compact spaces, compact cardinals, and elementary submodels

l

ehes

GCHy; seeoremialdefines

hat

Topology and its Applications 130 (2003) 99–109

www.elsevier.com/locate/topo

Compact spaces, compact cardinals,and elementary submodels

Kenneth Kunen1

University of Wisconsin, Madison, WI 53706, USA

Received 4 February 2002; received in revised form 11 July 2002

Abstract

If M is an elementary submodel andX a topological space, thenXM denotes the setX ∩M giventhe topology generated by the open subsets ofX which are members ofM . Call a compact spacsquashableiff for someM , XM is compact andXM �= X. The first supercompact cardinal is tleastκ such that all compactX with |X| � κ are squashable. The firstλ such thatλ2 is squashable ilarger than the first 1-extendible cardinal. 2002 Elsevier Science B.V. All rights reserved.

MSC:54D30; 03E55

Keywords:Elementary submodel; Supercompact cardinal; Extendible cardinal

1. Introduction

Elementary submodels were first used in set theory by Gödel [5,6] to prove thefrom V = L. They have now become a standard tool in combinatorics and topologDow [4]. In using them, one applies the downward Löwenheim–Skolem–Tarski Theto the universe,V , to get a smallM ≺ V . This technique simplifies many combinatorclosure arguments. When this technique is formalized in ZFC, one cannot actually“M ≺ V ”, so one replaces it by “M ≺ H(θ)”, where θ is a regular cardinal which ilarge enough thatH(θ) contains all objects under study.H(θ) can replaceV in mostapplications becauseH(θ) |= ZFC− P (that is, ZFC minus the power set axiom), so tmany elementary combinatorial arguments can be carried out withinH(θ).

E-mail address:[email protected] (K. Kunen).1 Author partially supported by NSF Grant DMS-0097881.

0166-8641/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0166-8641(02)00214-6

100 K. Kunen / Topology and its Applications 130 (2003) 99–109

Bandlow [2,3] used these methods prove theorems about Corson compacta. Moreting a

theact

since to

can

erlyfocustweeniven

recently, Junqueira and Tall [7–9,13] have proved a number of general results relacompact spaceX to X ∩M. We shall follow their notation here, and shall explorerelationship between properties ofX ∩M and large cardinals (primarily, supercompand extendible—see Kanamori [10, §§22,23]).

Formally, a topological space is a pair(X,T ), whereT is a topology (family of opensets) on the setX. ThenX = ⋃

T , soT determinesX, while X does not determineT .Nevertheless, we shall follow the usual abuse of notation and refer to(X,T ) asX. Thisabuse extends into elementary submodels, but we must exercise a bit of care here,applyM ≺H(θ) toX, we needT ∈M, not justX ∈M.

Definition 1.1. Suppose that(X,T ) ∈M ≺H(θ). Then:

• XM denotes the space(X ∩M,TM), whereTM is the topology onX ∩M which has,as a base,{U ∩M: U ∈ T ∩M}.

• X ∩M denotes the setX ∩M with the subspace topology, denoted byT �M.

Note thatTM ⊆ T �M (that is,XM is coarser thanX ∩M). This paper will focus oncompact Hausdorff spaces. For these, if(X,T ) ∈M andX ⊆M, thenTM = T �M = T ,sinceTM is coarser thanT and is Hausdorff. We shall investigate the question: Whenwe haveXM compact andX �⊆M? Observe (see [7]) that compactness ofXM impliescompactness ofX.

Consider the exampleX = α + 1, with the usual compact order topology. ThenX ∩Mis compact only in the trivial case thatα+1 ⊂M (since the leastγ �=M is a limit ordinal),butXM is always compact, and is homeomorphic to the order topology on the setX ∩M(whose type is some successor ordinal). More generally,

Theorem 1.2 (Junqueira and Tall [9]).Assume that(X,T ) ∈ H(θ), X is compactHausdorff, andθ is regular. Then the following are equivalent.

(1) X scattered.(2) XM is compact for allM such that(X,T ) ∈M ≺H(θ).(3) XM is compact for some countableM such that(X,T ) ∈M ≺H(θ).

