Open problems in topology

49 Pages·1990·1.004 MB·English

by  Jan Van Mill, George M. Reed

TopologyanditsApplications136(2004)37–85
www.elsevier.com/locate/topol
Open problems in topology
Elliott Pearl
AtlasConferencesInc.,Toronto,ON,Canada
Received22May2003;receivedinrevisedform5June2003
Abstract
Thisisacumulativestatusreporton the1100 problemslistedinthevolumeOpenProblemsin
Topology(North-Holland,1990),editedbyJ.vanMillandG.M.Reed.
2003ElsevierB.V.Allrightsreserved.
MSC:54-06
Keywords:Problems;Solutions
Introduction
This is a cumulative status report on the 1100 problems listed in the volume Open
ProblemsinTopology(North-Holland,1990),editedbyJ.vanMillandG.M.Reed[192].
Thebookisout-of-printbutthepublisherhasmadeitfreelyavailableonline.Thisreportis
acompleterevisionofthesevenstatusreportsthathaveappearedinthejournalTopology
anditsApplications[193–198,221].
Thisreportcontainsa matrix(Figs. 1 and 2) of numbersindicatingthe status of each
problem. On the matrix, a numbered box is shaded if the problem has been answered
absolutely or shown to be independent of ZFC. A numbered box is half-shaded if the
problemhasbeenansweredinpart,foraspecialcase,orconsistently,sincethevolumewas
published.Thereare199fullyshadedboxesand76half-shadedboxes.Itisremarkablethat
three-quartersof the problemsremain openafter thirteen years. We hopethat this report
andtheavailabilityofthebookwillregenerateinterestintheseproblems.
E-mailaddress:[email protected](E.Pearl).
0166-8641/$–seefrontmatter 2003ElsevierB.V.Allrightsreserved.
doi:10.1016/S0166-8641(03)00183-4
38 E.Pearl/TopologyanditsApplications136(2004)37–85
Dow’squestionsbyA.Dow
Problem2.FindnecessaryandsufficientconditionsonacompactspaceXsuchthatω×X
has remote points. In the notes to this problem, Dow conjectured that there is a model
satisfying that if X is compactand ω×X has remote points then X has an open subset
with countable cellularity. However, Dow [93] showed that there is a compact, nowhere
c.c.c.spaceXsuchthatω×Xhasremotepoints.
Problem5.Yes.J.BakerandK.Kunen[15]provedthatifκisaregularcardinal,thenthere
isaweakPκ+-pointinU(κ),thespaceofuniformultrafiltersonκ.Problem5onlyasked
forthecaseκ=ω1.TheweakPκ+-pointproblemisstillopenforsingularcardinalsκ.
Problem8.Isthereaclopensubsetofthesubuniformultrafiltersofω whoseclosurein
1
βω1 isitsonepointcompactification?Yes,underPFA(S.Todorcˇevic´[274,§8]).
Problem 9. A. Dow and J. Vermeer [100] provedthat it is consistent that the σ-algebra
ofBorelsetsoftheunitintervalisnotthequotientofanycompleteBooleanalgebra.By
Stoneduality,thereisabasicallydisconnectedspaceofweightcthatcannotbeembedded
intoanextremallydisconnectedspace.
Problem 13. Is every compact space of weight ω homeomorphicto the remainder of a
1
ψ-space?A.DowandR.Frankiewicz[94]showedthatanegativeanswerisconsistent.
Problem14(A.Błaszczyk).Istherea compactc.c.c.spaceofweightc whosedensityis
notlessthanc?M.RabusandS.Shelah[227]provedthateveryuncountablecardinalcan
bethetopologicaldensityofac.c.c.Booleanalgebra.
Problem 16. Does countable closed tightness imply countable tightness in compact
spaces? I. Juhász and Z. Szentmiklóssy [143] provedthat if κ is an uncountableregular
cardinaland a compactHausdorffspace X containsa free sequence of length κ, then X
alsocontainssuchasequencethatisconvergent.
