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1、TOPOLOGYWITHOUTTEARS1SIDNEYA.MORRISVersionofJune22,200121cCopyright1985-2001.Nopartofthisbookmaybereproducedbyanyprocesswithoutpriorwrittenpermissionfromtheauthor.2Thisbookisbeingprogressivelyupdatedandexpanded;itisanticipatedthattherewillbeaboutfifteenchaptersinall.Onlythosechapterswhichappea
2、rincolourhavebeenupdatedsofar.Ifyoudiscoveranyerrorsoryouhavesuggestedimprovementspleasee-mail:Sid.Morris@unisa.edu.auContentsIntroductioniii1TopologicalSpaces11.1Topology.................................21.2OpenSets.................................91.3Finite-ClosedTopology....................
3、.....131.4Postscript.................................20Appendix1:InfiniteSets21iiIntroductionTopologyisanimportantandinterestingareaofmathematics,thestudyofwhichwillnotonlyintroduceyoutonewconceptsandtheoremsbutalsoputintocontextoldoneslikecontinuousfunctions.However,tosayjustthisistounderstate
4、thesignificanceoftopology.Itissofundamentalthatitsinfluenceisevidentinalmosteveryotherbranchofmathematics.Thismakesthestudyoftopologyrelevanttoallwhoaspiretobemathematicianswhethertheirfirstloveis(orwillbe)algebra,analysis,categorytheory,chaos,continuummechanics,dynamics,geometry,industrialmathem
5、atics,mathematicalbiology,mathematicaleconomics,mathematicalfinance,mathematicalmodelling,mathematicalphysics,mathematicsofcommunication,numbertheory,numericalmathematics,operationsresearchorstatistics.Topologicalnotionslikecompactness,connectednessanddensenessareasbasictomathematiciansoftodaya
6、ssetsandfunctionsweretothoseoflastcentury.Topologyhasseveraldifferentbranches—generaltopology(alsoknownaspoint-settopology),algebraictopology,differentialtopologyandtopologicalalgebra—thefirst,generaltopology,beingthedoortothestudyoftheothers.Weaiminthisbooktoprovideathoroughgroundingingeneraltop
7、ology.Anyonewhoconscientiouslystudiesaboutthefirsttenchaptersandsolvesatleasthalfoftheexerciseswillcertainlyhavesuchagrounding.Forthereaderwhohasnotpreviouslystudiedanaxiomaticbranchofmathematicssuchasabstractalgebra,learningtowriteproof