If one wants the finerX ∩M to be compact for countableM, one needs the strongassertion thatX is scatteredand Corson compact; equivalently, by Alster [1], strongEberlein compact. See Section 3 for some further remarks on this. The mainof this paper will be on more general compact spaces and the relationships becompactness ofXM and large cardinals. We summarize our results now, with proofs gin Section 2.

Definition 1.3. Let (X,T ) be compact Hausdorff.

• (X,T ) is squashableiff for some regularθ and someM: (X,T ) ∈M ≺ H(θ), XMis compact, andX �⊆M.

K. Kunen / Topology and its Applications 130 (2003) 99–109 101

• (X,T ) is κ-squashableiff for some regularθ and someM: (X,T ) ∈M ≺H(θ), XM

not

can

ongert order

srther

t 2s and

ing

th0,

s

is compact, and|M|< κ .

See Definition 3.5 for the notion of “squashing”. Note thatκ-squashability is trivialwhen κ > |X| and implies squashability whenκ � |X|. By the following lemma,squashability is atopologicalproperty (i.e., invariant under homeomorphism), and issensitive to the specificθ used:

Lemma 1.4. Let (X,T ) be compact Hausdorff.

• (X,T ) is squashable iff for all(Y,U) homeomorphic to(X,T ), all regular θ , and allsetsa: If Y,U, a ∈ H(θ), then there is anM ≺ H(θ) such thatY,U, a ∈M, YM iscompact, andY �⊆M.

• (X,T ) is κ-squashable iff for all(Y,U) homeomorphic to(X,T ), all regular θ , andall setsa: If Y,U, a ∈H(θ), then there is anM ≺H(θ) such thatY,U, a ∈M, YM iscompact, and|M|< κ .

We never use the seta in this paper, but in applications of elementary submodels, itencode whatever else besides the topology is needed for the argument.

Theorem 1.5. If (X,T ) is compact Hausdorff andκ is |X|-supercompact then(X,T ) isκ-squashable.

This sharpens a theorem from [13], which used a 2-huge cardinal (a strassumption) to get similar results. By Theorem 1.6, supercompactness is the correcof largeness in Theorem 1.5, although in the special case|X| = κ , all that is needed iweak compactness, notκ-supercompactness (i.e., measurability). This is discussed fuin Section 2.

Theorem 1.6. Supposeσ is such thatλ2 is squashable for everyλ � σ . Then there is asupercompact cardinal� σ .

We writeλ2 for the product space here to distinguish it from the cardinal exponenλ.The firstλ such thatλ2 is squashable is related to the low end of supercompactnesextendibility:

Definition 1.7. For n < ω, let extn(κ,λ) assert that there is an elementary embeddj :R(κ + n)→ R(λ+ n) such thatj (κ)= λ > κ andj�κ is the identity.κ is n-extendibleiff extn(κ,λ) for someλ.

Clearly, this property gets stronger asn gets bigger. Extendibility is interleaved wisupercompactness. Ifκ is 1-extendible, thenκ is the κ th measurable cardinal (see [1Proposition 23.1]). Ifκ is 2κ -supercompact, then there is anA ∈ [κ]κ such that ext1(α,β)for all α,β ∈ A with α < β . If κ is 2-extendible, thenκ is theκ th element of the clas{σ : σ is 2σ -supercompact}.


Theorem 1.8. Let λ be an infinite cardinal and assume thatX is compact Hausdorffle

,

rdinal

ableple, by

all

at forut

.t

t

and squashable and thatχ(p,X) = λ for all p ∈X. Then there are strongly inaccessibcardinalsα < β < κ < λ such thatext1(α,β).

Theorem 1.9. If κ is the first1-extendible cardinal,λ is least such thatλ2 is squashableandσ is the least cardinal such thatσ is 2σ -supercompact, thenκ < λ < σ .

These theorems improve a result from [9], which obtains a strongly inaccessible ca� λ and the existence of 0# from the squashability ofλ2.