Stepra¯ns’sproblemsbyJ.Stepra¯ns
Problem 19. Yes, there is an ω-Toronto space. An α-Toronto space is a scattered space
ofCantor–Bendixsonrank α whichis homeomorphictoeach ofits subspacesofrankα.
G.GruenhageandJ.Moore[128]constructedcountableα-Torontospacesforeachα(cid:1)ω.
Gruenhagealso constructedconsistent examplesof countableα-Toronto spaces for each
α<ω .
1
Problem20.Yes,J.Stepra¯nsconstructedahomogeneous,idempotentfilteronω.
Problem23.SolvedbyA.Krawczyk.
E.Pearl/TopologyanditsApplications136(2004)37–85 39
Problem24.SolvedbyA.Krawczyk.
Problem26.No,S.Schuder[245]provedthatI(2)(cid:1)[0,1]A(2)doesnothold.Thegraph
G={(x,x− 1): 1 (cid:1)x(cid:1) 2}on[0,1]canbecolouredbyf :[0,1]→{0,1},f(x)=1iff
3 3 3
1 (cid:1)x< 2,butthereisnoA(2)-colouringg:[0,1]→A(2)forG.
3 3
Problem 30. If every autohomeomorphism of N∗ is somewhere trivial, is then every
autohomeomorphismtrivial?ThisisthesameasProblem205.S.Shelah[251,IV]proved
thatitisconsistentthatallautohomeomorphismsaretrivial.S.ShelahandJ.Stepra¯ns[258]
provedthatitisconsistentwithMAω1 thatthereisanontrivialautohomeomorphismofN∗,
yetallautohomeomorphismsaresomewheretrivial.In[259],theyprovethatMAdoesnot
implythatallautohomeomorphismsaresomewheretrivial.
Tall’sproblemsbyF.D.Tall
Problem 43. Yes, G. Gruenhage and P. Koszmider [126] constructed a consistent coun-
terexample to the Arhangel’ski˘ı–Tall problem: a locally compact, normal, metacompact
spacethatisnotparacompact.
Problem44.SeeS.Watson’sProblem92.
Problem 48. If κ is a singular strong limit cardinal and X is a < κ-collectionwise
Hausdorffand normal(or countablyparacompact)space of character <κ, is then X is
κ-collectionwiseHausdorff?UnderSCH,N.Kemoto[156]provedthisconjecture.
Problem49.P.Szeptycki[270]provedthatthereareEastonmodelswherefirstcountable,
ℵ -para-LindelöfspacesarecollectionwiseHausdorff.
1
Problem50.T.LaBergeandA.Landver[172]provedfromasupercompactcardinalthatit
isconsistentthatfirstcountable,<ℵ -collectionwiseHausdorffspacesareweakly(cid:1)ℵ -
2 2
collectionwiseHausdorff.
Problem 57. Yes, there is a ZFC example of a screenable normal space that is not
collectionwise normal. By a theorem of M.E. Rudin [233], it suffices to construct a
screenablenormalspacethatisnotparacompact.Z.Balogh[20]constructedsuchaspace.
Balogh’sexampleprovidesapositiveanswertoProblem119.
ℵ ℵ
Problem 63. Does 2 0 <2 1 imply there is an S-space? (or an L-space?) T. Eisworth,
ℵ ℵ
P.NyikosandS.Shelah[115]provedthatitisconsistentwith2 0 <2 1 thatthereareno
locallycompact,firstcountableS-spaces.
Problem 66. P. Larson and F.D. Tall [173] proved that there is a c.c.c. partial order
forcingthateveryhereditarilynormal,firstcountablespacesatisfyingthecountablechain
conditionishereditarilyseparable.
40 E.Pearl/TopologyanditsApplications136(2004)37–85
ProblemsIwishIcouldsolvebyS.Watson
Problem69. A. Dow [92]showed thatit is consistentandindependentof CH that every
+
normalspaceofcharacteratmostc iscollectionwiseHausdorff.Thisprovidesanegative
answertobothProblems69and70.
Problem70.No,seeProblem69.
Problem77.SeeF.D.Tall’sProblem48.
