Mathematics, Hungarian Academy of Sciences

dedicatedtothe60thbirthdayofIstv´anJuh´asz

organizedbytheJ´anosBolyaiMathematicalSocietyincooperationwiththePaul

Erd˝osSummerResearchCenterofMathematicsandtheAlfr´edR´enyiInstituteof

Mathematics,HungarianAcademyofSciences

Abstracts

Budapest,August8–13,2003

Supportedbythe”MathematicsinInformationSociety”project(intheframeworkoftheEuropeanCommunity’s”ConfirmingtheInternationalRoleofCommunityResearch”

programme)

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,20031

Contents

A.V.Arhangel’skii:Cardinalinvariantsintopologicalspaceswithsomealgebraicstructures3F.Azarpanah:Zero-DivisorGraphofC(X)

M´aty´asBogn´ar:Anewapproachtothedegreeofpoints

33

M.G.Charalambous:EveryEberleincompactspaceYhasadenseGδmetrizablesubspaceXwithdimX≤dimY4´AkosCs´asz´ar:GeneralizedopensetsingeneralizedtopologiesAlanDow:Compactspacesofcountabletightness

MaximilianGanster:Coveringpropertiesandpreopensets

1spacesD.N.Georgiou:AunifiedtheoryofT2455566

J´anosGerlits:Onleft-separatedspaces

WilliamD.Gillam:EmbeddabilityofCountableMetricSpaces

RafalGorak:Constructionofuniformhomeomorphismsbetweenspacesequipedwiththepointwisetopology6S.D.Iliadis:Containingspacesandactionsofgroups.SaeidJafari:Non-degeneratefunctionsinthesenseofBottI.Juh´asz:ResolvabilityversusmaximalresolvabilityMasaruKada:Hechler’stheoremforidealsoftherealsO.A.S.Karamzadeh:OnKashspaces

LjubiˇsaD.R.Koˇcinac:OnhyperspacetopologiesJanKraszewski:Transitivepropertiesofideals

779

KennethKunen:TheComplexStone-WeierstrassPropertyandElementarySub-models10OttoLaback:SomeremarksontheexistenceofTychonoff-Topologiesonspace-time-manifolds10G´aborLuk´acs:Onliftedclosureoperators

JuanCarlosMart´ınez:CardinalsequencesofscatteredBooleanspaces

1111

SeithutiMoshokoa:Ontheextensionproblemforuniformlycontinuousreal-valuedmaps12EvaMurtinova:Weakernotionsofregularityandnormality

12

2Abstracts

PeterJ.Nyikos:2-to-1closedpreimagesofω1

JanPelant:WeaklyWhyburnspacesofcontinuousfunctionsonordinalsSaharonShelah:Onthespectrumofχofultra-filtersonωLajosSoukup:CardinalsequencesofscatteredspacesJanvanMill:Erd˝osspace(s)

W.Weiss:PartitioningTopologicalSpaces-60TheoremsSzymonZeberski:Largecardinalsbelowfirstmeasurablecardinal

12131313141414

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,20033

Cardinalinvariantsintopologicalspaceswithsomealgebraicstructures

A.V.Arhangel’skii

(OhioUniversity&UniversityofMoscow)

(arhangel@bing.math.ohiou.edu)

Zero-DivisorGraphofC(X)

F.Azarpanah

(ChamranUniversity,Iran)

(azarpanah@ipm.ir)

jointresearchwithM.Motamedi

Wedenotethezero-divisorgraphofC(X)byΓ(C(X))andwewillassociatetheringpropertiesofC(X),thegraphpropertiesofΓ(C(X))andthetopologicalpropertiesofX.CyclesinΓ(C(X))areinvestigatedandanalgebraicandatopologicalcharacterizationisgivenforthegraphΓ(C(X))tobetriangulatedorhypertriangulated.WehaveshownthatthecliquenumberofΓ(C(X)),thecellularityofXandtheGoldiedimensionofC(X)coincide.ItturnsoutthatthedominatingnumberofΓ(C(X))isbetweenthedensityandtheweightofX.FinallywehaveshownthatΓ(C(X))istriangulatedandthesetofcentersofΓ(C(X))isadominatingsetifandonlyifthesetofisolatedpointsofXisdenseinXifandonlyiftheSocleofC(X)isanessentialideal.

