ThethirdauthorwouldliketoacknowledgethesupportofaNationalUniversityofIrelandTravellingStudentship 686GFaireyPGartsideAMarshIn4theauthorsinvestigatespacesthathaveuniversalsparametrisedbycompac ID: 392411
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Comment.Math.Univ.Carolin.46,4(2005)685{703685CardinalinvariantsofuniversalsGarethFairey,PaulGartside,AndrewMarshAbstract.WeexaminewhenaspaceXhasazerosetuniversalparametrisedbyametris-ablespaceofminimalweightandshowthatthisdependsonthe-weightofXwhenXisperfectlynormal.WealsoshowthatifYparametrisesazerosetuniversalforXthenhL(Xn)hd(Y)foralln2N.Weconstructzerosetuniversalsthathaveniceproperties(suchasseparabilityorccc)inthecasewherethespacehasaK-coarsertopology.ExamplesaregivenincludinganSspacewithzerosetuniversalparametrisedbyanLspace(andviceversa).Keywords:zerosetuniversals,continuousfunctionuniversals,SandLspaces,admissi-bletopology,cardinalinvariants,functionspacesClassi cation:54C30,54C50,54D65,54D80,54E351.IntroductionInthispaperwedealwithcontinuousfunctionuniversalsandzerosetuni-versals.Auniversalwillinsomeappropriatesenseparametriseallobjectsinacertainclass.Morespeci callywecande neacontinuousfunctionuniversalasfollows.GivenaspaceXwesaythataspaceYparametrisesacontinuousfunc-tionuniversalforXviathefunctionFifF:XY!Riscontinuousandforanycontinuousf:X!Rthereexistssomey2YsuchthatF(x;y)=f(x)forallx2X.WewilluseFytodenotethecorrespondingfunctiononX.Notethatiftheyaboveisuniquethen,ine ect,YisthesetofallcontinuousrealvaluedfunctionsonXwithanadmissibletopology(theevaluationmapiscontinuous,see[1]).GivenaspaceXwesaythataspaceYparametrisesazerosetuniversalforXifthereexistsU,azerosetinXYsuchthatforallAXwithAazerosetthereexistsy2YsuchthatUy=fx2X:(x;y)2Ug=A.ThiszerosetUmustbewitnessedbysomecontinuousfunctionF:XY!Randwewillrefertosuchafunctionastheparametrisingfunction.NotethattheparticularcasewhenXisperfectlynormal,andYisthesetofclosedsubsetshasbeenstudiedunderthenamecontinuousperfectnormality([15],[8]). ThethirdauthorwouldliketoacknowledgethesupportofaNationalUniversityofIrelandTravellingStudentship. 686G.Fairey,P.Gartside,A.MarshIn[4]theauthorsinvestigatespacesthathaveuniversalsparametrisedbycom-pactorLindelofspaces.Inthispaperwetackleanumberofotherquestionsregardingtheparametrisationofcontinuousfunctionuniversalsandzerosetuni-versals.FirstofallweinvestigatethecasewhereXhasazerosetuniversalparametrisedbyametrisablespaceofminimalweight.Parametrisationbymetrisablespacesistrivialaswecantakeasucientlylargesetwiththediscretetopology.WeshowthatwecanparametriseazerosetuniversalforaspaceXbyametrisablespaceYwherew(Y)=w(X)whenthe-Z-weightofXequalstheweightofX.The-Z-weightisde nedtobetheminimalsizeofacollectionBofopensubsetsofXsuchthateverycozerosubsetisthecountableunionofelementsfromB.ExamininghowthehereditarypropertiesoftheparametrisingspaceYboundthehereditarypropertiesofXleadsustotheinequalityhL(Xn)hd(Y)foralln2!.ThisismorethanwegetwhenYparametrisesanopenuniversalforX(see[7]).Weconstructanumberofexamplesincludingahereditarilyseparable,hereditarilyLindelofspaceXwithnw(X)=@1suchthatXhasacontinuousfunctionuniversalparametrisedbyaspaceYwithhL(Y!)=hd(Y!)=!.Thishasapplicationstoadmissibletopologies.Inthe nalsectionwelookatwhentheparametrisingspaceisseparable,cccorhascalibre!1.FirstwedescribehowtoconstructacontinuousfunctionuniversalforaspaceXwhenXhasaK-coarsertopology(acoarsertopology suchthateachpointofXhasaneighbourhoodbaseconsistingof -compactsets).Thisgeneralconstructionwillallowustoconstructcontinuousfunctionuniversalswith`nice'propertiesonceweknowthatthespaceinquestionhasaK-coarsertopologywiththeappropriateproperties.AsanexampleweshowthattheSorgenfreylinehasacontinuousfunctionuniversalparametrisedbyaseparablespace.Wewillusethefollowingtheoremfrequently.Foraproofsee[4].Theorem1.LetXbeTychono .IfYparametrisesazerosetuniversalforXthensomesubspaceofY!parametrisesacontinuousfunctionuniversalforX.2.ParametrisationbymetrisablespacesofminimalweightInthissectionweexaminethesituationwhereaspaceXhasazerosetuniversalorcontinuousfunctionuniversalparametrisedbyametrisablespaceofminimalweight.TheweightoftheparametrisingspaceisanupperboundfortheweightofthespaceXsotheminimalweightoftheparametrisingspaceistheweightofX.Thekeynotionisthatofthe-weightofX:theminimalsizeofabaseforXsuchthateveryopensetinXisthecountableunionofelementsfromthebase.WebeginbydealingwiththecasewhenXisperfectlynormal.WeuseD()todenotethediscretespaceofsize. Cardinalinvariantsofuniversals687Theorem2.LetXbeaperfectlynormalspacewithw(X)=.Thenthefollowingareequivalent:(i)XhasazerosetuniversalparametrisedbyD()!,(ii)Xhasazerosetuniversalparametrisedbya rstcountablespaceofweight,(iii)Xhas-weight,(iv)Xhasacontinuousfunctionuniversalparametrisedbyametrisablespaceofweight.Proof:LetXbeaperfectlynormalspacewithw(X)=.Clearly(i)implies(ii)and(iv)implies(ii).