Topology:AVeryShortIntroduction

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ABOLITIONISM RichardS.NewmanTHEABRAHAMICRELIGIONS CharlesL.CohenACCOUNTING ChristopherNobesADAMSMITH ChristopherJ.BerryADOLESCENCE PeterK.SmithADVERTISING WinstonFletcherAESTHETICS BenceNanayAFRICANAMERICANRELIGION EddieS.GlaudeJrAFRICANHISTORY JohnParkerandRichardRathboneAFRICANPOLITICS IanTaylorAFRICANRELIGIONS JacobK.OluponaAGEING NancyA.PachanaAGNOSTICISM RobinLePoidevinAGRICULTURE PaulBrassleyandRichardSoffeALEXANDERTHEGREAT HughBowdenALGEBRA PeterM.HigginsAMERICANCULTURALHISTORY EricAvilaAMERICANFOREIGNRELATIONS AndrewPrestonAMERICANHISTORY PaulS.BoyerAMERICANIMMIGRATION DavidA.GerberAMERICANLEGALHISTORY G.EdwardWhiteAMERICANNAVALHISTORY CraigL.SymondsAMERICANPOLITICALHISTORY DonaldCritchlowAMERICANPOLITICALPARTIESANDELECTIONSL. SandyMaiselAMERICANPOLITICS RichardM.ValellyTHEAMERICANPRESIDENCY CharlesO.JonesTHEAMERICANREVOLUTION RobertJ.AllisonAMERICANSLAVERY HeatherAndreaWilliamsTHEAMERICANWEST StephenAronAMERICANWOMEN’SHISTORY SusanWareANAESTHESIA AidanO’DonnellANALYTICPHILOSOPHY MichaelBeaneyANARCHISM ColinWardANCIENTASSYRIA KarenRadnerANCIENTEGYPT IanShawANCIENTEGYPTIANARTANDARCHITECTURE ChristinaRiggsANCIENTGREECE PaulCartledgeTHEANCIENTNEAREAST AmandaH.PodanyANCIENTPHILOSOPHY JuliaAnnasANCIENTWARFARE HarrySidebottomANGELS DavidAlbertJonesANGLICANISM MarkChapmanTHEANGLO-SAXONAGE JohnBlairANIMALBEHAVIOUR TristramD.WyattTHEANIMALKINGDOM PeterHollandANIMALRIGHTS DavidDeGraziaTHEANTARCTIC KlausDoddsANTHROPOCENE ErleC.EllisANTISEMITISM StevenBellerANXIETY DanielFreemanandJasonFreemanTHEAPOCRYPHALGOSPELS PaulFosterAPPLIEDMATHEMATICS AlainGorielyARCHAEOLOGY PaulBahnARCHITECTURE AndrewBallantyneARISTOCRACY WilliamDoyleARISTOTLE JonathanBarnesARTHISTORY DanaArnoldARTTHEORY CynthiaFreeland

ARTIFICIALINTELLIGENCE MargaretA.BodenASIANAMERICANHISTORY MadelineY.HsuASTROBIOLOGY DavidC.CatlingASTROPHYSICS JamesBinneyATHEISM JulianBagginiTHEATMOSPHERE PaulI.PalmerAUGUSTINE HenryChadwickAUSTRALIA KennethMorganAUTISM UtaFrithAUTOBIOGRAPHY LauraMarcusTHEAVANTGARDE DavidCottingtonTHEAZTECS DavídCarrascoBABYLONIA TrevorBryceBACTERIA SebastianG.B.AmyesBANKING JohnGoddardandJohnO.S.WilsonBARTHES JonathanCullerTHEBEATS DavidSterrittBEAUTY RogerScrutonBEHAVIOURALECONOMICS MichelleBaddeleyBESTSELLERS JohnSutherlandTHEBIBLE JohnRichesBIBLICALARCHAEOLOGY EricH.ClineBIGDATA DawnE.HolmesBIOGRAPHY HermioneLeeBIOMETRICS MichaelFairhurstBLACKHOLES KatherineBlundellBLOOD ChrisCooperTHEBLUES ElijahWaldTHEBODY ChrisShillingTHEBOOKOFCOMMONPRAYER BrianCummingsTHEBOOKOFMORMON TerrylGivensBORDERS AlexanderC.DienerandJoshuaHagenTHEBRAIN MichaelO’SheaBRANDING RobertJonesTHEBRICS AndrewF.CooperTHEBRITISHCONSTITUTION MartinLoughlinTHEBRITISHEMPIRE AshleyJacksonBRITISHPOLITICS AnthonyWrightBUDDHA MichaelCarrithersBUDDHISM DamienKeownBUDDHISTETHICS DamienKeownBYZANTIUM PeterSarrisC.S.LEWIS JamesComoCALVINISM JonBalserakCANCER NicholasJamesCAPITALISM JamesFulcherCATHOLICISM GeraldO’CollinsCAUSATION StephenMumfordandRaniLillAnjumTHECELL TerenceAllenandGrahamCowlingTHECELTS BarryCunliffeCHAOS LeonardSmithCHARLESDICKENS JennyHartleyCHEMISTRY PeterAtkinsCHILDPSYCHOLOGY UshaGoswamiCHILDREN’SLITERATURE KimberleyReynoldsCHINESELITERATURE SabinaKnightCHOICETHEORY MichaelAllinghamCHRISTIANART BethWilliamsonCHRISTIANETHICS D.StephenLongCHRISTIANITY LindaWoodheadCIRCADIANRHYTHMS RussellFosterandLeonKreitzmanCITIZENSHIP RichardBellamyCIVILENGINEERING DavidMuirWoodCLASSICALLITERATURE WilliamAllanCLASSICALMYTHOLOGY HelenMoralesCLASSICS MaryBeardandJohnHendersonCLAUSEWITZ MichaelHowardCLIMATE MarkMaslinCLIMATECHANGE MarkMaslinCLINICALPSYCHOLOGY SusanLlewelynandKatieAafjes-vanDoornCOGNITIVENEUROSCIENCE RichardPassingham

THECOLDWAR RobertMcMahonCOLONIALAMERICA AlanTaylorCOLONIALLATINAMERICANLITERATURE RolenaAdornoCOMBINATORICS RobinWilsonCOMEDY MatthewBevisCOMMUNISM LeslieHolmesCOMPARATIVELITERATURE BenHutchinsonCOMPLEXITY JohnH.HollandTHECOMPUTER DarrelInceCOMPUTERSCIENCE SubrataDasguptaCONCENTRATIONCAMPS DanStoneCONFUCIANISM DanielK.GardnerTHECONQUISTADORS MatthewRestallandFelipeFernández-ArmestoCONSCIENCE PaulStrohmCONSCIOUSNESS SusanBlackmoreCONTEMPORARYART JulianStallabrassCONTEMPORARYFICTION RobertEaglestoneCONTINENTALPHILOSOPHY SimonCritchleyCOPERNICUS OwenGingerichCORALREEFS CharlesSheppardCORPORATESOCIALRESPONSIBILITY JeremyMoonCORRUPTION LeslieHolmesCOSMOLOGY PeterColesCOUNTRYMUSIC RichardCarlinCRIMEFICTION RichardBradfordCRIMINALJUSTICE JulianV.RobertsCRIMINOLOGY TimNewburnCRITICALTHEORY StephenEricBronnerTHECRUSADES ChristopherTyermanCRYPTOGRAPHY FredPiperandSeanMurphyCRYSTALLOGRAPHY A.M.GlazerTHECULTURALREVOLUTION RichardCurtKrausDADAANDSURREALISM DavidHopkinsDANTE PeterHainsworthandDavidRobeyDARWIN JonathanHowardTHEDEADSEASCROLLS TimothyH.LimDECADENCE DavidWeirDECOLONIZATION DaneKennedyDEMOCRACY BernardCrickDEMOGRAPHY SarahHarperDEPRESSION JanScottandMaryJaneTacchiDERRIDA SimonGlendinningDESCARTES TomSorellDESERTS NickMiddletonDESIGN JohnHeskettDEVELOPMENT IanGoldinDEVELOPMENTALBIOLOGY LewisWolpertTHEDEVIL DarrenOldridgeDIASPORA KevinKennyDICTIONARIES LyndaMugglestoneDINOSAURS DavidNormanDIPLOMACY JosephM.SiracusaDOCUMENTARYFILM PatriciaAufderheideDREAMING J.AllanHobsonDRUGS LesIversenDRUIDS BarryCunliffeDYNASTY JeroenDuindamDYSLEXIA MargaretJ.SnowlingEARLYMUSIC ThomasForrestKellyTHEEARTH MartinRedfernEARTHSYSTEMSCIENCE TimLentonECONOMICS ParthaDasguptaEDUCATION GaryThomasEGYPTIANMYTH GeraldinePinchEIGHTEENTH‑CENTURYBRITAIN PaulLangfordTHEELEMENTS PhilipBallEMOTION DylanEvansEMPIRE StephenHoweENERGYSYSTEMS NickJenkinsENGELS TerrellCarverENGINEERING DavidBlockley

THEENGLISHLANGUAGE SimonHorobinENGLISHLITERATURE JonathanBateTHEENLIGHTENMENT JohnRobertsonENTREPRENEURSHIP PaulWestheadandMikeWrightENVIRONMENTALECONOMICS StephenSmithENVIRONMENTALETHICS RobinAttfieldENVIRONMENTALLAW ElizabethFisherENVIRONMENTALPOLITICS AndrewDobsonEPICUREANISM CatherineWilsonEPIDEMIOLOGY RodolfoSaracciETHICS SimonBlackburnETHNOMUSICOLOGY TimothyRiceTHEETRUSCANS ChristopherSmithEUGENICS PhilippaLevineTHEEUROPEANUNION SimonUsherwoodandJohnPinderEUROPEANUNIONLAW AnthonyArnullEVOLUTION BrianandDeborahCharlesworthEXISTENTIALISM ThomasFlynnEXPLORATION StewartA.WeaverEXTINCTION PaulB.WignallTHEEYE MichaelLandFAIRYTALE MarinaWarnerFAMILYLAW JonathanHerringFASCISM KevinPassmoreFASHION RebeccaArnoldFEDERALISM MarkJ.RozellandClydeWilcoxFEMINISM MargaretWaltersFILM MichaelWoodFILMMUSIC KathrynKalinakFILMNOIR JamesNaremoreTHEFIRSTWORLDWAR MichaelHowardFOLKMUSIC MarkSlobinFOOD JohnKrebsFORENSICPSYCHOLOGY DavidCanterFORENSICSCIENCE JimFraserFORESTS JabouryGhazoulFOSSILS KeithThomsonFOUCAULT GaryGuttingTHEFOUNDINGFATHERS R.B.BernsteinFRACTALS KennethFalconerFREESPEECH NigelWarburtonFREEWILL ThomasPinkFREEMASONRY AndreasÖnnerforsFRENCHLITERATURE JohnD.LyonsTHEFRENCHREVOLUTION WilliamDoyleFREUD AnthonyStorrFUNDAMENTALISM MaliseRuthvenFUNGI NicholasP.MoneyTHEFUTURE JenniferM.GidleyGALAXIES JohnGribbinGALILEO StillmanDrakeGAMETHEORY KenBinmoreGANDHI BhikhuParekhGARDENHISTORY GordonCampbellGENES JonathanSlackGENIUS AndrewRobinsonGENOMICS JohnArchibaldGEOFFREYCHAUCER DavidWallaceGEOGRAPHYJOHN MatthewsandDavidHerbertGEOLOGY JanZalasiewiczGEOPHYSICS WilliamLowrieGEOPOLITICS KlausDoddsGERMANLITERATURE NicholasBoyleGERMANPHILOSOPHY AndrewBowieGLACIATION DavidJ.A.EvansGLOBALCATASTROPHES BillMcGuireGLOBALECONOMICHISTORY RobertC.AllenGLOBALIZATION ManfredStegerGOD JohnBowkerGOETHE RitchieRobertsonTHEGOTHIC NickGroom

GOVERNANCE MarkBevirGRAVITY TimothyCliftonTHEGREATDEPRESSIONANDTHENEWDEAL EricRauchwayHABERMAS JamesGordonFinlaysonTHEHABSBURGEMPIRE MartynRadyHAPPINESS DanielM.HaybronTHEHARLEMRENAISSANCE CherylA.WallTHEHEBREWBIBLEASLITERATURE TodLinafeltHEGEL PeterSingerHEIDEGGER MichaelInwoodTHEHELLENISTICAGE PeterThonemannHEREDITY JohnWallerHERMENEUTICS JensZimmermannHERODOTUS JenniferT.RobertsHIEROGLYPHS PenelopeWilsonHINDUISM KimKnottHISTORY JohnH.ArnoldTHEHISTORYOFASTRONOMY MichaelHoskinTHEHISTORYOFCHEMISTRY WilliamH.BrockTHEHISTORYOFCHILDHOOD JamesMartenTHEHISTORYOFCINEMA GeoffreyNowell-SmithTHEHISTORYOFLIFE MichaelBentonTHEHISTORYOFMATHEMATICS JacquelineStedallTHEHISTORYOFMEDICINE WilliamBynumTHEHISTORYOFPHYSICS J.L.HeilbronTHEHISTORYOFTIME LeofrancHolford‑StrevensHIVANDAIDS AlanWhitesideHOBBES RichardTuckHOLLYWOOD PeterDecherneyTHEHOLYROMANEMPIRE JoachimWhaleyHOME MichaelAllenFoxHOMER BarbaraGraziosiHORMONES MartinLuckHUMANANATOMY LeslieKlenermanHUMANEVOLUTION BernardWoodHUMANRIGHTS AndrewClaphamHUMANISM StephenLawHUME A.J.AyerHUMOUR NoëlCarrollTHEICEAGE JamieWoodwardIDENTITY FlorianCoulmasIDEOLOGY MichaelFreedenTHEIMMUNESYSTEM PaulKlenermanINDIANCINEMA AshishRajadhyakshaINDIANPHILOSOPHY SueHamiltonTHEINDUSTRIALREVOLUTION RobertC.AllenINFECTIOUSDISEASE MartaL.WayneandBenjaminM.BolkerINFINITY IanStewartINFORMATION LucianoFloridiINNOVATION MarkDodgsonandDavidGannINTELLECTUALPROPERTY SivaVaidhyanathanINTELLIGENCE IanJ.DearyINTERNATIONALLAW VaughanLoweINTERNATIONALMIGRATION KhalidKoserINTERNATIONALRELATIONS PaulWilkinsonINTERNATIONALSECURITY ChristopherS.BrowningIRAN AliM.AnsariISLAM MaliseRuthvenISLAMICHISTORY AdamSilversteinISOTOPES RobEllamITALIANLITERATURE PeterHainsworthandDavidRobeyJESUS RichardBauckhamJEWISHHISTORY DavidN.MyersJOURNALISM IanHargreavesJUDAISM NormanSolomonJUNG AnthonyStevensKABBALAH JosephDanKAFKA RitchieRobertsonKANT RogerScrutonKEYNES RobertSkidelskyKIERKEGAARD PatrickGardiner