Observe that the character assumptions onX in Theorem 1.8 imply that|X| = 2λ (bythe Cech–Pospíšil Theorem and Arhangel’skiı’s Theorem), so thatX will be squashablewhenever some cardinal�λ is 2λ-supercompact. Note that there are many squashspaces of all sizes whose existence does not entail large cardinals. For examTheorem 1.2, a compact scattered spaceX is squashable iff|X| > ℵ0. Likewise, ifX isCorson compact and not scattered, thenX is squashable iff|X|> 2ℵ0, since wheneverMis countably closed,XM =X ∩M will be compact.

By Theorem 1.8, ifκ is the least strong inaccessible such thatR(κ) is a model for “1-extendible cardinals exist”, then alsoR(κ) is a model for “no compact Hausdorff spaceof whose points have the same character is squashable”.

2. Compactness and large cardinals

We begin by proving a strengthening of Lemma 1.4 which also makes it clear tha compact Hausdorff(X,T ), the squashability ofX is equivalent to a statement aboobjects of size|X|, although the original definition involved(X,T ) ∈H(θ), whereθ , andpossibly also|T |, exceeds|X|. To see this, we restate the definition ofXM , replacingT bya base or a subbase (noting that w(X)� |X|), and replacingH(θ) by an arbitrary transitivemodel of ZFC− P:

Definition 2.1. If X is any set andA ⊆ P(X), thenTA denotes the topology onX whichhasA for a subbase.(X,A)∼= (Y,B) means that(X,TA) and(Y,TB) are homeomorphicIf (X,A) ∈M ≺ N , whereN is a transitive model of ZFC− P, thenXM denotes the seX ∩M with the topology which has, as a subbase,{U ∩M: U ∈ A ∩M}.

This is the same topology we get by applying Definition 1.1 to(X,TA) in the case thaTA ∈N . Lemma 1.4 is immediate from the following:

Lemma 2.2. For compact Hausdorff(X,T ), the following are equivalent:

(1) (X,T ) is squashable.(2) For all transitiveN |= ZFC− P, all (Y,B) ∈ N such(Y,B) ∼= (X,T ), and all setsa ∈N , there is anM ≺N such thatY,B, a ∈M, YM is compact, andY �⊆M.

(3) (2) restricted toN with |N | = |X|.(4) (2) restricted toN of the formH(θ) for regularθ .


Also, for each fixedκ , we get the same four equivalents when “squashable” is replaced by

t

s3.uasha-odels

osure

“ κ-squashable” in(1) and “ Y �⊆M” is replaced by “|M|< κ” in (2).

Proof. We shall only prove the “squashable” version of the lemma, since the “κ-squash-able” proof is almost identical. Since(2)⇒ (4)⇒ (1) and (2)⇒ (3) are obvious, it issufficient to prove(3)⇒ (2) and(1)⇒ (3).

For (3)⇒ (2), fix N,Y,B, a as in(2). Let P ≺ N with |P | = |Y | = |X|, Y,B, a ∈ P ,and Y ⊂ P . Let Q be the transitive collapse ofP with j :Q → P the Mostowskiisomorphism, so thatj :Q→ N is an elementary embedding. Letj (Y ) = Y , j (B) = B,andj (a)= a. Since|Q| = |X| and(Y , B)∼= (X,T ) (by compactness ofX), we can apply(3) to Q to get M ≺ Q such thatY , B, a ∈ M, YM is compact, andY �⊆ M . Now, letM = j “M.

For (1)⇒ (3), first apply “squashable” to fix a regularθ and anM such that(X,T ) ∈M ≺ H(θ), XM is compact, andX �⊆M. Assume that(3) fails, and fix (N,Y,B, a, f )which is a counter-example in the sense that(N,Y,B, a) is a counter-example to(2), |N | =|X|, andf :X→ Y is a homeomorphism. Then(N,Y,B, a, f ) ∈H(θ), and the statementhat it is a counter-example can be expressed withinH(θ), so that by(X,T ) ∈M ≺H(θ),we may assume that(N,Y,B, a, f ) ∈ M. But thenM ∩ N ≺ N , Y,B, a ∈ M ∩ N ,YM∩N = YM is compact, andY �⊆M ∩ N (sincef ∈ M), so that(N,Y,B, a, f ) is nota counter-example.✷Proof of Theorem 1.5. Let j :V →W be a|X|-supercompact embedding. ThenW is atransitive class,j is an elementary embedding,j (κ) > |X|, j�κ is the identity, andW |X| ⊂W . Fix N,Y,B, a as in(2) of the “κ-squashable” version of Lemma 2.2, with|N | = |X|,as in (3). Abbreviate the conclusion of(2) as ∃MΦ(κ,M,N,Y,B, a). Now j “N ∈ WandΦW(j (κ), j “N,j (N), j (Y ), j (B), j (a)) holds, whereΦW denotes relativization tothe modelW . Hence, we have also∃MΦW(j (κ),M, j (N), j (Y ), j (B), j (a)), and thus∃MΦ(κ,M,N,Y,B, a). ✷