(ii)implies(iii):LetXhaveazerosetuniversalparametrisedbyYandletF:XY!Rbetherelevantparametrisingfunction.LetU=F 1(Rn0).AssumethatYis rstcountableandthatfC : 2gisabasisforY.Foreach 2wede neB =SfU:Uisopen,UC Ug.LetB=fB : 2g.WeshowthatBisa-basis.FixUXsuchthatUisopen.SinceXisperfectlynormalthenUisinfactacozeroset,sothereissomey2YsuchthatU=Uy.LetfC n:n2!gbeacountablebasisaty.ThenU=SfB n:n2!g.(iii)implies(i):LetB=fB:2gbea-baseforX.AsXisperfectlynormalwecanassumethateachB2Bisacozerosetaswitnessedbyf:X!I.Forall 2D()!let ndenotethen'thelementofthesequence .Wede neF:XD()!!RbyF(x; )=1Xn=0f n(x) 2n+1forall 2D()!andx2X.Wemustshowthat(a)Fiscontinuousand(b)thatforallUopeninXthereissome 2D()!suchthat(F ) 1(Rnf0g)=U.Toshow(b)we xUopeninX.NowsinceBisa-basewecan nd 2D()!suchthatSfB n:n2!g=U.Thenforallx2XweknowthatF(x; )=0ifandonlyifx=2SfB n:n2!g=U.Toshow(a)we xx2X, 2D()!andUopensuchthatF(x; )2U.FindN2!suchthat(F(x; ) 2 N;F(x; )+2 N)U.Foreachj2!suchthatjNde neUj=f 1 j(f j(x) 2 N 1;f j(x)+2 N 1).LetV=TNj=0UjandW=QNj=0f jgQ1j=N+1D().If(x0; )2VWthenjF(x; ) F(x0; )j1Xj=0jf j(x) f j(x)j 2j+1=NXj=0jf j(x) f j(x)j 2j+1+1Xj=N+1jf j(x) f j(x)j 2j+1 688G.Fairey,P.Gartside,A.MarshNXj=02 N 12 j 1+1Xj=N+12 j 11 2N:ThisshowsthatF(x0; )2Uandsowearedone.(i)implies(iv):WeknowthatifaspaceYparametrisesazerosetuniversalforXthenbyTheorem1somesubspaceofY!parametrisesacontinuousfunctionuniversalforX.Sincemetrisabilityandweightarecountablyproductiveandhereditarywearedone.IntheclassofTychono spacesthecrucialpropertyis-Z-weight:theminimalsizeofacollectionBofopensubsetsofXsuchthateverycozerosubsetisthecountableunionofelementsfromB.ThefollowingversionofTheorem2canbeprovedinmuchthesameway.Theorem3.LetXbeTychono withw(X)=.Thefollowingareequivalent:(i)Xhasacontinuousfunctionuniversalparametrisedbyametrisablespaceofweight,(ii)Xhasazerosetuniversalparametrisedbya rstcountablespaceofweight,(iii)Xhas-Z-weight.Corollary4.EveryLindelofT3spacehasacontinuousfunctionuniversalpara-metrisedbyametrisablespaceofminimalpossibleweight.In[5]itisshownthatifXisaT3LindelofspacecontaininganuncountablediscretespacethennometrisablespaceofminimalweightparametrisesanopenuniversalforX.Sowegetthefollowing.Corollary5.ThereexistsaTychono spaceforwhich-Z-weightisstrictlylessthan-weight.Wenowexaminethespecialcasewherethespaceinquestionisthediscretespaceofsize.ItisclearfromTheorem2thatD()hasacontinuousfunctionuniversalparametrisedbyametrisablespaceofweightifandonlyifthereisasubcollectionCofP()ofsizesuchthateverysubsetofistheunionofcountablymanyelementsofC.Wewillcallsuchacardinala-cardinal.Weknowthefollowingabout-cardinals(see[5]).Theorem6.Letbeacardinal.(i)Ifisa-cardinalthenhascountableco nality.(ii)Ifisastronglimitandcf()=!thenisa-cardinal.(iii)If(GCH)holdsandisa-cardinalthenisastronglimit.(iv)Itisconsistentthat@!isa-cardinalandthat2@0=@!+1.AlsoconnectedwiththequestionofparametrisationofD()isthenotionofsupermetrisability:aspaceXofweightissaidtobesupermetrisableifandonlythereisa nermetrisabletopologyonXofweight. Cardinalinvariantsofuniversals689Theorem7.Thefollowingareequivalent:(i)isa-cardinal,(ii)allspacesofweighthave-weight,(iii)allTychono spacesofweighthaveacontinuousfunctionuniversalparametrisedbyametrisablespaceofweight,(iv)Rissupermetrisable,(v)allTychono spacesofweightaresupermetrisable,(vi)D()hasazerosetuniversalandacontinuousfunctionuniversalpara-metrisedbyametrisablespaceofweight.Proof:(i)implies(ii):Letbea-cardinalandletCP()besuchthateverysubsetofisacountableunionofelementsfromC.LetB=fB : 2gbeabasisforaspaceX.ThenB0=fSfB : 2Cg:C2Cgisa-basisforX.(ii)implies(iii):ThisisshowninTheorem3.(iii)implies(iv):NotethatthesetofcontinuousfunctionsonthediscretespaceofsizeissimplyR.NowthereissomemetrisablespaceYofweightthatparametrisesacontinuousfunctionuniversalforD()(viathefunctionF).Foreachfunctionf2Rchooseyf2YsuchthatF(x;yf)=f(x)forallx2.Thespacefyf:f2RgwitnessesthatRissupermetrisable.(iv)implies(v):ThisfollowsfromthefactthateveryTychono spaceofweightembedsinR.