subset f the eal inewhich orms single omponentandcontains wonumbers,and b, must ontainvery umberetween andb. ThiscorollaryfBolzano's heoremeems o havebeenthe hiefmotive orJordan's efinitionfcomponent.Schoenflies'ontribution.n 1904,A. Schoenfliesublishedhefirst fhisfundamentalesearchesinto he opologicalspects fpoint et heory18].He was aware fCantor's efinitionfconnected,whichhe cited loc. cit.pp. 208-209), ut went n to commenthat venthoughheconcept fdistance ormedprimitiveeometricotion or he xiomatic asisofhiswork,twaspreferableogive purelyet-theoreticefinitionfconnected,hereuponegives he ollowing: perfectet scalledconnectedf t is notdecomposablento atleasttwononempty]9ubsets ach of which sperfect.This s,for oundedets, he quivalentfJordan'sefinitionfun eul enant hich,ccordingotheaccompanyingemarks,ecameknown o Schoenfliesnly fter e had announced isownversion. tating hatJordanntroducedhedefinitionnly o derive antor's ormulationf theconceptwithwhich eoperatedhereafter),choenfliesbserveshatconnectednesssan importantandfundamentalropertyorAnalysisitus s a whole." his tatementepresentsn mportanttepforwardntheevolution fthe onnectednessoncept.Whereas antor nly eparated henotionfromhe ther ropertiesf continuum,choenfliesow levatedt to theposition f fundamental

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1978] EVOLUTION OF THE TOPOLOGICAL CONCEPT OF "CONNECTED" 723propertyfTopology,ndwent ntoprove ts nvariancend tostudyhepropertyspeciallynthecontext fplanetopology.Despite choenflies'ecognitionf thefundamentalharacterfconnectedness,isviewwasstilllimited,nthat eexpressedhe pinionhat,while hedefinitionas formulatedor erfectets, tcould quallywellbe stated ormerely)losed ets;but since or losed etswhich renotperfect,connectednessannot ome nto uestion,t s sufficiento imit hedefinitionoperfectets" loc.cit., . 173)!Thus,whilemakingn mportanttep orward,choenflies ade nothertep ackward.

The work of W. H. and G. C. Young.Although hronologicallyhe workof W. H. and G. C.Youngvirtuallyoincides ithhat fLennes ndRiesz o bediscussedelow,t sinterpolatedereas a kind fcapstoneothework lready escribed,s well s of ntrinsicnterestor ts dumbrationof aterwork nthe heoryfconnectedness.Theclassic ook 22]ofW.H. and G. C. Youngl' ntroducesdefinitionfconnectedntermsfregions:" A setofpoints uch hat, escribingregionnanymanner ound achpoint ndeachlimitingoint fthe et s internaloint,hese egionslways eneratesingle egion,ssaidtobeaconnectedetprovidedtcontainsmore han nepoint.Hence fa set s connectedhe etgotbyclosingt sconnected,nd vice ersa."12From hisdefinition,heYoungsprove:A connectedet cannot e divided nto losedcompo-nents= subsets)withoutommon oints. onversely set which annot e dividedntoclosedcomponentsithoutommonointss, f losed, connectedet.We recall hat his ropositionasusedbyJordanndothersodefineonnectednthe asewherehe et nquestionsclosed.

