On a theorem of Shapiro

Sakaé Fuchino, Saharon Shelah and Lajos Soukup

September 26, 1996

Abstract

We show that a theorem of Leonid B. Shapiro which was proved under MA, is actually independent from ZFC. We also give a direct proof of the Boolean algebra version of the theorem under MA(Cohen).

1 Introduction

L.B. Shapiro [8] recently proved the following theorem:

Theorem 1.1 (L.B. Shapiro) (MA(Cohen)) For any compact Hausdorff space X of weight < 2^{ℵ_{0}} and ℵ_{0} ≤ τ < 2^{ℵ_{0}} the following assertions are equivalent:

i) There exists a continuous surjection from X onto ^{τ}II; ii) There exists a continuous injection from ^{τ}2 into X; iii) There exists a closed subset Y ⊆ X such that χ(y, Y) ≥ τ for every y ∈ Y.

The original proof of Theorem 1.1 by L.B. Shapiro in [8] was formulated under MA. However practically the same proof still works when we merely assume MA(Cohen) where MA(Cohen) stands for Martin’s Axiom restricted to the partial orderings of the form Fn(κ, 2).

A part of the theorem above can be translated into the language of Boolean algebras:

The first author would like to thank Professor L.B. Shapiro for calling his attention to [8], and Professor K. Eda for informing him of an example in [3]. The second author is partially supported by the Deutsche Forschungsgemeinschaft(DFG) grant Ko 490/7–1. He also gratefully acknowledges partial support by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundatiion (Germany). The present paper is the second author’s Publication No. 543. The third author is partially supported by the Hungarian National Foundation for Scientific Research grant No. 1908 and the Deutsche Forschungsgemeinschaft(DFG)

grant Ko 490/7–1.

Corollary 1.2 (Boolean algebra version of Shapiro’s theorem) (MA(Cohen)) For any infinite Boolean algebra B of cardinality < 2^{ℵ_{0}} and any infinite τ, the following are equivalent:

i′) There exists an injective Boolean mapping from Fr τ into B; ii′) There exists a surjective Boolean mapping from B onto Fr τ.

The implication from ii′) to i′) as well as the implication from ii) to i) can be proved already in ZFC. For the proof of ii) from i), let g : ^{τ}2 → X be a continuous injection. Note that g[^{τ}2] is a closed subset of X. For any fixed y_{0} ∈ ^{τ}2 let f′ : X → ^{τ}2 be defined by

f′(x) = \{ \begin{matrix}g^{−1}(x) & ; if x ∈ g[^{τ}2], \\ y_{0} & ; otherwise. \end{matrix}

Then f′ is a continuous surjection from X onto^{τ}2. Let f′′ be a continuous surjection from ^{τ}2 to ^{τ}II. E.g. let h : ^{ω}2 → II be the continuous surjection defined by u ↦ the real represented by the binary expression 0.u(0)u(1)u(2) · · ·. h^{κ} : ^{κ}(^{ω}2) → ^{κ}II is then a continuous surjection. Since^{κ}(^{ω}2) is homeomorphic to^{κ}2 we can find a continuous surjection f′′ from ^{τ}2 onto ^{τ}II corresponding to h^{κ}. The mapping g = f′′ ∘ f′ is then as desired. In the next section we shall give a direct proof of i′) ⇒ ii′). For iii) ⇒ i) we need some deep results by Shapiro on dyadic compactum (see [8]).

The equivalence of the assertions i′) and ii′) above is not true in general for Boolean algebras of cardinality ≥ 2^{ℵ_{0}}: For any σ-complete Boolean algebra B and any infinite κ, there exits no surjective Boolean mapping f : B → Fr κ (see Lemma 1.3 below). Hence e.g. for Boolean algebra B = \overline{Fr ω} we have that | B | = 2^{ℵ_{0}}; Fr 2^{ℵ_{0}} is embeddable into B (by Balcar-Fraňek-Theorem, see [1]) but there exists no surjective Boolean mapping from B onto Fr 2^{ℵ_{0}}. The non-existence of surjective Boolean mapping from a σ-complete Boolean algebra in the ground model onto Fr τ is preserved in a generic extension by a partial ordering of cardinality < τ though B may be no more σ-complete in such a generic extension:

Lemma 1.3 Let B be a σ-complete Boolean algebra and P a partial ordering. For any κ > | P | we have that

∥–_{P} “ there exists no surjective Boolean mapping from B onto Fr κ ”.