(v)implies(vi):Let bea nermetrictopologyonRofweight.Then(R; )parametrisesacontinuousfunctionuniversalforD()viatheevaluationmap.(vi)implies(i):ThisfollowsfromTheorem2.3.HereditarypropertiesAssumethatYparametrisesazerosetuniversalforX.WeexaminehowcardinalinvariantsofYboundcardinalinvariantsofX.Inparticularweareinterestedinhereditarypropertiessuchasthehereditarydensity.Thecaseisverymuchthesameasforopenuniversalsstudiedin[7].HoweverinthecasewhereYishereditarilyseparablewewillshowthatinfactXnishereditarilyLindelofforalln2!.WhendealingwithopenuniversalweonlygethL(X)hd(Y).Theorem8.LetXbeTychono andassumethatXhasazerosetuniversalparametrisedbyY.Thenthefollowingaretrue:w(X)nw(Y);hd(X)hL(Y);hc(X)hc(Y);hL(Xn)hd(Y)foralln2N:Proof:The rstthreestatementscanbeprovedbyminormodi cationsoftheproofsin[7].WewillprovethathL(Xn)hd(Y).We rstprovethefollowing.If 690G.Fairey,P.Gartside,A.MarshXisTychono andhaszerosetuniversalparametrisedbyYsuchthatd(Y),then9ZRanda1{1continuoussurjectionG:X!Z.Proofofclaim:WemustshowthatifDisadensesubsetofYsuchthatjDj=,thenfFd:d2DgseparatespointsinX.Fromthis,wecande neG:X!RDbyG(x)(d)=F(x;d),thentakeZ=ran(G).Letx1andx2betwodistinctpointsinX.Thereforethereisy2YsuchthatF(x1;y)=0andF(x2;y)=0,andhencedisjointopensetsU1andU2suchthatF(x1;y)2U1andF(x2;y)2U2.BycontinuityofF,thereareopensetsV1andV2inY,bothcontainingysuchthatfxigViF 1(Ui)fori=1;2.Hencefory02V1\V2,weseethatF(x1;y0)=F(x2;y0);also,sinceV1\V2isopeninY,itmustmeeteverydenseset,sothereissuchay0withinanydenseset.Proofofmainresult:Let=hd(Y).Thenfromtheabovecondition,weknow rstlythathL(X).Now,foracontradiction,wesupposethat,forsomen2N(wepicktheleastsuch),fx : +gisaright-separatedsubsetofXn,witnessedbythebasicopensetsfQni=1Vi : g.ByProposition21of[7],thiscanbedoneinamoresymmetricway,sothatfor ; +,wheneverx 2Qni=1Vi forsome2Sn(wherethegroupactionpermutesco-ordinates),then .Also,wecanassumethatallpointsareo thediagonal,sincehL(X)(andthediagonalishomeomorphictoX).Infact,wecanusethislastfacttoguaranteethateachpointx hasnotwoco-ordinatesequal.NowthediagonalofX,denotedX,ispreciselythesetf(x;y)2XX:G(x)=G(y)g,whereGisasde nedpreviously.Havingde nedS=f(i;j):1ijng,thesubsetcorrespondingtothediagonalinXiXjisT Tm!f(y;z)2XiXj:jG(y) G(z) j1=mg.Hence,de ningW( :!+m)=f(y;z)2XX:jG(y) G(z) j1=mg,weseethatfx2Xn:8(i;j)2S;x(i)=x(j)g=\(i;j)2Sfx2Xn:(x(i);x(j))2Xg=\(i;j)2S\ fx2Xn:(x(i);x(j))2W g=\ \(i;j)2Sfx2Xn:(x(i);x(j))2W g=\ W0 whereW0 isthebasicopensetfx2Xn:8(i;j)2S(x(i);x(j))2W g.Bythepigeon-holeprinciple,theremustbeonesuchW0 whichcontainsthediagonal,butwhichmisseseveryx (takingasub-familyofx sifnecessary).Also,wecanre neeachoftheopensetsVi sothatx 2Qni=1Vi still,butnow,(Vi )nW0 .SinceXisTychono ,wecanevenmakefVi :1ing Cardinalinvariantsofuniversals691pairwisedisjointcozerosets.Therefore,forall +,thereissomey inYsuchthatSni=1Vi =fx2X:F(x;y )=0g.Hence,thereareopensubsetsofY,fU : +g,sothat,foreach +,y 2U andfx (i):1ingU F 1(Rnf0g).Asaresult,whenevery 2U ,thenfori=1;:::;n,x (i)2Sni=1Vi .Inotherwords,thereissome2Snsuchthatx 2Qni=1Vi .Butthismeansthat ,sofy : +gisleft-separated,hencehd(Y)+whichcontradictsouroriginalhypothesis.4.SspacesandLspacesInthissectionweconstructthreerelatedexamplesof`bad'spaceswithcontin-uousfunctionuniversalsparametrisedbya`good'space.Forinstance,thereisanonmetrisablespacewithacontinuousfunctionuniversalparametrisedbyaspacewhosecountablepowerisbothhereditarilyLindelofandhereditarilyseparable.ThespacesareconstructedfromtheinteractionoftheBairemetrictopologyona xedsubset,A,of!!withvariousordersonA.Moreparticularly,givenapartialorderonA,then(asinTodorcevic[14])wede netheintersectiontopologyA[]fromacountablelocalbaseateacha2AmadeupofthesetsB(n;a)forn!whereB(n;a)=fb2A:(ba)^(an=bn)g.Thetwoordersthatweconsiderhereare:whichisde nedcomponent-wise(i.e.abpreciselyif,foralln,anbn),andthelexicographicorderL(whereaLbpreciselyif,atthesmallestnsuchthatan=bn,thenanbn).Assumingthatb=!1,wecan ndasubsetAof!!withordertype!1under(wherehanin!hbnin!preciselyifthereissomeN!suchthat,forallnN,anbn).Todorcevichasshownthat,inthiscase,variousintersectiontopologiesonAbehaveasfollows:([14,Theorem0.6])A[]isastrongSspace(i.e.thecountablepowerofA[]ishereditarilyseparable,butA[]itselffailstobeLindelof).([14,Theorem0.6])A[]isastrongLspace(i.e.itscountablepowerishere-ditarilyLindelof,butisitselfnon-separable).