Anewapproachtothedegreeofpoints

´tya´sBogna´rMa

(E¨otv¨osUniversityBudapest)(bognar@ludens.elte.hu)

Thedegreeofapoint(ortheorderofapoint)isafundamentalconceptinmender-Urysohn

curvetheory(seee.g.C.Kuratowski,TopologieII).Itcanbedefinedinarbitrarytopologicalspaces,howeversometimesitdoesnotcoincidewiththevisualrequirements.LetN0bethesetofnonnegativeintegers.

ForasubsetMofatopologicalspace(E,τ)letCompτ(M)denotethefamilyofthecompo-nentsofthesetMin(E,τ).

ThenumberofelementsofasetQwillbedenotedbyn(Q),wheren(Q)∈N0∪{∞}.

Let(E,τ)beatopologicalspace.Letp∈Eandletm∈N0.Wewritedgτ(p)|miftoeachneighbourhoodUofp(i.e.toeachU∈τ(p))thereisV∈τ(p)suchthatV⊂Uandn(Compτ(V\\{p}))≤m.

Wesaythatthedegreeofpwithrespectto(E,τ)isinfiniteandwewritedegτ(p)=∞ifthereisnom∈N0withdgτ(p)|m.Otherwisewesaythatthisdegreeisfiniteanditsdefinitionis

degτ(p)=min{m∈N0:dgτ(p)|m}.

4Abstracts

Linecomplexesarepracticallylocallyfinitegraphswithoutloops.IfWisalinecomplex,τitsnaturaltopology,τ󰀂thetopologyof∪WandA={p}isanodeofWthendegτ(A)=degτ󰀁(p),whiletheusualconceptmayhaveresultsordAW=ordp(∪W).

EveryEberleincompactspaceYhasadenseGδmetrizablesubspaceX

withdimX≤dimY

M.G.Charalambous

(UniversityoftheAegean,Greece)

(mcha@aegean.gr)

AnEberleincompactspaceisacompact󰀄subspaceofc0(S)forsomesetS.Herec0(S)isthesubspaceoftheproductofunitintervalss∈SIsconsistingofthepointsx=(xs)suchthattheset{s∈S:xs>󰀂}isfiniteforeachpositiverealnumber󰀂.

G.DimovhasprovedthataBairesubspaceYofc0(S)containsadenseGδsubspaceXwhichismetrizable,extendingaresultpreviouslyknownforYEberleincompact.WeprovethatdimX≤dimY.Thisfollowsfromcertainresultsthatweestablishconcerningthedimensionofuniformspaces.

ItshouldbenotedthatforadensemetrizablesubspaceXofacompactspaceY,wedonotnecessarilyhavedimX≤dimY.ForRoy’sexampleofametrizablespaceXwithdimX=1andindX=0hasacompactificationYwithdimY=0.

ItisalsoofinteresttonotethatT.KimuraandK.MorishitahaverecentlyprovedthateverymetrizablespaceXhasanEberleincompactificationYwithdimY=dimX.Infact,onecanevenshowthatIndY=dimX.

Generalizedopensetsingeneralizedtopologies

´´sza´rAkosCsa

(E¨otv¨osUniversityBudapest)

Inthetopologicalliterature,onefindsaseriesofclassesofsetsthataregeneralizationsofopen

sets:semi-opensets,preopensets,α-opensets,β-opensets.Eachoftheseclassesisageneralizedtopology(GT),i.eitcontains∅andtheunionofanarbitrarysubclass.

Thepurposeofthepresenttalkisamodificationofthedefinitionoftheaboveclassesinthefollowingsense.LetλbeaGTonasetX.Define,forA⊂X,

󰀂󰀃

iλA={L∈λ:L⊂A},cλA={X−L:L∈λ,A⊂X−L}.Nowletusreplace,inthedefinitionofsemi-open,preopen,α-open,β-opensets,theoperationint(A)byiλandcl(A)bycλ;moreprecisely,letussaythatA⊂Xisλ-semi-openiffA⊂cλiλA,λ-preopeniffA⊂iλcλA,λ-α-openiffA⊂iλcλiλA,λ-β-openiffA⊂cλiλcλA.Denotetheseclassesbyσ(λ),π(λ),α(λ),β(λ),respectively.EachoftheseclassesisinGT,soonecanspeakoftheclassofξ(η(λ))forξ,η=σ,π,α,β.

Weshowthatinsomecasestheclassξ(η(λ))coincideswithoneoftheclassesσ(λ),π(λ),α(λ),β(λ),but,ingeneral,oneobtainsnewclassesofsets.Asanapplication,wegetcharacterizationofthegeneralizedtopologiesoftheformσ(τ)whereτisatopologyonX.

