APPLICATIONS OF ANOTHER CHARACTERIZATION OF βℕ \ ℕ

ALAN DOW^{†} AND KLAAS PIETER HART^{‡}

Abstract. Steprāns provided a characterization of βℕ \ ℕ in the ℵ_{2}-Cohen model that is much in the spirit of Parovičenko’s characterization of this space under CH. A variety of the topological results established in the Cohen model can be deduced directly from the properties of βℕ \ ℕ or P(ℕ)/fin that feature in Steprāns’ result.

Introduction

Topological problems that involve the behaviour of families of subsets of the set of natural numbers tend to have (moderately) easy solutions if the Continuum Hypothesis (CH) is assumed. The reason for this is that one’s inductions and recursions last only ℵ_{1} steps and that at each intermediate step only countably many previous objects have to be dealt with.

An archetypal example is Parovičenko’s characterization, see [22], of the space ℕ^{∗} as the only compact zero-dimensional F -space of weight 𝔠 without isolated points in which non-empty G_{δ}-sets have non-empty interiors. The proof actually shows that P(ℕ)/fin is the unique atomless Boolean algebra of size 𝔠 with a certain property R_{ω} and then applies Stone duality to establish uniqueness of ℕ^{∗}. It runs as follows: consider two Boolean algebras A and B with the properties of P(ℕ)/fin just mentioned and well-order both in type ω_{1}. Assume we have isomorphism h between subalgebras A_{α} and B_{α} that contain {a_{β} : β < α} and {b_{β} : β < α} respectively. We need to define h(a_{α}) (if a_{α} ∉ A_{α}); consider S = {a ∈ A_{α} : a < a_{α}} and T = {a ∈ A_{α} : a < a′_{α}}. We must find b ∈ B such that

(1) h(a) < b if a ∈ S;

(2) h(a) < b′ if a ∈ T;
(3) h(a) ∧ b ≠ 0 and h(a) ∧ b′ ≠ 0 if a ∈ A_{α} but a ∉ S ∪ T.

Property R_{ω} says exactly that this is possible — its proper formulation can be found after Definition 2.9.

This paper grew out of observations that in the Cohen model the Boolean algebra P(ℕ)/fin retains much of the properties that were used above. In a sense to be made precise later, in Definition 2.3, P(ℕ)/fin contains many subalgebras that are like A_{α} and B_{α} above (ℵ_{0}-ideal subalgebras); even though these will not be countable the important sets S and T will be. We also define a cardinal invariant, 𝔪_{c}, that captures just enough of R_{ω} to allow a Parovičenko-like characterization of P(ℕ)/fin in the ℵ_{2}-Cohen model — this is Steprāns’ result alluded to in the abstract (Theorem 2.13). During the preparation of this paper we became aware of recent work on the weak Freese-Nation property in [14, 15, 13]. Although the weak Freese-Nation property is stronger than our properties the proofs of the consequences are very similar; therefore we restrict, with few exceptions, ourselves to more topological (and new) applications. Perhaps the difference in approach (weak Freese-Nation versus (ℵ_{1}, ℵ_{0})-ideal) is mostly a matter of taste but ours arose directly out of Steprāns original results and the essentially folklore facts about the effects of adding Cohen reals.

Date: Thursday 25-05-2000 at 15:43:53 (cest).

2000 Mathematics Subject Classification. Primary: 54A35; Secondary: 03E35, 06E05, 54D35, 54F65.

Key words and phrases. βℕ, Cohen forcing, Parovičenko’s theorem, characterizations.

†The research of the first author was supported in part by the Netherlands Organization for Scientific Research (NWO) — Grant B 61-408 and by NSERC.

‡The research of the second author was supported in part by the Netherlands Organization for Scientific Research (NWO) — Grant R 61-444.

In Section 2 we shall formulate the properties alluded to above and prove that in the Cohen model P(ℕ)/fin does indeed satisfy them. In Section 3 we select some results about P(ℕ)/fin (or ℕ^{∗}) that are known to hold in the Cohen model and derive them directly from the new properties — whenever we credit a result to some author(s) we mean to credit them with establishing that it holds in the Cohen model. In Sections 4 and 5 we investigate the properties themselves and their behaviour with respect to subalgebras and quotients. Finally, in Section 6 we investigate how much of an important phenomenon regarding ℕ^{∗} persists; we are referring to the fact that under CH for every compact zero-dimensional space X of weight 𝔠 or less the Čech-Stone remainder (ω × X)^{∗} is homeomorphic to ℕ^{∗}.

We would like to take this opportunity to thank the referee for a very insightful remark concerning our version of Bell’s example (see Definition 3.7), which enabled us to simplify the presentation considerably.

1. Preliminaries

Boolean algebras. Our notation is fairly standard: b′ invariably denotes the complement of b.

For a subset S of a Boolean algebra B, let S^{⊥} denote the ideal of members of B that are disjoint from every element of S, i.e., S^{⊥} = {b ∈ B : (∀s ∈ S)(b ∧ s = 0)}. For convenience, we use b^{⊥} in place of {b}^{⊥}. Also, let b^{↓} be the principal ideal generated by b, namely {a ∈ B : a ⩽ b}. Clearly b^{↓} is equal to (b′)^{⊥}. Also, for subsets S and T we let S ⊥ T abbreviate (∀s ∈ S)(∀t ∈ T)(s ∧ t = 0); in fact we shall often abbreviate s ∧ t = 0 by s ⊥ t.

Cohen reals. ‘The Cohen model’ is any model obtained from a model of the GCH by adding a substantial quantity of Cohen reals — more than ℵ_{1}. In particular ‘the ℵ_{2}-Cohen model’ is obtained by adding ℵ_{2} many Cohen reals. Actually, since we are intent on proving our results using the properties of P(ℕ)/fin only, many readers may elect to take Lemma 2.2, Theorem 2.7 and the remark made after Proposition 2.12 on faith or else consult [18] for the necessary background on Cohen forcing.