A similar argument shows that in the case|X| = κ , we do not needκ-supercompactnes(i.e., measurability), but only weak compactness; see Theorems 2.11, 2.12, and 2.1

To reverse Theorem 1.5 and obtain extendibility and supercompactness from sqbility assumptions, we need to obtain elementary embeddings between transitive mwith suitable closure properties. Now, starting from(X,T ) ∈M ≺H(θ), we can letN bethe transitive collapse ofM and get an elementary embeddingj :N → H(θ). If XM iscompact and all points ofX have large character, then we can deduce the correct clproperties ofN by applying the proof of theCech–Pospíšil Theorem:

Definition 2.3. A λ-Cech–Pospíšil treein a spaceX is a treeK = 〈Ks : s ∈ �λ2〉 satisfying:

(1) EachKs is non-empty and is closed inX.(2) s ⊆ t →Ks ⊇Kt .(3) Ks%0 ∩Ks%1 = ∅.(4) If lh(s)= γ , a limit ordinal, thenKs = ⋂

α<γ Ks�α .


Note that when discussing sequencess, t ∈ �λ2, lh(s)= dom(s), ands ⊆ t means that

n-

m

if

s is an initial segment oft .

Theorem 2.4 (Cech and Pospíšil).If X is compact Hausdorff andχ(p,X) � λ for allp ∈X, then there is aλ-Cech–Pospíšil tree inX, and hence|X| � 2λ.

Applied within a transitive modelN , these trees can be used to prove thatP(λ)⊂N :

Definition 2.5. Let N be a transitive model of ZFC− P, (X,T ) ∈ N , andN |= “T is atopology onX”. Then (X,T ) is truly compactiff, in V , the topology onX which hasTfor a base is compact.

Lemma 2.6. LetN be a transitive model ofZFC− P withX,T ,K ∈N . Assume thatN |=“ T is a topology onX andK is aλ-Cech–Pospíšil tree inX”. Assume that(X,T ) is trulycompact. ThenP(λ)⊂N .

Proof. Prove by induction onγ � λ that γ 2 ⊂ N . Assume thatγ is a limit (otherwisethe induction step is trivial), and fixs ∈ γ 2. Then forα < γ , induction gives uss�α ∈ N ,so thatKs�α is defined and is closed inX. Since(X,T ) is truly compact,

⋂α<γ Ks�α is

non-empty, so fixx ∈ ⋂α<γ Ks�α . Then,s = ⋃{t : x ∈Kt & lh(t) < γ } ∈N . ✷

Next, note that these truly compact(X,T ) occur naturally when we collapse an elemetary submodel:

Lemma 2.7. Suppose that(X,T ) ∈M ≺H(θ) andXM is compact. Letj :N →M be theMostowski isomorphism from a transitiveN ontoM, and letj (X) = X and j (T ) = T .ThenN is a transitive model ofZFC− P and(X, T ) is truly compact. Furthermore, ifXhas no isolated points,λ= min{χ(p,X): p ∈X}, andj (λ)= λ, thenP(λ)⊂N .

