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

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1、TopologyCourseLectureNotesAislingMcCluskeyandBrianMcMasterAugust1997Chapter1FundamentalConceptsInthestudyofmetricspaces,weobservedthat:(i)manyoftheconceptscanbedescribedpurelyintermsofopensets,(ii)open-setdescriptionsaresometimessimplerthanmetricdescr

2、iptions,e.g.continuity,(iii)manyresultsabouttheseconceptscanbeprovedusingonlythebasicpropertiesofopensets(namely,thatboththeemptysetandtheun-derlyingsetXareopen,thattheintersectionofanytwoopensetsisagainopenandthattheunionofarbitrarilymanyopensetsisop

3、en).Thispromptsthequestion:Howfarwouldwegetifwestartedwithacollec-tionofsubsetspossessingtheseabove-mentionedpropertiesandproceededtodefineeverythingintermsofthem?1.1DescribingTopologicalSpacesWenotedabovethatmanyimportantresultsinmetricspacescanbeprov

4、edusingonlythebasicpropertiesofopensetsthat²theemptysetandunderlyingsetXarebothopen,²theintersectionofanytwoopensetsisopen,and²unionsofarbitrarilymanyopensetsareopen.1WewillcallanycollectionofsetsonXsatisfyingthesepropertiesatopology.Inthefollowingsec

5、tion,wealsoseektogivealternativewaysofdescribingthisimportantcollectionofsets.1.1.1DefiningTopologicalSpacesDefinition1.1Atopologicalspaceisapair(X;T)consistingofasetXandafamilyTofsubsetsofXsatisfyingthefollowingconditions:(T1);2TandX2T(T2)Tisclosedunde

6、rarbitraryunion(T3)Tisclosedunderfiniteintersection.ThesetXiscalledaspace,theelementsofXarecalledpointsofthespaceandthesubsetsofXbelongingtoTarecalledopeninthespace;thefamilyTofopensubsetsofXisalsocalledatopologyforX.Examples(i)Anymetricspace(X;d)isato

7、pologicalspacewhereTd,thetopologyforXinducedbythemetricd,isdefinedbyagreeingthatGshallbedeclaredasopenwhenevereachxinGiscontainedinanopenballentirelyinG,i.e.;½GµXisopenin(X;Td),8x2G;9rx>0suchthatx2Brx(x)µG:(ii)Thefollowingisaspecialcaseof(i),above.LetR

8、bethesetofrealnumbersandletIbetheusual(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