The Lennesand Riesz definitions.t is remarkable hatthroughouthe perioddiscussedabove-fromhe ime fBolzano o1905, verhaLf century-theotion fconnected asconfinedto closed ets;and this n spite f thefactthatCantordivorcedhenotion rom losure n hisdefinitionf continuum. n the otherhand, t is not surprising,ince attention as devotedexclusivelyitherothe eal ontinuumr tothe ubsets feuclideanpace usuallyheplane), ndthe nly on-closedets f mportanceere f special haracter,uch s the et frationalsropensegmentsnthereal ine, ndthe ircularrtriangularegionsftheplane.Ofcourse, ordan,ndfollowingimSchoenflies,adproposed efinitionsfconnected hichvirtuallyegged or eneralizationonon-closedets.Andthis tepwasfinallyakenn 1905-06 ybothN. J. Lennes and F. Riesz. Lennesgave his definitiont a meetingf theAmericanMathematicalocietynDecember905,nd twaspublishednthe bstractfhis alk he ollowingyearntheBulletin f that ociety11].Riesz'sdefinitionaspresentedotheHungarian cademyofSciences nJanuary2,1906,ndpublishedater he ameyear15].Herewasclearly "multiple"-a case of ndependentnventionymore han ne nvestigator.Lennes' definition eads (loc. cit.): A set ofpoints s connectedf in every air ofcomplementarysubsets t least onesubset ontains limit ointof oints nthe theret.This is stated n such a fashionthatt smeaningfuln any pace nwhichimit oint sdefinedalthoughndoubtedlyhe uthor'sthinking,ike hat fmost opologistsfthe ime, as ofeuclideanpaces).Riesz'sdefinitionasseveralemarkableeatures.nthe irstlace, t sgivenn the ontextf nessay[16] devoted o therelations etween he"physical ontinuum"nd the "mathematicalcontinuum."'13n defininghephysicalontinuum,ieszusesthe elationunterscheidbar"etweenspacepoints, otthetopological otion flimit oint.The definitionroceeds s follows: Dasphysikalischeontinuumeisstusamenhangend,enn s nicht nzweiTeilmengenerlegt erdenkann, ass edesElement ereinenTeilmengenterscheidbareivon edemElementeeranderenTeilmenge." oticethestrikingesemblanceo theJordan-Schoenfliesefinition,lthough ieszmakes o referenceothe atter. owever,heres conclusivevidencenpreviousapers fRiesz'thathewasfamiliar ithJordan's ours 'Analysendhence, robably, ithJordan's efinition.'4In the econd lace, hedefinitionfconnectedor opologicalpaces hedoesnotusethe atter

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724 R. L. WILDER [Novemberterm)s givennitiallyorwhathecalls "mathematicalontinuum,"hich,nmodem erms,sanabstractopologicalpacedefinedyfour xioms dumbrativef uch ater ystemss were iven yHausdorffndKuratowski.hedefinitioneads s follows: Das mathematischeontinuumeissezusammenhangendeenn s nicht nzweioffene eilmengenerlegt erden ann, ieKomplemen-tarmengenur inander ein." For subsets f such a space he thendistinguisheswodegrees fconnectedness:set s called onnectedf tcannot edecomposednto wo ubsets hose losuresaredisjoint; t is calledabsolutelyonnectedffor very ecompositionf it intotwo nonempty]subsets,here xists t east neelement hich elongsoonesubset nd s a limit oint fthe ther.It is thesecondof these, fcourse-i.e., bsolutelyonnected-thats the modemdefinitionfconnected.If Rieszhad been familiar ith hemodemdeviceof relativizinghetopological otions f"closed" nd"open,"hewould, resumably,ave dentifiedhedefinitionsfconnectednessormathematicalontinuumndthat f absoluteonnectedness.Actually,hemultiple hichccurred hen hesewodefinitionsere ivenwas oinedbya third,viz.,F. Hausdorff'sefinition.pparently hen iving is definitionnhisbookof 1914 loc.cit.),Hausdorffas unaware feither ennes' r Riesz' definitions.n theother and,Hausdorffidproceed,n this ook, o studyomeof thepropertiesfconnectedets s topologicalonceptsntheirwnright.However,hefirst aperdevotedothe tudyf connectedetswas notpublishedntil 921;werefer ere othe lassicpaper ur es ensemblesonnexesfB. Knaster ndC. Kuratowski8].Thispaperwassignificantnthe volutionf the oncept fconnectednessecause: 1) itestablishedhefact hat onnectedpaces acking ompactnessropertiesavea varietyf nterestingopologicalproperties;2) it gave mpetuso a host fstudies, othnTopologynd ntheogical oundationsfset heory;3) itgave heultimatemphasiso Schoenflies'tatement,uoted bove, oncerninghefundamentalharacterfconnectedness."5