The weak Freese-Nation property. A partially ordered set P is said to have the weak Freese-Nation property if there is a function F : P → [P]^{ℵ_{0}} such that whenever p ⩽ q there is r ∈ F (p) ∩ F (q) with p ⩽ r ⩽ q.

Elementary substructures. Consider two structures M and N (groups, fields, Boolean algebras, models of set theory …), where M is a substructure of N. We say that M is an elementary substructure of N, and we write M ≺ N, if every equation, involving the relations and operations of the structures and constants from M, that has a solution in N has a solution in M as well.

The Löwenheim-Skolem theorem says that every subset A of a structure N can be enlarged to an elementary substructure M of whose cardinality is the maximum of |A| and ℵ_{0}. The construction proceeds in the obvious way: in a recursion of length ω one keeps adding solutions to equations that involve ever more constants.

We prefer to think of an argument that uses elementary substructures as the lazy man’s closing off argument; rather than setting up an impressive recursive construction we say “let θ be a suitably large cardinal and let M be an elementary substructure of H(θ)” and add some words that specify what M should certainly contain.

The point is that the impressive recursion is carried out inside H(θ), where θ is ‘suitably large’ (most of the time θ = 𝔠^{+} is a good choice as everything under consideration has cardinality at most 𝔠), and that it (or a nonessential variation) is automatically subsumed when one constructs an elementary substructure of H(θ).

In this paper we shall be working mostly with ℵ_{1}-sized elementary substructures, most of which will be ℵ_{0}-covering. The latter means that every countable subset A of M is a subset of a countable element B of M. This is not an unreasonable property, considering that the ordinal ω_{1} has it: every countable subset of ω_{1} is a subset of a countable ordinal.

An ℵ_{0}-covering structure can be constructed in a straightforward way. One recursively constructs a chain ⟨M_{α} : α < ω_{1}⟩ of countable elementary substructures of H(θ) with the property that ⟨M_{β} : β ⩽ α⟩ ∈ M_{α+1} for all α. In the end M = ⋃_{α<ω_{1}} M_{α} is as required: if A ⊆ M is countable then A ⊆ M_{α} for some α and M_{α} ∈ M.

For just a few of the results we indicate two proofs: a direct one and one via elementarity — we invite the reader to compare the two approaches and to reflect on their efficacy.

2. Two new properties of P(ℕ)/fin

In this section we introduce two properties that Boolean algebras may have. We shall prove that in the Cohen model P(ℕ)/fin has both and that in the ℵ_{2}-Cohen model their conjunction actually characterizes P(ℕ)/fin.

(ℵ_{1}, ℵ_{0})-ideal algebras. We begin by defining the ℵ_{0}-ideal subalgebras alluded to in the introduction.

Definition 2.1. For a Boolean algebra B, we will say that a subalgebra A of B is ℵ_{0}-ideal if for each b ∈ B \ A the ideal {a ∈ A : a < b} = A ∩ b^{↓} has a countable cofinal subset.

Of course, by duality, the ideal b^{⊥} ∩ A is countably generated as well; thus in ℵ_{0}-ideal subalgebras the phrase “S and T are countable” from the introduction is replaced by “S and T have countable cofinal subsets”.

The main impetus for this definition comes from following result.

Lemma 2.2 ([23, Lemma 2.2]). If G is Fn(I, 2)-generic over V then P(ℕ) ∩ V is an ℵ_{0}-ideal subalgebra of P(ℕ) in V [G].

Proof. Let X̃ be an Fn(I, 2)-name for a subset of ℕ. It is a well-known fact about Fn(I, 2) that there is a countable subset J of I such that X̃ is completely determined by Fn(J, 2). This means that for every p ∈ Fn(I, 2) and every n ∈ ℕ we have p ⊩ n ∈ X̃ (or p ⊩ n ∉ X̃) if and only if p ↾ J does.

For every p ∈ Fn(J, 2) define X_{p} = {n : p ⊩ n ∈ X̃}; the countable family of these X_{p} is as required. □

The factoring lemma for Cohen forcing ([18, p. 255]) implies that for every subset J of I the subalgebra A_{J} = P(ℕ) ∩ V [G ↾ J] is ℵ_{0}-ideal in the final P(ℕ). Using the fact, seen in the proof above, that names for subsets of ℕ are essentially countable one can verify that A_{⋃ J} = ⋃_{J∈J} A_{J} for every chain J of subsets of I of uncountable cofinality. This shows that in the Cohen model P(ℕ) has many ℵ_{0}-ideal subalgebras and also that the family of these subalgebras is closed under unions of chains of uncountable cofinality.

What we call ℵ_{0}-ideal is called ‘good’ in [23] and in [14] the term σ-subalgebra is used. In the latter paper it is also shown that if F : B → [B]^{ℵ_{0}} witnesses the weak Freese-Nation property of B then every subalgebra that is closed under F is an ℵ_{0}-ideal subalgebra; therefore an algebra with the weak Freese-Nation property has many ℵ_{0}-ideal subalgebras and the family of these subalgebras is closed under directed unions.

We are naturally interested in Boolean algebras with many ℵ_{0}-ideal subalgebras. Most of our results only require that there are many ℵ_{1}-sized ℵ_{0}-ideal subalgebras.

Definition 2.3. We will say that a Boolean algebra B is (ℵ_{1}, ℵ_{0})-ideal if the set of ℵ_{1}-sized ℵ_{0}-ideal subalgebras of B contains an ℵ_{1}-cub of [B]^{ℵ_{1}}. That is, there is a family A consisting of ℵ_{1}-sized ℵ_{0}-ideal subalgebras of B such that every subset of size ℵ_{1} is contained in some member of A and the union of each chain from A of cofinality ω_{1} is again in A.