KNOWLEDGE JenniferNagelTHEKORAN MichaelCookLAKES WarwickF.VincentLANDSCAPEARCHITECTURE IanH.ThompsonLANDSCAPESANDGEOMORPHOLOGY AndrewGoudieandHeatherVilesLANGUAGES StephenR.AndersonLATEANTIQUITY GillianClarkLAW RaymondWacksTHELAWSOFTHERMODYNAMICS PeterAtkinsLEADERSHIP KeithGrintLEARNING MarkHaselgroveLEIBNIZ MariaRosaAntognazzaLEOTOLSTOY LizaKnappLIBERALISM MichaelFreedenLIGHT IanWalmsleyLINCOLN AllenC.GuelzoLINGUISTICS PeterMatthewsLITERARYTHEORY JonathanCullerLOCKE JohnDunnLOGIC GrahamPriestLOVE RonalddeSousaMACHIAVELLI QuentinSkinnerMADNESS AndrewScullMAGIC OwenDaviesMAGNACARTA NicholasVincentMAGNETISM StephenBlundellMALTHUS DonaldWinchMAMMALS T.S.KempMANAGEMENT JohnHendryMAO DeliaDavinMARINEBIOLOGY PhilipV.MladenovTHEMARQUISDESADE JohnPhillipsMARTINLUTHER ScottH.HendrixMARTYRDOM JolyonMitchellMARX PeterSingerMATERIALS ChristopherHallMATHEMATICALFINANCE MarkH.A.DavisMATHEMATICS TimothyGowersMATTER GeoffCottrellTHEMEANINGOFLIFE TerryEagletonMEASUREMENT DavidHandMEDICALETHICS MichaelDunnandTonyHopeMEDICALLAW CharlesFosterMEDIEVALBRITAIN JohnGillinghamandRalphA.GriffithsMEDIEVALLITERATURE ElaineTreharneMEDIEVALPHILOSOPHY JohnMarenbonMEMORY JonathanK.FosterMETAPHYSICS StephenMumfordMETHODISM WilliamJ.AbrahamTHEMEXICANREVOLUTION AlanKnightMICHAELFARADAY FrankA.J.L.JamesMICROBIOLOGY NicholasP.MoneyMICROECONOMICS AvinashDixitMICROSCOPY TerenceAllenTHEMIDDLEAGES MiriRubinMILITARYJUSTICE EugeneR.FidellMILITARYSTRATEGY AntulioJ.EchevarriaIIMINERALS DavidVaughanMIRACLES YujinNagasawaMODERNARCHITECTURE AdamSharrMODERNART DavidCottingtonMODERNCHINA RanaMitterMODERNDRAMA KirstenE.Shepherd-BarrMODERNFRANCE VanessaR.SchwartzMODERNINDIA CraigJeffreyMODERNIRELAND SeniaPašetaMODERNITALY AnnaCentoBullMODERNJAPAN ChristopherGoto-JonesMODERNLATINAMERICANLITERATURE RobertoGonzálezEchevarríaMODERNWAR RichardEnglishMODERNISM ChristopherButler

MOLECULARBIOLOGY AyshaDivanandJaniceA.RoydsMOLECULES PhilipBallMONASTICISM StephenJ.DavisTHEMONGOLS MorrisRossabiMOONS DavidA.RotheryMORMONISM RichardLymanBushmanMOUNTAINS MartinF.PriceMUHAMMAD JonathanA.C.BrownMULTICULTURALISM AliRattansiMULTILINGUALISM JohnC.MaherMUSIC NicholasCookMYTH RobertA.SegalNAPOLEON DavidBellTHENAPOLEONICWARS MikeRapportNATIONALISM StevenGrosbyNATIVEAMERICANLITERATURE SeanTeutonNAVIGATION JimBennettNAZIGERMANY JaneCaplanNELSONMANDELA EllekeBoehmerNEOLIBERALISM ManfredStegerandRaviRoyNETWORKS GuidoCaldarelliandMicheleCatanzaroTHENEWTESTAMENT LukeTimothyJohnsonTHENEWTESTAMENTASLITERATURE KyleKeeferNEWTON RobertIliffeNIETZSCHE MichaelTannerNINETEENTH‑CENTURYBRITAIN ChristopherHarvieandH.C.G.MatthewTHENORMANCONQUEST GeorgeGarnettNORTHAMERICANINDIANS ThedaPerdueandMichaelD.GreenNORTHERNIRELAND MarcMulhollandNOTHING FrankCloseNUCLEARPHYSICS FrankCloseNUCLEARPOWER MaxwellIrvineNUCLEARWEAPONS JosephM.SiracusaNUMBERS PeterM.HigginsNUTRITION DavidA.BenderOBJECTIVITY StephenGaukrogerOCEANS DorrikStowTHEOLDTESTAMENT MichaelD.CooganTHEORCHESTRAD. KernHolomanORGANICCHEMISTRY GrahamPatrickORGANIZATIONS MaryJoHatchORGANIZEDCRIME GeorgiosA.AntonopoulosandGeorgiosPapanicolaouORTHODOXCHRISTIANITYA. EdwardSiecienskiPAGANISM OwenDaviesPAIN RobBoddiceTHEPALESTINIAN-ISRAELICONFLICT MartinBuntonPANDEMICS ChristianW.McMillenPARTICLEPHYSICS FrankClosePAUL E.P.SandersPEACE OliverP.RichmondPENTECOSTALISM WilliamK.KayPERCEPTION BrianRogersTHEPERIODICTABLE EricR.ScerriPHILOSOPHY EdwardCraigPHILOSOPHYINTHEISLAMICWORLD PeterAdamsonPHILOSOPHYOFBIOLOGY SamirOkashaPHILOSOPHYOFLAW RaymondWacksPHILOSOPHYOFSCIENCE SamirOkashaPHILOSOPHYOFRELIGION TimBaynePHOTOGRAPHY SteveEdwardsPHYSICALCHEMISTRY PeterAtkinsPHYSICS SidneyPerkowitzPILGRIMAGE IanReaderPLAGUE PaulSlackPLANETS DavidA.RotheryPLANTS TimothyWalkerPLATETECTONICS PeterMolnarPLATO JuliaAnnasPOETRY BernardO’DonoghuePOLITICALPHILOSOPHY DavidMillerPOLITICS KennethMinogue

POPULISM CasMuddeandCristóbalRoviraKaltwasserPOSTCOLONIALISM RobertYoungPOSTMODERNISM ChristopherButlerPOSTSTRUCTURALISM CatherineBelseyPOVERTY PhilipN.JeffersonPREHISTORY ChrisGosdenPRESOCRATICPHILOSOPHY CatherineOsbornePRIVACY RaymondWacksPROBABILITY JohnHaighPROGRESSIVISM WalterNugentPROJECTS AndrewDaviesPROTESTANTISM MarkA.NollPSYCHIATRY TomBurnsPSYCHOANALYSIS DanielPickPSYCHOLOGY GillianButlerandFredaMcManusPSYCHOLOGYOFMUSIC ElizabethHellmuthMargulisPSYCHOPATHY EssiVidingPSYCHOTHERAPY TomBurnsandEvaBurns-LundgrenPUBLICADMINISTRATION StellaZ.TheodoulouandRaviK.RoyPUBLICHEALTH VirginiaBerridgePURITANISM FrancisJ.BremerTHEQUAKERS PinkDandelionQUANTUMTHEORY JohnPolkinghorneRACISM AliRattansiRADIOACTIVITY ClaudioTunizRASTAFARI EnnisB.EdmondsREADING BelindaJackTHEREAGANREVOLUTION GilTroyREALITY JanWesterhoffTHEREFORMATION PeterMarshallRELATIVITY RussellStannardRELIGIONINAMERICA TimothyBealTHERENAISSANCE JerryBrottonRENAISSANCEART GeraldineA.JohnsonREPTILES T.S.KempREVOLUTIONS JackA.GoldstoneRHETORIC RichardToyeRISK BaruchFischhoffandJohnKadvanyRITUAL BarryStephensonRIVERS NickMiddletonROBOTICS AlanWinfieldROCKS JanZalasiewiczROMANBRITAIN PeterSalwayTHEROMANEMPIRE ChristopherKellyTHEROMANREPUBLIC DavidM.GwynnROMANTICISM MichaelFerberROUSSEAU RobertWoklerRUSSELL A.C.GraylingRUSSIANHISTORY GeoffreyHoskingRUSSIANLITERATURE CatrionaKellyTHERUSSIANREVOLUTION S.A.SmithTHESAINTS SimonYarrowSAVANNAS PeterA.FurleySCEPTICISM DuncanPritchardSCHIZOPHRENIA ChrisFrithandEveJohnstoneSCHOPENHAUER ChristopherJanawaySCIENCEANDRELIGION ThomasDixonSCIENCEFICTION DavidSeedTHESCIENTIFICREVOLUTION LawrenceM.PrincipeSCOTLAND RabHoustonSECULARISM AndrewCopsonSEXUALSELECTION MarleneZukandLeighW.SimmonsSEXUALITY VéroniqueMottierSHAKESPEARE’SCOMEDIES BartvanEsSHAKESPEARE’SSONNETSANDPOEMS JonathanF.S.PostSHAKESPEARE’STRAGEDIES StanleyWellsSIKHISM EleanorNesbittTHESILKROAD JamesA.MillwardSLANG JonathonGreenSLEEP StevenW.LockleyandRussellG.FosterSOCIALANDCULTURALANTHROPOLOGY JohnMonaghanandPeterJust

SOCIALPSYCHOLOGY RichardJ.CrispSOCIALWORK SallyHollandandJonathanScourfieldSOCIALISM MichaelNewmanSOCIOLINGUISTICS JohnEdwardsSOCIOLOGY SteveBruceSOCRATES C.C.W.TaylorSOUND MikeGoldsmithSOUTHEASTASIA JamesR.RushTHESOVIETUNION StephenLovellTHESPANISHCIVILWAR HelenGrahamSPANISHLITERATURE JoLabanyiSPINOZA RogerScrutonSPIRITUALITY PhilipSheldrakeSPORT MikeCroninSTARS AndrewKingStatisticsDavidJ. HandSTEMCELLS JonathanSlackSTOICISM BradInwoodSTRUCTURALENGINEERING DavidBlockleySTUARTBRITAIN JohnMorrillSUPERCONDUCTIVITY StephenBlundellSYMMETRY IanStewartSYNAESTHESIA JuliaSimnerSYNTHETICBIOLOGY JamieA.DaviesTAXATION StephenSmithTEETH PeterS.UngarTELESCOPES GeoffCottrellTERRORISM CharlesTownshendTHEATRE MarvinCarlsonTHEOLOGY DavidF.FordTHINKINGANDREASONING JonathanStB.T.EvansTHOMASAQUINAS FergusKerrTHOUGHT TimBayneTIBETANBUDDHISM MatthewT.KapsteinTIDES DavidGeorgeBowersandEmyrMartynRobertsTOCQUEVILLE HarveyC.MansfieldTOPOLOGY RichardEarlTRAGEDY AdrianPooleTRANSLATION MatthewReynoldsTHETREATYOFVERSAILLES MichaelS.NeibergTHETROJANWAR EricH.ClineTRUST KatherineHawleyTHETUDORS JohnGuyTWENTIETH‑CENTURYBRITAIN KennethO.MorganTYPOGRAPHY PaulLunaTHEUNITEDNATIONS JussiM.HanhimäkiUNIVERSITIESANDCOLLEGES DavidPalfreymanandPaulTempleTHEU.S.CONGRESS DonaldA.RitchieTHEU.S.CONSTITUTION DavidJ.BodenhamerTHEU.S.SUPREMECOURT LindaGreenhouseUTILITARIANISM KatarzynadeLazari-RadekandPeterSingerUTOPIANISM LymanTowerSargentVETERINARYSCIENCE JamesYeatesTHEVIKINGS JulianD.RichardsVIRUSES DorothyH.CrawfordVOLTAIRE NicholasCronkWARANDTECHNOLOGY AlexRolandWATER JohnFinneyWAVES MikeGoldsmithWEATHER StormDunlopTHEWELFARESTATE DavidGarlandWILLIAMSHAKESPEARE StanleyWellsWITCHCRAFT MalcolmGaskillWITTGENSTEINA.C. GraylingWORK StephenFinemanWORLDMUSIC PhilipBohlmanTHEWORLDTRADEORGANIZATION AmritaNarlikarWORLDWARII GerhardL.WeinbergWRITINGANDSCRIPT AndrewRobinsonZIONISM MichaelStanislawski

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RichardEarl

TOPOLOGYAVeryShortIntroduction

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Contents

Acknowledgements

Listofillustrations

Whatistopology?

Makingsurfaces

Thinkingcontinuously

Theplaneandotherspaces

Flavoursoftopology

Unknotorknottobe?

