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1、TOPOLOGYWITHOUTTEARS1SIDNEYA.MORRISVersionofJune22,200121cCopyright1985-2001.Nopartofthisbookmaybereproducedbyanyprocesswithoutpriorwrittenpermissionfromtheauthor.2Thisbookisbeingprogressivelyupdatedandexpanded;itisanticipatedthattherewillbeaboutfifteenchaptersinall.Onlythosechapterswhichappeari
2、ncolourhavebeenupdatedsofar.Ifyoudiscoveranyerrorsoryouhavesuggestedimprovementspleasee-mail:Sid.Morris@unisa.edu.auContentsIntroductioniii1TopologicalSpaces11.1Topology.................................21.2OpenSets.................................91.3Finite-ClosedTopology........................
3、.131.4Postscript.................................20Appendix1:InfiniteSets21iiIntroductionTopologyisanimportantandinterestingareaofmathematics,thestudyofwhichwillnotonlyintroduceyoutonewconceptsandtheoremsbutalsoputintocontextoldoneslikecontinuousfunctions.However,tosayjustthisistounderstatethesig
4、nificanceoftopology.Itissofundamentalthatitsinfluenceisevidentinalmosteveryotherbranchofmathematics.Thismakesthestudyoftopologyrelevanttoallwhoaspiretobemathematicianswhethertheirfirstloveis(orwillbe)algebra,analysis,categorytheory,chaos,continuummechanics,dynamics,geometry,industrialmathematics,ma
5、thematicalbiology,mathematicaleconomics,mathematicalfinance,mathematicalmodelling,mathematicalphysics,mathematicsofcommunication,numbertheory,numericalmathematics,operationsresearchorstatistics.Topologicalnotionslikecompactness,connectednessanddensenessareasbasictomathematiciansoftodayassetsandfu
6、nctionsweretothoseoflastcentury.Topologyhasseveraldifferentbranches—generaltopology(alsoknownaspoint-settopology),algebraictopology,differentialtopologyandtopologicalalgebra—thefirst,generaltopology,beingthedoortothestudyoftheothers.Weaiminthisbooktoprovideathoroughgroundingingeneraltopology.Anyone
7、whoconscientiouslystudiesaboutthefirsttenchaptersandsolvesatleasthalfoftheexerciseswillcertainlyhavesuchagrounding.Forthereaderwhohasnotpreviouslystudiedanaxiomaticbranchofmathematicssuchasabstractalgebra,learningtowriteproof