Proof of Theorem 1.8. We have(X,T ) ∈M ≺ H(θ) with XM compact,X �⊆M, andχ(p,X) = λ for all p ∈ X. Now, follow the notation in Lemma 2.7. Ifλ ⊂ M, wewould haveλ = λ and P(λ) ⊂ M, which would implyX ⊂ M because|X| = 2λ byArhangel’skiı’s Theorem. Thus, ifκ is the first ordinal not inM, thenκ < j (κ) � λ andκ � λ.j :N → H(θ) is an elementary embedding.P(κ) ⊆ P(λ) ⊂ N , so κ is measurable

and hence strongly inaccessible, so thatj (κ) is also strongly inaccessible. Thus, froP(κ)⊂ N andP(j (κ))⊂H(θ), we getR(κ + 1)⊂ N andR(j (κ)+ 1)⊂H(θ). Then,j �R(κ + 1) :R(κ + 1)→R(j (κ)+ 1) establishes that ext1(κ, j (κ)).

Now, j � R(κ + 1) ∈ H(θ) andR(j (κ) + 1) ∈ H(θ) (since|R(j (κ) + 1)| = 2j (κ) �2λ = |X|< θ ), soH(θ) |= ext1(κ, j (κ)), soN |= ∃α < κ [ext1(α, κ)]. Fixing one suchα,and usingR(κ + 1) ∈N , we have ext1(α, κ) is really true (inV , and inH(θ)), so usingjagain, there is aβ < κ such that ext1(α,β). ✷Proof of Theorem 1.9. We already haveκ < λ� σ by Theorems 1.8 and 1.5. But nowj :V →W is a 2σ -supercompact embedding first movingσ , then the squashability ofσ2


implies that(σ2 is squashable)W by Lemma 2.2, since this notion can be expressed with

2 of

sh-

objects of size only 2σ . Hence,λ2 is squashable for someλ < σ . ✷To prove Theorem 1.6, we need the following minor modification of Theorem

Magidor [12]:

Definition 2.8. If A,B are structures for the same language, thenB ≺2 A means thatB isan elementary substructure ofA with respect to all formulas of second order logic.

Lemma 2.9. For any cardinalσ , the following are equivalent.

(1) Some cardinal� σ is supercompact.(2) WheneverA is a structure for a finite language and|A| � σ , there is aB ≺2 A with

|B|< |A|.(3) WheneverA is a structure for a finite language and|A| � σ , there is aB ≺2 A with

B �= A.

Proof. (1)⇔ (2) is from [12] and(2)⇒ (3) is obvious. For(3)⇒ (2), assume(3), andfix A with |A| = λ� σ . WLOG,A = (λ;R1, . . . ,Rn).

Apply (3) to the structure(R(λ + ω); ∈,A) to getM ≺2 R(λ + ω) with A ∈M andM �= R(λ+ω). Then, ifN is the transitive collapse ofM, we getj :N → R(λ+ ω) withj an elementary embedding with respect to second order logic,A ∈ ran(j), andj not theidentity map. Letj (A)= A. ThenA = (λ; R1, . . . , Rn), wherej (λ)= λ.

Sincej is second-order elementary,N must really equalR(λ + ω). It follows thatλ < λ, since there is no non-trivial elementary embedding ofR(λ + ω) into itself(see [10], Corollary 23.14). Now, letB be the restriction ofA to j “λ. ThenB ≺2 A and|B| = λ < |A|. ✷Proof of Theorem 1.6. Assume thatλ2 is squashable for allλ � σ . By Lemma 2.9, itis sufficient to prove that for allλ � σ and all structuresA = (λ;R1, . . . ,Rn), there is aproper elementary substructureB ≺2 A. So, assume, for some fixedλ � σ , someA is acounter-example to this.

Applying squashability, fixθ,M with (X,T ) ∈ M ≺ H(θ), whereX = λ2, XM iscompact, andX �⊆M. Obtainj, λ,N as in the proof of Theorem 1.8. Note thatθ � (2λ)+.By M ≺ H(θ), we may assume that our counter-exampleA is inM. Now, fix B so thatB = j “ λ. SinceP(λ)⊂N , we haveB ≺2 A, andB �= A, sincej “ λ= λ would imply thatX ⊂M. ✷

Finally, we consider the case where|X| itself is inaccessible. Here, to prove squaability of |X|, weak compactness, or even justΠ1

1 -indescribability, suffices:

Definition 2.10. A cardinalκ is Π11 -indescribableiff wheneverϕ is aΠ1

1 sentence,A =(κ;R1, . . . ,Rn), andA |= ϕ, there is anα < κ such thatB := (α;R1�α, . . . ,Rn�α) ≺ A

andB |= ϕ.