Problem84.No,D.Shakhmatov,F.D. TallandS.Watson[248]constructeda consistent
exampleofanormalMoorespacewhichisnotsubmetrizable.Whetherapositiveanswer
canbeestablishedwithoutusingalargecardinalremainsopen.Also,Tall[271]hasshown
that under the assumption of a supercompact cardinal, there is a model of set theory in
whichallnormalMoorespacesaresubmetrizable,butinwhichthereexistnonmetrizable
normalMoorespaces.
Problem85.SeeF.D.Tall’sProblem43.
Problem86.Arecountablyparacompact,locallycompact,metacompactspacesparacom-
pact? The counterexampleof G. Gruenhage and P. Koszmider to the Arhangel’ski˘ı–Tall
problem(seeProblem43)iscountablyparacompact.
Problem87.Yes.G.GruenhageandP.Koszmider[127]showedthat,underMAℵ ,normal,
1
locallycompact.meta-Lindelöfspacesareparacompact.
Problem88.DoesZFCimplythatthereisaperfectlynormal,locallycompactspacewhich
isnotparacompact?P.LarsonandF.D.Tall[173]provedthatifitisconsistentthatthereis
asupercompactcardinal,thenitisconsistentthateveryperfectlynormal,locallycompact
spaceisparacompact.
Problem 92. Are normal, locally compact, locally connected spaces collectionwise
normal?Z.Balogh[18]showedthatitisconsistent,relativetotheexistenceofacompact
cardinal,thatlocally compact,normalspaces are collectionwisenormal.It remainsopen
whetherlargecardinalsareneededtoestablishapositiveanswer.
ℵ ℵ
Problem94.Does2 0 <2 1 implythatseparable,firstcountable,countablyparacompact
spaces are collectionwise Hausdorff? R. Knight [159] announced that there is model of
2ℵ0 <2ℵ1 such that there exists an uncountable subset of R that is a ∆-set. Note that
in such a model there must be dominatingfamily in ω1ω of cardinality 2ω. The cardinal
arithmeticofKnight’smodelseemstobeflexibleenoughtoobtainaone-to-onecontinuum
function.(The topologicalexample is the tangent-disk space over this subset of R; such
a tangent-disk space is countably paracompact iff the subset is a ∆-set.) This provides
negative answers to both Problems 94 and 96. In [158], Knight constructed a model of
ZFCinwhichthereisasubsetofRthatis∆-setbutnotaQ-set.
E.Pearl/TopologyanditsApplications136(2004)37–85 41
Problem96.Ifthecontinuumfunctionisone-to-oneandX isacountablyparacompact,
firstcountablespace,thenise(X)(cid:1)c(X)?No,seeProblem94.
Problem97.Does ♦∗ imply thatcountablyparacompact,first countablespacesare ℵ -
1
collectionwise Hausdorff? Yes, K. Smith and P. Szeptycki [264] showed that, assuming
♦∗, paranormal spaces of character (cid:1)ℵ are ω -collectionwise Hausdorff. A space is
1 1
paranormal if every countable discrete collection of closed sets can be expanded to a
locally finite collection of open sets. Both countably paracompact spaces and normal
spacesareparanormal.
Problem 99. No, P. Nyikos [212] claimed that if there is a Souslin tree then there is a
collectionwiseHausdorff,Aronszajntreewhichisnotcountablyparacompact.
Problem 104. W. Fleissner [119, §3] gave a repaired construction of the space Son of
George,whichiswhatWatsonreallywantedinProblem104.
Problem 110. Is it consistent that meta-Lindelöf, collectionwise normal spaces are
paracompact? No. Z. Balogh [21] constructed a hereditarily meta-Lindelöf, hereditarily
collectionwise normal, hereditarily realcompact Dowker space. Balogh also constructed
a meta-Lindelöf, collectionwise normal, countably paracompact space which is not
metacompact.
Problem113.Z.BaloghgaveaZFCconstructionofaDowkerspacewhichishereditarily
normal and scattered of height ω. This gives affirmative answers to Problems 113, 114,
and115,whichaskforaZFCexampleofaDowkerspacethatis,respectively,hereditarily
normal,σ-discrete,andscattered.See[23]foranexpositionofBalogh’stechnique.
