The cardinality of compact spaces satisfying the countable chain condition

Topology and its Applications 174 (2014) 41–55

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Topology and its Applications www.elsevier.com/locate/topol

The cardinality of compact spaces satisfying the countable chain condition Toshimichi Usuba Organization of Advanced Science and Technology, Kobe University, Rokko-dai 1-1, Nada, Kobe 657-8501, Japan

a r t i c l e

i n f o

Article history: Received 28 October 2013 Received in revised form 25 May 2014 Accepted 27 May 2014 Available online 3 July 2014

a b s t r a c t We prove that for a compact Hausdorff space X, if λc(X) < w(X) for every infinite cardinal λ < w(X) and λc(X) < cf(w(X)) for every infinite cardinal λ < cf(w(X)), then Tikhonov cube w(X) [0, 1] is a continuous image of X, in particular the cardinality of X is just 2w(X) . As an application of this result, we consider elementary submodel spaces and improve Tall’s result in [17]. © 2014 Elsevier B.V. All rights reserved.

MSC: 03E55 03G05 54A25 Keywords: Compact space Countable chain condition Dyadic system Elementary submodel space Independent family Precaliber

1. Introduction Throughout this paper, all topological spaces are assumed to be Hausdorff. A cardinal means an infinite cardinal. In this paper we study the cardinality of compact spaces which have relatively small cellular number, in particular the cardinality of compact spaces satisfying the countable chain condition (c.c.c., for short). There are many attempts to determine the cardinality of compact spaces by cardinal functions. The inequality w(X) ≤ |X| ≤ 2w(X) for a compact X is a classical non-trivial result. When X is compact and second countable, it is well known that either |X| is countable, or |X| = 2ω . A famous inequality for the cardinality of compact spaces is |X| ≤ 2χ(X) for a compact X, which is a special case of Arkhangel’ski˘ı’s inequality |X| ≤ 2χ(X)+L(X) for a space X. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2014.05.025 0166-8641/© 2014 Elsevier B.V. All rights reserved.

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T. Usuba / Topology and its Applications 174 (2014) 41–55

The cardinality of compact spaces has been also studied involving continuous maps onto Tikhonov cubes [0, 1] and Cantor cubes κ {0, 1}. It is known that if X is an extremally disconnected compact space, then there is a continuous surjection from X to w(X) {0, 1}, hence X has cardinality just 2w(X) . When X is totally disconnected and compact, the existence of a continuous surjection from X to w(X) {0, 1} is connected with calibers of X, and if w(X) satisfies certain assumptions, one can find such a surjection. See Comfort and Negrepontis (Chapter 5 of [3]) for this topic. We also have to mention Šhapirovski˘ı’s results: κ

Definition 1.1. Let κ be a cardinal and X a topological space. A κ-dyadic system in X is a sequence Fα0 , Fα1 : α < κ such that: (1) Fα0 and Fα1 are pairwise disjoint closed subsets of X.  g(α) (2) For every finite partial function g from κ to {0, 1}, the intersection {Fα : α ∈ dom(g)} is non-empty. Fact 1.2. (Šhapirovski˘ı [14]) Let κ be a cardinal and X a compact space. Then the following are equivalent: (1) X has a κ-dyadic system. (2) There is a closed set F ⊆ X such that πχ(x, F ) ≥ κ for every x ∈ F . (For simplicity, we put πχ(x, F ) = 1 if x is isolated in F .) (3) κ {0, 1} is a continuous image of some closed subset of X. (4) κ [0, 1] is a continuous image of X. Definition 1.3. Let X be a topological space. Define cardinal functions c(X) and i(X) as follows: (1) c(X), the cellular number of X, is the cardinal sup{|U| : U is a cellular family of X} + ℵ0 . (2) i(X), the index of X, is the cardinal sup{κ : κ [0, 1] is a continuous image of X} + ℵ0 . It is clear that i(X) ≤ w(X). Fact 1.4. (Šhapirovski˘ı [13–15]) Let X be a compact space. (1) For a cardinal λ < w(X), if λc(X) = λ, then (2) The inequality w(X) ≤ i(X)c(X) holds.

λ+

[0, 1] is a continuous image of X.

By Šhapirovski˘ı’s theorem, for instance, if X is compact, w(X) = λ+ , and λc(X) = λ, then w(X) [0, 1] is a continuous image of X, so |X| = 2w(X) . However, when the weight of the space is a limit cardinal, Šhapirovski˘ı’s inequality does not tell us the cardinality of the space. In this paper, we improve Šhapirovski˘ı’s theorem as follows: Theorem 1.5. Let X be a compact space. (1) If λc(X) < i(X) for every cardinal λ < i(X) and λc(X) < cf(i(X)) for every λ < cf(i(X)), then i(X) [0, 1] is a continuous image of X. (2) If λc(X) < w(X) for every cardinal λ < w(X) and λc(X) < cf(w(X)) for every λ < cf(w(X)), then w(X) [0, 1] is a continuous image of X. In particular |X| = 2w(X) . Part (2) of Theorem 1.5 can be obtained by (1) together with Šhapirovski˘ı’s inequality w(X) ≤ i(X)c(X) as follows: if i(X) < w(X), then i(X)c(X) < w(X) by the assumption of (2), this contradicts Šhapirovski˘ı’s inequality. Hence i(X) = w(X), and we have a required continuous map by (1).