Epilogue

Appendix

Historicaltimeline

Furtherreading

Index

Acknowledgements

ThanksgotoMartinGalpin,AndyKrasun,NatalieLane,MarcLackenby,KevinMcGertyfortheircommentsonandhelpwithdraftchapters.ThanksespeciallytoMarcforhisencouragementtowritethebook.

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Listofillustrations

Londonundergroundmaps(a)ArchivePL/AlamyStockPhoto(b),©TfLfromtheLondonTransportMuseumcollection.

Examplesofpolyhedra

ManipulationsofacubetofinditsEulernumber

ThePlatonicsolids

MattParkerandhisfootballMattParker/YouTube.

Diagonalsinasquaremustintersect

Twonon-planargraphs

ComplicatedJordancurves(a)GetDrawings(b)DavidEppstein/WikimediaCommons/PublicDomain.

OriginalfigurefromFlatlandshowinghowASphereisperceivedbyASquareTheHistoryCollection/AlamyStockPhoto.

Theunknotandthetrefoil

Thetorus

Makingacylinder

Makingatorus

Gluinginstructionsforatriangle,pentagon,andsquare

Validandinvalidsubdivisionsofasphere

Connectedsumswithtori

TheMöbiusstrip(b)DottedYeti/Shutterstock.com.

MovinganorientedlooparoundaMöbiusstrip

TheKleinbottleandprojectiveplane

Thereallineandcomplexplane

VisualizingRiemannsurfaces

Distance,speed,andaccelerationonajourney

Examplesofgraphs

Continuousanddiscontinuousfunctions

Therigorousdefinitionofcontinuousanddiscontinuous

Theintermediatevaluetheorem

Theblancmangefunction

Visualizingdifferentmetrics

Graphsoffunctionsoftwovariables

Openballsintheplane

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Opensetsintheplane

Visualizingconvergence

Relatingtoadisconnectedsubspace

Relatingtopath-connectedness

Examplesofaminimum,maximum,andsaddlepoint

Criticalpointsoffunctionsonatorusandsphere

Vectorfieldsonthesphereandtorus

Examplescalculatingindicesofvectorfields

Loopsonatorus

Functionsonthedisc

Theunknotandtrefoils

ThethreeReidemeistermoves

Thegrannyandreefknots

Primeknotswithsevenorfewercrossings

Theknotgroupofatrefoil

Linksinvolvedintheskeinrelation

Simpleexamplesoflinks

CalculatingtheAlexanderpolynomialofatrefoil

Chapter1Whatistopology?

Asyoureadthis,passengerstheworldoveraretravellingonmetro(orsubwayorunderground)trains.Therearearound60billionindividualjourneysmadeannuallyonsuchmetrosystems.ButwhetherthisbeinTokyo,London,SãoPaolo,NewYork,Shanghai,Paris,Cairo,Moscow,thosetravellersareperusingmapsfortheirjourneysthatarecruciallydifferentfrommapsinatlasesorseenongeographyclassroomwalls.Foremostinthemindsofthosepassengersaretheconnectionstheyneedtomake—gettingoutattherightstationandchangingtothecorrectnewline.Theyarenotinterestedinwhetherthemap’sleft–rightlinesdoindeedrunwest–east,orwhethertheyreallydidmakearightangleturnwhentheychangedlines,asdepictedonthemetromap.

TheoldestmetronetworkintheworldistheLondonUnderground.Whenfirstproduced,theundergroundmapssuperimposedthedifferenttrainlinesontoanactual(geographicallyaccurate)mapofLondon,asshowninFigure1(a).AfirstversionofthecurrentmapwasdesignedbyHarryBeckin1931asinFigure1(b).Beck’smap,andthecurrentundergroundmap,arenotwrong.Rathertheytransparentlyshowinformationimportanttotravellers—forexample,thevariousconnectionsbetweenlinesandthenumberofstopsbetweenstations.Itisanearlyexampleofatopologicalmapanddemonstratesthedifferentfocusoftopology—whichisallaboutshape,connection,relativeposition—comparedwiththatofgeometry(orgeography)whichisaboutmorerigidnotionssuchasdistance,angle,andarea.

1.Londonundergroundmaps(a)Geographicallyaccurate1908map,(b)Beck’stopological1931map.

Topologyisnowamajorareaofmodernmathematics,soyoumaybesurprisedtolearnthatanappreciationoftopologycamelateinthehistoryofmathematics.Thewordtopology—meaning‘thestudyofplace’—wasn’tevencoineduntil1836.(‘Geometry’bycomparisonisanancientGreekwordand‘algebra’isanArabicword,withitsmathematicalmeaningdatingbacktothe9thcentury.)Justwhythiswasthecaseisnotasimplequestiontoaddress,thoughwewillseesomeaspectsoftopologydevelopedasmathematicianssoughttoputtheirsubjectonamorerigorousfooting.Topologyisahighlyvisualsubjectthatlendsitselftoaninformaltreatmentandthisbookwillgiveyouasenseoftopology’sideasanditstechnicalvocabulary.

Atopologist’salphabetAsafirstexample,toconveyhowdifferentlytopologistsandgeometersseeobjects,considerwhatcapitallettersatopologistwoulddeemtobethe‘same’.Usingthesansseriffont,thefourletters

EFTY

arealltopologicallythesame.Theyarenotcongruent,meaningthatnoneoftheletterscanbe

pickedup,androtatedorreflected,andthenputdownasoneoftheotherletters.ButIhopeyoucanenvisage,ifallowedtobend,stretch,orshrinktheletters,howanyofthemmightbetransformedintooneoftheothers.

Toatopologistthesefourlettersarehomeomorphictooneanother.ThegeometerwouldnoticethattheanglemadebythearmsoftheYisdifferentfromanyanglefoundintheotherletters.Thetopologist,ontheotherhand,wouldbehappytoflattenthearmsoftheY,andstretchitsbodyalittle,togivetheTshape.LikewisetheEcouldhaveitsbottomrungbentaroundtothevertical,andthenshortenedsomewhat,tomaketheF.FinallydoingthesametothetopoftheFwouldmakeaTonitsside.Thesefourletterscanbecontinuouslydeformedintooneanotherandbackagain.Broadlyspeakingthisiswhatitistobehomeomorphic,tobetopologicallythesame.

Butwhatisitabouttheselettersthatmakesthemtopologicallydifferentfromotherletters?Anothercollectionoflettersthataretopologicallythesameasoneanotheris:

CGIJLMNSUVWZ.

Topologicallyalloftheseareequivalenttoalinesegmentandit’snothardtoimaginehoweachmightbeformedbybendingandstretchingasuitablymutableletterI.Sohopefullyyou’reconvincedthatthelettersinthesecondlistareallhomeomorphictooneanother,butwhatmakesthissecondcollectiontopologicallydifferentfromthefirstlist?

Note,foreachletterE,F,T,Y,thateverypointliesonadistortedbitoflinewithoneexception.IneachoftheselettersthereisasinglepointthatmightbedescribedasaT-junction.TheseT-junctionsarehighlightedbelow.

OnewayinwhichtheseT-junctionsarespecialisthat,ifremoved,theremainderoftheletterisdisconnectedintothreeparts;theremovalofanyotherpointwouldleavejusttwopartsremaining.InwhateverwayswemightbendanddeformanEthedeformedversionwouldstillincludeasingleT-junction.AsnoneofthesecondsetC…ZhassuchaT-junctionthennoneofthemcanbeadeformedversionofanE(oranF,T,orY ).

Thisgivesagenuinesenseofhowmathematiciansresolvethequestion:aretwoshapesthesametopologically?Thiseitheramountstofindingsomemeansofcontinuouslydeformingoneintotheother,orinvolvesfindingsometopologicalinvariantofonethatdoesnotapplytotheother.Thewordinvariantisusedindifferentcontextsinmathematics:forexample,ifyoushuffleapackofcards,therewillstillremainfifty-twocardsafterwardsandfoursuits,theseareinvariants;butthetopcardmayhavechangedandthejackofclubsmaynolongercomebeforetheeightofdiamonds,andsosuchfactsaren’tinvariantsofashuffle.Atopologicalinvariantissomethingimmutableaboutashape,nomatterhowwestretchanddeformit.IntheaboveexampleweusedthepresenceofaT-junctionasourtopologicalinvariant.YoumightnotethatanEincludesfourrightangleswhilstanFcontainsonlythree.ThepresenceoffourrightanglesisageometricinvariantandsoshowsthatEandFarenotcongruent(i.e.notgeometricallythesame),but—workingtopologically—wearepermittedtounbendtheserightanglesandsorightanglesarenotimportantfromatopologicalpointofview.Ratherthey’remutableaspectsofashapeandnottopologicalinvariants.

Theremainingtwenty-sixletters,groupedtopologically,breakdownas:

DO,KX,AR,B,PQ,H.

YoumightwanttotakeamomentthinkingaboutwhatmakesanAdifferentfromaPorOdifferentfromQ.Infact,theOintroducesanimportanttopologicalinvariantthatseparatesitfrombothIandE.TheshapeoftheOisdifferentasitmakesaloop.TechnicallyOisnotsimplyconnected,atopicwewilldiscussmoreinChapter5.

Euler’sformulaOneofthefirsttopologicalresultswasduetoLeonhardEuler(pronounced‘oil-er’),atitanof18th-centurymathematicsandoneofthemostprolificmathematiciansever;hisformuladatestoaround1750.Theresultrelates—atfirstglance—topolyhedra,three-dimensionalobjectssuchas

cubesandpyramids(Figure2).Itisalsosofundamental—astraightforwardobservationatleast—thatitissurprisingancientGreekmathematiciansmissedit.

2.Examplesofpolyhedra(a)Acube,(b)ASquare-basedpyramid.

Lookingatthecube,wecanseethatitismadeupofvertices(thecornersofthecube—thesingularis‘vertex’),theseverticesbeingconnectedbyedgesandthattheseedgesthenbound(square)faces.ForthecubethenumberofverticesVequals8,thereareE=12edgesandF=6faces.Forthe(square-based)pyramidwehaveV=5,E=8,andF=5.Nopatternmaybeevidentimmediatelybutifweincludethefourotherso-calledPlatonicsolids—tetrahedron,octahedron,dodecahedron,icosahedron(Figure4)—andotherfamiliarpolyhedra,wecreateTable1.

Table1. Vertices,edges,facesforvariouspolyhedra

4.ThePlatonicsolids(a)Tetrahedron,(b)Cube,(c)Octahedron,(d)Dodecahedron,(e)Icosahedron.

(Weshallseesoonthatthetruncatedicosahedronisfamiliartous—justnotbythatname!)

ForallthegeometrythattheancientGreeksknew,itseemsstrikingthatthispatterneludedthem,butwewillprovenow—ormorehonestlysketchaproofof—Euler’sformulawhichstates,forapolyhedronwithVvertices,Eedges,andFfaces,that

Intheproofouraimwillbetobeginwithapolyhedronandmanipulateitincertainways—forexample,wemightremoveorsubdividefaces—butinallcaseswewillcarefullytracktheeffectourmanipulationhas(ifany)onthenumber .If,aftersuchmanipulations,wearrive

••••

atasimplifiedsituationwhereweknowwhat equals,andweknowtheeffectsourmanipulationshadonthatnumber,thenwemaybeabletoworkbackwardstofindwhat

wasoriginally.

Webeginthenwithapolyhedron,andfirstremoveoneofthefaces.Thishastheeffectofreducingby1asFhasdecreasedby1.Nowthatthepolyhedronhasamissingface—

effectivelythepolyhedronhasbeenpunctured—itcanbeflattenedintotheplane,takingcarethatallthevertices,edges,facespresentonthepuncturedpolyhedronremainandareconnectedintheplaneinthesamemannertheywereonthepuncturedpolyhedron.Forexample,ifweremovedonefacefromacubeandflattenedtheremainingcubethenwewouldhavesomethinglikeFigure3(a).

3.ManipulationsofacubetofinditsEulernumber(a)Aflattened,puncturedcube,(b)Withflattenedfacestriangulated,(c)Removingatriangle,(d)Removingatriangle.

Thenextmanipulationistosubdivideeachoftheflattenedfacesintotriangles—ashasbeendonetotheflattenedcubeinFigure3(b).IntroducingasingletrianglehastheeffectofincreasingFby1—whatwasonefacebecomessplitintotwo—ofincreasingEby1—thenewedge,introducedtomakeatriangle—anddoesn’tchangeV.Sothereisnooveralleffectto aswekeepintroducingtriangles;theincreaseof1toF,atermthatisaddedintheformula,ispreciselybalancedbytheincreaseinEwhichisatermwesubtract.Whenthishasbeendoneforeachflattenedface(asinFigure3(b))then isstilljustonelessthanitwasoriginally.

Wenowremovethetrianglesoneatatime.Forexample,ifweremovethebottomtrianglefromFigure3(b)tomakeFigure3(c),thenweremoveoneedgeandonefaceand,bythesamereasoningasbefore,thishasnooveralleffecton .