We leave to the reader the verification that P(ℕ) is an (ℵ_{1}, ℵ_{0})-ideal algebra if and only if P(ℕ)/fin is an (ℵ_{1}, ℵ_{0})-ideal algebra (but see Corollary 5.12). It is also worth noting that P(ω_{1}) is not (ℵ_{1}, ℵ_{0})-ideal (see [14, Proposition 5.3]).

Since the definition of (ℵ_{1}, ℵ_{0})-ideal requires that we have some ℵ_{1}-cub consisting of ℵ_{0}-ideal subalgebras, it is a relatively standard fact that every ℵ_{0}-covering elementary substructure of size ℵ_{1} of a suitable H(θ) induces an ℵ_{0}-ideal subalgebra. We shall use the following lemma throughout this paper, not always mentioning it explicitly — it is an instance of the rule-of-thumb that says: if X, A ∈ M, where M is suitably closed and A some sort of cub in P(X), then X ∩ M ∈ A.

Lemma 2.4. Let B be an (ℵ_{1}, ℵ_{0})-ideal algebra, let θ be a suitably large cardinal and let M be an ℵ_{0}-covering elementary substructure of size ℵ_{1} of H(θ) that contains B. Then B ∩ M is an ℵ_{0}-ideal subalgebra of B.

Proof. Note first that M contains an ℵ_{1}-cub A as in Definition 2.3: it must contain a solution to the equation

x is an ℵ_{1}-cub in [B]^{ℵ_{1}} that consists of ℵ_{0}-ideal subalgebras.

Let f : ω_{1} → A ∩ M be a surjection, not necessarily from M. Because M is ℵ_{0}-covering we can find, for every α ∈ ω_{1}, a countable element X_{α} of M that contains f[α]. Consider the equation

x ∈ A and ⋃(X_{α} ∩ A) ⊆ x.

This equation has a solution in H(θ) and hence in M; we may take A_{α} ∈ A ∩ M such that ⋃(X_{α} ∩ A) ⊆ A_{α}. Thus we construct an increasing chain ⟨A_{α} : α < ω_{1}⟩ in A ∩ M that is cofinal in A ∩ M. It follows that ⋃(A ∩ M) = ⋃_{α<ω_{1}} A_{α} belongs to A. Now check carefully that B ∩ M = ⋃(A ∩ M) — use that A is unbounded in [B]^{ℵ_{1}}. □

The remarks preceding Definition 2.3 show that an algebra with the weak Freese-Nation property is (ℵ_{1}, ℵ_{0})-ideal. The converse is almost true — the difference is that we do not require closure under countable unions. In the notation used after Lemma 2.2 the family A = {A_{J} : J ∈ [I]^{ℵ_{1}}} witnesses that in the Cohen model P(ℕ) is always (ℵ_{1}, ℵ_{0})-ideal. However A is not closed under unions of countable chains. Indeed, in [12] one finds the theorem that if V satisfies the GCH and the instance (ℵ_{ω+1}, ℵ_{ω}) → (ℵ_{1}, ℵ_{0}) of Chang’s conjecture then after adding one dominating real d and then ℵ_{ω} Cohen reals P(ℕ) does not have the weak Freese-Nation property — as V [d] still satisfies the GCH the final model is a ‘Cohen model’.

Many properties of P(ℕ) that hold in the Cohen model can be derived from the weak Freese-Nation property — see [15, 13] for example — and many of these can be derived from the fact that P(ℕ) is (ℵ_{1}, ℵ_{0})-ideal. It is not our intention to duplicate the effort of [13]; we will concentrate on topological applications. However, to give the flavour, and because we shall use the result a few times, we consider Kunen’s theorem from [17] that in the Cohen model the Boolean algebra P(ℕ)/fin does not have a chain of order type ω_{2}.

It is quite straightforward to show that an algebra with the weak Freese-Nation property does not have any well-ordered chains of order type ω_{2}; with a bit more effort the same can be said of (ℵ_{1}, ℵ_{0})-ideal algebras.

Proposition 2.5. An (ℵ_{1}, ℵ_{0})-ideal Boolean algebra does not have any chains of order type ω_{2}.

Proof. Assume that {c_{α} : α < ω_{2}} is an increasing chain in B and let A be as in Definition 2.3. Recursively construct a chain {A_{α} : α ∈ ω_{1}} in A and an increasing sequence {γ_{α} : α ∈ ω_{1}} of ordinals in ω_{2} as follows. Let γ_{0} = 0 and, given A_{β} and γ_{β} for β < α, let A = ⋃_{β<α} A_{α} and γ = sup_{β<α} γ_{β}. Choose A_{α} ∈ A such that c_{γ} ∈ A_{α} and such that for every a ∈ A, if there is a β with a ⩽ c_{β} then there is a β such that a ⩽ c_{β} and c_{β} ∈ A_{α}; let γ_{α} be the first γ for which c_{γ} ∉ A_{α}.

In the end set A = ⋃{A_{α} : α ∈ ω_{1}} and λ = sup{γ_{α} : α ∈ ω_{1}}. Now we have a contradiction because although c_{λ}^{↓} ∩ A should be countably generated it is not. Indeed, let C be a countable subset of c_{λ}^{↓} ∩ A; by construction we have for every c ∈ C a β < λ such that c ⩽ c_{β}. Let γ be the supremum of these β’s; then γ + 1 < λ and so c_{γ+1} < c_{λ} but no c ∈ C is above c_{γ+1}. □

A proof using elementary substructures runs as follows: let M ≺ H(θ) be ℵ_{0}- covering and of cardinality ℵ_{1}, where θ is suitably large, and assume that B and the chain {c_{α} : α < ω_{2}} belong to M. Next let δ be the ordinal M ∩ ω_{2}; observe that cf δ = ℵ_{1}: if cf δ were countable then, because M is ℵ_{0}-covering, δ would be the supremum of an element of M and hence in M. Consider the element c_{δ}. By Lemma 2.4 there is a countable subset T of c_{δ}^{↓} ∩ B ∩ M that is cofinal in it. There is then (at least) one element a of T such that {α < δ : c_{α} ⩽ a} is cofinal in δ. However, δ is a solution to

x ∈ ω_{2} and a < c_{x}

hence there must be a solution β in M but then β < δ and a < c_{β}, so that {α < δ : c_{α} ⩽ a} is not cofinal in δ.

