湖南师范大学-拓扑学-Chapter-1-Set-Theory-and-Logic

Chapter1SetTheoryandLogic1FundamentalConcepts,1.BasicNotationSetsCapitalletters;elementsa,b,cLowercaseLetters;Anelementin:,else;,asubsetAofB,;BcontainsA;apropersubsetAofB:;bothand.2.Theunionandtheintersectionofsets,theemptysetorand,Thesethavingnoelementsiscalledtheemptyset,denotedby.3.Contrapositive,ConverseandNegationGivenastatementoftheform“IfP,thenQ”,itscontrapositiveisdefinedtobethestatement:“IfQisnottrue,thenPisnottrue”.Example.If,thenitscontrapositiv-eisthestatement:If，theitisnottruethat,Givenastatement“PQ”,thestatement“QP”Ifboththestatement“PQ”anditsco-nverse“QP”aretrue,weexpressthisfa-ctbythenotation“PQ”.4.TheDifferenceofTwosetsand.5.RulesofSetTheory,Commutativelaws,.AssociativelawsDistributivelaws,DeMorgansLaws6.CollectionsofsetsGivenaset,thesetofallsubsetsofiscalledthepowersetof,denotedby,7.ArbitaryUnionsandIntersectionforatleastone=x|=x|thereexistsAAsuchthatxA.forevery8.CartesianProductAB=(a,b)|aAandbB,2Functions,RulesofassignmentAruleofassignmentisasubsetrofthecartesianproductCDoftwosets,havi-ngthepropertythateachelementofCappearsasthefirstcoordinateofatmostoneorderedpairbelongingtor.Example.LetC=1,2,3,D=4,5,6.Thenr=(1,4),(2,4),isaruleofassign-ment,but=(1,4),(1,5),(2,4)isnot.,2.FunctionsAfunctionisaruleofassignmentr,togeth-erwithasetBthatcontainstheimageofr.ThedomainAoftherulerisalsocalledthedomainofthefunction;theimagesetofrisalsocalledtheimagesetoff;andthesetBiscalledtherangeoff.3.Restrictionsandcomposites,Iff:ABandif,wedefinetherestrictionofftotobethefunction,mappingintoBwhoseruleis.Itisdenotedbywhichisread“frestrictedto”.Example.Let,4.Injective,SurjectiveandBijectiveDefAfunctionissaidtobeinjective(orone-one)ifforeachpairofdistinctpointsofA,th-eirimagesunderfaredistinct.Itissaidtobesurjectiveoronto,ifeveryelementofBistheimageofsomeelementofAunderthefuncti-onf.Iffisbothinjectiveandsurjective,itissa-idtobebijective(orone-to-onecorresponde-nce).,Example.Letthenfisneitherinjectivenorsurjective,itsrestrictionisinjectivebutnotsurjective,thefunctionobtainedfrom,grestrictingthedomainandchangingthera-ngebychangingtherangeisbothinjectiveandsurjective,thatisbijective,soitshasaninverseAcriterionforshowingthatagivenfunctionfisbijective:,5.ImageandPreimageLetTheniscalledtheimageofunderf,thepreimageofunderf,isdenotedbyisdefinedtobethesubsetofB,satisfyingIff:ABisbijectiveandthehadtwomeanings:thepreimageofunderfandtheimageofunder.,3Relations,Relations.ArelationonasetAisasubsetCoftheCartesian,if,denoteitby.Example.LetPdenotethesetofallpeopleintheworld.Anddefinebytheeq-uationD=(x,y)|xisadescendentofy,2.EquivalenceRelationsandPartitionsAnequivalencerelationonasetAisrelationConAhavingthefollowingthereproperties:(1)Reflexivity.xCxforeveryxinA.(2)Symmetry.IfxCy,thenyCx.