Similarly,wemightthenremovetheright-mosttriangletocreateFigure3(d),themanipulationagainhavingnoeffecton .ButthebottomtriangleofFigure3(d)isconnecteddifferently.Ifweremovethattrianglethenweremove2edges,1face,and1vertex.Thealgebraisalittlemorecomplicatedthistime,butagainremoving1vertexand1facemeansgoesdown2butthisiscounteredbyremoving2edgesasEisatermwesubtract.Orifyouprefermoreformalalgebraicreasoning,wearejustsaying

Let’ssummarizewhat’shappenedsofar:

Weremovedafaceand decreasedby1.Weflattenedthepolyhedronintotheplane—allV,E,Fremain,sonochangeto .Wesubdividedtheflattenedfacesintotriangles—thishadnoeffecton .Wekeptremovingtrianglesfromtheedgeoftheflattenedpolygon—eachremovalhavingno

effecton

Eventuallyonlyasingletrianglewillremain,havingremovedallothers.Atrianglehasasingleface,threevertices,andthreeedges,sothat equals .Thisisthevalueof thatwefinishwith.Theonlymanipulationthateverchanged wasthatveryfirstremovalofafacewhichreduceditbyone;initiallythenitwasthecasethat

This‘sketchproof’wasgivenbyAugustin-LouisCauchyin1811.It’sworthhighlightingthereareseveral‘i’sstilltobedottedtomakeaproofwithwhichaprofessionalmathematicianwouldbehappy,butalsonotinghowmuchoftheideaoftheproofisgenuinelyhere.Wedidn’ttakecaredescribinghowweremovedtrianglesfromtheboundaryoftheflattenedpolygon;ifwe’dbeencarelesswemighthaveremovedatrianglethatdisconnectedtheflattenedpolygonintotwoseparatepolygons,andweshouldhavetakentimetomakesuresuchanoccasioncanalwaysbeavoided.OtherissueswillbecomemoreapparentinChapter2butthese‘i’scanindeedbedotted.InProofsandRefutations,theHungarianphilosopherImreLakatosusedthespecificexampleofEuler’sformula,andhistoricaleffortstoproveit,tohighlighthowharditcansometimesbetogenerateawatertightproofandtoalsoraisethequestionofwhenatheoremproperlybecomespartofmathematicsorhasmathematicalcontent.

It’salsoworthmentioningthatRenéDescarteshad,overacenturybeforeEuler,demonstratedatheoremforpolyhedrathatisequivalenttoEuler’sformula;histheoremwasintermsof‘angulardefects’atvertices.Allofasuddenwearebackinthegeometricalworldandit’slessthanclearthatthereisagenuinelynewsubject,animportantlydifferentwayofmathematicalthinking,thatEuler’sfingertipswerebrushingagainst.Euler’sformulationencouragesappreciationoftheresultassomethingalittlenew—inEuler’sterminology,ratherthanDescartes’s,it’smuchclearerthattheconnectionofthevertices,edges,andfacesiswhatcounts,buthistoricallywearestillalongtimefromadeeperappreciationoftopologyasafundamentalmodeofmathematicalthinking.

TherearefivePlatonicsolidsPlatonicsolidsarepolyhedrawithregularfacesthatareallcongruent(geometricallythesame)andwhichmeetinthesamemannerateachvertex.Thereareinfinitelymanyregularpolygons—equilateraltriangles,squares,regularpentagons,etc.—butin3Ditturnsoutthattherearejustfiveregularsolidswhichhavebeenknownsinceantiquity.TheseareshowninFigure4andwithEuler’sformulawecanshowtherearejustthesefive.

ConsideraregularpolyhedronwithVvertices,Eedges,andFfaces.Asthesolidisregulartheneachfaceisboundedbythesamenumberofedges;let’scallthisnumbern.Likewisethereisacommonnumbermforhowmanyedgesmeetateachvertex.So,withthecube, (thefacesaresquares)and (threeedgesmeetateachvertex).Continuingwiththecubeasourexamplefornow,thinkabouthowwecanmakeacubebygluingtogethertheedgesofsixsquares.

Webeginwith6separatesquaressothat,beforeanygluinghappens,thereare6squares,24edges,and24vertices.Notethattomakethecubeittakestwo‘unglued’edgestomakeeachedgeofthecube(whichagreeswiththerebeing24/2=12edges)andittakesthree‘unglued’verticestomakeasinglevertexofthecube(againthereare24/3=8vertices).Moregenerally,whenwehaveFfaceseachwithnedges,wewouldhavenFedgesbeforeanygluing.Ittakestwooftheseungluededgestomakeasingleedgeofthepolyhedronwhichthenhasedges.Thereareasmanyungluedverticesasungluededges,namely ,andthesewillbegluedtogethertomake verticesonthepolyhedronasittakesmungluedverticestomakeonegluedvertexonthesolid.

PuttingtheseexpressionsforVandFintoEuler’sformulaweget

(Theequation hasbeenrearrangedtomakeFthesubjectoftheequationsothat.)Wecanthendividebothsidesoftheaboveequationby2Eandrearrangetofind

As1/Eispositivethismeansthat

Somandncan’tbothbeverylargeasthen and wouldbeverysmallandtheirsumwouldnotexceed .Alsorecallmandnarepositivewholenumbers,sotherearen’tmanyoptionsandit’snothardtofindalltheirpossiblevalues.

It’simpossibleforbothmandntoexceed4asthen wouldbelessthan

.Soeither or or or (withperhapsmorethanoneof

thesebeingtrue).Forexample,if ,theonlynforwhichtheinequalityistrueare.If then equals orlessandtheinequalityis

nottrue.Forthethreecasesof and wehave

Asimilarcalculationfor leadsto , ,andwhen wefind ,.Inallthesefivecaseswecanusethepreviousformulastoworkoutthenumbersof

vertices andfaces .WecanputthefulldetailsintoTable2.

Table2. PossiblemandnvaluesforthePlatonicsolids

Ifweareseekingtoberigoroushere,weshouldreallypointoutthatthepreviouscalculationsshowthatthereareatmostfivepossiblepairsofvaluesthatm,ncantake.Thosecalculationslimitthepossibilities,butdonotnecessarilymeanthatthereisaPlatonicsolidforeachofthesecases,norprecludetherebeingmorethanonePlatonicsolidforpermittedmandn—itmightbethattherearetwodifferentPlatonicsolidswiththreepentagonsmeetingateachvertex.ListedinTable2arethefivePlatonicsolids,andsowecanseethatthereisatleastonesolidforpermittedm,n.Andit’snothardtoappreciatewhytherecanbeatmostone.Inthecasewhere ,

thenthreesquaresmeetateachvertex;seeminglythisonlytellsussomethingaboutpartsofthesolid,butifwefollowthisrecipeofattachingthreesquaresateachvertexthenthereisonlyonewaytoprogressbuildingupthesolid—it’snotclearthatthisrecipewillactuallyleadtoacompletesolid,butitdoesshowthattherecanbeatmostonePlatonicsolidforeachallowedm,n.

(Asanaside,youmayhavenoticedthat and , and , aresolutionsifwepermitEtobeinfinite.These‘solutions’correspondtotessellationsoftheplanewherefoursquaresmeetatavertex,wherethreeregularhexagonsmeetatavertex(aswith

honeycombs),andwheresixequilateraltrianglesmeetatavertex.TherearealsosomepatternsapparentinTable2forthevaluesofV,E,Fforthecubeandoctahedron,andlikewisethedodecahedronandicosahedron.Thisisbecausethesesolidsaredualtooneanother—thismeansthatthemidpointsofthefacesofacubemakeanoctahedron,andviceversa;likewisethedodecahedronisdualtotheicosahedronandthetetrahedronisself-dualinthissense.)

FootballsIn2017themathematicspopularizerMattParkerbeganapetitionseekingtogetroadsignstofootballstadiacorrected.

Youmaynothavenoticedtheinaccuracyofsuchsignsinthepast,perhapsbeinghappyjusttoknowyou’retravellingtherightwayforthegame.Butit’sclear(Figure5)thatthesign’sfootballdoesnotresembletheactualfootballthatMattiscarrying.Afootball’ssurfaceismadefrompentagonsandhexagonsandtheeverydayfootballismoreformallyknownasatruncatedicosahedron.(Itcanbecreatedfromanicosahedronbyplaningdown,aroundeachvertex,thefiveedgesmeetingthere.Ifweplanedownone-thirdofeachofthosefiveedges,wecreateanewpentagonalfaceandcontinuedplaningeventuallyshrinksallthetriangularfacestohexagons.)IthinkthoughtheirksomeprincipleforMattwasnotthatthesign’sfootballwasbadlydrawn,itwasinfactimpossiblydrawn.

5.MattParkerandhisfootball.

Thereisnowaythataspherecanbemadebystitchingtogetherhexagonsasshownonthesign.Thatwouldbeanexamplewhere and ,usingthepreviousnotation,andwhilstwecancovertheplaneinthisway—whichmaybewhythesignlooksplausibleatfirstglance—makingafootballthiswayismathematicallyimpossible.

Infact,Euler’sformulashowsushowmanypentagonsandhexagonsthereareonafootball.Recallinghowtotruncateanicosahedronweseethereareasmanypentagonsasoriginalvertices(12)andhexagonsasoriginalfaces(20),butEuler’sformulacanshowthisistheonlywaytoconstructsuchafootball.SayafootballhasPpentagonalfacesandHhexagonalfaces.Then,beforegluingthesetogether,wehave5P+6Hungluededgesandthesamenumberofungluedvertices.LookingatMatt’sballwecanseethat(i)twoungluededgesareneededtomakeanedgeonthefootballand(ii)threeungluedverticesmakeavertexwith(iii)twohexagonsandonepentagonmeetingatavertex;from(iii)weseetherearetwiceasmany‘unglued’verticescollectivelyonthehexagonsasonthepentagons.So

IfweputthesevaluesintoEuler’sformula wefindthat

whichsimplifiesandrearrangestoP=12andtheequation10P=6HyieldsH=20.This,then,istheonlywaytomakeafootballifwefollowtherules(i),(ii),(iii).

•••

GraphtheoryLet’schangetackalittleandconsiderthefollowingproblem.ThesquarePQRSinFigure6hasdiagonallyoppositeverticesPandR,QandS.IfweweretodrawcurvesfromPtoR,andfromQtoS,curveswhichremainwithinthesquareasinFigure6,thensurelythosecurveswouldhavetocrossatleastonce.(InFigure6therearethreeintersections.)Thisseemsobvious—andistrue—buthowwouldyougoaboutprovingthis?

6.DiagonalsinasquaremustintersectCurvesPRandQSinthesquarePQRS.

Beforemoreissaid,itmightbeworthstressinghowcharacteristicofatopologicalquestionthisis.ThecurvesPRandQSneedtoconnecttheirendpoints.Thosecurvesdon’tneedtobepolygonal,orhavewell-definedgradients,orbedefinedbyspecificfunctions.Theyneedtoconnecttheendpointsinsomecontinuoussense—fullerdetailsinChapter3—buttheyareotherwisegeneralpathsfromPtoRandfromQtoSthatremaininthesquare.

Atfirstglance,thisproblemmightseemquiteremovedfromthepolyhedrawewerejustdiscussing.However,Figure6doesn’tlookthatdifferentfromFigures3(a)–(d).Wehavevertices(P,Q,R,SandanypointswherethecurvesPRandQSmeet),edgesrunningbetweenthesevertices(thoughadmittedlythey’renowcurved),andwehavefacesboundedbythoseedges.ItwascrucialtotheproofofEuler’sformulathat foreachofFigures3(a)–(d).Ifwealsoincludetheoutsideregionasaface—essentiallytheoneremovedsowecouldflattenthepolyhedron—thenwearrivebackatEuler’sformula .(Bythisreckoning ,

, inFigure6.)

Sosuppose,somehow,wecoulddrawcurvesPRandQSinthesquarePQRSwhichdon’tintersect.We’dthenfind

thefourcornersP,Q,R,S.thesquare’sfoursidesandthecurvesPR,QS.theoutsideofthesquare,abovePS,belowQR,rightofPQ,leftofSR.

Butthisleavesuswith andsosuchascenarioisimpossiblebyEuler’sformula.

Graphtheoryisanareaofmathematicsthatmodelsnetworksinawidesense:physical,biological,andsocialsystems,variouslyrepresentingtransportnetworks,computernetworks,websitestructure,evolutionofwordsacrosslanguagesandtimeinphilology,migrationsinbiology,etc.Agraphisacollectionofpointscalledvertices,withtheseverticesconnectedbyedges.Wewillalsoassumethatgraphsareconnected,meaningthatthereisawalkbetweenanytwoverticesalongtheedges.Thisdefinitionmaybeextendedtoincludeone-wayedges—directedgraphsordigraphs—andweightsmightbeintroducedtoedgesrepresentingthedifficulty—intermsoftime,distance,orcost—oftravellingalongaparticularedge.

Somegraphsareplanar,meaningthattheycanbedrawnintheplanewithouttheiredgescrossing(atpointsthataren’tvertices).ThetwographsK5andK3,3inFigure7areimportantlynotplanar.Thecompletegraphon5vertices,denotedK5,hasasingleedgebetweeneachpairofthe5verticesmaking10edges.YoumightthinkthatK5isplanarasit’sdrawninFigure7(a)—thepointisthat,sodrawn,manyoftheedges’crossingsdon’toccuratverticesandtodeemthese

crossingsasverticeswouldmeanwewerenolongerconsideringK5whichhasonly5vertices.IfwecouldproperlydrawK5intheplanetherewouldbe10triangularfacesv1v2v3,v1v2v4,throughtov3v4v5.We’dthenhavethat equals andsoK5isnotplanar.

7.Twonon-planargraphs(a)ThecompletegraphK5,(b)ThecompletebipartitegraphK3,3.

K3,3isthecompletebipartitegraphbetweentwotriosofvertices.AsomewhatsubtlerargumentshowsK3,3isnotplanar.NotethatK3,3has verticesand edges.Ifdrawnintheplanethiswouldmean .ButafaceofK3,3wouldhaveatleastfouredgesasitsperimeternecessarilyrunsfromavtoawtoadifferentvandtoawandonlythenmayreturntotheoriginalv.So,countingtheedgesbygoingaroundallthefaces,wewouldgetatotalofatleast edges.However,asanedgecanboundatmosttwofacesthiswouldmeanwe’dhaveatleast edges,whichisourrequiredcontradiction.

ThePolishmathematicianKazimierzKuratowskiprovedin1930thatagraphisplanarpreciselywhenneitheracopyofK5norK3,3canbefoundwithinthegraph.WewillinduecourseseethatK3,3canbedrawnonothersurfacessuchasatorus(Figure13(c)).