Concludingemarks.roman evolutionaryointofview, he developmentf theconcept fconnectednessroves o be a revealing case study." ts roots, s in thecase of manyothermathematicaloncepts,reembeddedn the ontemplationfphysicalime, pace, nd "substance."Atthehands fCantor tfinallyplit ff romhilosophicalndphysicalonsiderationso becomepart fmathematicalheory.ut t was noteasily ivorcedrom heconcept fcontinuum ithinwhich t was first ormulated-a onsequencef itsmathematicalnvironment,hich onsistedchiefly f the study f curves nd surfaces,xamples f what Cantorcalled continua n themathematicalense.This was a case of theoperation f "environmentaltress," n thatthemathematicalnvironmentorked o confinehenotionwithin restrictedrea.Itfailedofind tsproper lace nmathematicalheoryntil choenfliesointedut ts nvarianceunder opologicalransformations,s well as its ndependenttatus s a topological roperty.utalthoughchoenflies,hodiscoveredssentiallyhe amedefinitionhatJordan ad given verdecade arlier, ade nimportanttep orward,opology adstill otgrownmuch eyond he tudyofconfigurationshose ompactnessroperties ade theJordan-Schoenfliesefinitionuite de-quate. ndeed, o much o, thatwhen heYoungswrote heirlassic The Theoryf Setsof Points"duringhedecade between ordan nd Schoenflies,hey eemto havedeliberatelyhrased heirdefinitionfregionallowing region o nclude oundaryoints reely)o that heCantor efinitionwould epreservedsee the emarkboveconcerningheYoungdefinition).Lennes'sgeneralizationf thesedefinitionsas apparently result f his considerationfnon-closedets. n the aper12]givingn detail he esultsnnouncedn the1906 bstractloc.cit.),hefirstefinedonnectednessor pen ets ineuclidean pace)byusing rokenines, n open etUbeing onnectedfevery airofpoints , b in U are oinedby a brokenine yingwhollyn U (loc.

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1978] EVOLUTION OF THE TOPOLOGICAL CONCEPT OF "CONNECTED" 725cit., . 293).He observedhat y theCantor efinitionfconnected,heunion fthe nteriorndexteriorf a planar ircle orms connectedet;moreover,hat iffromhe rdinaryontinuumnspace f nydimensionsny etwhateverhichsnowhereense sremoved,he esidue ould orma connectedet" loc. cit.,303,footnote).e thereuponave he ormfthedefinitionowgenerallyaccepted,emarkinghat t"applies ncaseswhereheformeroes not." In otherwords,trenderssetsconnected hich ur ntuitionellsus should e connectedndrules ut those hat,ikethecomplementf the ircle ntheplane, hould otbe termedonnected.)Oneofthe emarkableeaturesfRiesz'definition,swehave lready oticed,sthattwasgiveninthe ontext fan abstractopologicalpace.Thisaspect fRiesz'work eems lsotohavebeengenerallynnoticedor ome ime, espitehe act hat rechetalled ttention4,NoteB] toRiesz'abstractpaceaxioms s theywereater resentedtthe nternationalongressnRome,1908 17].In anyevent, is definition,lthoughgreeing ithLennes', chieves herebytsmostgeneralcharacter,reedromllmetriconsiderations.Althoughhe ackofdiffusionrom necountryoanother, hichharacterizedarliereriodsnmathematics,adbegun osubside,he eriod uring hich he opologicalonceptfconnectednesswasdevelopedtill hows onsiderableack ofdiffusion.ennes' ndRiesz'work, oth ublishednreputableournals uringhefirstecadeofthe entury,asgenerallynknownntil heournalFundamenta athematicaeommenced ublicationn 1920.The occurrence f a three-membermultipleuringhefirstuarterfthepresententurysquitenoteworthy.One furtheromment: ne of thenoteworthyeatures f theKnaster nd Kuratowskirticlereferredoabovewas tspresentationfparadoxicalxamplesfconnectedetshaving ocompact-nessproperties.havepointed utelsewhere21]thecontributionhat aradox anmaketo thedevelopmentfmathematicaloncepts. he examples iven yKnasterndKuratowskiloc.cit.)proved great timulationo the tudy fconnectedness.here nsued sizableiteratureevotedothe oncept,ndinrecent ears his asengenderednterestinguestionsntheFoundationsfSetTheory.