Problem114.SeeProblem113.
Problem115.SeeProblem113.
Problem116.Yes(Z.Balogh[20]).SeeProblem57.
Problem 132. Under CH, W.L. Saltsman [243] constructed a nondegenerate connected
CDHsubsetoftheplanewhichhasarigidopensubset.
Problem134.UnderCH,W.L.Saltsman[242]constructedaconnectedCDHsubsetofthe
planewhichisnotSLH.
Problem141.In the discussionbeforethe statementof Problem142,Watson stated that
ifoneforceswithaSouslintree,thenonecanmakeacollectionwisenormalspaceintoa
nonnormalspace.WatsonnowretractssuchaclaimandthisbecomesProblem1411.
2
Problem 142. Can Cohen forcing make a collectionwise normal space not normal?
W. Fleissner, T. LaBerge and A. Stanley [120] described a construction that takes any
42 E.Pearl/TopologyanditsApplications136(2004)37–85
normal space X and outputs a normal superspace T such that T becomes nonnormal
after adding one Cohen real if and only if X is a Dowker space. A similar construction
applied to Rudin’s box productDowker space yields a collectionwise normal space that
becomesnonnormalaftertheadditionofoneCohenreal.Thisprovidesnegativeanswers
toProblems142,143and144.R.Grunberg,L.JunqueiraandF.D.Tall[129]showedthat
ifXisnormalbutnotnormalafteraddingoneCohenrealthenXisaDowkerspace.
Problem143.CanoneCohenrealkillnormality?SeeProblem142.
Problem144.Isthere,inZFC,ac.c.c.partialorderwhichkillscollectionwisenormality?
Yes. R. Grunberg, L. Junqueira and F.D. Tall [129] showed that any strengthening of
the topology on the real line which is locally compact, locally countable, separable and
collectionwisenormalisanexampleofacollectionwisenormalspacewhichcanbemade
nonnormalbyc.c.c.forcing.TheEric(vanDouwen)Lineissuchastrengthening.
Problem145.Cancountablyclosed,cardinal-preservingforcingmakeanonnormalspace
normal?R.Grunberg,L.JunqueiraandF.D.Tall[129]gaveaconsistentanswer.Suppose
thereisanuncountableregularκ suchthatκ<κ =κ. Thenthereis anonnormalspace X
andacountablyclosed,cardinal-preservingP suchthatP forcesXtobenormal.
Problem146.Canc.c.c.forcingmake a nonnormalspace metrizable?Yes, W. Fleissner
produced,inZFC,ac.c.c.forcingwhichturnsanonnormalspaceintoametrizablespace.
Problem147. Is there, in ZFC, a cardinal-preservingforcing which makes a nonnormal
spacemetrizable?SeeProblem146.
Problem149.Doescountablyclosedforcingpreservehereditarynormality?No.R.Grun-
berg,L.JunqueiraandF.D.Tall[129]showedthataddingaCohensubsetofω withcount-
1
able conditions will destroy the normality of a non-ℵ -collectionwise Hausdorff space.
1
In particular,this countablyclosed forcingdoes notpreserve the hereditarynormality of
Bing’sExampleG.
Problem151.A.Dow[91,Theorem7.28]provedthatitispossibletolowerthedensityof
aspacewithcardinal-preservingforcing.Dow’smethodusesameasurablecardinal.
Problem153.A.Dow[91,Theorem7.29]provedthatitispossibletomakearegular(or
afirstcountableHausdorff)non-LindelöfspaceLindelöfwithcardinal-preservingforcing.
Dow’smethodusesameasurablecardinal.
Problem 160. D. Shakhmatov and M. Tkachenko [249] proved that the existence of a
ℵ
compactHausdorffspaceofsize2 0 thatisT1-complementarytoitselfisbothconsistent
with and independent of ZFC. They also constructed, in ZFC, a countably compact
ℵ
Tychonoffspaceofsize2 0 whichisT1-complementarytoitselfandacompactHausdorff
ℵ
space of size 2 0 which is T1-complementary to a countably compact Tychonoff space.
