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

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

2、.continuity,(iii)manyresultsabouttheseconceptscanbeprovedusingonlythebasicpropertiesofopensets(namely,thatboththeemptysetandtheun-derlyingsetXareopen,thattheintersectionofanytwoopensetsisagainopenandthattheunionofarbitrarilymanyopensetsisopen).Thispromptstheques

3、tion:Howfarwouldwegetifwestartedwithacollec-tionofsubsetspossessingtheseabove-mentionedpropertiesandproceededtodefineeverythingintermsofthem?1.1DescribingTopologicalSpacesWenotedabovethatmanyimportantresultsinmetricspacescanbeprovedusingonlythebasicpropertiesofop

4、ensetsthat²theemptysetandunderlyingsetXarebothopen,²theintersectionofanytwoopensetsisopen,and²unionsofarbitrarilymanyopensetsareopen.1WewillcallanycollectionofsetsonXsatisfyingthesepropertiesatopology.Inthefollowingsection,wealsoseektogivealternativewaysofdescri

5、bingthisimportantcollectionofsets.1.1.1DefiningTopologicalSpacesDefinition1.1Atopologicalspaceisapair(X;T)consistingofasetXandafamilyTofsubsetsofXsatisfyingthefollowingconditions:(T1);2TandX2T(T2)Tisclosedunderarbitraryunion(T3)Tisclosedunderfiniteintersection.Thes

6、etXiscalledaspace,theelementsofXarecalledpointsofthespaceandthesubsetsofXbelongingtoTarecalledopeninthespace;thefamilyTofopensubsetsofXisalsocalledatopologyforX.Examples(i)Anymetricspace(X;d)isatopologicalspacewhereTd,thetopologyforXinducedbythemetricd,isdefinedb

7、yagreeingthatGshallbedeclaredasopenwhenevereachxinGiscontainedinanopenballentirelyinG,i.e.;½GµXisopenin(X;Td),8x2G;9rx>0suchthatx2Brx(x)µG:(ii)Thefollowingisaspecialcaseof(i),above.LetRbethesetofrealnumbersandletIbetheusual(metric)topologydefinedbyagreeingthat;½G

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