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1、TOPOLOGYWITHOUTTEARS1SIDNEYA.MORRISVersionofJune22,200121cCopyright1985-2001.Nopartofthisbookmaybereproducedbyanyprocesswithoutpriorwrittenpermissionfromtheauthor.2Thisbookisbeingprogressivelyupdatedandexpanded;itisanticipatedthattherewillbeaboutfifteenchaptersinall.Onlythosechapt
2、erswhichappearincolourhavebeenupdatedsofar.Ifyoudiscoveranyerrorsoryouhavesuggestedimprovementspleasee-mail:Sid.Morris@unisa.edu.auContentsIntroductioniii1TopologicalSpaces11.1Topology.................................21.2OpenSets.................................91.3Finite-ClosedTo
3、pology.........................131.4Postscript.................................20Appendix1:InfiniteSets21iiIntroductionTopologyisanimportantandinterestingareaofmathematics,thestudyofwhichwillnotonlyintroduceyoutonewconceptsandtheoremsbutalsoputintocontextoldoneslikecontinuousfuncti
4、ons.However,tosayjustthisistounderstatethesignificanceoftopology.Itissofundamentalthatitsinfluenceisevidentinalmosteveryotherbranchofmathematics.Thismakesthestudyoftopologyrelevanttoallwhoaspiretobemathematicianswhethertheirfirstloveis(orwillbe)algebra,analysis,categorytheory,chaos,c
5、ontinuummechanics,dynamics,geometry,industrialmathematics,mathematicalbiology,mathematicaleconomics,mathematicalfinance,mathematicalmodelling,mathematicalphysics,mathematicsofcommunication,numbertheory,numericalmathematics,operationsresearchorstatistics.Topologicalnotionslikecompac
6、tness,connectednessanddensenessareasbasictomathematiciansoftodayassetsandfunctionsweretothoseoflastcentury.Topologyhasseveraldifferentbranches—generaltopology(alsoknownaspoint-settopology),algebraictopology,differentialtopologyandtopologicalalgebra—thefirst,generaltopology,beingthedo
7、ortothestudyoftheothers.Weaiminthisbooktoprovideathoroughgroundingingeneraltopology.Anyonewhoconscientiouslystudiesaboutthefirsttenchaptersandsolvesatleasthalfoftheexerciseswillcertainlyhavesuchagrounding.Forthereaderwhohasnotpreviouslystudiedanaxiomaticbranchofmathematicssuchasabs
8、tractalgebra,learningtowriteproof