Proof Suppose that there would be a P -name ḟ such that

∥–_{P} “ ḟ : B → Fr κ is a surjective Boolean mapping ”.

For each p ∈ P let

B_{p} = { b ∈ B : p ∥–_{P} “ ḟ(b) = c for some c ∈ Fr κ ” }

and

C_{p} = { c ∈ Fr κ : p ∥–_{P} “ ḟ(b) = c for some b ∈ B ” }.

Then B_{p} and C_{p} are subalgebras of B and Fr κ respectively. Since ⋃_{p∈P} C_{p} = Fr κ and κ > | P | there exists some p ∈ P such that C_{p} is infinite. Let c_{n}, n < ω be pairwise disjoint positive elements of C_{p}. By the definition of B_{p} and C_{p}, there exits pairwise disjoint positive elements b_{n}, n < ω of B_{p} such that p ∥–_{P} “ ḟ(b_{n}) = c_{n} ” holds for every n < ω. Let X ⊆ ω be such that there exists no c ∈ Fr κ such that c · c_{n} = c_{n} holds for all n ∈ X and c · c_{n} = 0 for all n < ω \ X. Let d = Σ_{n∈X}^{B}b_{n}. Then for any q ≤ p there can be no c ∈ Fr κ such that q ∥–_{P} “ ḟ(d) = c ”. This is a contradiction. (Lemma 1.3) The lemma above together with Corollary 1.2 yields the following:

Proposition 1.4 Let B be a complete Boolean algebra with | B | = τ ≥ ℵ_{0}. Then

∥–_{Fn(κ,2)} “ there exists no surjective Boolean mapping from B onto Fr τ ”

holds if and only if κ < τ.

Proof If κ < τ then | Fn(κ, 2) | = κ < τ. Hence by Lemma 1.3,

∥–_{Fn(κ,2)} “ there exists no surjective Boolean mapping from B onto Fr τ ”

holds.

Suppose now that κ ≥ τ. Then as in the proof of Proposition 2.1, we can show that

∥–_{Fn(κ,2)} “ there exists a surjective Boolean mapping from B onto Fr τ ”

holds. (Proposition 1.4)

( •| ) (read “stick”, see [2]) is the following principle:

( •| ): There exists a sequence (x_{α})_{α<ω_{1}} of countable subsets of ω_{1} such that for any y ∈ [ω_{1}]^{ℵ_{1}} there exists α < ω_{1} such that x_{α} ⊆ y.

Clearly ( •| ) follows from CH. Another combinatorial principle (♣), a strengthning of ( •| ), is introduced in Ostaszewski [7]. Let Lim(ω_{1}) = { γ < ω_{1} : γ is a limit }.

(♣): There exists a sequence (x_{γ})_{γ∈Lim(ω_{1})} of countable subsets of ω_{1} such that for every γ ∈ Lim(ω_{1}), x_{γ} is a cofinal subset of γ, otp(x_{γ}) = ω and for every X ∈ [ω_{1}]^{ℵ_{1}} there is γ ∈ Lim(ω_{1}) such that x_{γ} ⊆ X.

Clearly ( •| ) follows from (♣). Unlike ( •| ), (♣) does not follow from CH, since (♣)

+ CH is equivalent with ◇ (K. Devlin, see [7]). For more about the combinatorial principles ( •| ) and (♣), and independence results connected with them, see [4].