([14,Theorem3.0])A[L]andA[L]areeachhomeomorphictoasubspaceoftheSorgenfreylinesuchthatthecountablepowerofeachishereditarilyseparableandhereditarilyLindelof.Neitherspace,however,ismetrisableaseachfailstohaveacountablebase.Onceweknowthesebasicproperties,wecanusethem,alongwiththede nitionofthelocalbaseatapointtoproducethefollowingthreeexamples:Example9(b=@1).ThereisanLspacewithazerosetuniversalparametrisedbyastrongSspace.Hencethereisanon-hereditarilyseparablespacewithacontinuousfunctionuniversalparametrisedbyaspacewhosecountablepowerishereditarilyseparable.Proof:The`hence'partfollowsfromthe rstassertionandTheorem1. 692G.Fairey,P.Gartside,A.MarshLetX=A[]andY=(A[]!)!.WeshallshowthatXhasazerosetuniversalparametrisedbyY.SinceXisanLspaceandhencehereditarilyLindelofandregular,thenitmustindeedbeperfectlynormal.Therefore,everyopensetisacozeroset,andfromacoverbybasicopensetswecan ndacountablesub-cover.Inotherwords,thecozerosetsinXarepreciselythoseoftheformSfB(mn;an):n!gwhereeachmn!andan2A.Thus,wehaveawayofcodingupthezerosetsofXbypointsinY,whichweusetode neafunctionF:XY!2!byF(x;han;mnin!)=hbnin!wherebn=1ifx2B(mn;an)andbn=0otherwise.Now,F(x;han;mnin!)=h0ipreciselyif,foreachn,x=2B(mn;an),i.e.x=2SfB(mn;an):n!g.Since,withthecanonicalembeddingoftheCantorsetwithin[0;1](wherehbnin!mapston!2bn=3(n+1)),onlyh0imapsto0,itwillsucetoshowthatthisfunctionFiscontinuous.Thissimpli estocheckingcontinuitywithrespecttoeachofthetwotypesofsub-basicopensetsin2!:ForU(m;0)=fhbnin!:bm=0g,welet(x;han;mnin!)2F 1(U(m;0)).Hence,equivalently,x=2B(mm;am)=fb2A:bam^ammm=bmmg.Therefore,oneoftwocaseswillapply.Firstly,ifxmm=ammm,weletn=mm.Otherwise,xmm=ammmandx6am,sothereissomen&#x-1.6;鎕mmsuchthatx(n 1)am(n 1),inwhichcasexn=amn.Thus,ify2B(n;x)andhci;riii!issuchthatcm2B(n;am)andrm=mm,thenyn=xn=amn=cmn.Inthesecondcase,itmustalsobethecasethaty(n)cm(n)andymm=cmmm,soinbothcases,(y;hci;riii!)2F 1(U(m;0))whichisthereforeopen.ForU(m;1)=fhbnin!:bm=1g,welet(x;han;mnin!)2F 1(U(m;1)).Hence,equivalently,x2B(mm;am)=fb2A:bam^ammm=bmmg,so,inotherwords,xamandxmm=ammm.Now,ify2B(mm;x)andcm2B(mm;am),thenyxamcmandymm=xmm=ammm=cmmm,soy2B(mm;cm)whichisenoughtoguaranteethatF 1(U(m;1))isopentoo.Since,toprovecontinuityofafunction,itsucestoshowthattheinverseimageofeachopensetinasub-basisisopen,wehaveestablishedthatFisacontinuousfunction,soYparametrisesazerosetuniversalforX,asrequired.Example10(b=@1).ThereisanSspacewithazerosetuniversalparametrisedbyastrongLspace.Hencethereisanon-hereditarilyLindelofspacewithcontinuousfunctionuni-versalparametrisedbyaspacewithhereditarilyLindelofpower.Proof:The`hence'partfollowsfromthe rstassertionandTheorem1.LetX=A[]andY=2!(A[]!)!.WeshallshowthatXhasazerosetuniversalparametrisedbyY.First,weshowthatXisperfectlynormal,byprovingthatanyopensubsetisacozeroset.Usingthatinformation,wecreatea Cardinalinvariantsofuniversals693closed-setuniversal,asinGartsideandLo[7],andprovethatitis,infact,alsoazerosetuniversal.Let(Z; )besuchthatZRand is nerthantheEuclideantopologyonZ.Wesaythat(Z; )isa\Kunenline"-typespaceif,foreverysubsetSofX,j Sdn S j@0,wheredrepresentstheEuclideanmetrictopology.Todorcevic[14,Chapter2]constructsatopologyA[H]onAof\Kunenline"-typewhichisalocallycompactstrongSspace.Thistopologyisbasedon,andhas nertopologythan,A[],butheobserves(onpage24of[14]):\Notethat:::wecouldhaveaddedtheconditionabinthisde nitionofHinsteadofab.Butsincewedon'thaveauseforthis,wekeepthede nitionasitis."Ifthede nitionforA[H]ischangedinthisway,thenthetopologyofA[]issandwichedbetweenthoseofA[H]andtheEuclideantopologyonA(consideredasasubspaceoftheirrationals),whichshowsthatA[]isa\Kunenline"-typespace.Hence,everyopensubsetofXistheunionofaEuclidean-typeopensetwithacountableunionofA[]-typebasicopensets.Now,Euclidean-typeopensetsarecozerosets,asthetopologyis nerthantheEuclideantopology,andbasicopensetsareclopensets,socountableunionsofbasicopensetsarecozerosetsalso,sothereforeanyopensetinA[]isacozeroset.Asshownin[4],thereisazerosetuniversalfortheEuclideanzerosets,givenviathenon-negativecontinuousfunctionF1:X2!!R.WeusethesamemethodasinExample9toproduceazerosetuniversalforthecount-ableunionsofbasicopensets.Wede neafunctionF2:X(A[]!)!byF2(x;han;mnin!)=Pn!2bn=3n+1wherebn=1ifx2B(mn;an)andbn=0otherwise.Now,F2(x;han;mnin!)