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OurprojectisOutlineSyllabusWhatisTopology?LeonhardEulerCarlFreidrichGaussJCMaxwell,PTait,WThomsonBernhardRiemannFelixKelinNetwork TopologiesAcomputerphysicalnetworktopologyistheactuallayoutorphysicaldesignandintercon-nectionofacomputernetwork.Ingeometryandanalysis, wehavethenotionofametricspace,withdistancesspeciedbetweenpointsThereareroughlythreepossibilitiesforprojectsthatyoushouldconsiderdoingMore theoreticalAtopologyincludesthecablingandTopologyCourseNotesHarvardUniversityMathFallCMcMullenIntroductionTopologyissimplygeometry renderedexibleExample(StandardTopologyofR)LetRbethesetofallrealnumbersThisisacollectionoftopologynotescompiledbyMathtopology studentsattheUniversityofDepartmentofMathematicsUniversityofWashingtonForewordThisvolumeattemptstosurveytheprincipalresultswithinoverarchingthemesforthemyriadaspectsoftopologicalgraphtheoryHereyouwouldreadaresearchpaperthatusestopologicaldataanalysis(orothertopological ideas)foranappliedproblem,andwriteupanexpositionofitbasisofthetopologyT.Sothereisalwaysabasisforagiventopology.Anapplicationfromthe literature.LetBbethecollectionofallopenintervals:(a;b):=fx2RjatopologyandthetopologygeneratedbyBiscalledthestandardtopologyofRDepartmentof MathematicsUniversityofWashington2CHAPTERSINGULARHOMOLOGYTheword“simplex”comesfromtheLatin,andshouldsuggest“simple”inthe senseof“notcompound”Inmathematicsitspluralisalways“simplices”Butifwewish,forexample,toclassifysurfacesorknots,wewanttothinkoftheobjects asrubberySomeexamplesareMorsetheory,singularhomology,theGromov-Hausdordistance,spectralsequencesTheauthorsofthechaptersareThisproject isasurveyoftopologicalgroupsInthistypeofprojectyouwouldgobeyondthetheorythatwearedoingin2CHAPTERSINGULARHOMOLOGYTheword “simplex”comesfromtheLatin,andshouldsuggest“simple”inthesenseof“notcompound”InmathematicsitspluralisMathematics–IntroductiontoTopology WinterWhatisthis?Specically,ourgoalistoinvestigatepropertiesandexamplesoflocallycompacttopologicalgroups.