The reader is invited to supply a proof of the following proposition, which was established in [15] for algebras with the weak Freese-Nation property.

Proposition 2.6. An (ℵ_{1}, ℵ_{0})-ideal algebra contains no ℵ_{2}-Lusin families. □

An ℵ_{2}-Lusin family is a subset A of pairwise disjoint elements with the following property: for every x at least one of the sets {a ∈ A : a ⩽ x} or {a ∈ A : a ∩ x = 0} has size less than ℵ_{2}.

In the Cohen model P(ℕ)/fin is (ℵ_{1}, ℵ_{0})-ideal. We have already indicated that in the Cohen model P(ℕ)/fin is an (ℵ_{1}, ℵ_{0})-ideal algebra. We state it as a separate theorem for future reference.

Theorem 2.7. Let V be a model of CH and let κ be any cardinal. If G is generic on Fn(κ, 2) then in V [G] the algebra P(ℕ) is (ℵ_{1}, ℵ_{0})-ideal. □

As it is clear that ω_{2} is the union of an increasing sequence of ℵ_{1}-sized subsets the following Proposition, which is Steprāns’ Lemma 2.3, now follows.

Proposition 2.8. In the ℵ_{2}-Cohen model there is an increasing sequence of ℵ_{0}-ideal subalgebras of P(ℕ), each of size ℵ_{1}, which is continuous at limits of uncountable cofinality and whose union is all of P(ℕ). □

Generalizing R_{ω}. The following definition generalizes Parivičenko’s property R_{ω}. After the definition we discuss it more fully and indicate why it is the best possible generalization of R_{ω}. The new property is actually a cardinal invariant which somehow quantifies some, but not all, of the strength of MA_{countable} — see Proposition 2.12 and Remark 4.6.

Definition 2.9. For a Boolean algebra B, say that a subset A is ℵ_{0}-ideal complete, if for any two countable subsets S and T of A with S ⊥ T there is a b ∈ B \ A such that b^{↓} ∩ A is generated by S and b^{⊥} ∩ A is generated by T. We will let 𝔪_{c}(B) denote the minimum cardinality of a subset of B that is not ℵ_{0}-ideal complete. Also 𝔪_{c} denotes 𝔪_{c}(P(ℕ)/fin).

A remark about the previous definition might be in order. In the definition of ℵ_{0}- ideal completeness the set A is divided into three subsets: A_{S}, the set of elements a for which there is a finite subset F of S such that a ⩽ ⋁ F; the set A_{T}, defined similarly, and A_{r}, the rest of A. The element b must effect the same division of A: we demand that A_{S} = {a ∈ A : a < b}, A_{T} = {a ∈ A : a < b′} and A_{r} = {a ∈ A : a ⩽̸ b and a ⩽̸ b′}. Observe that one can also write A_{r} = {a ∈ A : b ∧ a ≠ 0 and b′ ∧ a ≠ 0}; one says that b reaps the set A_{r}. We see that every subset of size less than 𝔪_{c}(B) can always be reaped; we shall come back to this in Section 4.

Thus Parovičenko’s property R_{ω} has become the statement that countable subsets are ℵ_{0}-ideal complete, in other words that 𝔪_{c}(B) > ℵ_{0}.

Remark 2.10. In Definition 2.9 we explicitly do not exclude the possibility that S or T is finite or even empty. Thus if 𝔪_{c}(B) > ℵ_{0} then there is no countable strictly increasing sequence with 1 as its supremum: for let S be such a sequence and take T = {0}, then there must apparently be a b < 1 such that a < b for all a ∈ S.

Remark 2.11. In the case of P(ℕ)/fin one cannot relax the requirements on S and T: consider a Hausdorff gap; this is a pair of increasing sequences {a_{α} : α ∈ ω_{1}} and {b_{α} : α ∈ ω_{1}} such that a_{α} ∧ b_{β} = 0 for all α and β, and for which there is no x such that a_{α} ⩽ x for all α and b_{β} ⩽ x^{⊥} for all β. Thus there are cases with |S| = |T | = ℵ_{1} where no b can be found.

In an (ℵ_{1}, ℵ_{0})-ideal algebra with 𝔪_{c}(B) > ℵ_{0} this can be sharpened, as follows. By recursion one can construct a strictly increasing chain ⟨s_{α} : α < δ⟩ in B with 0 < s_{α} < 1 for all α, until no further choices can be made. Because B is (ℵ_{1}, ℵ_{0})- ideal this must stop before ω_{2} and because 𝔪_{c}(B) > ℵ_{0} we have cf δ = ℵ_{1}. Thus we have a situation where no b be found with |S| = ℵ_{1} and |T | = 1 (take T = {0}). This shows that if P(ℕ)/fin is (ℵ_{1}, ℵ_{0})-ideal then the cardinal number 𝔱 (see [6]) is equal to ℵ_{1}.

The following proposition shows why we are interested in 𝔪_{c}.

Proposition 2.12. MA_{countable} implies that 𝔪_{c} = 𝔠.

Proof. Let A, S and T be given, where, without loss of generality, we assume that S and T are increasing sequences of length ω and |A| < 𝔠. There is a natural countable poset that produces an infinite set b such that s < b and t < b′ for all s ∈ S and t ∈ T: it consists of triples ⟨p, s, t⟩, where p ∈ Fn(ω, 2), s ∈ S, t ∈ T and s ∩ t ⊆ dom(p). The ordering is ⟨p, s, t⟩ ⩽ ⟨q, u, v⟩ iff p ⊇ q, s ⊇ u, t ⊇ v and if n ∈ dom(p) \ dom(q) then p(n) = 1 if n ∈ u and p(n) = 0 if n ∈ v.