=0preciselyif,foreachn,x=2B(mn;an),i.e.x=2SfB(mn;an):n!g,sothateachzerosetwhichisthecomplementofacountableunionofbasicopensetsisparametrisedbythisfunction.TheproofthatthisfunctionF2iscontinuousfollowsthesamepatternastheproofthatthefunctionFinExample9iscontinuous:ForU(m;0)=fhbnin!:bm=0g,welet(x;han;mnin!)2F 12(U(m;0)).Hence,equivalently,x=2B(mm;am)=fb2A:bam^ammm=bmmg.Therefore,oneoftwocaseswillapply.Firstly,ifxmm=ammm,weletn=mm.Otherwise,xmm=ammmandx6am,sothereissomen&#x-1.6;鎕mmsuchthatx(n 1)&#x-1.6;鎕am(n 1),inwhichcasexn=amn.Thus,ify2B(n;x)andhci;riii!issuchthatcm2B(n;am)andrm=mm,thenyn=xn=amn=cmn.Inthesecondcase,itmustalsobethecasethaty(n)&#x-1.6;镈cm(n)andymm=cmmm,soinbothcases,(y;hci;riii!)2F 1(U(m;0))whichisseentobeanopenset.ForU(m;1)=fhbnin!:bm=1g,welet(x;han;mnin!)2F 1(U(m;1)).Hence,equivalently,x2B(mm;am)=fb2A:bam^ammm=bmmg,so,inotherwords,xamandxmm=ammm.Now,ify2B(mm;x)andcm2B(mm;am),thenyxamcmandymm=xmm=ammm= 694G.Fairey,P.Gartside,A.Marshcmmm,soy2B(mm;cm)whichisenoughtoguaranteethatF 1(U(m;1))isopentoo.Thus,bothF1andF2arecontinuousfunctions,sode ningF:XY!RbyF(x;(hinin!;han;mnin!))=F1(x;hinin!)+F2(x;han;mnin!)willalsogiveacontinuousfunction.Asde ned,F(x;(hinin!;han;mnin!))=0pre-ciselywhenF1(x;hinin!)=0andF2(x;han;mnin!)=0,soeveryopensetisthecozerosetcorrespondingtoFyforsomey2Y,astheunionofaEuclideancozerosetwithacountableunionofbasicopensets.Example11(b=@1).Thereisahereditarilyseparable,hereditarilyLindelofnon-metrisablespaceXwithnw(X)=@1,whichhasazerosetuniversalpara-metrisedbyaspacewhichisbothhereditarilyseparableandhereditarilyLindelof.HencethereisaspacewithuncountablenetweightdespitehavingacontinuousfunctionuniversalparametrisedbyaspacewhosecountablepowerishereditarilyseparableandhereditarilyLindelof.Proof:The`hence'partfollowsfromthe rstassertionandTheorem1.LetX=A[L]andY=(A[L]!)!.WeshallshowthatXhasazerosetuniversalparametrisedbyY.Now,sinceXishereditarilyLindelof,weknow,again,thatthecozerosetsinXarepreciselytheopensets,andthat,moreover,eachofthesesetsisacountableunionofbasicopensets.Hence,wecande neFinasimilarfashiontoourpreviousexample,byF(x;han;mnin!)=hbnin!,wherebn=1ifx2BL(mn;an)andbn=0otherwise.Withthisde nition,itisclearthatFhasallproperties(savecontinuity)thatareneeded.Itjustremainstocheckcontinuity.Thissimpli estocheckingcontinuitywithrespecttoeachofthetwotypesofsub-basicopensetsin2!:ForU(m;0)=fhbnin!:bm=0g,welet(x;han;mnin!)2F 1(U(m;0)).Hence,equivalently,x=2BL(mm;am)=fb2A:bLam^ammm=bmmg.Therefore,oneoftwocaseswillapply.Firstly,ifxmm=ammm,weletn=mm.Otherwise,xmm=ammmandx6Lam,soxLam,inwhichcase,picktheleastn&#x-5.1;䡣mmforwhichxn=amn.Thus,ify2BL(n;x)andhci;riii!issuchthatcm2BL(n;am)andrm=mm,thenyn=xn=amn=cmn.Inthesecondcase,itmustalsobethecasethatyLcmandymm=cmmm,soinbothcases,(y;hci;riii!)2F 1(U(m;0))whichisseentobeanopenset.ForU(m;1)=fhbnin!:bm=1g,welet(x;han;mnin!)2F 1(U(m;1)).Hence,equivalently,x2BL(mm;am)=fb2A:bLam^ammm=bmmg,so,inotherwords,xLamandxmm=ammm.Now,ify2BL(mm;x)andcm2BL(mm;am),thenyLxLamLcmandymm=xmm=ammm=cmmm,soy2BL(mm;cm)whichisenoughtoguaranteethatF 1(U(m;1))isopentoo.Hence,bythestandardargumentfromasub-basisoftherangespace,Fiscon- Cardinalinvariantsofuniversals695tinuous,soshowsthatYparametrisesazerosetuniversalforXasrequired.ObservethatthislastexampleisanuncountablesubspaceoftheSorgenfreyline.AlsonotethattheseexamplesalsogiverisetoSandLadmissibletopologies.5.Separabilityandchainconditions5.1Sucientconditions.Wewill ndsomesucientconditionsforaspacetohaveacontinuousfunctionuniversalparametrisedbyaseparablespace,acccspaceoraspacewithcalibre!1.TheserelyontheideaofaK-coarsertopologyonaspace.De nition12.Let ;betwotopologiesonasetXwith .Wesaythat isaK-coarsertopologyif(X;)hasaneighbourhoodbasisconsistingof -compactneighbourhoods.TheexistenceofaK-coarsertopology onaspace(X;)willallowustoconstructacontinuousfunctionuniversalfor(X;)byre ningthetopologyonCk(X)withoutadding\toomany"opensets.Fixaspace(X;).LetU=f(r;q):r;q2Q;rqgandUQ=U[ffqg:q2Qg.FixC=hC0;:::;CniwhereeachCiXandU=hU0;:::;UniwhereeachUiR.De neW0(C;U)=ff2RX:8in(f[Ci]Ui)gandW(C;U)=ff2C(X):8in(f[Ci]Ui)g.IfBP(R)and ;aretopologiesonXwede nethespaceCk ((X;);B)tohaveasitsunderlyingsetC(X;)andabasisS=fW(C;U):C2P(X)!;U2B!;jCj=jUj;8C2C(Cis {compact)g:ForanysetAthesetA!isthecollectionofall nitepartialfunctionsfrom!intoXwhosedomainconsistsofsomeinitialsegmentof!.NotethatCk((X;);U)issimplythespaceCk(X;).Let beaK-coarsertopologyon(X;).ThespaceCk ((X;);UQ)para-metrisesacontinuousfunctionuniversalfor(X;)viatheevaluationmap.InadditionthisspaceisT2and0-dimensional,andsothespaceisTychono .Al-thoughitmaybeeasiertoworkwiththespaceCk ((X;);U)itisdiculttoseehowonewouldshowthatthisspaceisTychono .Wesummarisewiththefollowingtheorem.Theorem13.Let beaK-coarsertopologyon(X;).