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Theorem 1.5 also can be seen as a partial answer of sup = max problem for the index number. In fact we prove a somewhat more general result than Theorem 1.5. Definition 1.6. Let X be a topological space. S(X), the Suslin number of X, is the least cardinal κ such that |U| < κ for every cellular family U of X. So X satisfies the c.c.c. ⇐⇒ c(X) = ω ⇐⇒ S(X) = ω1 . Definition 1.7. Let X be a topological space and κ a cardinal. X has precaliber κ if for every family U of open sets with |U| = κ, there is U  ∈ [U]κ such that U  has the finite intersection property. Theorem 1.8. Let κ be a cardinal and μ a regular uncountable cardinal. Suppose λ<μ < κ for every cardinal λ < κ. Let X be a regular space such that w(X) ≥ κ and S(X) ≤ μ. Suppose that either (1) λ<μ < cf(κ) for every cardinal λ < cf(κ), or (2) X has precaliber cf(κ). Then X has a κ-dyadic system. Now Theorem 1.5 is an immediate corollary of Theorem 1.8; suppose X is compact, λc(X) < i(X) for every λ < i(X) and λc(X) < cf(i(X)) for every λ < cf(i(X)). Let μ = c(X)+ . Since S(X) ≤ c(X)+ = μ, we have that λ<μ < i(X) for every λ < i(X) and λ<μ < cf(i(X)) for every λ < cf(i(X)). Applying Theorem 1.8, we have that X has an i(X)-dyadic system, and then there is a continuous surjection from X to i(X) [0, 1] by Fact 1.2. We use combinatorics on Boolean algebras to prove Theorem 1.8. As mentioned before, it is known that for every compact space X with w(X) = ω, either |X| = w(X) or |X| = 2w(X) hold. It is known that such a dichotomy does not hold if w(X) = ω1 . See §4 of Juhász [6]. In Section 3, however, we prove a similar dichotomy of the cardinality of compact spaces with strong limit singular weight: Theorem 1.9. Let X be a compact space. If w(X) is a strong limit singular cardinal and c(X) < w(X), then either |X| = w(X) or w(X) [0, 1] is a continuous image of X, so |X| = 2w(X) . In Section 4, we discuss some limitations of Theorems 1.5, 1.8, and 1.9, and we see that the cardinal arithmetic assumptions in Theorems 1.5 and 1.8 cannot be eliminated. In Section 5, as an application of Theorems 1.5 and 1.9, we will consider elementary submodel spaces, which were introduced by Junqueira and Tall [9]. From now on, θ will denote a sufficiently large regular cardinal. Let Hθ be the set of all sets with hereditary cardinality < θ. Definition 1.10. Let X = X, T  be a topological space. Let M ≺ Hθ be a model with X = X, T  ∈ M . The space XM is defined as the space X ∩ M with the topology generated by the family {O ∩ M : O ∈ T ∩ M } as an open base. See Junqueira [7], Junqueira, Larson and Tall [8], and Junqueira and Tall [9] for basic properties of elementary submodel spaces. Definition 1.11. (Kunen [11]) A compact space X is squashable if there is some M ≺ Hθ such that X ∈ M , XM is compact, but XM = X.

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Junqueira, Larson and Tall [8] showed that every scattered compact space is squashable. In contrast with this result, Kunen [11] showed that if the space κ {0, 1} is squashable for some cardinal κ, then there must exist a large cardinal: Fact 1.12. (Kunen [11]) Let κ be the least cardinal such that κ {0, 1} is squashable. Then κ is an inaccessible cardinal limit of 1-extendible cardinals. Here a cardinal κ is 1-extendible if there is an elementary embedding j : Vκ+1 → Vη+1 for some η > κ with the critical point κ. It is known that a 1-extendible cardinal is a large large cardinal. For instance, a 1-extendible cardinal is a measurable cardinal limit of measurable cardinals. On the other hand, Tall [17] showed that for every non-scattered space X and M ≺ Hθ with X ∈ M , if X has cardinality less than the first inaccessible, XM is compact and satisfies the c.c.c., then XM = X. Hence X is not squashable. The first inaccessible is much less than the least cardinal κ with κ {0, 1} squashable. Using Theorem 1.5, we improve the first inaccessible assumption in his result to the optimal one: Theorem 1.13. Let κ be the least cardinal such that κ {0, 1} is squashable. For every non-scattered space X and every M ≺ Hθ with X ∈ M , if X has cardinality less than κ, XM is compact and satisfies the c.c.c., then XM = X. We refer the reader to Kunen [12] for general background on Set theory, and Koppelberg [10] on Boolean algebras. 2. On Boolean algebras In order to prove Theorem 1.8, let us consider some properties of Boolean algebras. Recall: Definition 2.1. Let B = B, ∧, ∨, −, 0, 1 be a Boolean algebra and μ a cardinal. (1) (2) (3) (4)

I ⊆ B is an antichain in B if x ∧ y = 0 for every distinct x, y ∈ I. B satisfies the μ-c.c. if B has no antichain of cardinality ≥ μ. B has precaliber μ if for every A ∈ [B]μ , there is A ∈ [A]μ which has the finite intersection property. C ⊆ B is a complete subalgebra of B if C is a subalgebra of B and for every X ⊆ C, if the lower bound  X in B exists, then it belongs to C.

Definition 2.2. Let B be a Boolean algebra. F ⊆ B is an independent family in B if for every finite x, y ⊆ F   with x ∩ y = ∅, b∈x b ∧ c∈y −c = 0. Shelah [16] showed that under certain assumptions for κ and μ, for every Boolean algebra B satisfying the μ-c.c., every A ∈ [B]κ includes an independent family of cardinality κ. Now we introduce the following notion. Definition 2.3. Let κ be a cardinal. We say that B satisfies the κ-SDP (Selective Dyadic system Property)     if for every family {Iα , Jα : α < κ} of antichains of B such that Iα = − Jα and Iα = Iβ for α < β < κ, there are S ∈ [κ]κ , bα ∈ Iα , cα ∈ Jα for α ∈ S such that for every finite subsets x, y ⊆ S with   x ∩ y = ∅, α∈x bα ∧ β∈y cβ > 0. Note that if B satisfies the κ-SDP, then every subset of B of cardinality κ includes an independent family of cardinality κ; fix X ∈ [B]κ and a 1–1 enumeration bα : α < κ of X. For α < κ, let Iα = {bα }   and Jα = {−bα }. Iα and Jα are antichains with Iα = bα = −(−bα ) = − Jα . By κ-SDP, we have

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  that S ∈ [κ]κ such that for every finite subsets x, y ⊆ S with x ∩ y = ∅, α∈x bα ∧ β∈y −bβ > 0. Then {bα : α ∈ S} ⊆ X is an independent family of cardinality κ. We prove that under certain assumptions for κ and μ, every complete Boolean algebras satisfying the μ-c.c. also satisfies the κ-SDP. Before we prove this, we sketch the proof in the case where κ is regular. Given antichains {Iα , Jα : α < κ}, we will find a non-zero r ∈ B, a stationary set S ⊆ κ in κ and bα ∈ Iα , cα ∈ Jα for α ∈ S such that for every α ∈ S and non-zero a ∈ B, if a ≤ r and a is an element of the complete Boolean subalgebra generated by {bβ , cβ : β ∈ S ∩ α}, then a ∧ bα > 0 and a ∧ cα > 0. Then we can check that {cα , dα : α ∈ S} is a family as required. Definition 2.4. Let B be a complete Boolean algebra and C a complete subalgebra of B. For x, y ∈ B, let:  • lpr(x, C) = {c ∈ C : c ≤ x}.  • upr(x, C) = {c ∈ C : x ≤ c}. • upr(x, y, C) = upr(x, C) ∧ upr(y, C). Note that the following statements hold: (1) (2) (3) (4)

lpr(x, C), upr(x, C) ∈ C. lpr(−x, C) = − upr(x, C). lpr(x, C) ≤ x ≤ upr(x, C). lpr(x, C) = upr(x, C) if and only if x ∈ C.