MA(countable) — Martin’s axiom restricted to countable partial orderings — and MA(Cohen) both add a lot of Cohen reals over any small model of (a sufficiently large finite subset of) ZFC and in many cases where this property is needed, MA(countable) is just enough. Hence it seems to be quite natural to ask if these axioms are perhaps equivalent. However they are not. I. Juhász proved in an unpublished note that ¬CH + MA(countable) + (♣) is consistent (two other constructions of models of ¬CH + MA(countable) + (♣) are to be found in [5] and [4].). On the other hand, it is easy to see that the negation of MA(Fn(ℵ_{1}, 2)) follows from ¬CH + (♣): using ( •| ) we can obtain a Boolean algebra B of cardinality ℵ_{1} such that Fr ω_{1} is embeddable into B but there is no surjection from B onto Fr ω_{1} (see Theorem 4.4). By Proposition 2.1, this shows that m_{Fn(ℵ_{1},2)} = ℵ_{1} < 2^{ℵ_{0}}. It follows also that the assertions of Theorem 1.1 and Corollary 1.2 are independent from ZFC and MA(countable) is not enough to prove them.

Corollary 1.2 for other variety than Boolean algebras can be simply false. E.g., this is the case in the variety of abelian groups: in [3], an ℵ_{1}-free abelian group G in ℵ_{1} is constructed (in ZFC) which contains uncountable free subgroup but Hom(G, Z) = 0.

2 A proof of the Boolean algebra version of the theorem of Shapiro

In this section we shall prove Corollary 1.2. More precisely we prove the following Proposition 2.1. For any class C of partial orderings Let

m_{C} = min{ | D | : D is a family of dense subsets of P for some P ∈ C

such that there exists no D-generic filter over P }

If C is a singleton { P }, we shall write simply m_{P} in place of m_{{ P }}. Let us say that two partial orderings P, Q are coabsolute when their completions are isomorphic. It is easy to see that for any class C of partial orderings m_{C} = mC̃ where C̃ = { Q : Q is coabsolute with some P ∈ C }. If the class C is introduced by a property P of Boolean algebras, we also write m_{P} in place of m_{D}. We also write mcountable = m_{{ P : P is countable }} and mCohen = m_{{ P : P =Fn(κ,2) for some κ }}. Hence MA(Cohen) (MA(countable), MA etc. respectively) holds if and only if mCohen = 2^{ℵ_{0}} (mcountable = 2^{ℵ_{0}}, m_{ccc} = 2^{ℵ_{0}} etc. respectively) and we have m_{ccc} ≤ mCohen ≤ mcountable.

Proposition 2.1 Let B be a Boolean algebra containing Fr κ as a subalgebra. If | B | < m_{Fn(κ,2)}, then there exists a surjective Boolean mapping from B onto Fr κ.

Proof By Sikorski’s theorem, there is a Boolean mapping from B to \overline{Fr κ} — the completion of Fr κ, extending the inverse of the canonical embedding of Fr κ into B. Hence without loss of generality we may assume that B is a subalgebra of \overline{Fr κ}. Now let P = Fn(κ, 3). Note that P is coabsolute with Fn(κ, 2). We shall define a family D of dense subsets of P such that | D | < m_{Fn(κ,2)} so that among other things (see below), for D-generic set G, g = ⋃ G will be a function from κ to 3 and X = { α < κ : g(α) = 2 } will be of cardinality κ. Then we let f be the function on κ defined by:

f(α) = \{ \begin{matrix}0_{B} & ; if g(α) = 0, \\ 1_{B} & ; if g(α) = 1, \\ α & ; otherwise. \end{matrix}

Let f̄ be the Boolean mapping from Fr α to Fr X generated by f.

Now we are done, if we can show that f̄ extends to a Boolean mapping f̃ from B onto Fr X. But by the following Lemma 2.2, we can choose D appropriate for this purpose.

For p ∈ P, let B_{p} = Fr dom(p) (hence B_{p} ≤ B) and f_{p} : B_{p} → Fr (p^{−1}[{2}]) be the Boolean mapping generated by the mapping f_{p}^{0} on dom(p) defined by:

f_{p}^{0}(α) = \{ \begin{matrix}0_{B} & ; if p(α) = 0, \\ 1_{B} & ; if p(α) = 1, \\ α & ; otherwise. \end{matrix}

Lemma 2.2 For any b ∈ B and p ∈ P there exists q ≤ p and b_{1}, b_{2} ∈ B_{q} such that b_{1} ≤ b, b_{2} ≤ −b and f_{q}(b_{1}) + f_{q}(b_{2}) = 1 ( i.e, q “forces” f̃(b) = f_{q}(b_{1}) ).