(i)ThespaceCk ((X;);UQ)parametrisesacontinuousfunctionuniversalfor(X;)viatheevaluationmap.(ii)Ck ((X;);UQ)isT2and0-dimensional,andhenceisTychono .WewillnowshowthatCk ((X; );UQ)isadensesubspaceofCk((X;);UQ).Towardsthisendwehavethefollowingtheorem. 696G.Fairey,P.Gartside,A.MarshTheorem14.LetXbeaTychono space.LetC=hC0;:::;CniconsistofsubsetsofXandletU=hU0;:::;UniwhereeachUi2U.AssumeD=hD0;:::;DmiconsistsofsubsetsofXandthatV=hfq0g;:::;fqmgiwhereeachqj2Q.Ifeither(i)eachCiandDiiscompactor(ii)eachCiandDiisazerosetandifthereexistsf2W0(C;U)\W0(D;V)thenthereexistsg2W(C;U)\W(D;V).Thefollowinglemmawillsimplifythisproof.Lemma15.LetC=hCi:iniconsistofsubsetsofX.De neEA=Tj2ACjnSj=2ACjandforeachAn+1de neo(A)=j(n+1)nAj.LetIP(n+1)satisfy:thereexistsknsuchthato(A)k+1forallA2Iandifo(A)kthenA2I(wesaythatsuchanIisdownwardclosed).ThenSfEA:A2Ig=SfTj2ACj:A2Ig.NowwearereadytoproveTheorem14.Proof:Weonlygivetheproofforcase(i)ascase(ii)canbeprovedwithminormodi cationsofthesameargument.LetC;U;DandVbeasinthestatementofthelemma,case(i).Assumethatthereexistssomef2W0(C;U)\W0(D;V).Wedivideourproofintotwoparts.Firstweshowthat(a)thereexistsh2W(C;U)andthenweshowthat(b)thereexistsg2W(C;U)\W(D;V).Part(a):Foreachinwewillrecursivelyde neacontinuousfunctionhisatisfying:forallAn+1suchthato(A)iandforallx2EAwehavehi(x)2TfUj:j2Ag.Thiswillsuceasde ningh=hnwemusthavethath2W(C;U).Toconstructh0:thereisonlyoneAn+1suchthato(A)=0,thatisA=n+1.IfEA=;thenwecanchooser02TU.Wede neafunctionh0bysettingforeachx2Xthath0(x)=r0.IfEA=;thenanychoiceofh0willdo.Assumethatforsomeknandforallikwehavetherequiredfunctionhi.Toconstructhk+1:LethA0;:::;AlibeanorderingofthesetfA:o(a)=jk+1jg.Weclaimthatforeachslwecanrecursivelyde neacontinuousfunctionhsk+1satisfying:(1)forallAn+1suchthato(A)kandforallx2EAwehavehsk+1(x)2TfUj:j2Agand(2)forallisandforallx2EAiwehavehsk+1(x)2TfUj:j2Aig.Thende ninghk+1=hlk+1wewillhaveconstructedtherequiredhk+1.Allthatremainstobeshownisthattheclaimistrue.Leth 1k+1=hk.Assumethatforsomeslandalliswehavede nedtherequiredhik+1.LetZs+1k+1=fx:9jn(x2Cj^hsk+1(x)=2Uj)gandnotethatZs+1k+1isacompactset.ToseethiswecanrewriteZs+1k+1asZs+1k+1=[jn(Cj\(fsk+1) 1(RnUj)): Cardinalinvariantsofuniversals697Tode nehs+1k+1:ifEAs+1\Zs+1k+1=;thenleths+1k+1=hsk+1andnotethatthisfunctionsatis es(1)and(2)asdescribedinthepreviousparagraph.Ifnotthen ndrs+1k+12TfUj:j2As+1g.ByLemma15thesetSfEA:o(A)kg[SfEAi:isgiscompactandfromthede nitionsisdisjointfromZs+1k+1.Wecannow ndacontinuousfunctionps+1k+1suchthatps+1k+1Zs+1k+1=1andps+1k+1(SfEA:o(A)kg[SfEAi:isg)=0andps+1k+1[X][0;1].Wede nethefunctionhs+1k+1bysettingforeachx2Xthaths+1k+1(x)=hsk+1(x) hsk+1(x)ps+1k+1(x)+rs+1k+1ps+1k+1(x).Thisfunctioniscontinuousanditsatis es(1)and(2)asdescribedearlier.Part(b):Wewillnowrecursivelyde neforeachkmacontinuousfunctiongksuchthatgk2W(C;U)\W(hD0;:::;Dki;hfq0g;:::;fqkgi).Letg 1=h.Assumethatthereiskmsuchthatforeachikwehavede nedthere-quiredgi.Findacontinuousfunctionpk+1thatsatis es:forallx2Dk+1wehavepk+1(x)=1,forallikandx2Diwehavepk+1(x)=0andforalljnsuchthatDk+1\Cj=;andx2Cjwehavepk+1(x)=0.Nowwede nethefunctiongk+1bysettingforeachx2Xthatgk+1(x)=gk(x) pk+1(x)gk(x)+pk+1(x)qk+1.Itiseasilyveri edthatgk+12W(C;U)\W(hD0;:::;Dk+1i;hfq0g;:::;fqk+1gi).Nowde ningg=gmwehavecon-structedtherequiredfunction.ThenextresultfollowsalmostimmediatelyfromTheorem14.Corollary16.FixaTychono space(X;)andlet beaK-coarsertopology.ThenCk ((X; );UQ)isadensesubspaceofCk ((X;);UQ).NowwehavereducedtheproblemofshowingthatCk ((X;);UQ)isseparable,cccorhascalibre!1tothatofdemonstratingthatCk ((X; );UQ)hastheseproperties.FromnowonwewillwriteCk(X;UQ)andCk(X)astherewillbeonlyonetopologyconsideredonX.FixanarbitraryspaceX.WewillinvestigatewhenCk(X;UQ)isccc,separableorhascalibre!1.Lemma17.FixaTychono spaceX.Then(i)Ck(X;UQ)isseparableifandonlyifCk(X)isseparable,(ii)Ck(X;UQ)iscccifandonlyifCk(X)isccc,(iii)Ck(X;UQ)hascalibre!1ifandonlyifCk(X)hascalibre!1.Proof:TheidentitymapisacontinuousfunctionfromCk(X;UQ)ontoCk(X)andsooneimplicationisimmediatein(i),(ii)and(iii).Part(i):AssumethatCk(X)isseparableandsoXhasacoarserseparablemetrictopology .LetA=fAn:n2!gbeabasisfor .AssumethatAisclosedunder niteunions.Foreachn;m2!letfn;m:X!Rbea -continuousfunctionsatisfyingfn;m(x)=1whenx2 Amandfn;m(x)=0whenx2XnAn. 698G.Fairey,P.Gartside,A.MarshOfcoursethisisonlywell-de nedwhen AmAnandifthisdoesnotholdthenweletfn;m(x)=0forallx2X.Thelinearspanofffn;m:n;m2!goverQisacountablesetthatisdenseinCk(X;UQ).Part(iii):AssumethatCk(X)hascalibre!1.FixanuncountablecollectionW=fW : 2!1gofbasicopennon-emptysubsetsofCk(X;UQ).SowecanassumethateachW isoftheformW(C ;U )\W(D ;V )whereforall 2!1:C =fC i:in gconsistsofzerosetsofX,U =fU i:in gwhereeachUi2U,D =fD j:jm gconsistsofpairwisedisjointzerosubsetsofXandV =ffq jg:jm gwhereeachq j2Q.Bypassingtoanuncountablesubcollectionwecanassumethatforall ; 2!1wehaveU =U andV =V .WewilldropthesubscriptsanduseUandVtodenotethesesets.AssumethatjUj=nandjVj=m.WecanwriteVasffqjg:jmgwhereeachqj2Q.Choose�0suchthat4minfjqi qjj:i;jmg.Wede neanewcollectionV0byde ningforeachjmthesetV0j=(qj ;qj+)andlettingV0=fV0j:jmg.Notethatforeach 2!1weknowthatW(C ;U )\W(D ;V0 )isanon-emptysubsetofCk(X).Sothereissomef2C(X)andA!1suchthatjAj=!1andf2TfW(C ;U )\W(D ;V0 ): 2Ag.Wede netwocollectionsofzerosetsCandDbyde ningforeachin,Ci=f 1( Ui)andforeachjmwede neDj=f 1( V0i).Notethatifj;j0mthenDj\Dj0=;whenj=j0.Wecande neanewfunctionf0bysettingf0(x)=f(x)whenx=2SDandf0(x)=qjwhenx2Dj.Nowwehavethatf02W0(C;U)\W0(D;V)andsoapplyingLemma14weknowthatthereexistsg2W(C;U)\W(D;V).ButW(C;U)\W(D;V)\ 2AW andsowearedone.Part(ii)canbeprovedinmuchthesamewayaspart(iii).Corollary18.Let(X;)beaTychono spaceandlet beaK-coarsertopology.(i)If(X; )issecondcountablethen(X;)hasacontinuousfunctionuni-versalparametrisedbyaseparablespace.(ii)IfCk(X; )iscccthen(X;)hasacontinuousfunctionuniversalpara-metrisedbyacccspace.(iii)IfCk(X; )hascalibre!1then(X;)hasacontinuousfunctionuniversalparametrisedbyaspacewithcalibre!1.Wecanalsoderivesucientconditionsinthecccorcalibre!1casesthatdonotdependonthepropertiesofanexternalobject(suchasCk(X; )).In[13]necessaryandsucientconditionsonXforCk(X)tohavecalibre!1aredescribedandin[12]thesameisdonefortheccccase.Wecansummarisethese Cardinalinvariantsofuniversals699resultsinthefollowinglemma.NotethatCisann-chainofsetsifCisanorderedcollectionofn+1manysetshC0;:::;CniandCi\Cj=;ifji jj�1.Lemma19.LetXbeaTychono space.Ck(X)iscccifandonlyforalln�1andforeverycollectionofn-chainsofcompactsetsfhF0 ;:::;Fn i: 2!1gthereare 1; 22!1andann-chainofzerosetsfCi:ingsatisfying:(a)forj=1;2andinwehaveFi jCiand(b)forj=1;2andinwehaveCi\Ci+1=;ifandonlyifFi j\Fi+1 j=;.Ck(X)hascalibre!1ifandonlyifforalln&#x-402;&#x.571;1andeverycollectionofn-chainsofcompactsetsfhF0 ;:::;Fn i: 2!1gthereissomeA!1withjAj=!1andann-chainofzerosetsfCi:ingsatisfying:(a)for 2AandinwehaveFi Ciand(b)for 2AandinwehaveCi\Ci+1=;ifandonlyifFi \Fi+1 =;.Theseresultsareusefulwhendealingwithspacesthatarenotlocallycompact,asinthelocallycompactcaseCk(X)itselfwillparametriseacontinuousfunctionuniversalforX.ForexamplewecannowconstructaseparablespaceYthatparametrisesacontinuousfunctionuniversalfortheSorgenfreyline.IfweletXbethedisjointsumofc+manycopiesoftheSorgenfreylinethenweknowthatXhasnocontinuousfunctionuniversalparametrisedbyaseparablespaceasCp(X)isnotevenseparable.ButsinceXwillhaveaK-coarsermetrictopologywecanconstructacontinuousfunctionuniversalparametrisedbyacccspace.5.2Necessaryconditions.Wewilldeal rstwiththecasewhereaspaceXhasacontinuousfunctionuniversalparametrisedbyaseparablespace.Wesayaspace(X;)isco-SMifandonlyifthereisaseparablemetrictopology suchthat(X;)hasaneighbourhoodbasisof -closedsets.Lemma20.LetXbeaTychono space.IfXhasacontinuousfunctionuni-versalparametrisedbyaseparablespacethenXisco-SM.Proof:LetYbeaseparablemetricspacethatparametrisesacontinuousfunc-tionuniversalforXviathefunctionF:XY!R.LetDbeacountabledensesubsetofY.Eachd2DrepresentsthecontinuousfunctionFd.Let bethecoarsesttopologythatmakeseachFdcontinuousandnotethat isseparablemetric.FixxinopenU.Picky2YsothatF(x;y)=1andF[(XnU)fyg]=f0g.BycontinuityofFat(x;y)thereareopenVandWwithx2V,y2WandF[VW](2 3;4 3).Claim:Ifx0=2 Uthenthereisa -openTcontainingx0disjointfromV.Fromtheclaimitfollowsthat V U,andbyregularityofX,the -closedneighbourhoodsofxformalocalbase|asrequiredforco-SM. 700G.Fairey,P.Gartside,A.MarshIfweassumethatx0=2 UthenwemusthavethatF(x0;y)=0.SobycontinuityofFat(x0;y)thereareopenV0andW0withx02V0andy2W0sothatF[V0W0]( 1 3;1 3).Pickd2D\(W\W0).Thend2W0soF(x0;d)2( 1 3;1 3).Henceby -continuityofFdatx0,thereisa -openT3x0suchthatF[Tfdg]( 1 3;1 3).Sinced2W,F[Vfdg](2 3;4 3).HenceVandTaredisjoint|asrequired.Notethatthisfallsshortofthesucientconditiongivenpreviouslyleadingtothefollowingquestion.Problem21.IsthereaTychono spaceXsuchthatXisco-SMbutXcanhavenocontinuousfunctionuniversalparametrisedbyaseparablespace?Turningourattentiontoparametrisingspaceswhicharecccweintroducethefollowingtwoproperties.De