NastysurprisesEulerarrivedathisformulaacenturybeforethewordtopologywascoined.Hisformulaischaracteristicofavisualsideoftopologynaturallyalignedwithgeometry.Buttopology,asasubject,woulddevelopalongvariousthemesandinparticularhadanimportantroleinthefoundationalworkmathematiciansweredoingaroundthestartofthe20thcentury.AsIhintedearlier,topology’srisemayhavebeenhamperedbyatraditionalmindsetthatsomeofitsquestionshadobviousanswers.ForexampleCamilleJordan,aslateasthe19thcentury,provedthefollowing:acurveintheplane,whichdoesnotcrossitselfandwhichfinishesbackwhereitbegan—acurvewhichwewouldnowcallaJordancurve—splitstheplaneintotworegions,thetechnicalphrasefortheseregionsbeingconnectedcomponents.Oneoftheseregionsisbounded,theinside,andtheotherisunbounded,theoutside,andthisistheJordancurvetheorem.Earliermathematicianswouldhavehappilythoughtthisobviousandthefirstrigorousproofdidn’tappearuntil1887.Youmayagreewiththoseearliermathematiciansthattheresultcanbesafelyassumed.MaybeeventhePollock-likeJordancurveinFigure8(a)doesnotswayyourviewoftheintuitivenessoftheresult.

8.ComplicatedJordancurves(a)AmorecomplicatedJordancurve,(b)AJordancurvewithpositivearea.

InFigure8(b)istheKnopp–Osgoodcurvewhich,forallitsfractal-likeappearance,isaJordancurve.Astonishinglyithasapositivearea—thatisthecurveitselfhaspositivearea,we’renotreferringtosomeregionthatitbounds.Wouldyouhavesaidamomentagothatit’sobviousthatcurvescan’tthemselveshavearea?

Youshouldn’tworrytoomuchinthesensethatmostthingsthoseearlymathematiciansthoughttobetrueturnedouttobetrue,onceproperlyunderstoodandqualified,butmathematicianstowardstheendofthe19thcenturyweregettingnervousabouttherulesandassumptionsthatmathematicsreliedon.

Arelatedproblemwithintopologyatthattimewasrigorouslydefiningwhatdimensionmeans.Againthishadpreviouslybeentreatedasanintuitiveconcept,onlyformathematicianstobeginfindingspace-fillingcurvesthatpassthrougheverypointintheplaneorotherweird-and-wonderfulspacesthatcanreasonablybeassigneddimensionsthatarenotwholenumbers—spacesthatwouldnowbecalledfractals.

Anearlythemeoftopologywasthisgeneraltopologyorpoint-settopologyseekingtoaddresswhatitmeanstobeaset,tobeaspace,etc.Metricandtopologicalspaceswereintroduced—tobediscussedinChapter4—eachbeingattemptstodescribegeneralstructureswherecontinuitycouldbedefined.Settheorydealswithcollectionsthatareessentiallyjustthings-in-a-bag.Thisgeneraltopologysoughttodefinewaysinwhichobjectsmightbeconsidered‘close’toone

another,withtheaimbeingtodefinecontinuityinabroadsetting.

AFlatlandmindsetThenovellaFlatland:ARomanceofManyDimensions,writtenin1884byEdwinAbbott,isasatireonVictorianmores.Thenarratoris‘ASquare’,aninhabitantofFlatland,aplanarworldhavingjusttwodimensions.ThecultureofFlatlandandthelogisticsoflivingintwodimensionsarefullydescribed,implicitlyhighlightingsomeofthenarrow-mindednessofVictorianculture—forexample,womenareone-ratherthantwo-dimensionalbeings.Thestorydoesn’texplicitlydiscusstopology,butinitsdescriptionofworldswithdifferentdimensionsandimplicationsfortheinhabitants,itprovidesausefulmetaphorforunderstandingcertainaspectsoftopology.

Forexample,ASquareisvisitedatonepointbyASphere.Beingathree-dimensionalobject,ASpherecanonlybeperceivedbyFlatlandersasacircularcross-section(Figure9).Bymovingupanddown—relativetoFlatland’splane—ASpherecangrow,shrink,andevendisappearentirely.Inasimilarmanner,totrulyunderstandthetopologyofaspace,wehavetobeginthinkinglikeinhabitantsofthatspace.

9.OriginalfigurefromFlatlandshowinghowASphereisperceivedbyASquare.

Topologyisoftencharacterizedasrubber-sheetgeometry.It’sasomewhatclichédmetaphor,butit’salsoslightlyinaccurate.Itgivesacorrectsenseoftopologybeingmoreaboutshapeandlessrigidthangeometryinitsfocus.Ontheotherhand,inChapter6wediscussknots,andasa(genuine)knot—likethetrefoil—andthe(unknotted)circle(Figure10)cannotbecontinuouslydeformedintooneanotherin3Dthenyoumightbetemptedtosaythecircleandtrefoilarenothomeomorphic,buttheyare.

10.Theunknotandthetrefoil(a)Theunknot,(b)Thetrefoilknot.

Theknottednessofthetrefoilsayssomethingaboutitspositionin3D.Infact,allknotsarehomeomorphictoacircle.Tobetterappreciatethis,youmightimaginelifeasanantlivingoneitherthecircleortrefoil.Astheantmovesaroundeithertheunknotortrefoilithasasenseofbeingonaloop,buttheanthasnonotionofwhetheritislivingonaknot.Itisonlybybeingabletoviewthingsfromoutsidethetwoloops,andlookingonfromapositionintheambientspace,thatweareabletorecognizeoneloopasknottedascomparedwiththeother.ThisFlatlandmindsetwillproveusefulagainlaterwhenwemeetsubspaces.

Topologywouldadvanceonvariousfrontsinthe19thand20thcenturies.Inparticular,BernhardRiemannwouldearlyonshowtheusefulnessofa‘topologicalmindset’,introducingRiemannsurfacesintothestudyofpolynomialequationsanddemonstratingsomedeepconnectionsbetweentopologyandmanyotherareasofmathematics.

Chapter2Makingsurfaces

TheshapeofsurfacesRecallEuler’sformulastates forapolyhedron.Variousdetailsoftheproofwerebrushedunderthecarpet,themostsignificantofthesebeingtheclaimthat,onceafaceisremovedfromapolyhedron,theremainingpolyhedroncanbeflattenedintotheplane.Thiswastrueforthepolyhedrawewereconsidering,buttheclaimsayssomethingimportantabouttheshapeoftheremainingpolyhedronthatwasperhapsunintentional.Inanycase,thenextexamplewilleithermakeusquestionwhatwemeanbyapolyhedronorhaveuslookingtogeneralizeEuler’sformula.

ForFigure11(a)’s‘polyhedron’,acountofvertices,edges,andfacesshowsthat ,, ,giving whichseemstodisproveEuler’sformula.Weareleft

withafewalternatives:eithertheobjectinFigure11(a)shouldnotbeconsideredapolyhedron,orweneedtorestrictEuler’sformulatoacertaintypeofpolyhedron,orweneedtoadaptandgeneralizeEuler’sformulaintoaversionthatremainstrueforabroaderfamilyofpolyhedra.

11.Thetorus(a)Apolyhedronwithonehole,(b)Atorus.

Themostobviousissuewiththisnew‘polyhedron’istheholethroughitsmiddle.Thisisnotimmediatelyreasonenoughtoexcludeitasapolyhedron,butthisshape,onceafaceisremoved,doesnotleavearemainderthatcanbeflattenedintotheplane,makingourearlierproofinvalid.WeneedeithertorestrictEuler’sformulatopolyhedrawithoutholes,orweneedtoworkoutthecorrect valuesforpolyhedrawithholes.

Recallingtherubberynatureoftopology,wemightrecognizethatthepolyhedraofChapter1allhadthesameunderlyingsphericalshape.Ifallowedtosmoothoutthosepolyhedra—thepointyverticesandtheridgyedges—wecouldtransformeachofthosepolyhedratoasphere,coveredwithapatchworkofcurvedfaces,justlikeMattParker’sfootball(Figure5)wascoveredwithcurvedpentagonsandhexagons.Buthoweverwesmoothdownournewpolyhedronwecan’tmakeasphere,ratherwewouldmakeatorus,theshapeofadoughnutwithaholethroughit(Figure11(b)).

PerhapsthenalloftheexamplesofChapter1—includingtheproofofEuler’sformula—pointtotheEulernumberofthespherebeing2.AndFigure11(a)isafirstexamplesuggestingtheEulernumberofthetorusis0;thiswouldmeanthenumber equals0howeverwedivideupthetorus.Allthiscouldbecomequiteinvolvedunlesswehaveawayofefficientlydescribingsurfaces—includingmorecomplicatedonesthanthesphereortorus—andforsystematicallycalculatingtheirEulernumbers,thatisthe valuecommontoallsurfacesofacertainunderlyingshape.

Gluingsurfacestogether

Ausefulwayofconstructingsurfacesistobeginwithapolygonandpairwisegluetogethertheedgesofthepolygon,thewayamodelkitmightdirectyouto‘gluetabAtotabB’.Howmightwemakeatorusinthismanner?Ifwebeginwitha(suitablyelastic)square(Figure12(a)),benditaround(Figure12(b)),andgluetheoppositeedgese1ande3sothattheverticesv1,v2getgluedrespectivelytov4,v3,andlikewiseforallotheroppositepointsofe1ande3—assignifiedbythetwoarrows—thenwewillmakethecylinderdrawninFigure12(c).Notethattheedgese2ande4ontheoriginalsquarehavebecomethetwocircularendsofthiscylinder.Wecanthengluetogetherthesecircularends;ifwedothissothattheoppositepointsoftheoriginale2ande4aregluedtogether,thenwemakeatorusasinFigure13(a).

12.Makingacylinder(a)Asquarewithidentifiededges,(b)Makingthecylinder,(c)Acylinder.

13.Makingatorus,(a)Toruswithvande1,e2drawn,(b)Squarewithgluinginstructions,(c)K3,3onthetorus.

Notethatthefourcornersoftheoriginalsquare—denotedv1,v2,v3,v4—haveallbeengluedtogethertomakeasinglepointvonthetorus.Similarly,theedgese1ande3havebeengluedtogethertoformacirclegoingaroundtheoutsideofthetorusande2ande4havebeengluedtogethertoformadifferentcirclegoingthroughtheholeofthetorus,thesetwocirclesmeetingatthepointv.

Importantly,thetorus’sshapeisfullydescribedbyasquarewithdirectionsforhowtheedgesaretobegluedtogether.Mathematicianswoulddrawthissquare-with-gluing-instructionsasshowninFigure13(b).Thesinglearrows—andimportantlythedirectionsinwhichthey’redrawn—showhowthosetwoedgesareglued,andthedoublearrows(againnotingdirections)tellushowtheotherpairofedgesisglued.

Asanaside,thegraphK3,3,whichwesawcan’tbedrawnintheplane(Figure7(b)),canbedrawnonthetorus(Figure13(c)).ThereasonK3,3canbedrawnonthetorusisbecauseatorushasalowerEulernumberof0and,arguingasbefore,itcanthenbeshownthatF=3.IfyoulookcarefullyatFigure13(c)youwillseethatthereareindeedthreefaces(quadrilateralsv1w2v2w1andv3w3v2w2andasinglehexagonalfacev1w2v3w1v2w3).

HowdoesallthishelpwithdeterminingEulernumbers?Withasinglesquareandgluingdirections,wehavebeenabletomakeanobjectwiththeshapeofatorusandthisiscertainlyaconciserdescriptionofatorusthanFigure11(a)forwhich , , .ButisthisgluedsquareenoughtoworkouttheEulernumberofatorus?Theanswerisyesifwe’recarefulwhenthinkingaboutjusthowtheoriginalverticesandedgesgluetogether.Ontheoriginalsquaretherewasjustoneface—thesquare’sinterior—fourungluededges(labellede1,e2,e3,e4),and

fourungluedvertices(labelledv1,v2,v3,v4).However,oncewe’vefollowedthegluingdirections,thosefouredgeshavebecomethetwocircularedgesonthetorusandthefourverticeshavebecomeonepointv,asinFigure13(a).Andthereisstillone‘face’onthetorus—thesquare’sinteriorhasbeenstretchedtobecomeallofthetorusexceptthosecirclesandv.SowhenweusethisgluedsquaretocalculatetheEulernumberofthetoruswegettheanswer

whichagreeswithourcalculationfromFigure11(a).

IfyoupreferFigure13(a)showingthetoruswiththefourgluedverticesbecomingone,andthefouredgesbecometwoloopsonthetorus,thenFigure13(b)willonlyappearasanunfinishedDIYjob.Butthesinglesurfaceobtainedfromgluingthetriangle,pentagon,andsquareinFigure14,followingallthegluingdirectionsa, b,…faccordingtothearrows,mightreasonablystartstretchingyourvisualizationskills.ButwecanstillworkouttheEulernumberofthischimeraandseektounderstandjustwhatsurfacewearelookingat.Thistimetherearethreefaces(thetriangle,pentagon,andsquare)andthetwelveungluededgesofthepolygonsmakesixgluededgesa, b, c,…fonthesurface.Howmanyverticeswillweultimatelyhave?Thereweretwelveungluedverticesoriginallybutvariousofthesegetgluedtogetheraswemakethesurface.Forexample,v1andv5aregluedtogetherastheyarebothattherearendoftheedgemarkeda.Infact,wecanchasearoundthesegluingstoseejusthowmanyverticeswehave:

14.Gluinginstructionsforatriangle,pentagon,andsquare.

v1andv5areglued(rearendofa)v5andv9areglued(rearendoff )v9andv4areglued(rearendofd)v4andv12areglued(frontendoff )v12andv2areglued(rearendofc)v2andv7areglued(rearendofb)v7andv11areglued(frontendofe)v11andv1areglued(frontendofc)

Soeightdifferent(unglued)verticesv1,v2,v4,v5,v7,v9,v11,v12allgetgluedtogetherasasinglevertexonthesurface.Inasimilarfashionwecanseethattheremainingfourverticesv3,v6,v10,v8getgluedtogether(inthatorder).Sooncemade,thesurfacehas2vertices,6edges,and3facesgivinganEulernumber of .Justwhatsurfacehavewemade?