For the proof of the Lemma 2.2 we use the following Lemma whose proof is left to the reader:

Lemma 2.3 Let b ∈ \overline{Fr κ} and let Y ⊆ κ be a countable set such that b ∈ \overline{Fr Y} holds. Let Y = { α_{n} : n < ω }. Then there exist an increasing sequence (l_{n})_{n<ω} with l_{n} < ω for n < ω and a sequcence (i_{n})_{n<ω} with i_{n} ∈ ^{l_{n}}{ −1, 1 } for n < ω such that, letting p_{n} = Σ_{k<l_{n}}i_{n}(k) · α_{k} for n < ω,

i) either p_{n} ≤ b or p_{n} ≤ −b and ii) Σ_{n<ω}p_{n} = 1. In particular we have b = Σ{ p_{n} : n < ω, p_{n} ≤ b }.

Proof of Lemma 2.2 Let Y = { α_{n} : n < ω }, (l_{n})_{n<ω}, (i_{n})_{n<ω} and p_{n}, n < ω be as in Lemma 2.3 for our b ∈ B. Without loss of generality we may assume that dom(p) ∩ Y = { α_{n} : n < k } for some k < ω. Let ^{k}{ −1, 1 } = { τ_{m} : m < 2^{k} }.

By induction we can take n_{m} < ω for m < 2^{k} such that

a) i_{n_{m}} is compatible (as an element of Fn(Y, { −1, 1 })) with τ_{m} and

b) { i_{n_{m}}|` (dom(i_{n_{m}}) \ k) : m < 2^{k} } is pairwise compatible.

Let ñ = max{ n_{m} : m < 2^{k} }, l̃ = lñ and ĩ = ⋃{ i_{n_{m}}|` (dom(i_{n_{m}}) \ k) : m < 2^{k} }. Let q ≤ p be such that dom(q) = dom(p) ∪ { α_{k}, …, α_{l̃−1} }, q|` dom(p) = p and

q(α_{m}) = \{ \begin{matrix}1 & ; if ĩ(α_{m}) = 1, \\ 0 & ; if ĩ(α_{m}) = −1. \end{matrix}

Then q as above together with b_{1} = Σ{ p_{n} : n < ñ, p_{n} ≤ b } and b_{2} = Σ{ p_{n} : n < ñ, p_{n} ≤ −b } is as desired. (Lemma 2.2) Now by the lemma above

D = { { p ∈ P : α ∈ dom(p) } : α < κ }
∪{ { p ∈ P : ∃β > α p(β) = 2 } : α < κ }

∪{ { q ∈ P : f_{q}(b_{1}) + f_{q}(b_{2}) = 1 for some b_{1} ≤ b, b_{2} ≤ −b } : b ∈ B }

is a family of dense subsets of P. Clearly the mapping f̄ defined as above with respect to this D can be extended to a Boolean mapping f̃ from B onto Fr X.

(Proposition 2.1)

3 Pcf and the theorem of Shapiro

Proposition 3.1 Assume that

⊕_{μ,κ,λ} for any F ⊆ [λ]^{ℵ_{0}} with | F | < μ, there is Y ∈ [λ]^{κ} such that a ∩ Y is finite for all a ∈ F.

Then, for any Boolean algebra B of cardinality < μ, if Fr λ is embeddable into B then there is a surjective Boolean mapping from B onto Fr κ.

Proof As in the proof of Proposition 2.1, we may assume without loss of generality that Fr λ ≤ B ≤ \overline{Fr λ} holds. Let | B | = i^{∗} (< μ) and let (y_{i})_{i<i^{∗}} be an enumeration of B. Let y_{i} = ∑_{n<ω} τ_{i}^{n}(α(i, n, 0), …, α(i, n, m_{i,n})) where τ_{i}^{n} is a Boolean term with m_{i,m} + 1 variables and α(i, n, 0), …, α(i, n, m_{i,n}) < λ for i < i^{∗} and n < ω. For i < i^{∗}, let w_{i} = { α(i, n, l) : n < ω, l ≤ m_{i,n} }. By the assumption, there exists a Y ∈ [λ]^{κ} such that w_{i} ∩ Y is finite for everly i < i^{∗}. Let g : B → Fr Y be defined by

g(y_{i}) = ∑_{n<ω} τ_{i}^{n}(α^{∗}(i, n, 0), …, α^{∗}(i, n, m_{i,n}))

where

α^{∗}(i, n, l) = \{ \begin{matrix}α(i, n, l) & ; if α(i, n, l) ∈ Y \\ 0_{B} & ; otherwise. \end{matrix}