nition22.AspaceXhasthepropertyP1ifandonlyifforeverypairofdisjointcompactsubsets(K;L)thereexistsapairofopensetsU(K;L);V(K;L)withKU(K;L),LV(K;L)and U(K;L)\ V(K;L)=;satisfyingthefollowing:foranycollectionf(K ;L ): 2!1gofpairsofdisjointcompactsetsthereexists 1; 1suchthat [i=1;2U(K i;L i)\ [i=1;2V(K i;L i)=;:De nition23.AspaceXhasthepropertyP2ifandonlyifforeverypairofdisjointcompactsubsets(K;L)thereexistsapairofopensetsU(K;L);V(K;L)withKU(K;L),LV(K;L)and U(K;L)\ V(K;L)=;satisfyingthefollowing:foranycollectionf(K ;L ): 2!1gofpairsofdisjointcompactsetsthereexists 1; 1suchthat[i=1;2K i\i=1;2U(K i;L i)and[i=1;2L i\i=1;2V(K i;L i): Cardinalinvariantsofuniversals701Lemma24.LetXbeaTychono space.IfXhasazerosetuniversalpara-metrisedbyacccspacethenXhaspropertyP1andeverycompactsubspacehaspropertyP2.Proof:LetYbecccandassumethatYparametrisesazerosetuniversalforXviathecontinuousfunctionF:XY!R.LetZbethedisjointsumof!manycopiesofYandletYndenotethenthcopyofYthatisasubspaceofZ.De neafunctionF0:XZ!RbylettingF0(x;z)=nF(x;z)whenz2Yn.FinallyletG=jF0j.NotethatZparametrisesazerosetuniversalforXviaG,thatZiscccandthatforanypairofdisjointcompactsetsK;LXthereexistsz2ZsuchthatGz[K]=0andGz[L][1;1).WesaythatsuchazseparatesKandL.Wewill rstshowthatXhaspropertyP2onitscompactsubspaces.FixacompactsubspaceC.LetK;LbedisjointcompactsubsetsofC.WeshowhowtoconstructtherequiredU(K;L)andV(K;L).SinceKandLarecompactwecan ndz(K;L)2ZthatseparatesKandL.LetU(K;L)=fx2C:G(x;z(K;L))1 4gandV(K;L)=fx2C:G(x;z(K;L))�3 4g.FindopenW(K;L)suchthatz(K;L)2W(K;L)andforall(x;z1);(x;z2)2CW(K;L)wehavejG(x;z1) G(x;z2)j1 8.Nowtakeacollectionf(K ;L ): 2!1gofpairsofdisjointcompactsubsetsofC.LookatthecorrespondingcollectionfW(K ;L ): 2!1g.SinceZisccctheremustbez2Zand 1; 22!1suchthatz2W(K 1;L 1)\W(K 2;L 2).Weclaimthat[i=1;2K i\i=1;2U(K i;L i)and[i=1;2L i\i=1;2V(K i;L i)asrequired.WewillonlyshowthatK 1U(K 2;L 2)astheothercasescanbedealtwithsimilarly.Fixx2K 1.NotethatG(x;z)1 8sinceG(x;z(K 1;L 1))=0andz2W(K 1;L 1).ButthenG(x;z(K 2;L 2))1 8+1 8=1 4andsox2U(K 2;L 2).NowwewillshowthatXhaspropertyP1.TheproofissimilartotheP1caseandsowewillonlyshowhowtoconstructU(K;L)andV(K;L).LetK;Lbedis-jointcompactsubsetsofX.Findz2ZthatseparatesKandL.Usingthecom-pactnessofKandLandthecontinuityofG ndopenU(K;L);V(K;L);W(K;L)suchthatKU(K;L),LV(K;L)andz2W(K;L)satisfying:forall(x;z0)2U(K;L)W(K;L),G(x;z0)1 4andforall(x;z0)2V(K;L)W(K;L),G(x;z0)�3 4. 702G.Fairey,P.Gartside,A.MarshLemma25.LetXbeacompactHausdor space.IfXhaspropertyP2thenXismetrisable.Proof:Itsucesto ndacountableT1-separatingcollectionofopensubsetsofX(seeforexample[9]).LetC=f(K ;L ): 2IgbeacollectionofdisjointpairsofcompactsubsetsofXthatsatis es():forall 1; 22Ieither[i=1;2K i6\i=1;2U(K i;L i)or[i=1;2L i6\i=1;2V(K i;L i):AssumethatS=fU(K ;L ): 2Ig[fV(K ;L ): 2IgisnotaT1-separatingcollection.Wewillshowthatwecan nd(K;L)suchthatC[f(K;L)gsatis esthesameproperty()asC.SinceSisnotaT1-separatingcollectionthereexistx1;x22XsuchthatforallC2Swehavex12Cimpliesx22C.LetK=fx1gandletL=fx2g.Fix 2I.Ifx12U(K ;L )andx22V(K ;L )thenx2=2U(K ;L ),contradictingthechoiceofx1;x2.Socondition()holdsforC[f(K;L)g.NowletCbeacollectionofdisjointpairsofcompactsubsetsofXthatismaximalwithrespectto()(i.e.Csatis es(),butforanycollectionD,ifC(DthenDdoesnothaveproperty()).SinceXhasP2wemusthavethatCiscountable.ButSasdescribedabovemustbeaT1-separatingcollection,andsowearedone.Problem26.DoesthepropertyP1implythepropertyP2?IfnotisthepropertyP1equivalenttometrisabilityincompactspaces?References[1]ArensR.,DugundjiJ.,Topologiesforfunctionspaces,Paci 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Cardinalinvariantsofuniversals703[11]HodelR.,CardinalfunctionsI,inHandbookofSet-theoreticTopology,NorthHolland,Amsterdam,1984,pp.1{61.[12]MarshA.,Topologyoffunctionspaces,PhD.Thesis,Univ.Pittsburgh,2004.[13]NakhmansonL.B.,TheSuslinnumberandcalibresoftheringofcontinuousfunctions,Izv.Vyssh.Uchebn.Zaved.Mat.(1984),no.3,49{55.[14]TodorcevicS.,PartitionProblemsinTopology,ContemporaryMathematics84,Amer.Math.Soc.,Providence,RI,1989.[15]ZenorP.,Somecontinuousseparationaxioms,Fund.Math.90(1975/1976),no.2,143{158.SchoolofPhysicsandAstronomy,TheUniversityofManchester,ManchesterM139PL,UnitedKingdomDepartmentofMathematics,301ThackerayHall,UniversityofPittsburgh,Pittsburgh,PA15260,USAE-mail:gartside@math.pitt.eduDepartmentofMathematics,NUI,Galway,NewcastleRoad,Galway,Ireland(ReceivedNovember5,2004,revisedAugust29,2005)