The ideas which will be used in the proofs of a series of lemmas are due to Shelah ([16], or Chapter 4, §10, Theorem 10.1 in [10]). Lemma 2.5. Let B be a complete Boolean algebra and C ⊆ B be a complete subalgebra. Fix a ∈ B. Fix   I, J ⊆ B such that I = a and J = −a. (1) For every b ∈ I and c ∈ J, if upr(b, c, C) > 0, then for every x ∈ C with 0 < x ≤ upr(b, c, C), x ∧ b > 0 and x ∧ c > 0. (2) If a ∈ / C, then there are b0 ∈ I and c0 ∈ J such that upr(b0 , c0 , C) > 0. Proof. (1) Let x ∈ C with 0 < x ≤ upr(b, c, C). If x ∧ b = 0, then b ≤ −x, so upr(b, C) ≤ −x since −x ∈ C. Then we have that x ≤ − upr(b, C) ≤ − upr(b, c, C), which contradicts that 0 < x ≤ upr(b, c, C). The same argument shows that x ∧ c > 0 for every x ∈ C with 0 < x ≤ upr(b, c, C). (2) First we claim: Claim 2.6. upr(a, C) =

 b∈I

upr(b, C) and upr(−a, C) =

 c∈J

upr(c, C).

  Proof. We prove only upr(a, C) = b∈I upr(b, C). Let r = b∈I upr(b, C). Note that r ∈ C. The inequality upr(a, C) ≥ r follows from the definitions. If a ∧ −r = 0, then a ≤ −(−r) = r and upr(a, C) ≤ r since r ∈ C. Thus we assume the contrary that a ∧ −r > 0. Note that       −r = − upr(b, C) = − upr(b, C) = lpr(−b, C) . b∈I

lpr(−b, C) ≤ −b for every b ∈ I, so we have

b∈I

b∈I

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0


 lpr(−b, C)

≤a∧

b∈I



On the other hand, since

I = a, we have 0 = a ∧ −



−b.

b∈I



I =a∧

 b∈I

−b. This is a contradiction. 2

Let r = upr(a, −a, C). Then r > 0; since a ∈ / C, we have lpr(a, C) < upr(a, C), and upr(a, C) ∧ upr(−a, C) = upr(a, C) ∧ − lpr(a, C) > 0.  Since upr(a, −a, C) = b∈I,c∈J (upr(b, C) ∧ upr(c, C)), there are b0 ∈ I and c0 ∈ J with upr(b0 , c0 , C) > 0. 2 For a complete Boolean algebra B and C ⊆ B, let B(C) be the minimal complete subalgebra of B with C ⊆ B(C). Lemma 2.7. Let μ be a regular uncountable cardinal, and B be a complete Boolean algebra which satisfies the μ-c.c. Let κ be a regular uncountable cardinal such that λ<μ < κ for every cardinal λ < κ. Let {Iα , Jα :     α < κ} be a family of antichains of B such that Iα = − Jα and Iα = Iβ for α < β < κ. Then there are non-zero r∗ , s∗ ∈ B, S ∈ [κ]κ , bα ∈ Iα , cα ∈ Jα for α ∈ S such that r∗ ∧ s∗ > 0 and for every C ⊆ B of cardinality < κ, if r∗ , s∗ ∈ C then the set



 α ∈ S : r∗ ∧ s∗ = upr bα , cα , B C ∪ {bβ , cβ : β ∈ ∩α}

has cardinality κ. Proof. First note that |Iα |, |Jα | < μ since B satisfies the μ-c.c. For α < κ, fix enumerations bα i : i < |Iα | α of Iα and cj : j < |Jα | of Jα . Fix a sufficiently large regular θ. Since λ<μ < κ for every λ < κ, we can find M ≺ Hθ such that |M | < κ, M ∩ κ ∈ κ, [M ]<μ ⊆ M , and M contains all relevant objects. Let B ∗ = B ∩ M . B ∗ is a subalgebra of B, and since B satisfies the μ-c.c. and M is closed under < μ-sequences, B ∗ is a complete subalgebra of B. Let α∗ = M ∩ κ. Claim 2.8.



/ B∗. Iα∗ ∈

  Proof. Suppose to the contrary that Iα∗ ∈ B ∗ . Let b = Iα∗ . Note that b ∈ B ∗ = B ∩ M . Now, by the    elementarity of M , there is some α ∈ M ∩ κ such that b = Iα . Hence Iα = Iα∗ but α < M ∩ κ = α∗ , which contradicts the choice of the Iα ’s. 2 ∗

∗

∗

α ∗ ∗ α ∗ By Lemma 2.5, there are i∗ < |Iα∗ | and j ∗ < |Jα∗ | such that upr(bα i∗ , cj ∗ , B ) > 0. Let r = upr(bi∗ , B ) ∗ α∗ ∗ ∗ ∗ ∗ ∗ ∗ and s = upr(cj ∗ , B ). Note that r , s ∈ B = M ∩ B and i , j ∈ M . Let S be the set of all α < κ such α that i∗ < |Iα | and j ∗ < |Jα |. For α ∈ S, let bα = bα i∗ and cα = cj ∗ . Note that S, bα , cα : α ∈ S ∈ M . We ∗ ∗ will check that r , s , S, bα , cα : α ∈ S are as required. Suppose to the contrary that there is C ⊆ B of size < κ such that r∗ , s∗ ∈ C, but the set





 α ∈ S : r∗ ∧ s∗ = upr bα , cα , B C ∪ {bβ , cβ : β ∈ S ∩ α}

is non-stationary in κ. We may assume that C ∈ M . Hence we can find a club D ∈ M in κ such that r∗ ∧ s∗ = upr(bα , cα , B(C ∪ {bβ , cβ : β ∈ S ∩ α})) for every α ∈ D. Since D ∈ M and α∗ = M ∩ κ, we have ∗ α∗ ∗ ∗ α∗ ∈ D. Note that bα∗ = bα i∗ and cα∗ = cj ∗ . Moreover, we have B(C ∪ {bβ , cβ : β ∈ S ∩ α }) ⊆ B because C, {bβ , cβ : β ∈ S ∩ α∗ } ⊆ M . Thus we have r∗ = upr(bα∗ , B ∗ ) = upr(bα∗ , B(C ∪ {bβ , cβ : β ∈ S ∩ α∗ })) and s∗ = upr(cα∗ , B ∗ ) = upr(cα∗ , B(C ∪ {bβ , cβ : β ∈ S ∩ α∗ })). This is a contradiction. 2

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Lemma 2.9. Let μ be a regular uncountable cardinal, and κ a regular uncountable cardinal such that λ<μ < κ for every cardinal λ < κ. Then every complete Boolean algebra satisfying the μ-c.c. satisfies the κ-SDP. Proof. Let B be a complete Boolean algebra satisfying the μ-c.c. Take antichains {Iα , Jα : α < κ}. By Lemma 2.7, we can find non-zero r∗ , s∗ ∈ B, S  ∈ [κ]κ , bα ∈ Iα , cα ∈ Jα for α ∈ S  such that r∗ ∧ s∗ > 0 and the set

 S = α ∈ S  : r∗ ∧ s∗ = upr bα , cα , B r∗ , s∗ ∪ bβ , cβ : β ∈ S  ∩ α has cardinality κ. Claim 2.10. For every finite subsets x, y ⊆ S with x ∩ y = ∅,



α∈x bα

∧



β∈y cβ

> 0.