The function g is well-defined since, for every n < ω, τ_{i}^{n}(α^{∗}(i, n, 0), …, α^{∗}(i, n, m_{i,n})) is an element of Fr (w_{i} ∩ Y) and Fr (w_{i} ∩ Y) is finite. Clearly this g is as desired.

(Proposition 3.1)

Corollary 3.2 For any Boolean algebra of cardinality < a (where a is the minimal cardinality of a maximal almost disjoint family in [ω]^{ℵ_{0}}), if Fr ω is embeddable into B then there is a surjection from B onto Fr ω.

Proof By Proposition 3.1 for ⊕_{a,ℵ_{0},ℵ_{0}}. (Corollary 3.2)

Theorem 3.3 Assume that

(∗)_{μ,λ,κ} there are _{∘}a_{i} ∈ [Reg ∩ (λ^{+} \ κ^{+})]^{<ℵ_{0}} for i < κ such that for every a ∈ [κ]^{ℵ_{0}}, max pcf(⋃_{i∈a} _{∘}a_{i}) ≥ μ holds.

Then for any Boolean algebra B of cardinality < μ, if Fr κ is embeddable into B then there is a surjective Boolean mapping g from B onto Fr κ.

( For more about (∗)_{μ,λ,κ} see [10]. For pcf theory in general, the reader may consult [11].) The theorem follows from Proposition 3.1 and the following:

Lemma 3.4 Assume that (∗)_{μ,λ,κ} (as in Theorem 3.3) holds. Then ⊕_{μ,κ,κ} holds.

Proof Since max pcf is always regular, we may assume that μ is regular. Let _{∘}a = ⋃_{i<κ} _{∘}a_{i}. In place of [κ]^{ℵ_{0}}, we consider [Z]^{ℵ_{0}} for Z = ⋃^{·}_{i<κ} Z_{i} where Z_{i} = { i } × ∏ _{∘}a_{i}. Hence we assume that F ⊆ [Z]^{ℵ_{0}} and | F | < μ.

For each a ∈ F, let g_{a} ∈ ∏ _{∘}a be defined by

g_{a}(θ) = sup{ η(θ) : η ∈ a, θ ∈ dom(η) }

for each θ ∈ _{∘}a, where we put sup ∅ = 0. Since ∏ _{∘}a/J_{<μ}[_{∘}a] is μ-directed and | F | < μ, there is f^{∗} ∈ ∏ _{∘}a such that g_{a} <_{J_{<μ}[_{∘} a]} f^{∗} holds for all a ∈ F. For i < κ, let z_{i} = { (0, i) } ∪ (f^{∗}| _{` ∘}a_{i}). Then z_{i} ∈ Z_{i} for i < κ. We show that Y = { z_{i} : i < κ } is as required. Suppose not. Then Y ∩ a is infinite for some a ∈ F. By the assumption, it follows that ⋃_{z_{i}∈Y ∩a} _{∘}a_{i} ∉ J_{<μ}[_{∘}a]. But for z_{i} ∈ Y ∩ a we have { (0, i) } ∪ (f^{∗}| _{` ∘}a_{i}) ∈ a. It follows that for θ ∈ _{∘}a_{i} we have f^{∗}(θ) ≤ g_{a}(θ). This is a contradiction to g_{a} <_{J_{<μ}[_{∘} a]} f^{∗}. (Lemma 3.4)

4 Independence of the theorem of Shapiro

The principle ( •| ) suggests the following cardinal invariant •| :

•| = min{ | X | : X ⊆ [ω_{1}]^{ℵ_{0}}, ∀y ∈ [ω_{1}]^{ℵ_{1}} ∃x ∈ X x ⊆ y }.