Gettingtherightanswer:subdivisionsWeneednowtomakeclearjustwhatsurfacesweareconsidering—closedsurfaces—andhowtheycanbedividedupintovertices,edges,andfaces.Aclosedsurfaceisonewithoutaboundary,suchasatorusorsphere,butnotthecylinderofFigure12(c).Ourprocessofmakingatorusbeginswithasquareandatthatpointoursurfacehasaboundaryconsistingofitsfouredges;whenwegluetwoedgestomakeacylinderthenthesurfacestillhasaboundary,namelyitstopandbottomcircles.Oncethetorushasbeenmade,noboundarypointsremainunglued.

Secondly,wecanonlycalculatethecorrectEulernumberofaclosedsurfaceifwearecarefuldividingitup.Cubes,footballs,dodecahedra,andpyramidsareallvalidwaysof‘subdividing’thesphereintovertices,edges,andfaces.IneachofthesecasesweobtainedanEulernumber

•••

of2.HoweverhereareotherwayswemightsubdividethespherethatseeminglyproducedifferingEulernumbers(Figure15).

15.Validandinvalidsubdivisionsofasphere(a)V=0,E=0,F=1,(b)V=0,E=1,F=2,(c)V=1,E=0,F=1,(d)V=1,E=1,F=2,(e)V=1,E=2,F=3,(f)V=2,E=0,F=1.

Inorderthevaluesof forthesixspheresinFigure15are1,1,2,2,2,3andweknowthecorrectEulernumberequals2.Soanyoldsubdivisionwillnotleadtoacorrectcalculationofthesphere’sEulernumber.Foracollectionofvertices,edges,andfacestomakeapermissiblesubdivisionthefollowingmustbetrue:

anedgemuststartandfinishinavertex;whentwoedgesmeet,theymustmeetinavertex;facesmustbe(distorted)polygons.

Lookingattheseso-calledsixsubdivisions,onlytwooftheseareinfactpermissible,15(c)and15(d).In15(a),15(e),15(f),thereisafacethatisnotadistortedpolygon;neitherthewholesphere(15(a))northepuncturedcummerbund(15(e))northetwice-puncturedsphere(15(f))aretopologicallythesameasapolygonandsonotpermissiblefaces.In15(b)and15(e),theedgesdonotbeginandendinavertex.15(e)isdeliberatelygiventoshowthatthecorrectEulernumbercanbeincorrectlycalculated.

LookingbackatthetorusinFigure13(a)andthesurfaceinFigure14,wecalculatedtheirEulernumbersusingsubdivisionsconsistentwiththeabovethreerules.Therefore,wecorrectlycalculatedtheirEulernumbersas0and–1respectively.

ConnectedsumsGiventwoclosedsurfacesS1andS2,thenwecancreatetheirconnectedsumS1#S2.Thisisawaytogluesurfacestogetherandausefulmeansofmakingnewsurfacesfromthefewwehavesofarmet.SayS1andS2eachhasasubdivisionthatincludesa‘triangular’faceboundedbythreeedges.(Theinvertedcommasherehintthat,thisbeingtopology,thefacesmaynotbethat

recognizablytriangularintermsofhavingstraightedges.)TheconnectedsumS1#S2isthencreatedbyremovingthesetwotriangularfaces,somakingtwoholesinthesurfaces,andthengluingthetwosurfacestogetheralongtheboundariesoftheholes,pairingupthethreeverticesandthreeedgeswiththoseontheboundaryofthesecondremovedfaceas,forexample,inFigure16.

16.Connectedsumswithtori(a)Onetoruswitha“triangle”missing,(b)Atoruswithtwoholesasaconnectedsum.

HelpfullythereisaformulafortheEulernumberofS1#S2.Inmakingtheconnectedsum,weremovetwotriangularfaces,thesixdifferentverticesonthesetrianglesaregluedtomakethreeverticesontheconnectedsum,andlikewisesixedgesaregluedtomakethree.Sothetotalnumberoffaceshasgonedownby2andthetotalnumbersofedgesandverticeshaveeachgonedownby3.AsVandFareaddedintheformulafortheEulernumber,andEissubtracted,overallwehave

Or,ifyoupreferamorecarefulalgebraicproof,saytheoriginalsubdivisionofS1hasV1vertices,E1edges,andF1facesanddefineV2,E2,F2similarlyforS2.ThenumberofverticesV#,edgesE#,andfacesF#ontheconnectedsumisgivenby

Finally

ThinkingintermsofconnectedsumshelpsusworkouttheEulernumbersofsomemorecomplicatedsurfaces.Weknowthatatorus hasanEulernumberof0.Theconnectedsum

isatoruswithtwoholes(Figure16(b))andwesee

andsimilarlythetoruswiththreeholes, ,hasEulernumber

Infact,wecanseethateverytimewemakeaconnectedsumwith thesurfacegainsonemoreholeandtheEulernumberreducesby2.Sothetoruswithgholes—whichcanbeconsideredas

,theconnectedsumofgcopiesofthetorus —hasEulernumber

Thenumbergofholesinthesurface iscalledthegenusofthesurface.

One-sidedsurfacesAtthispoint,westillcan’tidentifythepeculiarsurfacefromFigure14whichhasanEulernumberof–1.Sofarwe’veonlyconstructedsurfaceswithevenEulernumbersand–1isodd.Infact,withthetori ,we’veonlymethalfthestoryandhalfoftheclosedsurfaces.RecallhowinFigure12wemadeacylinderbygluingtwoedgesofasquare.Wecould,instead,havegluedthosetwosidesusingreversedarrows(Figure17(a)),introducingasingletwist.Sothepointsnearv2one1aregluedtothepointsnearv4one3andthosenearv1one1aregluedtothepointsnearv3one3.ThiswouldhavecreatedaMöbiusstrip,namedafterAugustMöbiuswhodiscovereditin1858.

17.TheMöbiusstrip(a)Asquarewithidentifiededges,(b)RunnersonaMöbiusstrip.

TheMöbiusstripisunusualinonlyhavingoneside—thisisapparentinFigure17(b)astherunnerscovertheentiretyofthestripratherthanjustonesideofitastheywouldifrunningaroundjusttheoutside(orinside)ofacylinder.Oryoucanimaginepaintingtheoutsideofacylinderblackandtheinsidewhite,butshouldyoubeginpaintingaMöbiusstriponecolouryouwouldfindyourselfcoveringtheentirestripinthatcolour.TheMöbiusstripisanexampleofanon-orientablesurface.Likethecylinderitisasurfacewithboundary,butnoteitsboundaryisasinglecircleratherthantwoseparateonesaswiththecylinder.

InFigures18(a)–(d)weseeanorientedloop—hereacircle—movingaroundaMöbiusstrip.ByanorientedloopImeanaloopwithagivensenseofdirection,hereinitially(18(a))appearingasclockwisetothereader.Butasthisloopmovesaroundthestrip(orequivalentlymovesleftinthesquare)weseethatwhenthecirclereturnstoitsoriginalposition(18(d))thatsensehasnowreversedandappearsanti-clockwise.Ifyouarehavingalittletroublevisualizingwhat’shappeningtotheloop,notein18(b)and18(c)howthepointslabelledParegluedtogetherandlikewisetheQs.In18(b)mostoftheloop(ontheleft)lookstobeclockwiserunningfromPtoQ,butastheloopappearsontherightandcontinuesfromQtoPthatsenseisbeginningtoappearasanti-clockwise.

18.MovinganorientedlooparoundaMöbiusstrip.

Anysurfaceonwhichitispossibletoreversethesenseofanorientedloopiscallednon-orientable.Ifitisimpossibletoreversealoop’ssense,thenthesurfaceiscalledorientable.AnysurfacethatcontainsaMöbiusstripisnon-orientableaswecouldjustsendanorientedlooponcearoundthatstriptoreverseitssense.Asurfacewithaninsideandanoutsideisorientable.Toappreciatethis,imaginewalkingaroundtheoutsideofsuchasurface.Lookingdowntoyourfeetonthesurfaceyoucoulddrawacircleinaclockwisemanner.Asyouwanderaroundtheoutsideofthesurfaceyoucanconsistentlytakeyournotionofclockwiseacrossthewholesurface.Thismeans,inparticular,thatthetori ,whichwemetearlierandwhicheachhaveaninsideandoutside,areallexamplesoforientablesurfaces.

ReturningtoFigure17(a),apartlygluedsquaremakingaMöbiusstrip,thereremaintwoungluededgese2ande4.WecouldgluethesetogetherasinFigure19(a),butwhatsurfacewouldwemake?Certainlyanon-orientableoneasitcontainsaMöbiusstrip(theshadedregion).Ifinsteadwemakethissurfacebygluinge2ande4first,wefirstcreateacylinderwithe1ande3asitscircularends.Buttocompletethesurface,ratherthanbringingthosecircularendstogetheraswithatorus,onecircularendhastobegluedbackwardsontotheothercircularend—thisisbecauseofthereversearrowsone1ande3.Figure19(b)showshowwemighttrytodothis;wecouldtakeonecircularendbackintothecylinderandglueittotheotherendfrominside,andthiswaythereversearrowslineupproperly.ThesurfacemadeiscalledaKleinbottle,afterFelixKleinwhofirstdescribeditin1882.Beingnon-orientable,theKleinbottledoesnothaveaninsideandoutside.

19.TheKleinbottleandprojectiveplane(a)AKleinbottle,(b)3DdepictionofaKleinBottle,(c)Aprojectiveplane.

ThereisasubtleproblemwiththeKleinbottleinFigure19(b).Whenwetakethecylinderbackintoitself,somesinglepointsinspaceactuallyrepresenttwodistinctpointsontheKleinbottle.SothisimageisnotaproperrepresentationorembeddingoftheKleinbottlein3D.Infact,itisimpossibletoconstructaKleinbottlein3Dwithoutsuchself-intersectionsasoccurwherethecylindercutsbackintoitself.TherelevantresultdemonstratingthisimpossibilitycanbeviewedasageneralizationoftheJordancurvetheorem.ThattheoremconcernedembeddingcirclesintheplanewithaJordancurvehavinganinsideandanoutside.Inalikemannerwhenaclosedsurfaceisembeddedin3D,thesurfaceagaindividestheremainingspaceintoaninsideandanoutsideandsotheclosedsurfacemustbeorientable.AstheKleinbottleisnon-orientable,itcannotbeembeddedin3D.

However,theKleinbottlecanbeembeddedin4Dandthisisn’ttoohardtoimagineifwetreatthefourthdimensionastime.TheKleinbottleistwo-dimensional(assurfacesare)andsofromthis4DviewpointitisimportanttoconsidertheKleinbottleasonlyexistingforaninstant,acertain‘now’;forittohaveapastorfuturewouldgiveitathirddimension.Sowhenfacedwithbringingthecylinderbackintoitself—whichwouldnormallycauseself-intersections—wecaninsteadmovethatbitofcylindergraduallyintothefuture(thefourthdimension),wheretheremainderoftheKleinbottledoesn’texistandthen,oncethecylinderhaspassedthroughthespaceitspresentselfoccupies,wecangraduallybringthatbitofthecylinderbackintothepresent.Theself-intersectionsnolongeroccur,asthedistinctpointsoftheKleinbottlethatbecamemergedinFigure19(b)insteadsitinthesamepointofspacebutcruciallyatdifferenttimes.

WecanalsodeterminetheEulernumberoftheKleinbottle,againbeingcarefultonotehowedgesandverticesaregluedtogether.Thesquareisouronlyface;e1ande3aregluedtogether,asaree2ande4,makingtworatherthanfouredges;finallyv1isgluedtov2whichisgluedtov4whichisgluedtov3andsowehavejustonevertex,giving ,theEulernumberoftheKleinbottle.Unfortunately,0isalsotheEulernumberofthetorus,soanyhopewemighthavehadthattheEulernumberaloneisinformationenoughtorecognizetheshapeofasurfacewassimplistic.ThetorusandKleinbottlearedifferentsurfaces—theformerisorientable(two-sided),thelatternot—andyettheybothhavethesameEulernumber.

Anotherimportantnon-orientablesurface,whichcanbeformedfromgluingasquare’sedgestogether,istheprojectiveplaneℙ.InFigure19(c)weassigne2ande4reversearrows(incontrastto19(a)).Thesurfaceformedisnon-orientable,asitagaincontainsaMöbiusstrip(theshadedregion),andwecancalculatetheEulernumberasbefore:again and butthistimev1andv3aregluedtogetherandseparatelyv2andv4areglued,sothat .HenceℙhasEulernumber .

TheclassificationtheoremClassificationisanimportantthemeinmathematics.Amathematicaltheoryoftenbeginswithdefinitionsandrulesaboutcertainmathematicalobjectsorstructures(sayfunctionsorcurves)andseekstoproveresultsaboutthemusingthoserules.It’snaturaltosearchforexamplessatisfyingthoserules,preferablyproducingacompletelistorclassificationofsuchobjects.

Wearenowclosetoclassifyingclosedsurfaces.Explicitly,weareseekingtogiveacompletelistofalltheclosedsurfaces,sothateveryclosedsurfaceishomeomorphicto(i.e.topologicallythesameas)oneofthesurfacesonthelist,andthelistcontainsnoduplicates—eachsurfaceonthe

•

•

listcanbeshowntobetopologicallydifferentfromallothersonthelist.

ItturnsoutthattheEulernumbergoesalongwaytoseparatingoutthedifferentsurfaces,butwehaveseenthatthiscannotbethewholestoryasthetorusandKleinbottlehavethesameEulernumberwhilstbeingdifferentsurfaces—thefirstisorientable,thesecondnot.Theonlymissingingredientintheclassificationisthatnotionoforientability.

Sothefirsthalfoftheclassificationtheoremfortwo-sidedsurfacesstates:

Anorientableclosedsurfaceishomeomorphictopreciselyoneofthetori whereThesetoriarenottopologicallythesameasoneanotherastheyhavedifferent

Eulernumbers—theEulernumberof is2–2g.