Proof. For α ∈ S, let d0α = bα and d1α = cα . For every n < ω, α(0) < · · · < α(n) from S, and k(0), . . . , k(n) ∈ k(0) k(n) {0, 1}, we prove that r∗ ∧s∗ ∧dα(0) ∧· · ·∧dα(n) > 0 by induction on n. The case n = 0 follows from Lemma 2.5. Suppose r∗ ∧ s∗ ∧ dα(0) ∧ · · · ∧ dα(n) > 0 and let α(n + 1) ∈ S be such that α(n + 1) > α(n). Note that k(0)

k(n)

α(0), . . . , α(n) ∈ S∩α(n+1). Thus r∗ ∧s∗ ∧dα(0) ∧· · ·∧dα(n) ∈ B({r∗ , s∗ }∪{bβ , cβ : β ∈ S∩α(n+1)}) and 0 < k(0)

k(n)

r∗ ∧s∗ ∧dα(0) ∧· · ·∧dα(n) ≤ r∗ ∧s∗ . By Lemma 2.5 again, we have r∗ ∧s∗ ∧dα(0) ∧· · ·∧dα(n) ∧dα(n+1) > 0. 2 k(0)

k(n)

k(0)

k(n)

k(n+1)

Lemma 2.11. Let μ be a regular uncountable cardinal, and κ a singular cardinal such that λ<μ < κ for every cardinal λ < κ. Let B be a complete Boolean algebra satisfying the μ-c.c. If B has precaliber cf(κ), then B satisfies the κ-SDP. Proof. By our assumption, we can take a strictly increasing sequence of regular cardinals κi : i < cf(κ) of limit κ such that λ<μ < κi for every λ < κi and i < cf(κ). Fix i < cf(κ). By Lemma 2.7, there are non-zero ri∗ , s∗i ∈ B, Si ∈ [κi ]κi , biα ∈ Iα , ciα ∈ Jα for α ∈ Si such that ri∗ ∧ s∗i > 0 and for every C ⊆ B of size < κi , if ri∗ , s∗i ∈ C then the set



 α ∈ Si : ri∗ ∧ s∗i = upr biα , ciα , B C ∪ biβ , ciβ : β ∈ Si ∩ α



has cardinality κi . For i < cf(κ), let Ci = {rj∗ , s∗j : j < cf(κ)} ∪ {bjβ , cjβ : j < i, β ∈ Sj }. The cardinality of Ci is less than κi . Then let Si be the set of all α ∈ Si such that ri∗ ∧ s∗i = upr(biα , ciα , B(Ci ∪ {biβ , ciβ : β ∈ Si ∩ α})). Note that Si has cardinality κi . Now we may assume that Si ∩ Sj = ∅ for every distinct i, j < cf(κ). Moreover, since B has precaliber cf(κ), we also may assume that {ri∗ ∧ s∗i : i < cf(κ)} has the finite intersection property.

Let S = i


α∈x bα

∧



β∈y cβ

> 0.

Proof. As before, for α ∈ S, let d0α = bα and d1α = cα . Take n < ω, α(0) < · · · < α(n) from S, and k(0) k(n) k(0), . . . , k(n) ∈ {0, 1}. We show that dα(0) ∧ · · · ∧ dα(n) > 0.

 We can find a finite z ⊆ cf(κ) with {α(0), . . . , α(n)} ⊆ {Si : i ∈ z}. Let t∗ = i∈z ri∗ ∧ s∗i . Note that t∗ > 0 and t∗ ∈ B(Ci ∪ {biβ , ciβ : β ∈ Si ∩ α(i)}) for every i ∈ z. Now let i(0) ∈ z be such that α(0) ∈ Si(0) . ∗ Since 0 < t∗ ≤ ri(0) ∧ s∗i(0) , we have t∗ ∧ dα(0) > 0 by Lemma 2.5. Let i(1) ∈ z be such that α(1) ∈ Si(1) . k(0)

Then t∗ ∧ dα(0) ∈ B(Ci(1) ∪ {bβ , cβ n times. 2 k(0)

i(1)

i(1)

 : β ∈ Si(1) ∩ α(1)}), hence t∗ ∧ dα(0) ∧ dα(1) > 0. Repeat this argument k(0)

k(1)

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Now we are ready to prove Theorem 1.8. Proof of Theorem 1.8. Let X be a regular space with S(X) ≤ μ and w(X) ≥ κ. Let B = ro(X) be the set of all regular open subsets of X. B is a complete Boolean algebra, and B satisfies the μ-c.c. First suppose κ is regular. B satisfies the κ-SDP by Lemma 2.9. Since X is regular, we have κ ≤ w(X) ≤ |B|, hence we can take κ many regular open subsets {Oα : α < κ}. For α < κ, let Vα = int(X \ Oα ). Note that Vα = −Oα in the complete Boolean algebra B. Fix α < κ. We can find families of pairwise disjoint regular open subsets Oα , Vα such that: (1) (2) (3) (4)

O ⊆ O ⊆ Oα for every O ∈ Oα . V ⊆ V ⊆ Vα for every V ∈ Vα .

O is dense in Oα .

α Vα is dense in Vα .