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,20035

Compactspacesofcountabletightness

AlanDow

(UniversityifNorthCarolinaatCharlotte)

(adow@uncc.edu)

Coveringpropertiesandpreopensets

MaximilianGanster

(GrazUniversityofTechnology)(ganster@weyl.math.tu-graz.ac.at)jointresearchwithM.Sarsak

Weconsidervariouscoveringpropertieswhichinvolvepreopensetsintheirdefinitions.Specialattentionwillbepaidtothecaseofmetacompactness,-arecentjointresearchprojectwithProf.M.SarsakfromHashemiteUniversity,Jordan.

AunifiedtheoryofT1spaces

2D.N.Georgiou

(DepartmentofMathematics,UniversityofPatras,26500Patras,Greece)

(georgiou@math.upatras.gr)

jointresearchwithM.Caldas,S.JafariandT.Noiri

In1970,Levine[4]introducedthenotionofT1spaceswhichlieproperlybetweenT1-spaces2andT0-spaces.Dunham[3]obtainedthefollowingcharacterizationofT1-spaces:atopological21ifandonlyifeachsingletonofXisopenorclosed.Moreover,Arenasetal.[1]space(X,τ)isT2showedthatatopologicalspace(X,τ)isT1ifandonlyifeverysubsetofXisτ-closed.In1987,2semi-T1spacesareintroducedbyBhattacharyyaandLahiri[3].Sundarametal.[5]showedthat2atopologicalspace(X,τ)issemi-T1ifandonlyifeachsingletonofXissemi-openorsemi-closed.2Inthispaper,weintroducethenotionscalledm-structureswhichareweakerthantopologicalstructures.Usingthem-structures,weinvestigateaunifiedtheoryofweakseparationaxiomscontainingT1spacesandsemi-T1spaces.22References

[1]F.G.Arenas,J.DontchevandM.Ganster,Onλ-setsanddualofgeneralizedcontinuity,QuestionsAnswersGen.Topology,15(1997),3-13.

6Abstracts

[2]P.BhattacharyyaandB.K.Lahiri,Semi-generalizedclosedsetsintopology,IndianJ.Math.29(1987),375-382.[3]W.Dunham,T1-spaces,KyungpookMath.J.,17(1977),161-169.2[4]N.Levine,Generalizedclosedsetsintopology,Rend.Circ.Mat.Palermo19(2)(1970),-96.[5]P.Sundaram,H.MakiandK.Balachandran,semi-generalizedcontinuousmapsandsemi-T1

2spaces,Bull.FukuokaUniv.Ed.PartIII40(1991),33-40

Onleft-separatedspaces

´nosGerlitsJa

(Alfr´edR´enyiInstituteofMathematics)

(gerlits@renyi.hu)

´nJuha´sz,LajosSoukupandZolta´nSzentmiko´ssyjointresearchwithIstva

EmbeddabilityofCountableMetricSpaces

WilliamD.Gillam

(WesleyanUniversity)(wgillam@wesleyan.edu)

Wediscusssomegeneralresultsaboutcountablemetricspaces(allofwhichare,uptohome-omorphism,subspacesoftherationals)withanemphasisonscatteredspaces(i.e.spaceswith

nocrowdedsubspaces)andpropertiesrelatedtohomeomorphicembeddability.Usinginvariantsforrelatively”simple”spaces(thoseoffiniteCantor-Bendixsonrank),togetherwithatechnicalresultreducingquestionsofembeddabilitytothiscase,weprovetwotheorems:IfFisasetofcountablemetricspacessuchthatnospaceinFembedsinanyotherspaceinF,thenFisfinite.(Arbitrarilylargesuchfinitesetsexist.)IfGisanynon-emptysetofcountablemetricspacesthenthereissomeXinGsothat,foranyYinG,ifYembedsinXthenXembedsinY.

Constructionofuniformhomeomorphismsbetweenspacesequiped

withthepointwisetopology

RafalGorak

(INSTITUTEOFMATHEMATICSofthePolishAcademyofSciences,Warszawa,Poland)

(gorakrafal@yahoo.com)

InmytalkIwillpresentthemethodofconstructinguniformhomeomorphismsbetweenreal

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,20037

valuedfunctionspacesequippedwiththetopologyofpointwiseconvergence(denotedusuallybyCp(X))inventedbyGulkoanddevelopedbymyselftosolvethefollowingproblems:

i)IwillshortlypresentmyresultconcerningcharacterizationofspacesXsuchthatCp(X)andCp(In)areuniformlyhomeomorphic(Inisthestandardn-thcube).ii)IwillalsofocusonspacesCp([0,α])whereαisanordinaland[0,α]={β:β≤α}endowedwiththestandardordertopology.Iwillgivetheuncompleteclassificationofsuchspacesupto(uniform)homeomorphismsandIwillstatetheconjecturewhatthecompleteclassificationis.