This implies thatκ is weakly inaccessible, but not necessarily strongly inaccessible.Many

ive

ly

f

fort toous

f all

ng

slast

Π11 -indescribability plus strong inaccessibility is equivalent to weak compactness.

authors include strong inaccessibility as part of the definition ofΠ11-indescribability;

see [10, §6] for more details.

Theorem 2.11. Assume thatκ = |X| isΠ11 -indescribable and thatX,T , a ∈H(θ), where

(X,T ) is compact Hausdorff andθ is regular. Then there is anM with X,T , a ∈M ≺H(θ) such that|M|< κ ,M ∩ κ ∈ κ , andXM is compact.

Proof. As in (3) ⇒ (2) of Lemma 2.2, it is sufficient to show that for any transitN |= ZFC− P with |N | = κ andY,B, a ∈ N such(Y,B) ∼= (X,T ), there is anM ≺ Nsuch thatY,B, a ∈M, YM is compact,|M|< κ , andM ∩ κ ∈ κ . But since(N,∈) may becoded by relations onκ , and compactness ofY expressed by aΠ1

1 sentence, we may appΠ1

1 -indescribability to getM. ✷Note that if κ = 2ρ , thenκ cannot beΠ1

1 -indescribable, and the spaceX = ρ2 is acounter-example to the theorem. That isρ2 ∈M implies thatρ ∈M (byM ≺H(θ)), andhenceρ ⊂M (byM ∩ κ ∈ κ). But then|M| � 2ρ = κ would follow from compactness oXM (see Lemma 2.7). Without the “M ∩ κ ∈ κ” in the conclusion, we would get anM asin Theorem 2.11 by Theorem 1.5 wheneverκ is above the first supercompact cardinal.

It is consistent for aκ < 2ℵ0 to beΠ11 -indescribable (see [11]), but the theorem

theseκ is trivial by Theorem 1.2, since thenX must be scattered. It is also consistenhave aΠ1

1 -indescribableκ with 2ℵ0 < κ < 2ℵ1, and the theorem does have non-vacucontent in this case. For strongly inaccessibleκ , the theorem fails wheneverκ is not weaklycompact:

Theorem 2.12. Assume thatκ is strongly inaccessible and notΠ11 -indescribable. Then

there is a compact HausdorffX of sizeκ such that noM satisfies(X,T ) ∈M ≺ H(θ),M ∩ κ ∈ κ , andXM is compact.

Proof. Sinceκ is strongly inaccessible and not weakly compact, there is aκ-Aronszajntree, T ⊂ <κ2. Let X be the corresponding Aronszajn line, which is the space omaximal chains inT ; this is a compact LOTS under its lexical ordering. ThenX is compact,|X| = κ (sinceκ is strongly inaccessible), and there are no increasing or decreasiκ-sequences inX. Now, assume that(X,T ) ∈ M ≺ H(θ), γ = M ∩ κ ∈ κ , andXM iscompact. We shall derive a contradiction.

Note thatγ is strong limit cardinal and|X ∩M| = γ . SinceX is a LOTS, the topologyofXM is just the order topology induced by the natural ordering onX∩M, so compactnesof XM implies thatX ∩ M is Dedekind-complete. Let 0 and 1 be the first andelements ofX. X ∩M is not dense inX (since 2γ < κ), so fix p ∈ X \ (X ∩M). Leta = sup([0,p)∩M) andb = inf((p,1] ∩M). Thena < p < b.

If a, b /∈M, then[0,p) ∩M has no least upper bound in the setX ∩M, contradictingthe Dedekind completeness ofX ∩M. So,a ∈M or b ∈M.