Asimilarresultholdsforone-sidedclosedsurfaces.Justasthetorus isabuildingblockfortheorientablesurfaces,socantheprojectiveplaneℙbeusedtomakethenon-orientablesurfaces.RecallthattheprojectiveplaneℙhasEulernumber1.Sotheconnectedsumsℙ#ℙandℙ#ℙ#ℙhave

andmoregenerallykcopiesofℙinaconnectedsum,asurfacedenotedℙ#k,hasEulernumber2–k.

Andthesecondhalfoftheclassificationtheoremforone-sidedsurfacesstates:

Anon-orientableclosedsurfaceishomeomorphictopreciselyoneofℙ#kwhereThesesurfacesarenottopologicallythesameastheyhavedifferentEulernumbers—theEulernumberofℙ#kis2–k.

MakingaconnectedsumwithℙisequivalenttosewingaMöbiusstripintothesurface.ℙitselfcanbemadebyintroducingaMöbiusstripintoasphere;todothiswemightmakeatearinthesphereandthen,ratherthangluingthetearbacktogether,wecouldinsteadassignreversearrowstothetwosidesofthetear,thusintroducingaMöbiusstrip.Sothesurfaceℙ#kcanbethoughtofasaspherewithkMöbiusstripssewedin.

Overallthen,theclassificationtheoremsaysthatifweknowtheEulernumberofaclosedsurfaceandwhetheritisone-ortwo-sided,thenweknowitstopologicalshape.Ifyouwerewondering,wheretheKleinbottleisonthislist,weknowitsEulernumbertobe0andweknowittobeone-sided.Theonlysurfaceintheclassificationmatchingthesefactsisthe surfaceℙ#ℙandthisistopologicallythesameastheKleinbottle.Wemightcreateayetmorecomplicatedconnectedsumsuchas whichatfirstglanceisnotonourlist.Thissurfaceisone-sidedanditsEulernumberequals

sotopologicallyit’sthesamesurfaceasℙ#7.AndatlonglastweareabletoidentifythesurfaceweformedinFigure14.ThatsurfacehadEulernumber–1andsothesurfaceisℙ#ℙ#ℙ,thisbeingtheonlysurfaceonourlistwiththatEulernumber.

ComplexnumbersSurfacesareanaturaltwo-dimensionalextensionofone-dimensionalcurveswhichmathematicianshadlongbeeninterestedinbut,historically,surfacesandtheirtopologybecameofparticularimportancebecauseoftheworkof19th-centurymathematicians,mostnotablyBernhardRiemann.

TounderstandRiemann’smotivationforstudyingsurfaces,weneedtotakeabriefforayintothe

worldofcomplexnumbers.Complexnumbershave,atfirstglance,nothingtodowithtopology,buttheneedtointroducethemhereisaconsequenceofthedeepinterconnectednessofmathematics.Inthemid-19thcenturymathematiciansfoundworthwhilereasonstothinkaboutoldermathematicsinnewtopologicalways.Itmightthenseemasthoughtopologywassomehowbornofpracticalnecessityforaddressingtheseolderproblems.HoweverI’dliketosuggestarosierpictureofhowmathematiciansthink:nothingwillputaglintintheeyesofagenerationofmathematicians,anitchtobethinkinghardabouttheessenceofmathematics,somuchasasenseoftherebeingsomethingprofoundjustaroundthecornerandadeeperunderstandingoftheirsubjecttantalizinglybeyondtheirfingertips.Andsoitwastoprove.

Theseso-called‘complex’numbersarose—somewhatuncertainly—fromtheworkofItalianmathematiciansduringtheRenaissance.Foralongtimemathematicianshadbeeninterestedinthesolutionsofpolynomialequations.Theseareequationsinvolvingpowersandmultiplesofanunknownquantity,sayx,suchas

Thisisadegree3equation,thatbeingthehighestpowerofx.Asolutionofanequationisavalueofxwhichmakesbothsidesequal.Wecanseethat solvesthisequationbecause

Youmightcheckthat isasolutionandsois .Andthat’sallofthem!Threesolutions.Otherpolynomials,though,seemtohavenosolutions.Forexample,thedegree2

equation

hasnorealnumbersassolutions.Ifyoutakeapositivenumberxthenitssquarex2isalsopositive(andsocannotequal–1);ifyoutakeanegativenumberthenitssquareisalsopositive;finally

.Sotherearenosolutions.Ifyoupreferamorepictorialapproachthenyoumightdrawthegraphsof and ,andthefactthatthesegraphsdon’tmeet(Figure20(a))isagainanotherwayofshowingthatnonumberxsolvestheequation .Basicallytheproblemisthatnegativenumbersdon’thaverealsquareroots.

20.Thereallineandcomplexplane(a)Graphsofy=x2andy=–1,(b)Therealline,(c)Thecomplexplane.

AndtherethestorymighthaveendedexceptthoseRenaissancemathematiciansfoundgoodreasonsto‘imagine’that doeshavesolutions,denotingasolutionasi.Thismayseemsomewhatludicrousatfirst,butaround1530amethodwasfoundforsolvingdegree3equations.Oneproblemwasthatthismethodnecessitatedcalculationswithsquarerootsofnegativenumbers,evenwhenalltheequation’ssolutionswererealnumbers.Theworthofthenumberibecametrulyapparentwiththeproofofthefundamentaltheoremofalgebrain1799byCarlGauss.Thistheoremshowsallthesolutionsofanypolynomialequationhavetheformwhereaandbarerealnumbers.Forexample,thenumber solvestheequation

asshownbythecalculation

Numbersofthisform, whereaandbarerealnumbersand ,arecalledcomplexnumbersandthefundamentaltheoremofalgebrasaysthatapolynomialofdegreenhas(countingpossiblerepeats)nsolutionsamongstthecomplexnumbers.

Inthesamewaythatrealnumbersarecommonlyrepresentedontherealline(Figure20(b))thecomplexnumberscanberepresentedasaplane,thecomplexplane(Figure20(c)).Acomplexnumbersuchas canthennaturallybeidentifiedwiththepoint(1,2)asshown.Therealnumbersoccupythehorizontalaxis—denoted‘Re’—andtheverticalaxis‘Im’iscalledtheimaginaryaxis.

Complexnumbershavearichtheoryoftheirownwhich,formathematiciansatleast,isreasonenoughtowarranttheirstudy.Youmay,though,besurprisedtofindthatquantumtheory,thephysicaltheorythatsuccessfullymodelssubatomicphysics,isnaturallydescribedusingthelanguageofcomplexnumbersandsophysicists,chemists,andengineersallneedtobewellversedintheuseofcomplexnumbers.

RiemannsurfacesTheintroductionofcomplexnumbersledtoamuchrichertheoryconnectingalgebraandgeometry.InFigure20(a)weseethatthecurves and don’tmeet;iftheydidmeetatapoint(x,y)intherealxy-planethenwe’dhave andnosuchxexists.Butusingcomplexnumberstheydointersectattwopoints,at andat .Thefactthatadegree2curveandadegree1curvemeetin pointsinthiscaseisnotentirelycoincidental.Moregenerallyitisthecasethat,ifproperlycounted,adegreemcurveandadegreencurveintersectin points.Multiplecontactsneedtobecountedproperly—soalinetangentiallymeetingthecurve wouldcountasadoublecontact,sothattherearestill

intersections.Thefinalfinesse,whencountingintersections,istoincludepointsatinfinity.Forexample,twoparallellines—eachdegree1curves—arestilldeemedtomeetatapointatinfinitysothatthereis intersectionasexpected.

Usingrealnumbers,thegraphof isaone-dimensionalcurvelyinginthetwo-dimensional

xy-plane.Inthiscasexandyareeverydayrealnumbersandthecurveconsistsofallpoints(x,x2)wherexisarealnumber.Wemightinsteadconsiderthesameequationwherexandycannowbecomplexnumbers.Againallthepointssatisfying areoftheform(x,x2)butthistimexcanbeanycomplexnumber.Whenusingrealnumbers,theinputxrepresentssomepointofthex-axisandthecorrespondingoutputx2canbeplotteddistancex2abovethepoint(x,0)inthexy-plane(Figure20(a)).However,whenitcomestousingcomplexnumbers,theinput isitselftwo-dimensional.Thex-‘axis’isaversionofthecomplexplane,they-‘axis’asecondversion,andthecomplexxy-‘plane’isinfactfour-dimensional.‘Above’thepoint(x,0)isapoint(x,x2),andtogetherthepoints(x,x2)makeatwo-dimensionalsurfacesituatedinthefour-dimensionalcomplexxy-space.Allthepointssuchas(2,4)thatwereontheoriginalrealcurvearestillpresent,andmakeupacross-sectionofthecomplexsurface;presenttoonowarepointslike(i,–1)and(2+i,3+4i).Ifweseparateoutthesecomplexnumbersintotheirrealandimaginarydimensions,thenwemightinsteadrepresentthesepointsas

andtheirfour-dimensionalnatureisalittleclearer.

Thecurve sitsintherealxy-planeasacurvedversionofthex-axis(Figure20(a));thecurveandaxisaretopologicallythesamewithahomeomorphismbetweenthetwojustpushingeachpoint(x,0)uptothepoint(x,x2).Ifweincludealsothecurve’spointatinfinitybringingtogetherthecurve’s‘ends’thenthecurvetopologicallybecomesacircle.Whenusingcomplexnumbers,thecurve sitsincomplexxy-spaceasacurvedversionofthex-axiswhich,remember,isitselfatwo-dimensionalcomplexplane.Whenweincludethepointatinfinitythisbringstogetherthiscurvedplaneasasphere.(ThisisthereverseprocessofpuncturingaspheretogettheplanethatwemetearlierintheproofofEuler’sformula.)

So,thecomplexversionof ,ifweincludeitspointatinfinity,istopologicallyasphere,asurface;thisiscalledtheRiemannsurfaceof .WemightsimilarlyconsidertheRiemannsurfacesofhigherdegreeequations.

InFigure21(a)wehavetherealcross-sectiondescribedbythedegree3equation.Ontheleftisaloop,andwhenweaddthepointatinfinitytothecurve

ontherightthentopologicallythisrealcross-sectionbecomestwoloops;soitmightnotbesurprisingthatthewholecomplexversion,theRiemannsurface,inthiscaseisatoruswithFigure21(a)justbeingacross-sectionofthattorus.InFigure21(b)wehavearealcross-sectionwithasingularpointwherethecurvecrossesitself.TopologicallythecomplexversionofthiscurveisapinchedtorusasinFigure21(c).

21.VisualizingRiemannsurfaces(a)Graphofy2=x(x–1)(x–2),(b)Graphofy2=x(x–1)2,(c)Apinchedtorus.

Providedtherearenosingularpoints,thenadegreedequationdefinesaRiemannsurfacewhichistopologicallyatoruswithgholes.Thereisaprofoundbuteasilydescribedconnectionbetweenthedegreeofacurve’sequationdandthegenusgofitsRiemannsurface.Thisisgivenbythedegree-genusformulawhichstatesthat

wheregisthegenusoftheRiemannsurfaceanddisthedegreeofthecurve’sequation.Rememberingtheexampleswehavemet,notethat for gives ,asphere,and

for gives ,atorus.Forcurveswithsingularpoints,theformulacanbegeneralizedincludingacorrectiontermforeachsingularity,asshownbyMaxNoetherin1884.

Thisisafirstglimpseatsomeofthedeepconnectionsbetweentopology,algebra,geometry,andcalculusthatwouldbeuncoveredinmathematics.QuotingtheFrenchmathematicianJeanDieudonné,‘Inthehistoryofmathematicsthetwentiethcenturywillremainasthecenturyoftopology.’Thecenturywouldseeafledglingsubject,intuitivelybutinformallyunderstood,goontobecomeoneofthecentralpillarsofmathematics.

Itisworthnotingthatthoseearlytopologists—Möbius,Klein,Riemann—didnotintheirtimehaveavailabletherigorousdefinitionsnecessarytoprovetheirresultstomodernstandards.In1861Möbiusgaveanearlysketchproofoftheclassificationtheoremfororientablesurfaces,andWaltherVonDyckgaveasketchproofforallclosedsurfacesin1888.Butwithouthavinganyformaldefinitionofwhatasurfaceis,theseproofscanatbestbeconsideredincomplete.Thisisnottorelegatesuchproofstothedustbin,nortoconsiderthemsimplywrong,assuchproofsoftencontainmostorallofthecrucialideasofaproof.Somewhatdifferentlyexpressedrigorous

versionsoftheclassificationtheoremwouldbeprovedbyMaxDehnandPoulHeegaardin1907andbyRoyBrahanain1921.

Curvesandsurfacesareone-andtwo-dimensionalexamplesofmanifolds,spacesthatlook‘upclose’liketherealline,theplane,orsomehigherdimensionalequivalent.Itwasn’tuntil1936thatHasslerWhitneygavethemoderndefinitionofamanifold,andprovedanimportanttheoremshowingwhenmanifoldscanbeembeddinginspace(recall,forexample,howtheKleinbottlecannotbemadein3Dspacewithoutself-intersectionsbutcanbemadein4D).Animportantaspectofmoderngeometryconcernsthedifferenttypesofmathematicalstructure—continuous,smooth,complex,metric—thatcanbeputonthesemanifoldsandIwillsayalittlemoreonthisinChapter5whenwediscussdifferentialtopology.

Chapter3Thinkingcontinuously

Givenjustonesentenceforthetask,manytopologistsmightchoosetodescribetheirsubjectasthestudyofcontinuity.Theword‘continuous’appearedafewtimesinChapter1butitwaslefttothereader’sintuitionastoquitewhatthewordentailed.Inmanywaysthisreflectshowmathematiciansusedtoregardcontinuity—historicallyitwasjustconsideredevidentwhatwasmeantby‘continuous’andasmany(butnotall)oftheresultsaboutcontinuousfunctionsthatare‘obvious’alsohappentobetrue,relativelylittleeffortwasspentprovidingfurtherclarity.Arigorousdefinitionofcontinuitydidnotappearuntilthe19thcentury.