Containingspacesandactionsofgroups.

S.D.Iliadis

(DepartmentofMathematics,UniversityofPatras,26500Patras,Greece)

(iliadis@math.upatras.gr)

AllconsideredspacesareassumedtobeT0-spacesofweightlessthanorequaltoagiveninfinitecardinalτ.LetGbeanarbitrarytopologicalgroup.InparalleltothenotionsofasaturatedclassofspacesandasaturatedclassofmappingsweintroducethenotionofasaturatedclassofG-spaces.Wegivethebasicpropertiesoftheseclasses,thatis,inanysuchclassthereexistsuniversalelementsandtheintersectionofsaturatedclassesisalsosuchaclass.WeprovethatforanysaturatedclassPofspacestheclassofallG-spacesofPisasaturatedclassofG-spaces.Inparticular,wehavethefollowingconsequence.ThefollowingclassesofG-spacesaresaturated:

(1)Theclassofallregular(completelyregular)G-spaces.

(2)Theclassofallregular(completelyregular)countable-dimensionalG-spaces.

(3)Theclassofallregular(completelyregular)stronglycountable-dimensionalG-spaces.(4)Theclassofallregular(completelyregular)locallyfinite-dimensionalG-spaces.

(5)Theclassofallregular(completelyregular)G-spacesofdimensionindlessthanorequaltoanordinalα∈τ+(inparticular,αmaybeannon-negativeinteger).

Non-degeneratefunctionsinthesenseofBott

SaeidJafari

(DepartmentofMathematicsandPhysics,RoskildeUniversity,Postbox260,4000Roskilde,

Denmark)

(sjafari@ruc.dk)

Inthistalk,wediscusstheconnectivityofthenonemptylevelsofnon-degeneratefunctionsinthesenseofBottonacompactconnectedmanifold.

8Abstracts

Resolvabilityversusmaximalresolvability

´szI.Juha

(Alfr´edR´enyiInstituteofMathematics)

(juhasz@renyi.hu)

´ssyjointresearchwithL.SoukupandZ.Szentmiklo

InthisjointworkwithL.SoukupandZ.Szentmikl´ossyweproveseveralnewresultsconcerning

resolvability.OurmaintoolisaverygeneralconstructionmethodinZFCthatallowsustorefinetopologiesinsuchawaythatintheresultingfinertopologyasetwillbedenseinanopensetonlyifsomeprescribedfamilyofdensesets”forces”this.Asacorollary,foreveryuncountableregularcardinalκweconstructa0-dimensionalT2spaceXwith∆(X)=κthatisλ-resolvableforallλ<κbutisnotκ-resolvable.ThissolvesanoldquestionofCederandPearson.

WealsostrengthenarecentresultofO.PavlovbyprovingthatforeveryregularcardinalκifthespaceXsatisfies∆(X)≥κandXhasnodiscretesubspaceofsizeκ,thenXisκ-resolvable.References

[1]J.CederandT.Pearson,Onproductsofmaximallyresolvablespaces,PacificJ.Math.22(1967),31–45.

[2]O.Pavlov,Onresolvabilityoftopologicalspaces,TopologyanditsApplications,126(2002),37–47.

Hechler’stheoremforidealsofthereals

MasaruKada

(KitamiInstituteofTechnology,Kitami,JAPAN)

(kada@math.cs.kitami-it.ac.jp)

Hechler’stheorem,whichisaclassicalresultofthetheoryofforcing,isthefollowingstatement:ForapartiallyorderedsetQsuchthateverycountablesubsethasastrictupperbound,thereisaforcingnotionsatisfyingcccsuchthat,intheforcingmodel,thereisacofinalsubsetof(ωω,≤∗)(thesetofallfunctionsfromωtoωorderedbyeventuallydominatingorder)whichisorder-isomorphictoQwithrespecttoset-inclusion.

Weshowthatstatementssimilartotheaboveholdforthemeageridealandthenullidealofthereals,orderedbyset-inclusion.BothofthemareprovedusingamethodofforcingconstructionsimilartotheoneinHechler’soriginalproof.

