topology course lecture notes - a. mccluskey, b. mcmaster

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1、TopologyCourseLectureNotesAislingMcCluskeyandBrianMcMasterAugust1997Chapter1FundamentalConceptsInthestudyofmetricspaces,weobservedthat:(i)manyoftheconceptscanbedescribedpurelyintermsofopensets,(ii)open-setdescriptionsaresometimessimplerthanmetricdescriptions,e.g.continuity,(iii)manyresultsaboutthe

2、seconceptscanbeprovedusingonlythebasicpropertiesofopensets(namely,thatboththeemptysetandtheun-derlyingsetXareopen,thattheintersectionofanytwoopensetsisagainopenandthattheunionofarbitrarilymanyopensetsisopen).Thispromptsthequestion:Howfarwouldwegetifwestartedwithacollec-tionofsubsetspossessingthese

3、above-mentionedpropertiesandproceededtodefineeverythingintermsofthem?1.1DescribingTopologicalSpacesWenotedabovethatmanyimportantresultsinmetricspacescanbeprovedusingonlythebasicpropertiesofopensetsthat²theemptysetandunderlyingsetXarebothopen,²theintersectionofanytwoopensetsisopen,and²unionsofarbitr

4、arilymanyopensetsareopen.1WewillcallanycollectionofsetsonXsatisfyingthesepropertiesatopology.Inthefollowingsection,wealsoseektogivealternativewaysofdescribingthisimportantcollectionofsets.1.1.1DefiningTopologicalSpacesDefinition1.1Atopologicalspaceisapair(X;T)consistingofasetXandafamilyTofsubsetsofX

5、satisfyingthefollowingconditions:(T1);2TandX2T(T2)Tisclosedunderarbitraryunion(T3)Tisclosedunderfiniteintersection.ThesetXiscalledaspace,theelementsofXarecalledpointsofthespaceandthesubsetsofXbelongingtoTarecalledopeninthespace;thefamilyTofopensubsetsofXisalsocalledatopologyforX.Examples(i)Anymetri

6、cspace(X;d)isatopologicalspacewhereTd,thetopologyforXinducedbythemetricd,isdefinedbyagreeingthatGshallbedeclaredasopenwhenevereachxinGiscontainedinanopenballentirelyinG,i.e.;½GµXisopenin(X;Td),8x2G;9rx>0suchthatx2Brx(x)µG:(ii)Thefollowingisaspecialcaseof(i),above.LetRbethesetofrealnumbersandletIbet

7、heusual(metric)topologydefinedbyagreeingthat;½GµXisopenin(R;I)(alternatively,I-open),8x2G;9rx>0suchthat(x¡rx;x+rx)½G:(iii)DefineT0=f;;XgforanysetX—knownasthetrivialoranti-discretetopology.(iv)DefineD=fGµX:GµXg—known