Say b ∈ M. Note that[0, b) cannot have a largest element (usingM ≺ H(θ) and[p,b) ∩M = ∅). Let σ be the cofinality of the order type[0, b). Thenσ ∈M andσ < κ


(sinceX is Aronszajn), soσ < γ . Sinceσ ∈M, there is aσ -sequence,�c= 〈cα : α < σ 〉 ↗

t

2.

call

b, with �c ∈M. But then, eachcα ∈M, contradicting[p,b)∩M = ∅. ✷Theorem 2.13. Assume thatκ is strongly inaccessible and notΠ1

1 -indescribable, and thathere are no1-extendible cardinals less thanκ . Then there is a compact HausdorffZ ofsizeκ which is not squashable.

Proof. Let Y0 be the disjoint sum of allλ2 for λ a cardinal less thanκ , and letY be the 1-point compactification ofY0. LetX be aκ-Aronszajn line, as in the proof of Theorem 2.1LetZ be the disjoint sum ofX andY .

If (Z,T ) ∈M ≺ H(θ) andZM is compact, thenλ ⊂M wheneverλ < κ andλ ∈M(sinceλ2 is not squashable by Theorem 1.9). It follows thatM ∩ κ is an ordinal� κ . ButM ∩ κ ∈ κ would yield a contradiction, as in the proof of Theorem 2.12, andM ∩ κ = κimplies thatZ ⊂M. ✷

3. Remarks on Corson compacta

A Corson Compactumis a spaceX homeomorphic to some closedY ⊆ λ[0,1] suchthat {α: yα �= 0} is countable for ally ∈ Y . A Strong Eberlein Compactumis a spaceXhomeomorphic to some closedY ⊆ λ{0,1} such that{α: yα = 1} is finite for all y ∈ Y .By Alster [1], this is equivalent toX being a scattered Corson compactum. We now reBandlow’s characterization of Corson compacta:

Definition 3.1. If (X,T ) ∈M ≺H(θ), thenM separatesX iff for all a, b ∈ X ∩M witha �= b, there is anf ∈C(X) ∩M such thatf (a) �= f (b).

Here,C(X)= C(X,R). Note that for Tychonov spaces, we trivially get such anf whena, b ∈X∩M. ForX compact Hausdorff, Bandlow’s terminology for “M separatesX” was“ϕXM is anM-retraction”.

Theorem 3.2 (Bandlow [2]).Let (X,T ) be compact Hausdorff and(X,T ) ∈H(θ). Thenthe following are equivalent:

(1) X is Corson compact.(2) M separatesX for all M such that(X,T ) ∈M ≺H(θ).(3) {M ∈ [H(θ)]ω: M separatesX} contains a club.

As usual,C ⊆ [I ]ω is club iff C is closed (⋃n∈ω An ∈ C whenever eachAn ∈ C andA0 ⊆

A1 ⊆ · · ·) andunbounded (∀A ∈ [I ]ω ∃B ∈ C[A⊆ B]). Note that{M ∈ [H(θ)]ω: (X,T ) ∈M ≺H(θ)} is always a club.

Corollary 3.3. Assume that(X,T ) ∈H(θ), whereX is compact Hausdorff andθ is regularand uncountable. Then the following are equivalent.


(1) X is strongly Eberlein compact.

.2,

ct

r

t

f

(2) X ∩M is compact for allM such that(X,T ) ∈M ≺H(θ).(3) {M ∈ [H(θ)]ω: X ∩M is compact} contains a club.

Proof. (2) ⇒ (3) is trivial and (1) ⇒ (2) is easy from the definition. For(3) ⇒ (1),observe that wheneverX ∩M is compact, we haveX ∩M =X ∩M, so thatM separatesX. Thus,X is Corson compact by Theorem 3.2. SinceX is also scattered by Theorem 1we have thatX is strongly Eberlein compact by Alster [1].✷

Note that in (3), we cannot replace “contains a club” by the weaker “�= ∅”, or even “isstationary”, as we had in the corresponding (3) of Theorem 1.2:

Example 3.4. There is a scattered compactumX which is not strongly Eberlein compasuch that{M ∈ [H(θ)]ω: X ∩M is compact} is stationary.