Notes1.Myattention asfirstalled oRiesz'worknthis onnectionyProfessor. E. Aull, owhom amindebtedor eferenceshereto.2. That s, f realpolynomial(x) isnegativeor = a andpositiveor = b,then t szero t some aluebetween andb. (Bolzano tated, amely,hat f (x) andp(x) arecontinuousealfunctionsver n interval

a < x < g, and f(a) T(b), then there exists a real number c such a

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726 R. L. WILDER [November11. nmodem erms, region as connectedpen etwithrwithoutn arbitraryet f tsboundaryoints.The Youngs efinedt s generatedy uccessivelyverlappingriangles.hedefinitionfconnecteds onp.204,loc. cit.12. Ifthewords andeach imitingoint" reomittedrom his efinitionnd "region" estrictedo nteriorsof he egionss definedy heYoungs,hen he bovedefinition$equivalentothe simple hain efinition"f

connected.eeR. L. Wilder,oc. cit.,p. 34,Corollary2.5.13. "Ich suchenurden Weg,dervon denraumlichenorstellungu dem mathematischen]aumbegriffefiihrt"loc. cit.).14.See, fornstance,n the ollected orks14], aperA2 (1905), nwhich e mentionsordan'secart" nthefirst age;paperA3 (1905), eferenceso theCours 'Analyse;nd paperA5 (1905),n the irstentencefwhich e definesd'un eul enant,"he ame erm hatwas usedby Jordan.15. Of course, choenflies as not trictlypeaking fthe ype fconnectednessxploited y KnasterndKuratowski,hichwas theLennes-Riesz efinitionowgenerallyccepted.References

1. B. Bolzano,Rein analytischereweis es Lehrsatzes,ass zwischene zweiWerten,ieeinentgegenge-setzesResultat ewihren, enigstensinereeleWurzel erGleichungiege, d. Fr. Prihonsky,rag, 817.2. , Paradoxes f the nfinite,r.of theParadoxienby F. Prihonsky,outledgend Kegan Paul,London, 950.3. G. Cantor,Ueber unendlicheineare unktmannigfaltigkeiten,. Fortsetzung,ath.Ann.,21 (1883)545-591.4. M. Fr&het, es espaces bstraits,authier-Villars,aris, 928.5. F. Hausdorff,rundzige erMengenlehre,onWeit, eipzig, 914.6. , Mengenlehre,over,NewYork, 944.7. C. Jordan,ours 'Analyse,nded., 1893, ol. 1.8. B. Knaster ndC. Kuratowski,ur esensemblesonnexes,und.Math., (1921)206-255.9. C. Kuratowski,opology,ol.2, Academic ress, ewYork, 968.10. S. Lefschetz,lgebraicopology, merican athematicalociety, ewYork, 942.

11. N. J. Lennes, urvesn non-metricalnalysis itus, ull.Amer.Math. oc., 12 1905-06) 84, bstract?10.12. , Curves n non-metricalnalysis ituswith pplicationsnthecalculus f variations,mer. .Math., 3 1911)287-326.13. R. L. Moore, oundationsf Point et Theory, merican athematicalociety,rovidence,.I., 1962.14. F. Riesz,Oeuvres ompletes,authier-Villars,aris, 960, ol.1.15. , A terfolgalomenesise, ath. . Phys. apok, 5 1906)97-122;16 1907) 145-161paperA6 in[14]).16. , Die Genesis es Raumbegriffes,ath.u. Naturwiss. erichteus Ungarn,4 (1907)309-353(paperA7 in 14]).17. , Stetigkeitsbegriffnd bstrakt engenlehre,tti el V CongressonternazionaleiMatematici,Romavol.2, p. 18.18. A. Schoenflies,eitrageurTheorie erPunktmengen,, Math.Ann., 8 1904) 195-238.19. W. SierpiAski,eneral opology,rans. yC. C. Krieger,niversityfToronto ress, oronto, 952.20. R. L. Wilder, opologyf Manifolds,merican athematicalociety, rovidence,.I.,1949.21. , Hereditarytress s a culturalorcen mathematics,istoriaMath., 1974)29-46.22. W. H. Young nd G. C. Young, he theoryf sets f points, ambridge niv.Press, ambridge,906.DEPARTMENT OF MATHEMATICS,UNIVERSITYOF CALIFORNIA,SANTA BARBARA,CA 93106.