(3)Transitivity.IfxCy,yCz,thenxCz.LemmaTwoequivalenceclassesEandareeitherdisjointorequal.,LemmaGivenaequivalencerelationonsetA,thenthecollectionofalltheequivalen-ceclassesdeterminedbysatisfiesandforanyE,EDef.ApartitionofasetAisacollectionofdisjointnonemptysubsetsofAwhoseunionisallofA.Examples.LetA=therealplane(1)PQOP=OQ.,(2)L=thecollectionofallstraightlinesintheplaneparalleltotheline.Lisapartionof.3.OrderRelationsArelationConasetAiscalledanorderrelationifithasthefollowingproperties:(1)Comparability.Forallx,yA,xy,eit-herxCyoryCx.(2)Nonreflexivity.FornoxinAdoestherelationxCxhold.(3)Transitivity.IfxCyandyCz,thenxCz.,Example.A=R,C=(x,y)|xyisanorderrelationtheusualorderrelation.D=(x,y)or(andxy)isalsoanorderrelation.Def.LetXbeaset,beanorderrelationonX,andab，openinterval(a,b)=x|axb,axbaxandxb.If(a,b)=,thenwecallatheimmediatepredecessorofb,callbtheimmediatesuccessorofa.,Def.SupposethatAandBaretwosetswithorderrelationsandrespectively.WesaythatAandBhavethesameordertype.Ifthereisabijectivecorrespondancebetweenthemthatpreservesorder,thatis,iftheseabijectivefunctionf:ABsuchthatExample.andRhasthesameordertype.,Discussthepropertiesoff.(1)(2),y.Example.Thedictionaryorderon.GivenapointP(x,y),Example.,thedictionaryorder.Def.AnorderedsetAissaidtohavetheleastupperboundproperty(thegreatestlowerboundproperty)ifeverynonemptysubsetofAthatisboundedabove(below)hasaleastupperbound(agreatestlowerbound).,Example.(-1,1)=ARintheusualorderhastheleastupperboundproperty.Example.B=(-1,0)0,1)doesnothavetheleastupperboundproperty.,4IntegersandtheRealNumbers,AbinaryoperationonaSetAisafun-tionfmappingAAintoA.3.IntegersARissaidtobeinductiveifitcontains1andifthenx+1A.Letbethe,collectionofallinductivesubsetsofR.ThenthesetofpositiveintegersisdefinedbytheequationTheorem(Well-orderedproperty)Everynonemptysubsetofhasasmallestelement.Theorem(Stronginductionprincipal)Let.Supposethat,Then,Theorem(Archimedeanorderingproperty)hasnoupperboundinR.,5CartesionProducts,6Finitesets,IndexingFunctions,IndexsetDef.,anonemptycollectionofsets.AnAnindexingfunctionforisasurjectivefunct-onffromsomesetJ,calledtheindexset,to.Thecollection,togetherwiththeindexingfunctionfiscalledanindexedfamilyofsets.,2.CartesianProductofindexedfamilyDef.Letmbeapositiveinteger.GivenasetX,wedefineanm-tupleofelementsofXtobefunction.Ifxisam-tuple,itisoftendenotedbyIfisafamilyofsetsindexedwiththeset1,2,m,.TheCartes-ianproductofthisindexedfamily,denotedby,or,isdefinedtobethesetofofallm-tuplesofelementsofXsuchthatforeachi.Def.GivenasetX,an-tupleofelementsofXisafunction.,itisalsocalledasequence,oraninfinitesequenceofelementsofX.Letbeafamilyofsets,indexedwiththepositiveintegers.Thecartesionproductofthisindexedfamilyofsets,denotedbyor=thesetofall-tuples,ofelementsofXst.foreachi.,.Example.IfR=thesetofrealnumbers,theninfinitedimensioneuclideanspace.,3.FinitesetsAnfinitesetAorabijection.