Clearly ℵ_{1} ≤ •| ≤ 2^{ℵ_{0}} and ( •| ) holds if and only if •| = ℵ_{1}. We can also consider the following variants of •| :

•|′ = min{ κ : κ ≥ ℵ_{1}, there is an X ⊆ [κ]^{ℵ_{0}}
such that | X | = κ and ∀y ∈ [κ]^{ℵ_{1}} ∃x ∈ X x ⊆ y },
•|′′ = min{ κ : κ ≥ ℵ_{1}, there is an X ⊆ [κ]^{ℵ_{0}}
such that | X | = κ and ∀y ∈ [κ]^{κ} ∃x ∈ X x ⊆ y }.

We have ℵ_{1} ≤ •|′′ ≤ •|′ ≤ 2^{ℵ_{0}} and ( •| ) holds if and only if •| = •|′ = •|′′ = ℵ_{1} holds.

It can be easily shown that •| ≤ •|′ holds. Moreover if •| < ℵ_{ω_{1}}, then •| = •|′ holds. The question, if •| < •|′ is consistent, is connected with some very fundamental unsolved problems on cardinal arithmetics while we can show •|′′ < •| is consistent. For more, see [4] and [10].

Proposition 4.1 There exists a Boolean algebra B such that | B | = •|′, Fr •|′ is embeddable into B but there is no surjective Boolean mapping from B onto Fr ω_{1}.

Proof Let Φ : κ → κ; α ↦ ξ_{α} be the continuously increasing function defined inductively by ξ_{0} = ω and ξ_{α+1} = ξ_{α} + | ξ_{α} |. Let κ = •|′ and let X ⊆ [κ × Fr ω_{1}]^{ℵ_{0}} be such that | X | = κ, ω × Fr ω ∈ X and ∀y ∈ [κ × Fr ω_{1}]^{ℵ_{1}} ∃x ∈ X x ⊆ y holds. Let (x_{α})_{α<κ} be an enumeration of X such that x_{α} ⊆ ξ_{α} × Fr ω_{1} for all α < κ.

Now let (B_{α})_{α<κ} be a continuously increasing sequence of Boolean algebras such that for all α < κ

1 ) the underlying set of B_{α} is ξ_{α}; 2 ) there exits a b_{α} ∈ B_{α+1} such that b_{α} is free over B_{α}; 3 ) if x_{α} generates a Boolean mapping f_{α} from a subalgebra of B_{α} onto an infinite subalgebra of Fr ω_{1} then B_{α+1} contains an element c_{α} of the form Σ_{n∈Z_{α}}^{B_{α+1}}b_{n}^{α} where Z_{α} ⊆ ω, b_{n}^{α}, n < ω are pairwise disjoint elements in dom(f_{α}), f_{α}(b_{n}^{α}) ≠ 0 for all n < ω and there is no d ∈ Fr ω_{1} such that d · f_{α}(b_{n}^{α}) = f(b_{n}^{α}) for all n ∈ Z_{α} and d · f_{α}(b_{n}^{α}) = 0 for all n < ω \ Z_{α} holds.

Let B = ⋃_{α<κ} B_{α}. We show that this B is as desired. By 1 ) the underlying set of B is κ. By 2 ) { b_{α} : α < κ } is an independent subset of B. Hence Fr κ is embeddable into B.

Suppose now that there would be a surjective Boolean mapping f from B onto Fr ω_{1}. Then there is a bijection g ⊆ f from a subset of B onto Fr ω_{1}. Since g is uncountable there is an α < κ such that x_{α} ⊆ g. Since x_{α} ⊆ f, x_{α} satisfies the condition in 3 ). Hence there is a c_{α} ∈ B_{α+1} such that c_{α} = Σ_{n∈Z_{α}}^{B_{α+1}}b_{n}^{α} for Z_{α} and b_{n}^{α}, n < ω as un 3 ). But then f(c_{α}) · f_{α}(b_{n}^{α}) = f(b_{n}^{α}) for all n ∈ Z_{α} and f(c_{α}) · f_{α}(b_{n}^{α}) = 0 for all n < ω \ Z_{α} holds. This is a contradiction to the choice of Z_{α}. (Proposition 4.1)

Corollary 4.2 m_{Fn(ω_{1},2)} ≤ •|′.