ThisprovidescompletesolutionstoProblems160and161.
E.Pearl/TopologyanditsApplications136(2004)37–85 43
Problem161.SeeProblem160.TheexistenceoftwoinfiniteT -complementarycompact
1
HausdorffspaceswasannouncedtoWatsonbyB.Aniszczykin1989,buttheexamplehas
neverbeenpublished.
Problem162.M.Tkacˇenko,V.Tkachuk,R.WilsonandI.Yashchenko[273]provedthat
proved that no T -complementarytopology exists for the maximal topology constructed
1
byE.K.vanDouwenontherationalnumbers.
Problem167.Whichtopologyonasetofsize nhasthelargestnumberofcomplements?
The natural conjecture is that the partial order (T topology) with the least number of
0
complementsis the partialorder made up of an antichainand two comparableelements,
the partial order (T topology) with the greatest number of complements is the partial
0
order made up an antichain and a maximum and a minimum. These conjectures remain
openalthoughJ.I.BrownandS.Watsonhaveshowntheyareasymptoticallycorrect.See
[44–46].
Problem172.Yes,J.HardingandA.Pogel[135]provedthateverylatticewith1and0is
embeddableinthelatticeoftopologiesofsomeset.
Problem 175. The problem should have stated “open intervals” instead of “open sets”.
Y.-Q.QiaoandF.D.Tall[272]showedthattheexistenceofalinearorderingascorrectly
statedisequivalenttotheexistenceofaperfectlynormalnonmetrizablenon-Archimedean
space(i.e.,anarchvillain).See Problem374.Qiao[225]showedthatthereisa modelof
MA+¬CH in which there is an archvillain (and yet no Souslin lines); this answers the
secondhalfoftheproblemnegatively.
Problem 176. Is there a topological space in which the nondegenerate connected sets
are precisely the cofinite sets? G. Gruenhage [125] gave several consistent examples.
AssumingMA,therearecompletelyregularaswellascountableexamples.AssumingCH,
thereisaperfectlynormalexample.
Weiss’squestionsbyW.Weiss
Problem179.P.Koszmider[161]provedthatthereisanuncountableproductofnontrivial
compactconvexsubsetsofnormedlinearspacesthatfailstohavethecompleteinvariance
property.
Problem180.Isthereaboundonthesizeofcountablycompact,locallycountable,regular
spaces? Such spaces are called good; if, in addition, separable subspaces are at most
countable then they are called splendid. I. Juhász, Z. Nagy and W. Weiss [141] proved
that there are splendid spaces of size ℵn for each n∈ω, and if V= L then there are
arbitrarilylargesplendidspaces.P.NyikosobservedthattheV=Lresultsonlyrequirethat
thecoveringlemmaholdsovertheCoreModelandvariousotherweakhypotheses.Onthe
otherhand,I.Juhász,S.ShelahandL.Soukup[142]showedthatiftheChangConjecture
44 E.Pearl/TopologyanditsApplications136(2004)37–85
variant(ℵω+1,ℵω)→(ω1,ω)holdsthentherearenosplendidspacesofcardinalitygreater
than ℵω. Also, ZFCis enoughtoshow thatthere arenoneofcardinalityexactlyequalto
ℵ oranyothersingularcardinalofcountablecofinality.Infactthisisthehurdleforwhich
ω
somethingbeyondZFCisneeded:inZFConecaneasilygofromℵn toℵn+1 butthejump
atℵ causesmajorcomplications.TheChangConjecturevarianthasnothingtosayabout
ω
goodspacesandtheproblemforgoodspacesisstillopen.Itwouldentailsolvinganother
problem:Istherearegularseparable,locallycountable,countablycompact,noncompact
space?ThisisaspecialcaseofthetitleproblemofthesectionbyP.Nyikosandnowcarries
arewardofaUS$1000:IsthereaZFCexampleofaseparable,firstcountable,countably
compact,noncompactHausdorffspace?