  In the complete Boolean algebra B, Oα and Vα are antichains of B, Oα = Oα , and Vα = Vα . By the κ-SDP of B, we can find A ∈ [κ]κ and Oα ∈ Oα , Vα ∈ Vα for α ∈ A such that for every finite   x, y ⊆ A with x ∩ y = ∅, α∈x Oα ∧ β∈y Vβ > 0. By reenumerating the sequences, we may assume that A = κ. For α < κ, put Fα0 = Oα and Fα1 = Vα . Then Fα0 , Fα1 : α < κ is a κ-dyadic system of X. Next suppose κ is singular. If X has precaliber cf(κ), then B also has precaliber cf(κ). Hence we can get the conclusion by combining Lemma 2.11 with the previous argument. If λ<μ < cf(κ) for every λ < cf(κ), then by Lemma 2.9, B has precaliber cf(κ). Hence we can get the same conclusion. 2 Under Martin’s axiom with 2ω > ω1 , for every regular uncountable κ < 2ω , every space X satisfying the c.c.c. has precaliber κ. Thus we have: Corollary 2.13. Suppose Martin’s axiom and 2ω > ω1 . Let κ be a singular cardinal such that ω1 ≤ cf(κ) < 2ω and λω < κ for every cardinal λ < κ. For every compact space X satisfying the c.c.c., if w(X) = κ then w(X) [0, 1] is a continuous image of X, so |X| = 2w(X) . 3. Dichotomy of cardinality In this section we prove Theorem 1.9. To do this, we use the following cardinal function instead of the index number: Definition 3.1. (Tall [17]) Let X be a topological space. m(X), the mapping number of X, is the least cardinal κ such that for every non-empty closed F ⊆ X there is x ∈ F with πχ(x, F ) < κ. Note that the following statements hold: (1) m(X) = ω if and only if X is scattered. (2) For a compact X, m(X) is the least cardinal κ such that κ [0, 1] cannot be a continuous image of X. Hence i(X) ≤ m(X) ≤ w(X)+ . (3) By Šhapirovski˘ı’s inequality, we have w(X) ≤ i(X)c(X) ≤ m(X)c(X) for a compact X. Lemma 3.2. Let X be a regular space. Let κ = S(X). For a cardinal λ, let Eλ = {x ∈ X : πχ(x, X) ≤ λ}, and D = {x ∈ X : πχ(x, X) < m(X)}. (1) Eλ has a subset of cardinality ≤ λ<κ which is dense in Eλ .

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(2) Suppose X is compact. Let U be a family of less than m(X) many open subsets with non-empty inter section. Then D ∩ U = ∅. Proof. (1) This proof is based on an argument of Juhász (see 2.15 in Juhász [5]). For x ∈ Eλ , fix a local π-base Vx at x with |Vx | ≤ λ. Take M ≺ Hθ such that |M | = λ<κ , λ ⊆ M , <κ M ⊆ M , and M contains all relevant objects. Note that Vx ⊆ M for every x ∈ M ∩ Eλ . We prove Eλ ⊆ M ∩ Eλ . Suppose to the contrary that there is a ∈ Eλ with a ∈ / M ∩ Eλ . Fix an open ∗ ∗ neighborhood O of a with O ∩ M ∩ Eλ = ∅. Let V = {V : V ∈ Vx , x ∈ M ∩ Eλ , V ∩ O∗ = ∅}. Note that V ⊆ M . Take a maximal pairwise disjoint subfamily V  of V. Claim 3.3. M ∩ Eλ ⊆

V .

Proof. Take y ∈ M ∩ Eλ and an open neighborhood W of y. We may assume W ∩ O∗ = ∅ since y ∈ M ∩ Eλ . Vy is a local π-base at y, hence there is V ∈ Vy with V ⊆ W . By the maximality of V  , there is V  ∈ V 

with V ∩ V  = ∅. Then we have V ∩ V  ⊆ W ∩ V  = ∅. 2 Note that |V  | < κ, and, since M is closed under < κ-sequences, we have V  ∈ M . Then, by the

 elementarity of M together with Claim 3.3, we have Eλ ⊆ V . Thus we have O∗ ∩ V  = ∅, and O∗ ∩ V = ∅ for some V ∈ V  ⊆ V. This contradicts the choice of V.  (2) It is clear when m(X) = ω. Suppose m(X) > ω. Suppose to the contrary that D ∩ U = ∅. Fix   x ∈ U. For U ∈ U, take a closed Gδ set FU with x ∈ FU ⊆ U . Let F = {FU : U ∈ U}. F is a non-empty  closed set with F ⊆ U. Moreover F is an intersection of |U| many open sets. Since X is compact, we have χ(F, X) ≤ |U|. For each y ∈ F , we have m(X) ≤ πχ(x, X) ≤ πχ(x, F ) + χ(F, X) ≤ πχ(x, F ) + |U|. Since |U| < m(X), we have πχ(x, F ) ≥ m(X) for every x ∈ F . On the other hand, by the definition of m(X), there is z ∈ F with πχ(z, F ) < m(X). This is a contradiction. 2 The following proposition immediately yields Theorem 1.9. Proposition 3.4. Let X be a compact space. (1) If m(X) is a strong limit singular cardinal and S(X) < m(X), then |X| = m(X). (2) If w(X) is a strong limit singular cardinal and S(X) < w(X), then either |X| = w(X) or a continuous image of X (so |X| = 2w(X) ).

w(X)

[0, 1] is

Proof. (1) The following argument is due to Junqueira, Larson and Tall [8]. Let κ = m(X). For each cardinal λ, let Eλ = {x ∈ X : πχ(x, X) ≤ λ}. By Lemma 3.2, Eλ has a dense |Y |

λ
subset Y in Eλ such that |Y | ≤ λ
|Fi | ≤ 22 i < κ. Hence | i
 For i < cf(κ), let Oi = X \ Fi . Oi is open. If X \ i
D ∩ i w(X), then w(X) [0, 1] is a continuous image of X. 2

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4. Some remarks In this section, we discuss some limitations of Theorems 1.5, 1.8, 1.9, and other relevant topics. 1. If κ ≤ 2ω or κ is a strong limit singular cardinal of countable cofinality, we can find a compact space X which satisfies the c.c.c. and |X| = w(X) = κ. Lemma 4.1. For each cardinal κ ≤ 2ω , there is a compact space X satisfying the c.c.c. such that |X| = w(X) = κ. Proof. Consider the ψ-space. Fix an almost disjoint family A of [ω]ω with |A| = κ. Let Y be the space ω ∪ A with the topology defined by: • Each n ∈ ω is an isolated point. • For each A ∈ A, an open neighborhood of A is of the form {A} ∪ (A \ n) for some n < ω. Let X be the one-point compactification of Y . X is compact, satisfies the c.c.c., and w(X) = |X| = κ.