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,20039

OnKashspacesO.A.S.Karamzadeh

(ChamranUniversity,Iran)(karamzadeh@cua.ac.ir)jointresearchwithA.A.Staji

Email:karamzadeh@cua.ac.irCoauthors:Title:Abstract:Letcbeacardinalnumber,thenacommutativeringRissaidtobeac-Kash-ringifidealswithgeneratingsetofcardinalitylessthancarenon-essentialandaremaximalwithrespecttothisproperty.AtopologicalspaceXiscalledc-KashifC(X)isac-Kashring.WeobservethatXisanalmostP-spaceifandonlyifXisℵ0-Kash.ItisalsoshownthatXisℵ1-KashifandonlyifXisapseudocompactalmostP-space.XandβXareshowntobec-Kashforthesamec.Someotherpropertiesofc-Kashspacesarediscussed.

OnhyperspacetopologiesˇinacLjubiˇsaD.R.Koc

(FacultyofPhilosophy,UniversityofNiˇs,18000Niˇs,Yugoslavia)

(kocinac@archimed.filfak.ni.ac.yu)

ForaspaceXconsiderthesetofallclosedsubsetsofXendowedwithdifferent(upper)

topologies.WediscussdualitybetweenXanditshyperspacesconsideringsomepropertiesgivenintermsofselectionprinciples.

Transitivepropertiesofideals

JanKraszewski

(UniversityofWroclaw,Poland)(kraszew@math.uni.wroc.pl)

Weconsideraninvariantσ-idealofsubsetsofaninfiniteAbeliangroup(G,+).Forsuchanidealwedefinethefollowingcardinalcoefficients.

addt(J)=min{|A|:A⊆J&¬(∃B∈J)(∀A∈A)(∃g∈G)A⊆B+g},add∗t(J)=min{|T|:T⊆G&(∃A∈J)A+T∈J},covt(J)=min{|T|:T⊆G&(∃A∈J)A+T=G},

coft(J)=min{|B|:B⊆J&BisatransitivebaseofJ},

whereafamilyB⊆JiscalledatransitivebaseifforeachA∈JthereexistsB∈Bandg∈GsuchthatA⊆B+g.Firsttwoonesarebothcalledtransitiveadditivity.Thelattertwoonesarecalledtransitivecoveringnumberandtransitivecofinality,respectively.

10Abstracts

WesaythatanidealJisκ−translatableif

(∀A∈J)(∃B∈J)(∀T∈[G]κ)(∃g∈G)A+T⊆B+g.

WedefineatranslatibilitynumberofJasfollows

τ(J)=min{κ:Jisnotκ−translatable}.

WeintroducealsosomeoperationsonJ.s(J)={A⊆G:(∀B∈J)A+B=G},g(J)={A⊆G:(∀B∈J)A+B∈J},

Weinvestigatepropertiesoftheseoperationsandcardinalcoefficients.Inparticular,weshowwhathappensforJ=S2,whereS2denotestheleastnontrivialσ-idealofsubsetsoftheCantorspace2ω.

TheComplexStone-WeierstrassPropertyandElementarySubmodels

KennethKunen

(UniversityofWisconsin)(kunen@math.wisc.edu)

ThecompactHausdorffspaceXhastheComplexStone-WeierstrassProperty(CSWP)iffXsatisfiesthecomplexversionoftheStone-WeierstrassTheorem(thatis,everyalgebraofcontinuouscomplex-valuedfunctionsonXwhichseparatespointsandcontainstheconstantfunctionsisdenseinC(X)).

ByresultsofW.Rudinandothers,theCSWPwasknowntobetrueofallcompactscatteredspaces,andfalseofspacescontainingacopyofeithertheCantorsetorβN.Itwasnotknowntobetrueofanynon-scatteredspace.

Byapplyingelementarysubmodeltechniques,onecanshowthatinfacttheCSWPholdsforanumberofnon-scatteredcompactspaces.Forexample,acompactLOTSsatisfiestheCSWPiffitdoesnotcontainacopyoftheCantorset.

SomeremarksontheexistenceofTychonoff-Topologieson

space-time-manifolds

OttoLaback

Name:

UniversityofTechnologyGrazDept.mathematic

Therearenaturalsemimetrictopologiesonspace-timemanifoldsbutuntilnownosuchcom-pletelyregulartopologies.Wetrytoexplainthisopenproblemandwillgivesomeremarksonmodellingthecausalityofspace-timemanifolds.