Problem181.LetXbearegularspaceandletλbetheleastcardinalsuchthatsomeopen
coverofsizeλhasnofinitesubcoverandletκ betheleastcardinalsuchthateverypoint
hasaneighbourhoodofsize<κ.Howareλ,κ and|X|related?P.Nyikosremarksthatif
κ =λ isstronglyinaccessible,|X| canbearbitrarilylarge.TakeD discreteasbigasyou
wishandletS bethespaceofultrafiltersµ∈βD forwhichthereisE∈µwith|E|<λ.
Asimilarexampleworkswheneverκ>22<λ.
Problem185(A.HajnalandI.Juhász).DoeseachLindelöfspaceofcardinalityℵ have
2
a Lindelöf subspace of cardinality ℵ ? P. Koszmider and F.D. Tall [164]used countably
1
closedforcingtoconstructanexampleofasubspaceofthecountableboxtopologyonthe
product of ℵ copies of the two-point discrete space. Their example is an uncountable
2
Lindelöf T P-space with no Lindelöf subspaces of cardinality ℵ (actually, with no
2 1
convergentω1-sequences).TheyshowedthattheirconstructionwillnotworkZFC.
Perfectlynormalcompacta,cosmicspaces,andsomepartitionproblemsby
G.Gruenhage
Therehavebeennosolutionstothesixproblemsinthissection.
OpenproblemsonβωbyK.P.Hart,J.vanMill
Problem 201. Yes. S. Shelah and J. Stepra¯ns [259] showed that it is consistent with
MA+¬CHthatatotallynontrivial(=nowheretrivial)automorphismexists.
Problem205.Theanswerisindependent.ThisisthesameasProblem30.
Problem210.Theanswerisindependent.OntheonehanditisconsistentwithMA+¬CH
∗
thateveryautohomeomorphismofω istrivial[257].Inthismodeltheanswerisnegative:
there are c autohomeomorphismsbut 2c P -points. On the other hand, J. Stepra¯ns [268]
c
showedtheconsistencyofapositiveanswer.
E.Pearl/TopologyanditsApplications136(2004)37–85 45
Problem216.No.A.Dow[95]showedthatitisconsistentwith¬CHthatclosedsubsets
ofthespaceω∗ areexactlythecompactzero-dimensionalF-spacesofweight(cid:1)c.
Problem 217. This is a special case of Problem 9, which was solved in the negative by
A.DowandJ.Vermeer.SeeProblem9.
∗
Problem221.Iseverynowheredensesetinω ac-set?SeeProblem222.
∗
Problem222.Isthereamaximalnowheredensesubsetin ω ?Inthebook,itwasnoted
thatnoc-setcanbeamaximalnowheredensesetandthatitisconsistentthateverynowhere
densesetisac-set(see[16]).P.Simon[262]showedthatProblem221andProblem222
∗
areequivalent:everynowheredensesetinω isac-setifandonlyiftherearenomaximal
∗
nowheredensesubsetsin ω . In1975,A.I.Veksler[286]hadshownthatnowheredense
P-setsarenotmaximal.
Problem223.A.Bella,A.BłaszczykandA.Szyman´ski[33]provedthatifXiscompact,
extremallydisconnected,withoutisolatedpointsandofπ-weightℵ orlessthenX isan
1
ARforextremallydisconnectedspacesiff X istheabsoluteofoneofthefollowingthree
spaces:theCantorset,theCantorcubeω12,orthesumofthesetwospaces.Thisprovides
anegativeanswertoProblem223underCH.
∗
Problem226.Isitconsistentthatthereis,uptopermutation,onlyoneP-pointinω ?Yes.
SeeS.Shelah[251,XVIII§4].
Problem228.Is thereis a p in ω∗ suchthateverycompactificationofω∪{p} contains
ω∗? Yes (A. Dow [101]).Take a map f from βω onto Ic, take a closed set A such that
f (cid:2)AisirreducibleandfinallytakeanypinA.
Problem 229. This problem was partially solved by E. Coplakova and K.P. Hart [69].