(CarA=0orCarA=n.)LemmaLetnbeapositiveinteger.LetAbeaset,.Thenabijectivecorresponde-ncefofthesetAwiththesetabijectivecorrespondencegofthesetA-withtheset1,，n.,TheoremLetBbeapropersubsetofA.LetAbeaset.Supposethatabijectionf:A,.Thenthereexistsnobijectiong:B1,2,n，butif,abijectionh:B1,2,mforsome,7,9,10Countableanduncountablesets,Infinitesets,countablyinfinitesets,uncountablesets.aninfiniteset:notfinite;acountablyinfinitesetA:abijectivecor-respondence;acountablesetA:eitherfiniteorcountablyinfinite;anuncountableset:notcountable.,Examples.(1)Ziscountablyinfinite.(2)iscountablyinfinite.2.Acriterionofcountablesets.TheoremLetB0，ThenBiscountable.asurjectivefn.aninjectivefn.,4.ThecharacteristicsofinfinitesetsTheoremLetAbeaset.Thefollowingsate-mentsareequivalent:(1)aninjectivemap.(2)abijectionofAwithapropersubsetofitself.(3)Aisinfinite.,5.AxiomofchoiceGivenacollectionAofdisjointnonemptysets,asetCconsistingofexactlyoneelementfromeachelementof,thatis,onlyoneelementforallItmeansthatCisobtainedbychoosingoneelementforeachofhesetsin.,Lemma(Existanceofachoicefunction)Givenacollectionofnonemptysets(notnecessarilydisjoint),afunction.s.t.foreach.ThefunctionCiscalledthechoicefunctionforthecollection.,6.Well-orderedsetsDef.Let(A,).IfeverynonemptysubsetofAhasasmallestelement,then(A,)issaidtobewell-ordered.Example11,2Z+，thedictionaryorder,Chasasmallestelement.isawellorderedset.,Example2thedictionaryorder,Example3(Z,)isnotwell-ordered.Thesetofintegersisnotwell-orderedintheusualorder.,7.Constructingwell-orderedsets,Cor.Afiniteorderedsethasonlyonepossibleordertype.,9.Contructingaparticularwell-orderedset.,Def.X:awell-orderedset.GiveniscalledthesectionofXby,Lemma10.2.Thereexistsawell-orderedsetAhavingalargestelements.t.thesectionofAbyisuncountablebuteveryothersectionofAiscountable.,11TheMaximumPrinciple(1hour)ReadingMaterials,Def.Thestrictpartialorderrelation:arelationonAhavingthefollowingtwoproperties:(i)(Nonreflexivity),Therelationaaneverholds(ii)(Transitivity)Ifabandbcac.,Theorem2.Themaximumprinciple:LetAbeaset,beastrictpartialorderonA.ThenthereexistsamaximalsimplyorderedsubsetBofA.,Ch.2TopologicalspacesandContinuousFunctions12TopologicalSpaces(3hours),DefinitionDef.LetXbeaset,beacollectionofsubsetsofX.Ifhasfollowingproperties:,(1)and(2)Ifthen(2)If,then,herenisafinitepositiveinteger.),2.ExamplesExample1LetXbeanyset.ThenisatopologyonX,itiscalledthediscretetopology.isalsoatopologyonX,itiscalledtheindiscretetopology.,(3),.,ThenTiscalledatopologyonX,andthepair(X,T)iscalledatopologicalspace.,Example2Let,Example3Thefinitecomplementtopology.LetXbeaset,then(X,)isatopologyspaceandiscalledthefinitecomplementtopologyonX.,Example4Thecountablecomplement,topology.LetXbeaset,3.Thecomparisonbetweentwotopologiesonthesameset.,Def.LetandbetwotopologiesonagivensetX.Ifthenwesaythatisfiner(bigger)than,oriscoarser(smaller)than.issaidtobecomparablewithifeitheror.,Example4Inexample2,iscomparablewithbutandisnotcomparable.,Example5LetbeafamilyoftopologiesonX.ThenthereisauniquelargesttopologycontainedinallandthereisauniquesmallesttopologyonXcontainingallthecollections.,13BasisforaTopology,Def.LetXbeaset,beacollectionofsubsetsofX.ifhasfollowingtwoproperties:(1)(2)If