2

Lemma 4.2. Suppose κ is a strong limit singular cardinal. Then there is a compact space X with S(X) = cf(κ)+ and |X| = w(X) = κ. Proof. Fix an increasing sequence κi : i < cf(κ) of cardinals with limit κ. For i < cf(κ), let Xi = κi {0, 1}. Let X be the one-point compactification of the sum of the Xn ’s. Then X is a compact space, w(X) = |X| = κ, and S(X) = cf(κ)+ . 2 2. We see that the cardinal arithmetic assumption of Theorem 1.5 is optimal under GCH. In fact the following lemmas are immediate corollaries of Argyros’ results [1]. Fact 4.3. ([1]) Let κ be a regular uncountable cardinal and X a compact space. Suppose that there is a family O of clopen sets of X such that: (1) |O| = κ. (2) {O, X \ O : O ∈ U} is a subbase for X. (3) Every O ∈ [O]κ does not have the finite intersection property. Then X does not have a κ-dyadic system Fα0 , Fα1 : α < κ such that each Fαi is clopen. Lemma 4.4. Let κ be a regular uncountable cardinal and X a compact space. Suppose that there is a family O of clopen sets of X such that: (1) |O| = κ. (2) {O, X \ O : O ∈ U} is a subbase for X. (3) Every O ∈ [O]κ does not have the finite intersection property. Then κ [0, 1] is not a continuous image of X. Proof. First note that X is zero-dimensional. Suppose to the contrary that there is a continuous surjection f : X → κ [0, 1]. Fix α < κ. Since X is zero-dimensional and compact, we can find clopen sets Fα0 , Fα1 such that f −1 “{s ∈ κ [0, 1] : s(α) ≤ 1/3} ⊆ Fα0 ⊆ f −1 “{s ∈ κ [0, 1] : s(α) < 1/2} and f −1 “{s ∈ κ [0, 1] : s(α) ≥

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3/2} ⊆ Fα1 ⊆ f −1 “{s ∈ κ [0, 1] : s(α) > 1/2}. Then Fα0 , Fα1 : α < κ is a κ-dyadic system consisting of clopen sets, which contradicts Fact 4.3. 2 Lemma 4.5. Let κ be an uncountable cardinal and suppose that if κ is singular then κ is strong limit. Suppose that there is a regular space X which does not have precaliber cf(κ). Then there is a zero-dimensional compact space Y such that S(Y ) ≤ S(X) and w(Y ) = κ ≤ |Y | ≤ 2<κ . Moreover if κ is regular then κ [0, 1] is not a continuous image of Y . Proof. First, we may assume that X is zero-dimensional and compact. Indeed, let B be the Boolean algebra consisting of all regular open sets in X, and X  be the Stone space of B. X  is zero-dimensional (in fact it is extremally disconnected), compact, S(X  ) = S(X), and does not have precaliber cf(κ). Then replace X by X  .  Fix {Oi : i < cf(κ)} such that each Oi ⊆ X is clopen and for every A ∈ [cf(κ)]cf(κ) , {Oi : i ∈ A} = ∅. First we deal with the case that κ is regular. Define an equivalence relation ≡ on X by x ≡ y if and only if ∀α < κ (x ∈ Oα ⇐⇒ y ∈ Oα ). Then there are at most 2<κ many equivalence classes of ≡, since {α < κ : x ∈ Oα } is bounded in κ for every x ∈ X. Hence |X/ ≡| ≤ 2<κ . Let π : X → X/ ≡ be the natural projection, and Y be the space X/ ≡ with the topology generated by {π“Oα , π“(X \ Oα ) : α < κ} as a subbase. Y is compact Hausdorff, zero-dimensional, S(Y ) ≤ S(X), w(Y ) ≤ κ, and |Y | ≤ 2<κ . We can check that {π“Oα : α < κ} witnesses that Y does not have precaliber κ, so w(Y ) must be κ. And by Lemma 4.4, there is no continuous surjection from Y to κ [0, 1]. Next we deal with the case that κ is singular. Consider the product space Z = X × κ {0, 1}. Z is compact, zero-dimensional, and S(Z) = S(X). Fix a strictly increasing sequence κi : i < cf(κ) of ordinals with limit κ, and a 1–1 enumeration {Wα : α < κ} of a clopen base for κ {0, 1} such that for every β < κ there is α < κ with {s ∈ κ {0, 1} : s(β) = 1} = Wα . Then put Vα = Oi × Wα , where i < cf(κ) is minimal with α < κi . Each Vα is clopen in Z, and for every x ∈ Z, {α < κ : x ∈ Vα } is bounded in κ. As before, define an equivalence relation ≡ on Z by x ≡ y if and only if ∀α < κ (x ∈ Vα ⇐⇒ y ∈ Vα ). Let π : Z → Z/ ≡ be the natural projection, and Y be the space Z/ ≡ with the topology generated by {π“Vα , π“(Z \ Vα ) : α < κ} as a subbase. Y is compact Hausdorff, zero-dimensional, S(Y ) ≤ S(Z) ≤ S(X), w(Y ) ≤ κ, and |Y | ≤ 2<κ = κ. We will see that |Y | = κ, and then w(Y ) must be κ; if w(X) < κ, then |Y | ≤ 2w(X) < κ since κ is strong limit. This is impossible. For β < κ, let sβ : κ → {0, 1} be such that sβ (β) = 1 and sβ (γ) = 0 for every γ = β. For β < κ, fix xβ ∈ X so that xβ , sβ  ∈ Vα = Oi × {s ∈ κ {0, 1} : s(β) = 1} for some α and i. Then xβ , sβ  ≡ xγ , sγ  for every β < γ < κ. This means that there are at least κ many equivalence classes of ≡, so |Y | = κ. 2 Corollary 4.6. Let κ be a strong limit singular cardinal. Then the following are equivalent: (1) (2) (3) (4)

For every compact space X satisfying the c.c.c., if w(X) = κ then w(X) [0, 1] is a continuous image of X. For every compact space X satisfying the c.c.c., if w(X) = κ then |X| > w(X). Every compact space satisfying the c.c.c. has precaliber cf(κ). Every regular space satisfying the c.c.c. has precaliber cf(κ).

Proof. (4) ⇒ (3) is trivial, and (3) ⇒ (1) is Theorem 1.8. (1) ⇐⇒ (2) follows from Proposition 3.4, and (2) ⇒ (4) is Lemma 4.5. 2 When λ is singular, under GCH, Argyros [1] constructed an extremally disconnected compact space X such that S(X) = cf(λ)+ and X does not have precaliber λ+ . For a strong limit cardinal λ with countable cofinality and 2λ = λ+ , Argyros and Tsarpalias [2] constructed an extremally disconnected compact space X such that X satisfies the c.c.c. and X does not have precaliber λ+ . Combining these spaces with Lemma 4.5, we have the following:

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Corollary 4.7. Let κ be an uncountable cardinal with 2<κ = κ. Suppose cf(κ) is the successor cardinal of a strong limit cardinal with countable cofinality and 2 ω1 then |X| = 2w(X) . Proof. Starting with the existence of a 3-hypermeasurable cardinal, Cummings [4] constructed a model of ZFC in which 2κ = κ+ for every regular κ but 2κ = κ++ for every singular κ. We check that the assertion holds in this Cummings’ model. Let X be a compact space satisfying the c.c.c. and w(X) = κ is regular uncountable > ω1 . If λω < κ for every cardinal λ < κ, then w(X) [0, 1] is a continuous image of X by Theorem 1.5. If λω ≥ κ for some cardinal λ < κ, then κ must be the successor cardinal of a strong limit singular cardinal μ of countable cofinality. Since μ+ = w(X) ≤ m(X)ω , m(X) must be equal or greater than μ. If m(X) = μ, then |X| = μ + by Proposition 3.4, this is impossible. Thus m(X) > μ, and |X| ≥ 2μ = 2μ = 2w(X) . 2 In the model of Corollary 4.9, for a strong limit cardinal λ of countable cofinality, every compact space + X satisfying the c.c.c. and having weight λ+ has cardinality 2λ . However, we do not know that whether λ+ [0, 1] is a continuous image of X or not. Question 4.10. Is it consistent that there is a strong limit cardinal λ of countable cofinality such that for every compact space X satisfying the c.c.c., if w(X) = λ+ then w(X) [0, 1] is a continuous image of X? By Lemma 4.5, for a strong limit cardinal λ of countable cofinality, if there is a regular space satisfying the c.c.c. which does not have precaliber λ+ , then we can construct a compact space X satisfying the c.c.c. such that w(X) = λ+ but w(X) [0, 1] is not a continuous image of X. Hence Question 4.10 is connected with the following question: Question 4.11. Let λ be a strong limit singular cardinal of countable cofinality. Is there (in ZFC) a regular space satisfying the c.c.c. which does not have precaliber λ+ ? Or, is the existence of such a space consistent with 2λ > λ+ ?

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Kunen (see §4 of Juhász [6]) showed that if 2ω = ω1 , 2ω1 > ω2 , and ω2 is not inaccessible in the constructible universe L, then there is a compact space X with ω1 = w(X) < |X| < 2ω1 . On the other hand, the following is still unknown: Question 4.12. Is it consistent that there is a compact space X satisfying the c.c.c. such that 2ω ≤ w(X) < |X| < 2w(X) ? Note that we have the following answers to the question for spaces with weight < 2ω : • If κ < 2ω = 2κ , it is easy to check that for each compact space X with w(X) = κ, either |X| = w(X) or |X| = 2w(X) . • If κ < 2ω < 2κ , then fix a compact space Y satisfying the c.c.c. and |Y | = w(Y ) = κ. Then the space X = Y × ω {0, 1} is a compact space satisfying the c.c.c. such that κ = w(X) < |X| = 2ω < 2κ . If Question 4.12 can be solved affirmatively, then we may settle Question 4.11 as follows: Suppose that + λ is a strong limit singular cardinal of countable cofinality such that λ+ < 2λ < 2λ . Suppose also that there is a regular space X satisfying the c.c.c. which does not have precaliber λ+ . By Lemma 4.5, we can + construct a compact space Y satisfying the c.c.c. such that w(Y ) = λ+ and λ+ ≤ |Y | ≤ 2<λ = 2λ . Then m(Y ) > λ, so |Y | ≥ 2λ . Thus we have that Y is a compact space satisfying the c.c.c., but λ+ = w(Y ) < + |Y | = 2λ < 2λ = 2w(Y ) . 5. Elementary submodel spaces In this section we prove Theorem 1.13. See [7,8], and [9] for the following facts. Fact 5.1. Let M ≺ Hθ . Let κ ∈ M be an ordinal with M ∩ κ ∈ κ. Then κ is a regular uncountable cardinal, and for every x ∈ M , x ⊆ M ⇐⇒ |x| < κ. Fact 5.2. Let X be a topological space and M ≺ Hθ be with X ∈ M . (1) (2) (3) (4) (5)

X is Hausdorff ⇐⇒ XM is Hausdorff. For an open base B ∈ M for X, the set {O ∩ M : O ∈ B ∩ M } is an open base for XM . If XM is compact, then so is X. If XM is compact, then X = XM ⇐⇒ w(X) ⊆ M ⇐⇒ |X| ⊆ M . If X satisfies the c.c.c., then XM also satisfies the c.c.c. The converse also holds if ω1 ⊆ M .

Fact 5.3. (Kunen [11], Tall [17]) Let X be a topological space and M ≺ Hθ be with X ∈ M . Suppose XM is compact. For each cardinal λ < m(X) with λ ∈ M , if λ ⊆ M then 2λ ⊆ M . In particular if m(X) > ω then ω1 ⊆ M . Theorem 1.13 is immediate from the following proposition. Proposition 5.4. Let X be a non-scattered space. Let M ≺ Hθ be with X ∈ M and suppose XM is compact, satisfies the c.c.c., and XM = X. Let κ ∈ M be a cardinal such that M ∩ κ ∈ κ. Then the following hold: (1) m(X) > κ, hence |X| ≥ 2κ . (2) (κ {0, 1})M is compact, so the space κ {0, 1} is squashable.