It is relatively straightforward to determine a family D of fewer than 𝔠 dense sets so that any D-generic filter produces an element b as required. □

It is well-known that MA_{countable} holds in any extension by a ccc finite-support iteration whose length is the final value of the continuum and hence in any model obtained by adding 𝔠 or more Cohen reals.

So in the Cohen model P(ℕ)/fin is an (ℵ_{1}, ℵ_{0})-ideal algebra in which 𝔪_{c} is 𝔠. Note that this is then consistent with most cardinal arithmetic. However if only ℵ_{2} Cohen reals are added then this provides our characterizations of P(ℕ)/fin and ℕ^{∗} (see also the results 5.3 through 5.5).

Theorem 2.13. In the ℵ_{2}-Cohen model the algebra P(ℕ)/fin is characterized by the properties of being an (ℵ_{1}, ℵ_{0})-ideal Boolean algebra of cardinality 𝔠 in which 𝔪_{c} has value 𝔠. □

The proof is quite straightforward: we use Proposition 2.8 to express any algebra with the properties of the Theorem as the union of a ω_{2}-chain of (ℵ_{1}, ℵ_{0})-ideal subalgebras and we apply 𝔪_{c} = 𝔠 to construct an isomorphism between it and P(ℕ)/fin by recursion. This result and its proof admit a topological reformulation that is quite appealing.

Theorem 2.14. In the ℵ_{2}-Cohen model ℕ^{∗} is the unique compact space that is expressible as the limit of an inverse system ⟨{X_{α} : α < ω_{2}}, {f_{α}^{β} : α < β < ω_{2}}⟩ such that

(1) each X_{α} is a compact zero-dimensional space of weight less than 𝔠;

(2) for each limit λ < ω_{2}, X_{λ} is equal to \underleftarrow{lim}_{β<λ} X_{β} and f_{α}^{λ} = \underleftarrow{lim}_{α<β<λ} f_{α}^{β};

(3) for each α < β < ω_{2}, f_{α}^{β} sends zero-set subsets of X_{β} to zero-sets of X_{α} (i.e. clopen sets are sent to G_{δ}-sets;

(4) for each α < ω_{2} and each pair, C_{0}, C_{1} of disjoint cozero-sets of X_{α} (possibly empty), there are a β < ω_{2} and a clopen subset b of X_{β} such that

f_{α}^{β}(b) = X_{α} \ C_{0} and f_{α}^{β}(X_{β} \ b) = X_{α} \ C_{1}. □

Remark 2.15. It is our (subjective) feeling that the (ℵ_{1}, ℵ_{0})-ideal property together with 𝔪_{c} captures the essence of the behaviour of P(ℕ) and P(ℕ)/fin in the Cohen model. By Theorem 2.13 this is certainly the case for the ℵ_{2}-Cohen model. Evidence in support of our general feeling will be provided in the next section, where we will derive a number of results from “P(ℕ) is (ℵ_{1}, ℵ_{0})-ideal” that were originally derived in the Cohen model. Apparently it is unknown whether these properties characterize P(ℕ)/fin in Cohen models with 𝔠 > ℵ_{2}.

Other cardinals. We may generalize Definition 2.3 to cardinals other than ℵ_{1}: we can call a Boolean algebra (κ, ℵ_{0})-ideal if the family of κ-sized ℵ_{0}-ideal subalgebras contains a κ-cub, meaning a subfamily closed under unions of chains of length at most κ (but of uncountable cofinality). Similarly we can define B to be (∗, ℵ_{0})-ideal if it is (κ, ℵ_{0})-ideal for every (regular) κ below |B|.

The discussion after Lemma 2.2 establishes that every Boolean algebra with the weak Freese-Nation property (∗, ℵ_{0})-ideal and in any Cohen model the algebra P(ℕ)/fin is (∗, ℵ_{0})-ideal. One can also prove a suitable version of Lemma 2.4.

Lemma 2.16. Let B be an (κ, ℵ_{0})-ideal algebra, let θ be a suitably large cardinal and let M be an elementary substructure of size κ of H(θ) that contains B. Then B ∩ M is an ℵ_{0}-ideal subalgebra of B, provided M can be written as ⋃_{α<κ} M_{α}, where ⟨M_{β} : β ⩽ α⟩ ∈ M_{α+1} for all α. □

In applications one also needs M to be ℵ_{0}-covering; this is possible only if the structure ([κ]^{ℵ_{0}}, ⊆) has cofinality κ. This accounts for the assumption cf[κ]^{ℵ_{0}} = κ in Theorem 5.6.

3. The Axiom “P(ℕ) is an (ℵ_{1}, ℵ_{0})-ideal algebra”

Throughout this section we assume that P(ℕ) is an (ℵ_{1}, ℵ_{0})-ideal algebra and show how useful this can be as an axiom in itself. We fix an ℵ_{1}-cub A in [P(ℕ)]^{ℵ_{1}} that consists of ℵ_{0}-ideal subalgebras.

Mappings onto cubes. In order to avoid additional definitions we state, in the rest of this section, some of the results in their topological, rather than Boolean algebraic, formulations. We shall also use elementary substructures to our advantage; we shall use the phrase ‘by elementarity’ to indicate that a judicious choice of equation would give the desired result.

The first result we present is due to Baumgartner and Weese [2].

Theorem 3.1. If X is a compact space with a countable dense set D such that every infinite subset of D contains a converging subsequence, then X does not map onto [0, 1]^{ω_{2}}.