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,200311

Onliftedclosureoperators

´borLuka´csGa

(DepartmentofMathematicsStatistics,YorkUniversity,4700KeeleStreetToronto,Ontario,M3J1P3Canada)

(lukacs@mathstat.yorku.ca)

ForatopologicalgroupG,wedenotebyκG:G→bGtheBohr-compactificationofG,andonesetsG+=κG(G);G+isadensesubgroupofbG.TheBohr-closureofasubsetA⊆Gistheclosure

ofAintheinitialtopologyinducedonGbythemapκG.Inotherwords,cb(A)=

AgroupKiscalledcb-compactiftheprojectionπH:K×H→Hmapscb-closedsubgroupsoftheK×Hontocb-closedsubgroupsofH.Weobservethefollowingpropertyofcb:

Theorem.AtopologicalgroupGiscb-compactifandonlyifitG+=bG(i.e.κGissurjective).AclosureoperatoronX([1])isafamilyofmapscX:sub(X)−→sub(X),X∈X,suchthat:(1)m≤cX(m)forallm∈sub(X)(extensive);(2)m≤nimpliescX(m)≤cX(n)forallm,n∈sub(X)(monotone);(3)f(cX(m))≤cY(f(m))forallm∈sub(X),foreverymorphismf:X→Y(morphismsarecontinuous).

Amorphismf:X→YinXisc-preservingifcY(f(m))=f(cX(m))foreverym∈sub(X).Following[2],anobjectX∈Xisc-compactifforeveryY∈XtheprojectionπY:X×Y→Yisc-preserving.

NoticethattheclosureoperatorcbwasobtainedastheliftingoftheKuratowskiclosureoperatorthroughtheBohr-reflection.Itturnsoutthatwecanliftanyclosureoperatorfromaniceenoughsubcategory.LetAbeanepireflectivesubcategoryofXwithreflectorRandunitρ.GivenaclosureoperatorconA,ccanbeliftedtoaclosureoperatorcρonXdefinedby

−1

cρX(m)=ρX(c(ρX(m))).WeshowananalogueoftheTheorem:undermildconditions,X∈Xiscρ-compactifandonlyifRXisc-compact.1.M.M.Clementino,E.Giuli,andW.Tholen.Portugal.Math.,53(4):397–433,1996.

Topologyinacategory:compactness.

G+

−1

κG(κG(A)).

2.DikranDikranjanandEraldoGiuli.Compactness,minimalityandclosednesswithrespecttoaclosureoperator.InCategoricaltopologyanditsrelationtoanalysis,algebraandcombinatorics(Prague,1988),pages284–296.WorldSci.Publishing,Teaneck,NJ,19.

CardinalsequencesofscatteredBooleanspaces

JuanCarlosMart´ınez

(FacultatdeMatem`atiques,UniversitatdeBarcelona,GranVia585,08007Barcelona,Spain)

(martinez@mat.ub.es)

TheCantor-Bendixsonprocessfortopologicalspacesisdefinedasfollows.SupposethatXis

atopologicalspace.Then,foreveryordinalαwedefinetheα-derivativeofXby:X0=X;if

12Abstracts

󰀁

α=β+1,XαisthesetofaccumulationpointsofXβ;andifαisalimit,Xα={Xβ:β<α}.Then,Xisscattered,ifXα=∅forsomeordinalα.

TheCantor-Bendixsonprocesspermitsustosplitascatteredspaceintolevels.SupposethatXisascatteredspace.WedefinetheheightofXbyht(X)=theleastordinalαsuchthatXαisfinite.ForαThen,weshallexposeaseriesofresultsoncardinalsequencesofscatteredBooleanspacesaswellasalistofopenproblemsinthesubject.

Ontheextensionproblemforuniformlycontinuousreal-valuedmaps

SeithutiMoshokoa

(UniversityofSouthAfrica,PRETORIA,SouthAfrica)

(moshosp@unisa.ac.za)

Itiswellknownthatuniformlycontinuousmapsbetweenmetricspacesadmitsauniqueuni-formlycontinuousextensiontothecompletion.However,therearemapsonmetricspaceswithextensionstothecompletionwhicharenotuniformlycontinuous.Wegiveasimpleresultconcern-inguniformlycontinuousmapsonboundedsubsetswhichextendstheclassicalTheoremreferredtoabove.FinallyadiscussionofananalogueoftheclassicalTietzeextensionTheoreminthiscontextwillalsobepresented.