They proved that if the bounding number b equals c then there exists a point p in Q∗
(the Cˇech–Stone remainder of the space of rational numbers) such that p generates an
ultrafilterintheset-theoreticsenseonQandsuchthatphasabaseconsistingofsetsthat
arehomeomorphictoQ.
Problem231.(MA+¬CH)ArethereaHausdorffgapG=(cid:17)(fα,gα): α∈ω1(cid:18)anda(P-
point,selectiveultrafilter)p suchthatp⊆IG+?HereIG istheinducedgap-ideal,i.e.,the
ideal on ω of subsets over which the gapis filled, IG={M: (∃h∈Mω)(∀α) fα(cid:2)M <∗
h<∗gα(cid:2)M}.S.Kamo[145]provedthatifV isobtainedfromamodelofCHbyadding
CohenrealstheninV anidealisagap-idealiffitis(cid:1)ω1-generated.Also,CHimpliesthat
anynontrivialidealisagap-ideal.Inapreprint,Kamo[146]showed,underMA+¬CH,that
foreveryHausdorffgapGtherearebothselectiveultrafiltersandnon-P-pointsconsisting
of positive sets (with respect to the gap-ideal IG). Also, under MA+¬CH, there is a
selectivenon-P -pointthatmeetseverygap-ideal.
ω2
46 E.Pearl/TopologyanditsApplications136(2004)37–85
Problem 237. D. Strauss [269] showed that (cid:17)βN,+(cid:18) cannot be embedded in (cid:17)N∗,+(cid:18).
Specifically,ifφ:βN→N∗ isacontinuoushomomorphismthentheimageofφ mustbe
finite.
Problem 240. Yes. I. Farah [117] proved a generalization of Problems 240 and 241:
AssumeZ isaβN-space,X iscompact,κ isanarbitrarycardinalandf :Xκ →Z.Then
Xκ canbecoveredbyfinitelymanyclopenrectanglessuchthatf dependsonatmostone
coordinateoneachoneofthem.
Problem241.SeeProblem240.
Problem244.S.ShelahandO.Spinas[255]provedthatforeverynonecanhaveamodel
inwhichwn((ω∗)n)>wn((ω∗)n+1).ThisprovidessomeinformationaboutProblem244.
Problem245.Yes,tothesecondpartoftheproblem.S.ShelahandO.Spinas[256]showed
thatwn(ω∗)>wn(ω∗×ω∗)isconsistent.
Problem264.Thisproblemissolved.AresultduetoA.Dow[89]showsthatunder¬CH
there are always p and q for which I and I are not homeomorphic.A. Dow and K.P.
p q
Hart[96]showedthatunderCHanytwocontinuaIp andIq arehomeomorphic.Itfollows
thatthestatementallcontinuaIp arehomeomorphicisequivalenttoCH.
Problem 265. Are there cutpoints in I other than the points f for f :ω →I? This
p p
problem is solved; as indicated in the paper the answer is yes under MAcountable [17].
A.DowandK.P.Hart[97]confirmedtheconjecturethattheanswerisnoinLaver’smodel
fortheBorelConjecture.
Problem 266. A. Dow and K.P. Hart [98] have shown that there are least 14 different
subcontinuaof βR\R: 10 in ZFC alone,fourmoreunderCH or atleast six moreunder
¬CH.
Onfirstcountable,countablycompactspacesIIIbyP.J.Nyikos
Problem 286. No. T. Eisworth and J. Roitman [116] showed that CH is not enough to
implytheexistenceofanOstaszewskispace.
Problem 287. Yes. T. Eisworth [114] showed that it is consistent with CH that first
countable, countably compact spaces with no uncountable free sequences are compact.
Consequently,itisconsistentwithCHthatperfectlynormal,countablycompactspacesare
compact.
Problem 292. M. Rabus [226] proved that it is consistent with MA and t=ℵ2 =c that
every ⊂∗-increasing ω -sequence in P(ω) is the bottom part of some tight (ω ,ω∗)-
1 1 2
gap. In the discussion after Axiom 5.6 (p. 151), P. Nyikos wrote: “Of course, the really