Proof. LetS ⊆ ω1 be a stationary set of limit ordinals such thatω1\S is also stationary. Foγ ∈ S, letEγ ⊂ γ be a cofinal set of successor ordinals of order typeω. Define the topologyT0 onω1 by:U ∈ T0 iff Eγ \U is finite for allγ ∈ S∩U . Observe thatT0 is locally compactHausdorff and finer than the usual order topology onω1. Also, γ < ω1 is closed inT0 iffγ /∈ S. Now, letT be the one-point compactification topology onω1 + 1. IfM is countableandM ≺H(θ), thenM∩X = γM ∪{ω1} for some limitγM < ω1. If γM ∈ S, thenX∩M isnot compact, soX is not a strong Eberlein compactum by Corollary 3.3. However,X ∩Mis compact wheneverγM /∈ S, and{M ∈ [H(θ)]ω: γM /∈ S} is stationary. ✷

Finally, we comment on squashings:

Definition 3.5. If (X,T ) is compact Hausdorff,E ⊆ X, andσ is a function fromX ontoE, thenσ is asquashingof X ontoE iff σ ◦ σ = σ andσ is continuous with respect toTonX and some compact Hausdorff topologyT ′ ⊆ T �E onE.

Equivalently,E is a section of some (continuous) map fromX onto some compacHausdorff space. A retraction is the special case whereT ′ = T �E.

Now, suppose that(X,T ) ∈M ≺ H(θ). For x, y ∈ X, definex ∼ y iff f (x) = f (y)for all f ∈ C(X) ∩M, and let[x] = {y ∈ x: x ∼ y}. Note that|[x] ∩ X ∩M| � 1 �|[x] ∩X ∩M|. For the second inequality, useM ≺H(θ) to show that the family of sets othe form{y ∈X ∩M: |f (y)− f (x)|< ε}, whereε > 0 andf ∈C(X) ∩M, has the finiteintersection property.

Following Bandlow [2],M separatesX iff ∀x ∈ X[|[x] ∩X ∩M| = 1], in which casewe have a retractionρ :X → X ∩M , whereρ(x) is the (unique)y ∈ X ∩M such thaty ∼ x. Following Junqueira and Tall [8],XM is compact iff∀x ∈X[|[x] ∩X∩M| = 1], inwhich case we have a squashingσ :X→XM , whereσ(x) is the (unique)y ∈X ∩M suchthaty ∼ x.

If M separatesX andXM is compact, thenρ andσ agree, so thatX∩M is closed inXand is homeomorphic toXM . In particular, this applies whenX is Corson compact andXMis compact. However, for Corson compacta, it is easy to see directly thatX ∩M ∼= XM ,


whether or notXM is compact. Other examples whereX ∩ M ∼= XM are discussed

) 545–

) 347–

1988)

Proc.

SA 25

.6.

rlag,

Pure

) 147–

in [8, §2].

References

[1] K. Alster, Some remarks on Eberlein compacts, Fund. Math. 104 (1979) 43–46.[2] I. Bandlow, A characterization of Corson-compact spaces, Comment. Math. Univ. Carolin. 32 (1991

550.[3] I. Bandlow, On function spaces of Corson-compact spaces, Comment. Math. Univ. Carolin. 35 (1994

356.[4] A. Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (

17–72.[5] K. Gödel, The consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis,

Nat. Acad. Sci. USA 24 (1938) 556–557.[6] K. Gödel, Consistency-proof for the Generalized Continuum-Hypothesis, Proc. Nat. Acad. Sci. U

(1939) 220–224.[7] L.R. Junqueira, Upward preservation by elementary submodels, Topology Proc. 25 (2000) 225–249[8] L.R. Junqueira, F.D. Tall, The topology of elementary submodels, Topology Appl. 82 (1998) 239–26[9] L.R. Junqueira, F.D. Tall, More reflections on compactness, to appear.

[10] A. Kanamori, The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings, Springer-VeBerlin, 1994.

[11] K. Kunen, Indescribability and the continuum, in: Axiomatic Set Theory, Part I, in: Proc. Sympos.Math., Vol. 13, American Mathematical Society, Providence, RI, 1971, pp. 199–203.

[12] M. Magidor, On the role of supercompact and extendible cardinals in logic, Israel J. Math. 10 (1971157.

[13] F.D. Tall, Reflecting on compact spaces, Topology Proc., in press.

compact spaces, compact cardinals, and elementary submodels

Documents