Proof. If f were a mapping of X onto [0, 1]^{ω_{2}} then f[D] would be a countable dense subset of [0, 1]^{ω_{2}} with the same property as D. Therefore we are done once we show that [0, 1]^{ω_{2}} has no countable dense subset every infinite subset of which contains a converging sequence. So we take a countable dense subset of [0, 1]^{ω_{2}}, which we identify with ℕ, and exhibit an infinite subset of it that does not contain a converging sequence.

To this end we fix a suitably large cardinal θ and consider an ℵ_{1}-sized ℵ_{0}-covering elementary substructure M of H(θ). We put δ = M ∩ ω_{2} and let c = ℕ ∩ π\overleftarrow{δ}[[\frac{1}{4}, 1]] and d = ℕ ∩ π\overleftarrow{δ}[[0, \frac{3}{4}]], where, generally, π_{α} denotes the projection onto the α-th coordinate.

Let C ∈ M be a countable set such that c^{↓} ∩ M is generated by C_{1} = c^{↓} ∩ C; similarly choose a countable element D of M for d and put D_{1} = d^{↓} ∩ D. This can be done because M is ℵ_{0}-covering.

For x ∈ C let S_{x} = \{α : x ⊆^{∗} π\overleftarrow{α}[[\frac{1}{4}, 1]]\}. Observe that if δ ∈ S_{x} then S_{x} is cofinal in ω_{2} because, apparently, there is then in M no solution η to (η ∈ ω_{2}) ∧ (∀α ∈ S_{x})(α < η) in M and hence not in H(θ) either. It follows that C_{1} is contained in the set

C_{2} = {x ∈ C : S_{x} is cofinal in ω_{2}},

which, by elementarity, is in M. We define T_{y}, for y ∈ D, in an analogous way and find the set

D_{2} = {y ∈ D : T_{y} is cofinal in ω_{2}},

which is in M and which contains D_{1}. We claim that the ideal generated by C_{2}∪D_{2} does not contain a cofinite subset of ℕ.

Indeed, take x_{1}, …, x_{k} in C_{2} and y_{1}, …, y_{k} in D_{2}. We can find distinct α_{1}, …, α_{k}, β_{1}, …, β_{k} with α_{i} ∈ S_{x_{i}} and β_{i} ∈ T_{y_{i}} for all i. The set U = ⋂_{i=1}^{k}(π\overleftarrow{α_{i}}[[0, \frac{1}{4})] ∩ π\overleftarrow{β_{i}}[(\frac{3}{4}, 1]]) is disjoint from ⋃_{i=1}^{k}(x_{i} ∪ y_{i}) and its intersection with ℕ is infinite.

Because C_{2} ∪ D_{2} is countable we can, by elementarity, find an infinite subset a of ℕ in M that is almost disjoint from every one of its elements. Now if a had an infinite converging subset then, again by elementarity, it would have one, b say, that belongs to M. However, if b ⊆^{∗} c then b ⊆^{∗} x for some x ∈ C_{1}, which is impossible; likewise b ⊆^{∗} d is impossible. It follows that π_{δ}[b] does not converge in [0, 1]. □

Remark 3.2. A careful study of the proof of Theorem 3.1 shows how one can reach δ by a traditional recursion. Build an increasing sequence ⟨A_{α} : α < ω_{1}⟩ in A and a sequence ⟨δ_{α} : α < ω_{1}⟩ in ω_{2} by doing the following at successor steps. Enumerate A_{α} as ⟨a_{β} : β < ω_{1}⟩ and choose, whenever possible, a subset b_{β} of a_{β} that converges in [0, 1]^{ℵ_{2}}. Next choose, for each β < ω_{1}, a subset d_{β} as follows: let C = {γ < β : S_{a_{γ}} is cofinal in ω_{2}} and D = {γ < β : T_{a_{γ}} is cofinal in ω_{2}} (here S_{x} and T_{x} are defined as in the proof); as in the proof we can find a nonzero d_{β} in (C ∪ D)^{⊥}. Let A_{α+1} be an element of A that contains A_{α} ∪ {b_{β}}_{β∈ω_{1}} ∪ {d_{β}}_{β∈ω_{1}} and choose δ_{α+1} so large that sup S_{a} < δ_{α+1} or sup T_{a} < δ_{α+1} whenever a ∈ A_{α+1} and S_{a} or T_{a} is bounded in ω_{2}. The rest of the proof is essentially the same.

The next result, from [8], provides a nice companion to Theorem 3.1.

We prove the result for the case 𝔠 = ℵ_{2} only — basically the same proof will work when 𝔠 = ℵ_{n} for some n ∈ ω. For larger values of 𝔠 we need assumptions like □ to push the argument through.

Theorem 3.3 (2^{ℵ_{1}} = 𝔠 = ℵ_{2}). If X is compact, separable and of cardinality greater than 𝔠 then X maps onto I^{𝔠}.

Proof. Suppose that X is compact and that ℕ is dense in X. Fix a suitably large cardinal θ and construct an increasing sequence ⟨M_{α} : α < ω_{2}⟩ of ℵ_{1}-sized elementary substructures of H(θ) that are ℵ_{0}-covering and that always ⟨M_{β} : β < α⟩ ∈ M_{α+1}; put M = ⋃_{α<ω_{2}} M_{α}. Furthermore by the cardinality assumptions we can ensure that M^{ω} and M^{ω_{1}} are subsets of M.

Fix any point x in X \ M (because |M| < |X|). For each α < ω_{2} let I_{α} = {F ⊆ ℕ : F ∈ M_{α} and x ∉ cl F }. Because |I_{α}| < 𝔠 we have I_{α} ∈ M and so by elementarity there is a point x_{α} ∈ X ∩ M such that I_{α} = {F ⊆ ℕ : F ∈ M_{α} and x_{α} ∉ cl F }. Fix a function f_{α} : X → [0, 1] so that f_{α}(x_{α}) = 0 and f_{α}(x) = 1, and set a_{α} = \{n : f_{α}(n) < \frac{1}{4}\} and b_{α} = \{n : f_{α}(n) > \frac{3}{4}\}. There is a g(α) < ω_{2} such that x_{α}, f_{α}, a_{α} and b_{α} belong to M_{g(α)}. Finally, fix a cub C in ω_{2} such that α < λ implies g(α) < λ whenever λ ∈ C. Set S = {λ ∈ C : cf λ = ω_{1}}.