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Proof. First note that κ is regular by Fact 5.1, and that X is compact, w(X) ≥ κ, m(X) > ω, and X satisfies the c.c.c. (see Tall [17]). (1) Suppose to the contrary that κ ≥ m(X). We will repeat the proof in [17] with slight modifications. Case 1: 2μ ≥ m(X) for some μ < m(X). We can pick such a μ from M . Then μ < κ, m(X), so 2μ ⊆ M and 2μ < κ. Thus w(X) ≤ m(X)ω ≤ (2μ )ω = 2μ < κ. This is a contradiction. Case 2: m(X) is a strong limit singular cardinal. Note that m(X) < κ since κ is regular. By Proposition 3.4, we have |X| = m(X) < κ. Then |X| < κ ≤ w(X), this is impossible. Case 3: m(X) is a strong limit regular cardinal. In this case, we have w(X) ≤ m(X)ω = m(X). Since w(X) ≥ κ ≥ m(X), we have w(X) = m(X). Hence w(X) is inaccessible. By Theorem 1.5, however, we have m(X) > w(X), this is impossible. (2) By (1), there are a closed set F ⊆ X and a continuous surjection f : F → κ {0, 1} with F, f ∈ M . F ∩M is a closed subset of XM and (κ {0, 1})M is a continuous image of F ∩ M . Thus (κ {0, 1})M is compact. 2 Corollary 5.5. Let κ be the least cardinal such that κ {0, 1} is squashable. For every non-scattered compact space X satisfying the c.c.c., if |X| < 2κ , then X is not squashable. Proof. Suppose to the contrary that there is M ≺ Hθ such that X ∈ M and XM is compact but XM = X. By Facts 5.2 and 5.3, we have that ω1 ⊆ M and XM satisfies the c.c.c. Let λ ∈ M be a cardinal with M ∩ λ ∈ λ. By Lemma 5.4, we have that |X| ≥ 2λ . By Lemma 5.4 again, the space λ {0, 1} is squashable. Then we have that κ ≤ λ by the minimality of κ, so |X| ≥ 2κ . This is a contradiction. 2 Corollary 5.6. The following are equivalent: (1) There is a non-scattered squashable compact space satisfying the c.c.c. (2) There is a cardinal κ such that κ {0, 1} is squashable. We also note the following universality of the squashability of κ {0, 1}. Lemma 5.7. Let M ≺ Hθ and let κ ∈ M be a cardinal. Then the following are equivalent: (1) For every compact X ∈ M , if w(X) = κ then XM is compact. (2) The space (κ {0, 1})M is compact. (3) For every a ⊆ M ∩ κ, there is A ∈ M ∩ P(κ) such that a = M ∩ A. Proof. (1) ⇒ (2) is trivial. (2) ⇒ (3). Fix a ⊆ M ∩ κ. For each α ∈ M ∩ κ and i ∈ {0, 1}, we have that Oαi = {f ∈ κ {0, 1} : f (α) = i} ∈ M and Oαi ∩ M is a non-empty closed subset of (κ {0, 1})M . Then the family {Oα0 ∩ M : α ∈ (M ∩ κ) \ a} ∪ {Oα1 ∩ M : α ∈ a} has the finite intersection property. The space (κ {0, 1})M is compact, so we can take an element f ∈ M from the intersection of the family. Then f (α) = 1 ⇐⇒ α ∈ a for every α ∈ M ∩ κ. Let A = {α < κ : f (α) = 1} ∈ M . We have A ∩ M = a. (3) ⇒ (1). Fix a compact X ∈ M with w(X) = κ. Fix an open base B ∈ M for X of cardinality κ and a bijection π ∈ M from B to κ. To see that XM is compact, take an open cover U of XM . We may assume that every O ∈ U is of the form V ∩ M for some V ∈ B ∩ M . Put a = {π(V ) : V ∩ M ∈ U} ⊆ M ∩ κ. By (3), there is A ∈ M ∩ P(κ) with a = A ∩ M . Let V = {V : π(V ) ∈ A} ∈ M . Note that V is an open cover of X. If not, there is x ∈ X ∩ M such that x ∈ / V for every V ∈ V. x ∈ X ∩ M , hence there is V ∩ M ∈ V with V ∩ M ∈ U and x ∈ V ∩ M . This is a contradiction. X ∈ M is compact and V ∈ M , hence there are finite

V0 , . . . , Vn ∈ V ∩ M such that X = i≤n Vi . Clearly XM = i≤n Vi ∩ M . Now, for each i ≤ n, we have π(Vi ) ∈ A ∩ M = a, so Vi ∩ M ∈ U. Thus {Vi ∩ M : i ≤ n} is a finite subcover of U. 2

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Corollary 5.8. For a cardinal κ, the following are equivalent: (1) κ {0, 1} is squashable. (2) Every compact space satisfying the c.c.c. of weight κ is squashable. (3) Every compact space of weight κ is squashable. Proof. (3) ⇒ (2) ⇒ (1) are clear. (1) ⇒ (3). Take a compact space X of weight κ. By (1), we can take M ≺ Hθ such that κ, X ∈ M and (κ {0, 1})M is compact but (κ {0, 1})M = κ {0, 1}. Note that XM is compact by Lemma 5.7. Since (κ {0, 1})M = κ {0, 1} and w(κ {0, 1}) = κ, we have κ  M . Hence |X|  M , and XM = X. Therefore M witnesses that X is squashable. 2 Acknowledgements The author is grateful to Stevo Todorčević for suggesting the use of independent families of Boolean algebras. The author also thanks Dmitri Shakhmatov, Franklin D. Tall, and the kind referee for their useful comments and suggestions. References [1] S. Argyros, Boolean algebras without free families, Algebra Univers. 14 (1982) 244–256. [2] S. Argyros, A. Tsarpalias, Calibers of compact spaces, Trans. Am. Math. Soc. 270 (1982) 149–162. [3] W.W. Comfort, S. Negrepontis, Chain Conditions in Topology, Cambridge Tracts in Mathematics, vol. 79, Cambridge University Press, 1982. [4] J. Cummings, A model in which GCH holds at successors but fails at limits, Trans. Am. Math. Soc. 329 (1) (1992) 1–39. [5] I. Juhász, Cardinal Functions in Topology: Ten Years Later, Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, 1980. [6] I. Juhász, Cardinal Functions II in Handbook of Set-Theoretic Topology, North-Holland, 1984. [7] L.R. Junqueira, Upward preservation by elementary submodels, Topol. Proc. 25 (2000) 225–249, Spring. [8] L.R. Junqueira, P. Larson, F.D. Tall, Compact spaces, elementary submodels, and the countable chain condition, Ann. Pure Appl. Log. 144 (2006) 107–116. [9] L.R. Junqueira, F.D. Tall, The topology of elementary submodels, Topol. Appl. 82 (1998) 239–266. [10] S. Koppelberg, General theory of Boolean algebras, in: J. Donald Monk, Robert Bonnet (Eds.), Handbook of Boolean Algebras, vol. 1, North-Holland, 1989. [11] K. Kunen, Compact spaces, compact cardinals, and elementary submodels, Topol. Appl. 130 (2003) 99–109. [12] K. Kunen, Set Theory, Studies in Logic, vol. 34, College Publications, 2011. [13] B.É. Šhapirovski˘ı, Canonical sets and character, density and weight in compact spaces, Dokl. Akad. Nauk SSSR 218 (1974) 58–61 (in Russian); also: Sov. Math. Dokl. 15 (1974) 1282–1287 (in English). [14] B.É. Šhapirovski˘ı, Maps onto Tikhonov cubes, Usp. Mat. Nauk 35 (3) (1980) 122–130 (in Russian); also: Russ. Math. Surv. 35 (3) (1980) 145–156 (in English). [15] B.É. Šhapirovski˘ı, The Suslin number in set-theoretic topology, Acta Univ. Latvensis Ser. Math. 552 (1989) 76–96 (in Russian); also: Topol. Appl. 57 (1994) 131–150 (in English). [16] S. Shelah, Remarks on Boolean algebras, Algebra Univers. 11 (1980) 77–89. [17] F.D. Tall, Compact spaces, elementary submodels, and the countable chain condition, II, Topol. Appl. 153 (2006) 273–278.