Weakernotionsofregularityandnormality

EvaMurtinova

(CharlesUniversity,Prague,CzechRepublic)

(murtin@karlin.mff.cuni.cz)

2-to-1closedpreimagesofω1

PeterJ.Nyikos

(UniversityofSouthCarolina,Columbia,SC29208)

Althoughtheyareaveryspecializedclassofspaces,2-1closedpreimagesofω1haveplayedanimportantheuristicroleinthetheoryoflocallycompactandcountablycompactspaces.Twicealreadythesolutionofaprobleminvolvingsuchspaceshasmadepossiblemajoradvancesinourunderstandingofcountablycompactspaces.In1986,Fremlin’sproof

thatsuchspacesalwayscontaincopiesofω1undertheProperForcingAxiom(PFA)wasthekeytoafloodofresultsculminatinginBalogh’sproofsthatthePFAimpliesthat(1)everyfirstcountable,countablycompactHausdorffspaceiseithercompactorcontainsacopyofω1,and(2)everycompactspaceofcountabletightnessissequential.Tenyears

later,statement(1)wasshowncompatiblewithCH,with2-1closedpreimagesofω1againturningouttocapturemostofthedifficultiesinvolved.Nowthereisathirdsetofproblems

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,200313

towhichitishopedthatthesespecialspaceswillprovidetheneededinsight:Isitconsistentthateverylocallycompact,hereditarilystronglycollectionwiseHausdorffspaceisnormal?col-lectionwisenormal?countablyparacompact?Theanswersarenegativeunder♦butthePFAisareasonablecandidateforpositiveanswers.

WeaklyWhyburnspacesofcontinuousfunctionsonordinals

JanPelant

ˇa25,11567Prague1,CzechRepublic)(MathematicalInstituteofAcad.Sci.ofCzechRep.,Zitn´

(pelant@math.cas.cz)

jointresearchwithAngeloBella

WecharacterizethoseordinalsξforwhichCp(ξ)isaweaklyWhyburnspace.Asabyproduct

weobtainthecoincidenceoftheFr´echetpropertyandtheWhyburnoneonCp(ξ).

Onthespectrumofχofultra-filtersonω

SaharonShelah

(HebrewUniversityofJerusalem)

(shelah@math.huji.ac.il)

Weinvestigatewhatthissetofcardinalscanbe(whenthecontinuumislarge,naturally)aswellasπχcontinuingBrendleShelah[BnSh2].Inparticular,thosesetsmaybenon-convex(seeproblem5there).

Cardinalsequencesofscatteredspaces

LajosSoukup

(Alfr´edR´enyiInstituteofMathematics)

(soukup@math-inst.hu)

´szandZ.Szentmiklo´ssyjointresearchwithI.Juha

WeshowthatifweaddanynumberofCohenrealstothegroundmodelthen,inthegeneric

extension,alocallycompactscatteredspacehasatmost(2ℵ0)Vmanylevelsofsizeω.

WealsogiveacompleteZFCcharacterizationofthecardinalsequencesofregularscatteredspaces.Althoughtheclassesoftheregularandofthe0-dimensionalscatteredspacesaredifferent,weprovethattheyhavethesamecardinalsequences.

14Abstracts

Erd˝osspace(s)JanvanMill

(VrijeUniversiteit,Amsterdam)

(vanmill@cs.vu.nl)

LetEdenoteErd˝osspace,i.e.,thesetofallvectorsin󰀪2allcoordinatesofwhicharerational.

TheirrationalversionofE,i.e.,thesetofallvectorsin󰀪2allcoordinatesofwhichareirrational,isdenotedbyEc.ItisknownthatbothEandEcare1-dimensionalandtotallydisconnected.Theorem(Dijkstra,vanMill,Stepr¯ans).Ecisnothomeomorphictoitscountableinfiniteproduct.

Theorem(Dijkstra,vanMill).Eishomeomorphictoitscountableinfiniteproduct.IfXisaspacethenH(X)isthegroupofallhomeomorphismsofXwiththecompact-opentopology.

Theorem(Dijkstra,vanMill).LetX=Rn,n≥2,andletDbeacountabledensesubsetofX.Thesubgroup{h∈H(X):h(D)=D}ofH(X)ishomeomorphictoE.

PartitioningTopologicalSpaces-60Theorems

W.Weiss

(UniversityofToronto)(weiss@math.toronto.edu)

Welookbackatsomeofthetheoremsaboutpartitioningtopologicalspacesandconsidersomeoftheremainingopenproblems.Afterabrieflookatthehigherdimensionalcasesweconcentrateontheonedimensionalcase.WeshowhowtouseCohenrealstopartitionmetricspaces.

Largecardinalsbelowfirstmeasurablecardinal

SzymonZeberski

(InstituteofMathematics,Wroc󰀊lawUniversity,,Poland)

(szebe@math.uni.wroc.pl)

´jointresearchwithJacekCichon

Weusethefollowingnotation.M(κ)denotesκisameasurablecardinal;SI(κ)–κisstrongly

inaccesible;WC(κ)–κisweaklycompact.