,Thetopologygeneratedby=?,TheniscalledabasisforatopologyonX.,Example1Letbethecollectionofallcircularregionsintheplane.Thenisabasison,thetopologygeneratedbyis,Example2GivenasetX.isabasisforthediscretetopologyonX.,3.Relationsbetweenagiventopologyanditsbasis.,LemmaLetXbeaset,letbeabasisforatopologyonX.Then(EveryopensetUinXcanbeexpressedasaunionofbasiselements.ButtheexpressionforUisnotunique.),LemmaLetXbeatop.sp.SupposethatisacollectionofopensetsofXs.t.foreachopensetUofXandeachxinU,s.t.,thenisbasisforthetop.ofX.,LemmaLetandbebasisforthetopsandrespectivelyonX.Thenthefollowingareequivalent:,(1)isfinerthan.(2),abasiselements.t.,Example.On,allrectangularregions,generatedby,thebasis,=generatedby,Then,generatedby,=thetop.generatedby,=thetop.generatedby.Then=.,Def.LetSbeacollectionofsubsetsofX.IfThenSiscalledasubbasisforatop.onX.,Example1isasubbasisforthestandardtop.onR.,Example2isasubbasisforatop.onR.whatisthetop.generatedbythesubbasis.,14TheOrderTop.(1.5hours),Def.LetXbeasetwithsimpleorderrelation,assumeXhasmorethanonelement.Letbethecollectionofallsetsofthefollowingtypes:(1)Allopenintervals(a,b)inX.(2)Allintervalsoftheformwhere0isthesmallestelementofX.(3)AllintervalsoftheformwherebisthelargestelementofX.,Thenisabasisforatop.onX.Thetop.generatedbyiscalledtheordertop.,Example.RR,dictionaryorder,theordertopologyisgeneratedbythebasis.,isalsoabasisfortheordertop.onRR,Since,Example.,anorderedsetwithasmallestelement.Theordertoponhediscretetop,open;(1,3)=2open，(n-1,n+1)=n,openforn2.,Example.itsordertop.isthedictionaryorder,hasasmallestelement.,15Theproducttop.,1.Def:LemmaLetandbetopspaces.LetThenisbasisforatop.on.,Def.Thetop.ongeneratedbythebasisiscalledtheproducttop.on.,TheoremIfisabasisforthetopof;thenthecollectionisabasisforthetopof.,2.ProjectionsLetThen,arecalledtheprojectionsofXYontoitsfirstandsecondfactors,respectively.Theorem,aresurjectiveandopen.,TheoremThecollectionisasubbasisfortheproducttop.OnXY.,16TheSupspaceTop.,LemmaLetbeatop.sp.Letthenisatop.onY.,Def.Thetop.oniscalledthesubspacetop.iscalledasubspaceofX.,LemmaIfisabasisforthetop.ofX,thenthecollectionisabasisforthesubspaceonY.,Example.LetRbetopologicalspacewiththestandardtopology.Consideropensubsetsofthesubspace.,Example.thesubspacetop.theordertop.,Example.LetI=0,1.ThedictionaryorderontherestrictiontoofthedictionaryorderonRR.Thedictionaryordertop.onthesubspacetop.onobtainedfromthedictionaryordertop.onRR.,Def.GivenanorderedsetIf(theintervalinX),thenYissaidtobeconvex.,TheoremLetXbeanorderedsetintheordertop.isconvex.Thentheordertop.onYisthesameasthetop.Yinheritsasasubsp.ofX.,17closedSetsandLimitPoints,1ClosetSetsDef.LetXbeatop.sp.,AissaidtobeclosedifX-Aisopen.,Examples(1)and.,(2)X,withfinitecomplementtop.isfinite.(3).,Theorem(Thepropertiesofclosedsets).LetXbeatop.sp.Then(1)andareclosed.(2)Arbitaryintersectionsofclosedsetsareclosed.(3)Finiteunionofclosedsetsareclosed.,TheoremLetYbeasubsp.ofX.s.t.TheoremLetYbeasubsp.ofX.IfAis