Inyoureverydayroutinetherearecontinuousanddiscontinuousfunctionsaroundyou.Forexample,ifyoudrivetowork,thedistanceyouhavetravelledafteracertaintimewillbeacontinuousfunctionoftime—forthisnottobethecasewouldmeanthatatonemomentyourcarwasinacertainplaceonlyforittoimmediatelyafterwardsbeatanotherplacesomedistanceaway.Yourspeedonthejourneywillsimilarlybecontinuous.However,theaccelerationneednotbe;ifyouweresatatrest(sayattrafficlights)theaccelerationwouldbezerobutthenwouldjumptoacertainvalueonceyourfootwasontheaccelerator.ThegraphsinFigure22giveaplausible(ifsimplistic)modelforsomeone’sdrivetowork.

22.Distance,speed,andaccelerationonajourney(a)Distance,(b)Speed,(c)Acceleration.

FromFigure22(b)wecanseethatthecarstopsatt3—wherethespeeds(t)becomeszero—andaftert4increasestothespeedlimit.Thedistancetravelledd(t)inFigure22(a)isacontinuousfunctionoftimet.Historicallythiswouldhavebeenunderstoodasmeaningitsgraphcouldbedrawnwithouttakingpenfrompaper,butwewillseektoprovideafullerunderstanding.Buttheaccelerationfunctiona(t)isnotcontinuousbecauseofthejumpsinthegraphinFigure22(c).Thetimest1,t2,…t6ofdiscontinuityintheaccelerationrelatetothedriver’sfootcomingofftheaccelerator,beingputonthebrake,comingoffthebrake,andthenthepatternrepeatsagain.

Myaiminthischapteristoprovideamorerigoroussenseofjustwhatcontinuityentailsforreal-valuedfunctionsofarealvariable.Thismeanswewillfocusonfunctionshavingasinglenumericalinputandasinglenumericaloutput.

Functions

Theideaofafunctionisacentralonetomathematics,thoughthishasonlybeentruesincearoundthe17thcentury.OnceDescartesandFermatindependentlyintroducedtheideaofCartesiancoordinatesxandytodescribepositioninaplane,acurvecouldjustaseasilybedescribedbyanequationasbyitsgeometry.Forexample,thecurve isaparabola,acurvetheancientGreekswouldhaveinvestigatedsolelyusinggeometry.AsketchofthiscurveisgiveninFigure23(a)—thecurve’sequationgivesaruleforplotting,aboveeachpoint(x,0)ofthex-axis,apoint(x,x2).Notehowcertainalgebraicpropertiesofthefunctionarerepresentedintheshapeandpositionofthecurve—asx2⩾0forallx,thecurveliesentirelyonorabovethex-axis;as

,thecurveissymmetricaboutthey-axis.

23.Examplesofgraphs(a)Graphof ,(b)Graphofy=.

Forsomefunctions,wemightnaturallyhavetolimittheallowedinputs—orwemightchoosetodosoanyway.Forexample,if thenweatleastneedtoensurethatxisnon-zeroas

divisionbyzeroismeaningless;forthefunction ,thenwecannotpermitxtobenegative,asnorealnumberhasanegativesquare(Figure23(b)).

Moregenerally,afunctioncomeswithasetofinputs,knownasthedomain,andthereislikewisethecodomain,asetcontainingtheoutputs.Itisanimportant,ifsubtle,pointtoappreciatethatafunctionisthiswholepackage:thedomain,thecodomain,andtheruleassigningvalues.

SomefirstthoughtsaboutcontinuityLet’sfirsttrytounderstandwhatitmeansforafunction,withrealinputsandoutputs,tobecontinuous.Currentlywesortofintuitivelyknowcontinuitywhenweseeit.Certainly,lookingattwofunctionsinFigures24(a)and24(b),itseemsreasonabletosayf(x)iscontinuousandg(x)isnotcontinuous,andfurtherthatg(x)isdiscontinuousonlyat .(Thefulldisconthegraphshowswherethefunctiontakesitsvalue,sothat .)ButwhatdoesintuitionsayaboutFigure24(c)?Ish(x)continuousornot?Itseemsthat,ifh(x)isdiscontinuous,theonlypointofdiscontinuityis ,butthefunctionoscillatessowildlythere,wemaynowbethinkingthatourintuitiondidn’thavealltheanswers.

24.Continuousanddiscontinuousfunctions(a)y=f (x)=sin x,(b) ,(c)

.

BacktoFigure24(b),whatisitaboutthefunction’sbehaviourat thatmakesusthinkg(x)isdiscontinuous?Forinputxalittlemorethan1,theng(x)hasmuchthesamevalueasg(1);howeverforinputxalittlelessthan1theng(x)isnoticeablydifferentfromg(1).Itisthisjumpinoutputat1thatiscrucialtog(x)beingdiscontinuousat .

Atfirst,wemightbetemptedtothinkthisisbecauseg(1)isdifferentfromthevalueofg(x)achievedimmediatelybeforewegettoxequalling1.Butthereareallsortsofproblemswiththisthinking.First,thereisnorealnumberxthatis‘immediatelybefore’1.Givenanumberlike0.999,closeto1,thenwecanalwaysimproveonthatandsee0.9999isalittlecloser.Orwemightsuggestusing0.999…(wheretheellipsismeansthatthereareinfinitelymanyrecurring9s)butthisisjustanotherdecimalexpansionfor1.Morerigorously,foranyinputx<1then(1+x)/2islessthan1butcloserto1thanxis.Insteadwemightbetemptedtotalkaboutaninputthatisinfinitesimallycloseto1butthen—whateverwemeanbythis—wearenolongertalkingabouttherealnumbersandhavejustreplacedresolvingonedefinitionwithresolvingadifferentone.

Weneedanotherapproachthatcanbecomfortablyexpressedentirelyintermsofrealnumbers.

Thisproblemwasindependentlyresolvedinthe19thcenturybyBernardBolzanoandKarlWeierstrass.Wefeelthatg(x)isdiscontinuousat becauseg(x)isnoticeablydifferentfromg(1)forsomeinputsxnearbyto1.

Thereisstillquiteabitofsubtletyneededtofullycapturewhatthismeans.Inourexample,g(x)hasajumpof1fromoutputvaluesnear2(justbefore )tooutputvaluesnear1(justafter

).Thesizeofthatjumpwasunimportant,thepresenceofanyjumpatallwassufficient.Andthenotionof‘nearbyinputs’shouldnotbeinterpretedasseveralinputsthatareinsomesensecloseto1;ratherwemeanthereareinputsxarbitrarilycloseto1suchthatg(x)isnoticeablydifferentfromg(1).Necessarilythismeansthatwearetalkingaboutinfinitelymanysuchinputsx,notjustseveralx.Bywayofexample,itisenoughtonotethat:

Thisrigorouslyshowsthatg(x)isdiscontinuousat .Thesequenceofinputs0.9,0.99,0.999,0.9999,…getsarbitrarilycloseto1.Whatthismeansis:howeverdemanding‘nearbyto1’isrequiredtobe,thereareinputsfromthissequencethatareatleastthatclose.

Whilstwestillhaven’tquitedefinedjustwhatwemeanbydiscontinuous,wehavemadesomeprogresswithregardtothefunctionh(x)(Figure24(c)).Thisfunctiondoesnotappeartohaveanynoticeable‘jump’inoutputs,butitdoesseemtomeetthedefinition

h(x)isnoticeablydifferentfromh(0)forsomeinputsxarbitrarilynearto0.

Near thefunctionh(x)isvaryingcrazily.Fromthegraphwecanseethatthereareinputsx,arbitrarilycloseto0,where whilstwehave .Itnowseemsclearbyouremergingsenseofcontinuitythath(x)isdiscontinuousat .

AnexampleindetailWestillneedtobecarefulturningthesenascentthoughtsintoarigorousdefinition.We’llconsiderindetailthefunction whichiscontinuousforallinputsx.Iff(x)iscontinuousataninput then—basedonourpreviousthoughts—weneedthat

f(x)isnotnoticeablydifferentfromf(a)forallinputsxsuitablyneartoa.

Takeamomenttoappreciatewhyweneedallinputsxsuitablyneartoatoproducenotnoticeablydifferentoutputsf(x)tof(a).Ifsome—butonlyone—nearbyinputxtoaresultedinnoticeablydifferentoutputsf(x)andf(a),thenwecouldjusttightenournotionof‘suitablynear’toexcludetheprobleminputx.Infact,ifwecanneverget‘suitablynear’withourinputs,thenthismeansthattherewerearbitrarilycloseproblematicinputsxtoawheref(x)wasnoticeablydifferentfromf(a)—so,f(x)wouldbediscontinuousat .

Tobegin,whatdoesitmeanfor tobecontinuousat ?Isittruethat

x2isnotnoticeablydifferentfrom forallinputsxsuitablynearto0?

WetryoutsomevaluesinTable3.

Table3. Sampleinputandoutputvaluesfor

Itseems—admittedlyonlyonthebasisoffivechoicesofx—thatx2iscloserto0thanxisto0,andsomequickalgebrachecksthatsmallnumbersgenerallysquaretosmallernumbers(inmagnitude).Wecannotfindinputsxcloseto0wheretheoutputsx2and0arenoticeablydifferent.

Nowwecanhangsomerigorousmathematicsontheseinitialthoughts:whateverpotential‘noticeabledifference’intheoutputsx2and weconsider,representedbyapositivenumbere,thenthereneedtobe‘suitablyclose’inputsxto0,representedbyapositivenumberd,suchthat

ifinputsxand0differbylessthandthenoutputsx2and0differbylessthane.

Astheoutputshereareclosertooneanotherthantheinputsare—thatis,asx2iscloserto0thanxisto0—thenwecanjustchoosedtoequale.Soifinputsdifferbyeorlesssodotheoutputs.Wehavethenshownthat iscontinuousat .

Whataboutcontinuityatadifferentinput,say ?WecancreateasimilartabletoTable3(seeTable4).

Table4. Moresampleinputandoutputvaluesfor

Thefunction isgrowingmuchmorerapidlyat thanitisat .Achangeofaround0.1intheinputsleadstoachangeintheoutputsofaround200;achangeof0.01intheinputsstillleadstoadifferenceofaround20intheoutputs.Thismayleadyoutothinkthattheoutputsare‘noticeablydifferent’here,butamorecarefulcheckofotherinputswouldshowthattheselargedifferenceshavebeenincrementallyachieved.Allthisisaconsequenceofthefunctionchangingmorerapidlynear ,andwhatneedstighteningisournotionof‘suitablynear’.Asthefunctionisgrowingmorerapidly,smallchangesintheinputwillleadtorelativelylargechanges,butstillinacontinuousfashion.Ifweconsidertheinput,alittlelargerthantheinput1000,thenthedifferenceintheoutputsequals

asd2<dwhend<1.Soashiftininputsbydresultsinashiftofoutputsroughly2000timeslarger.(NotesimilarbehaviourinTable4.)Thisis,initself,notaproblembutitdoesmeanthatifwewanttheoutputstodifferbynomorethanethenweshouldonlyallowtheinputstodifferbynomorethate/2001.Thisstillshowsthecontinuityof at ,wejustneeded

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atightersenseof‘suitablynear’withtheinputsasthefunctionwasgrowingsofast.Forcontinuityatyetlargerinputsthatnotionwouldhavetobecomeyetmorestringent,butwewouldalwaysbeabletofindsomesmallwiggleroomaboutaninputforwhichtheoutputsdon’tdifferbeyondthedesiredamounte.

ArigorousdefinitionPuttingallthisthinkingtogethergivesusarigorousdefinitionofcontinuity.I’dsuggestreadingthedefinitionandseekingtounderstandhowthismeansthatthefunctioninFigure25(a)iscontinuousandtheoneinFigure25(b)isn’t,butifyoufindthegeneralityofthedefinitionandthetechnicallevelofthelanguagedifficultthenmoveontothenextsectiononthepropertiesofcontinuousfunctions.Andbereassured,asittookgenerationsofmathematicianstofinallygetthisdefinitionright,andcurrentandpastgenerationsofmathematicsundergraduatesstillwrestlewithproofsinvolvingthisdefinitionintheiranalysiscourses.

25.Therigorousdefinitionofcontinuousanddiscontinuous(a)Acontinuousfunction,(b)Adiscontinuousfunction.

Formally,then,afunctionwithrealinputsxandrealoutputsf(x)iscontinuousataninputif:

foranypositiveethereissomepositived

suchthatthedifferencebetweentheoutputsf(x)andf(a)islessthane

whenthedifferencebetweentheinputsxandaislessthand.

InFigure25(a),wearefocusingondemonstratingthecontinuityoff(x)atinput .Aparticularchoiceofe>0hasbeenmadeandourtasknowistomakesurethattheoutputsdon’tdifferfromf(a)bymorethanthise.Sotheoutputshavetoremainbelowf(a)+eandabovef(a)–e(asshownonthey-axis).Andthishastohappenforinputsxinsomerangea–d<x<a+d.WecanseefromFigure25(a)thatsomesuchintervalhasbeenfound,asshownonthex-axis—therangeofoutputsonthisintervalareboundedbythedashedlinesandthesefallwithinthepermittedrangefortheoutputs.Toshowcontinuityoff(x)ataninput we’dhavetoshowthatthiscanbedoneforalle>0,howeversmall;toshowcontinuityofthefunctionf(x)we’dhavetodothisforallinputsx.

Thereareseveralimportantpointstonotehere:

werequirethattheoutputscanbeconstrainedinacertainwayiftheinputsareappropriatelyconstrained;weneedtobeabletodothisforallconstraintseintheoutputs;foreachchoiceofewewillneedachoiceofdthatmeetstherequirement;forasmallerchoiceofethendwillusuallyneedtobesmalleraswell;givenapositivee,thenanypositivedthatmeetstherequirementisfine—we’renotlookingforalargestsuchd,say;thefasterthefunctionf(x)ischangingata,thesmallerdwillneedtoberelativetoe.

Afunctionisthensaidtobecontinu