Weareinterestedinthreeclassesofcardinals:indescribablecardinals,subtlecardinalsandpartitioncardinals.Letusrecallthefolowingdefinition.

ColloquiumonGeneralandSet-TheoreticTopology,Budapest,200315

1

Definition.CardinalnumberκisΠ1n–indescribable(Πn−IND(κ))iffforeveryφwhichisΠ1=φn–sentence(ofalanguagewithonepredicateA),foreverysubsetAofRκif(Rκ,A,∈)|thenthereexistsα<κsuchthat(Rα,A∩Rα,∈)|=φ.

ItiswellknownthatΠ1⇒SI(κ)and(Π1⇒WC(κ).0−IND(κ)⇐1−IND(κ)⇐1

ForΠn–indescribableκletDΠ1(κ)={X⊆κ:(∀φ∈Π1=φ−→n)(∀A⊆Rκ)((Rκ,A,∈)|n

(∃α∈X)(Rα,A∩Rα,∈)|=φ)}.

Thesecondclassofcaldinalsweareinterestedinaresubtlecardinals.

Definition.LetA⊆P(κ).WesaythatκisA–subtleiffforeverysequence(Aα)α∈κsuchthat(∀α∈κ)(Aα⊆α),foreveryclosedunboundedsetC⊆κthereexistsasetX∈AsuchthatX⊆Cand

(∀α,β∈X)(α<β→Aα=Aβ∩α).

Definition.Letκbeacardinalnumber.1.κissubtle(ST(κ))iffκis[κ]2–subtle.

2.κisalmostineffable(AINF(κ))iffκis[κ]κ–subtle.

11

3.κisΠ1–subtle.n–subtle(Πn−SB(κ))iffΠn−IND(κ)andκisDΠ1n

Thethirdclassarepartitioncardinals.Foranordinalnumberαlet

ηα=min{λ:λ→(α)<ω}.

Definition.Letκbeacardinalnumber.

1.κisanErd¨oscardinal(E(κ))iffthereexistsalimitordinalαsuchthatκ=ηα.2.κisaRamseycardinal(R(κ))iffκ→(κ)<ω.

113.κisaΠ1n–Ramseycardinal(Πn−R(κ))iffΠn−IND(κ)and

(κ))(∀n∈ω)(|F([H]n)|=1).(∀F:[κ]<ω→2)(∃H∈DΠ1

n

WewriteA−→BifA(κ)impliesB(κ).A−󰀂→BdenotesthatifA(κ)then{λ<κ:B(λ)}isstationaryinκ.

∗A−󰀂→BmeansthatifA(κ)then{λ<κ:B(λ)}hascardinalityκ.Ourresultssumupthefollowingtheorem.Theorem.Wehavethefollowingdiagram

16Abstracts

rrhMr

rjrjrr...

¤󰁚󰁚

¤h~¤...¤

󰁚󰁚¤¤󰁚󰁚hh¤󰁚¤h~~󰁚¤¤cccc..1¤Π0−RΠ1−SB.¤1

󰁚󰁚¤¤󰁚󰁚¤hhh¤󰁚󰁚

~󰁚~¤󰁚¤cccccc¤1¤−INDΠ1RΠ0−SB3

󰁚¤󰁚¤

󰁚¤󰁚¤󰁚h*hh󰁚¤~~󰁚¤󰁚

cccccc¤¤1EAINFΠ−IND2¤󰁚¤󰁚󰁚¤󰁚󰁚¤hh󰁚󰁚󰁚¤¤󰁚~󰁚~󰁚

cccc¤1STΠ−IND1¤󰁚󰁚¤󰁚

h󰁚󰁚󰁚¤~󰁚

ccΠ1−IND0hcc1

Π1−R

References

[1]W.Boos,Lecturesonlargecardinalaxioms,ProccedingsoftheInternationalSummerInstituteandLogicColloquium,Kiel1974.

[2]F.Drake,Settheory,anintroductiontolargecardinals,NorthHolland1974.[3]A.Kanamori,TheHigherInfinite,Springer-VerlagBerlinHeidelberg1994.

[4]K.Kunen,Indescribabilityandthecontinuum,ProccedingsofSymposiainPureMathematics,VolumeXIII,PartI,pp.199-203.

Keywordsandphrases.Measurablecardinal;Π1n–indescribablecardinal;stationaryset.2000AMSSubjectClassification.03E55,03E10.