Proof By Proposition 2.1 and Proposition 4.1. (Corollary 4.2) With almost the same proof as in Proposition 4.1 we can also prove the following:

Proposition 4.3 There exists a Boolean algebra B such that | B | = •|′′, Fr •|′′ is embeddable into B but there is no surjective Boolean mapping from B onto Fr •|′′.

Since we have •|′ = ℵ_{1} under ( •| ), we obatin the following theorem:

Theorem 4.4 If ( •| ) holds then there exists a Boolean algebra B of cardinality ℵ_{1} such that Fr ω_{1} is embeddable into B but there is no surjection from B onto Fr ω_{1}.

Hence if ¬CH and ( •| ) holds, by Theorem 4.4, there exists a counter-example to the theorem of Shapiro. This shows that we cannot just drop MA(Cohen) from Theorem 1.1. Since MA(countable) + ¬CH + ( •| ) is consistent (see e.g. [5] or [4]), we see that MA(countable) is not enough for Theorem 1.1.

Corollary 4.5 mCohen ≤ •|′′.

Proof By Proposition 2.1 and Proposition 4.3. (Corollary 4.5)

If a Boolean algebra B is atomless then Fr ω can be embdded into B. By Proposition 2.1, if MA(countable) holds and B is of cardinality < 2^{ℵ_{0}}, there exists a surjection from B onto Fr ω. Here again we cannot simply drop the assumption

of MA(countable):

Proposition 4.6 It is consistent that there is an atomless Boolean algebra B of cardinality ℵ_{1} < 2^{ℵ_{0}} such that there is no surjective Boolean mapping from B onto Fr ω.

Proof By [9, Theorem 5.12], there is a model of ZFC+ ¬CH in which there is an endo-rigid atomless Boolean algebra B of cardinality ℵ_{1}. In particular there is no surjection from B onto Fr ω. (Proposition 4.6)

Note that, since ( •| ) is consistent with ¬CH and MA(countable), ( •| ) cannot supply such a Boolean algebra as in the proposition above.

References

[1] B. Balcar, F. Franěk: Independent families in complete Boolean algebras, Trans. Amer. Math. Soc., Vol. 274 (1982), 607–618.

[2] S. Broverman, J. Ginsburg, K. Kunen and F. Tall: Topologies determined by σ-ideals on ω_{1}, Can. J. Math., 30 No. 6 (1978), 1306–1312.

[3] K. Eda: Cardinality restrictions on preradicals, Contemporary Math. Vol. 87 (1989), 277–283.

[4] S. Fuchino, S. Shelah and L. Soukup: Sticks and Clubs, in preprint.

[5] P. Komjáth: Set systems with finite chromatic number, European Journal of Combinatorics, 10 (1989), 543–549.

[6] K. Kunen, F. Tall: Between Martin’s axiom and Souslin’s hypothesis, Fundamenta Mathematicae, Vol. 102 (1979), 173–181.

[7] A.J. Ostaszewski: On countably compact perfectly normal spaces, Journal of London Mathematical Society (2), 14 (1976), 505–516.

[8] L.B. Shapiro: On Šapirovskiĭ’s Theorem, preprint.

[9] S. Shelah: Constructions of many complicated uncountable structures and Boolean algebras, Israel Journal of Mathematics, Vol. 45 (1983), 100–146.

[10] : PCF and infinite free subsets, in preparation.

[11] : Cardinal Arithmetic, Oxford University Press, in print.

Authors’ addresses:

Institut of Mathematics, Hebrew University of Jerusalem 91904 Jerusalem, Israel and Institut für Mathematik II, Freie Universität Berlin 14195 Berlin, Germany

fuchino@math.fu-berlin.de

Institut of Mathematics, Hebrew University of Jerusalem 91904 Jerusalem, Israel and Department of Mathematics, Rutgers University New Brunswick, NJ 08854, USA

shelah@math.huji.ac.il

Mathematical Institute of the Hungarian Academy of Sciences

soukup@math-inst.hu