Now apply the Pressing-Down lemma to find a stationary set T ⊆ S and a β ∈ ω_{2} so that, for every λ ∈ T, each of a_{λ}^{↓} ∩ M_{λ}, a_{λ}^{⊥} ∩ M_{λ}, b_{λ}^{↓} ∩ M_{λ} and, b_{λ}^{⊥} ∩ M_{λ} is generated by a countable subset of M_{β}.

By induction on λ ∈ T we prove that

{(a_{α}, b_{α}) : α ∈ T ∩ λ + 1}

is a dyadic family. In fact, if H and K are disjoint finite subsets of T then

⋂_{α∈H} a_{α} ∩ ⋂_{α∈K} b_{α}

is not in the ideal generated by I_{β}. Let λ = max(H ∪ K) and suppose first that λ ∈ K. Put y = ⋂_{α∈H} a_{α} ∩ ⋂_{α∈K\{λ}} b_{α}; then y is not contained in any member of I_{β}.

Assume there is an I ∈ I_{β} such that y ∩ b_{λ} ⊆ I; then y \ I belongs to M_{λ} ∩ b_{λ}^{⊥} and hence it is contained in a c ∈ M_{β} ∩ b_{λ}^{⊥}. Because c ⊥ b_{λ} we have f_{λ}[c] ⊆ [0, \frac{3}{4}] and so x ∉ cl c whence c ∈ I_{β}. We have a contradiction since it now follows that y ⊆ I ∪ c ∈ I_{β}.

Next suppose λ ∈ H and put y = ⋂_{α∈H\{λ}} a_{α} ∩ ⋂_{α∈K} b_{α}; again, y is not contained in any element of I_{β}. Assume there is I ∈ I_{β} such that y ∩ a_{λ} ⊆ I; now y \ I belongs to M_{λ} ∩ a_{λ}^{⊥} and hence it is contained in a c ∈ M_{β} ∩ a_{λ}^{⊥}. Because c ⊥ a_{λ} we have f_{λ}[c] ⊆ [\frac{1}{4}, 1] and so x_{λ} ∉ cl c; because c ∈ M_{λ} this means x ∉ cl c whence c ∈ I_{β}. Again we have a contradiction because we have y ⊆ I ∪ c ∈ I_{β}.

It now follows that △{f_{λ} : λ ∈ T } is a continuous map from X into I^{T} and that the image of X contains {0, 1}^{T}, which in turn can be mapped onto [0, 1]^{T}. □

This result is optimal: in [11] Fedorčuk constructed, in the ℵ_{2}-Cohen model, a separable compact space of cardinality 𝔠 = 2^{ℵ_{1}} that does not map onto I^{𝔠} because its weight is ℵ_{1}.

The size of sequentially compact spaces. The next result arose in the study of compact sequentially compact spaces (see [9] for the applications). Recall that a filter(base) of sets in a space X is said to converge to a point if every neighbourhood of the point contains an element of the filter(base).

Lemma 3.4. If X is a regular space and ℕ ⊆ X has the property that every infinite subset contains a converging sequence then for each ultrafilter u on ℕ that converges to some point of X there is an ℵ_{1}-sized filter subbase, v, that converges (to the same point).

Proof. Let u be an ultrafilter on ℕ that converges to a point x of X. Let M ≺ H(θ) be any ℵ_{1}-sized ℵ_{0}-covering model such that u, X and x are in M. We shall prove that v = M ∩ u also converges to x.

Since M is ℵ_{0}-covering and u ∈ M, there is an increasing chain {u_{α} : α ∈ ω_{1}} of countable subsets of u such that each u_{α} is a member of M and u ∩ M = ⋃{u_{α} : α ∈ ω_{1}}. For each α ∈ ω_{1}, there is an a_{α} ⊆ ℕ such that a_{α} ∈ M and a_{α} \ U is finite for each U ∈ u_{α}. By the assumption on the embedding of ℕ in X, we may assume that a_{α} converges to a point x_{α} ∈ X. Observe that for each b ∈ P(ℕ) ∩ M we have b ∈ u if and only if a_{α} ⊆^{∗} b for uncountably many α.

Suppose that x is an element of some open subset W of X. Let {b_{n} : n ∈ ω} ⊆ M ∩ P(ℕ) generate (W ∩ ℕ)^{⊥} ∩ M. Since u converges to x, the set W ∩ ℕ is a member of u. Therefore ℕ\b_{n} is a member of u for each n, hence there is an α such that {ℕ \ b_{n} : n ∈ ω} ⊆ u_{α}. It follows, then, that a_{β} is almost disjoint from each b_{n} for all β ⩾ α. Thus, for α ⩽ β < ω_{1} we have a_{β} ∉ (W ∩ ℕ)^{⊥}, which means that W ∩ a_{β} is infinite for each β ⩾ α. It follows that {x_{β} : α ⩽ β < ω_{1}} is contained in the closure of W. Since W was an arbitrary neighbourhood of x and X is regular, it follows that there is an α′ such that {x_{β} : α′ ⩽ β < ω_{1}} is contained in W. Since W is open, it follows that a_{β} is almost contained in W whenever α′ ⩽ β < ω_{1}.

Now suppose that {c_{n} : n ∈ ω} ⊆ M ∩ P(ℕ) generates (W ∩ ℕ)^{↓} ∩ M. By the above, it follows that, whenever α′ ⩽ β < ω_{1}, there is an n such that a_{β} is almost contained in c_{n}. Fix n such that a_{β} is almost contained in c_{n} for uncountably many η. As we observed above, it follows that c_{n} ∈ u. Therefore, as required, we have shown that W contains a member of v. □

Theorem 3.5. If X is a regular space in which ℕ is dense and every subset of ℕ contains a converging sequence, then X has cardinality at most 2^{ℵ_{1}}.

Proof. Each point of X will be the unique limit point of some filter base on ℕ of cardinality ℵ_{1}. □

Compare this theorem with Theorem 3.1, which draws the conclusion that X cannot be mapped onto [0, 1]^{ℵ_{2}}. In fact if 2^{ℵ_{1}} < 2^{ℵ_{2}} then Theorem 3.1 becomes a consequence of Theorem 3.5.

ℕ^{∗} minus a point. It was shown by Gillman in [16], assuming CH, that for every point u of ℕ^{∗} one can partition in ℕ^{∗} \ {u} into two open sets, each of which has u in its closure. Clearly this show that ℕ^{∗} \ {u} is not C^{∗}-embedded in ℕ^{∗}. Here we present Malykhin’s result, from [20], that establishes the complete opposite.

Theorem 3.6 (𝔪_{c} > ℵ_{1}). ℕ^{∗} minus a point is C^{∗}-embedded in ℕ^{∗}.

Proof. Assume that ℕ^{∗} \ {u} is not C^{∗}-embedded; so there is a continuous function f : ℕ^{∗} \{u} → [0, 1] such that u is simultaneously a limit point of f^{←}(0) and f^{←}(1).

Fix an increasing sequence {c_{n} : n ∈ ω} in P(ℕ) \ u such that in the case that u is not a P -point every member of u meets some c_{n} in an infinite set. Now define I = {a ∈ {c_{n}}_{n}^{⊥} : a^{∗} ⊆ f^{←}(0)} and J = {a ∈ {c_{n}}_{n}^{⊥} : a^{∗} ⊆ f^{←}(1)}. The ideals I and J are P -ideals: if I is a countable subset of I then apply 𝔪_{c} > ℵ_{0} to find a ∈ {c_{n}}_{n}^{⊥} with I ⊆ a^{↓}. Because a ∉ u the function f is defined on all of a^{∗}; it then follows that there is a b ⊆ a such that I ⊆ b^{↓} and b^{∗} ⊆ f^{←}(0). Claim 1. If U ∈ u then there is a ∈ I such that a ⊆ U (similarly there is b ∈ J with b ⊆ U). Proof. For every n the set a_{n} = U \ c_{n} belongs to u, hence a_{n}^{∗} meets f^{←}(0) and there is a subset b_{n} of a_{n} with b_{n}^{∗} ⊆ f^{←}(0) — here we use the well-known fact that f^{←}(0) is regularly closed. Now take an infinite set a such that b ⊆^{∗} ⋃_{m⩾n} a_{m} for all n; then a ⊆ U and a^{∗} ⊆ f^{←}(0). △

Let M be an ℵ_{0}-covering elementary substructure of H(θ), of size ℵ_{1}, that contains u, f and {c_{n} : n ∈ ω}. Claim 2. If b ∈ P(ℕ)/fin is such that I ∩ M ⊆ b^{↓} (or J ∩ M ⊆ b^{↓}) then there is U ∈ u ∩ M such that U ⊆^{∗} b. Proof. Let C ⊆ b^{↓} ∩ M be a countable cofinal set and choose for every c ∈ C, whenever possible, an i_{c} ∈ I ∩M such that c ⩽̸ i_{c}. Let I ∈ M be a countable subset of I that contains all the possible i_{c}; because I is countable there is i ∈ I ∩ M such that a < i for all a ∈ I. Note that i ∈ b^{↓} ∩ M, hence there is c ∈ C such that i < c; it follows that I ∩ M ⊆ c^{↓}. Note that in M there is no solution to “x ∈ I and x ⩽̸ c” hence there is none in H(θ); it follows that I ⊆ c^{↓}. But this implies that c ∈ u. △

The claim implies that if b ∈ P(ℕ) meets every U ∈ u ∩ M in an infinite set then there are I ∈ I∩M and J ∈ J ∩M that meet b in an infinite set. This in turn implies that the closed set F = ⋂{U^{∗} : U ∈ u ∩ M} is contained in cl f^{←}(0) ∩ cl f^{←}(1). The inequality 𝔪_{c} > ℵ_{1} implies that u ∩ M does not generate an ultrafilter, so that F consists of more than one point. This contradicts our assumption that u is the only point in cl f^{←}(0) ∩ cl f^{←}(1). □

A first-countable nonremainder. The final result in this section is due to M. Bell [3]. He produced a compact first countable space which is not a continuous image of ℕ^{∗} (equivalently: not a remainder of ℕ). We will show that such a space can be taken to be a subspace of the following space, which is an image of ℕ^{∗}. The space is, in hindsight, easy to describe. In the first version of this paper we started out with a generalization of Alexandroff’s doubling procedure; the referee rightfully pointed out that we were simply working with the square of the Alexandroff double of the Cantor set. In private correspondence, Bell points out that his original space is not embeddable in the square of the Alexandroff double.

Definition 3.7. Let 𝔻 be the Alexandroff double of the Cantor set, i.e., 𝔻 = ℂ×2, toplogized as follows: all points of ℂ × {1} are isolated and basic neighbourhoods of a point ⟨x, 0⟩ is of the form (U × 2) \ {⟨x, 1⟩}, where U is a neighbourhood of x in ℂ. It is well-known that this results in a compact first countable space.

We let 𝕂 = 𝔻 × 𝔻. We shall show that 𝕂 is a continuous image of ℕ^{∗} and that it contains a closed subspace that is not a continuous image of ℕ^{∗}.

In proving that 𝕂 is a continuous image of ℕ^{∗} we use results from [4]. We let W = {⟨k, l⟩ ∈ ℕ^{2} : l ⩽ 2^{k}} and we let π : W → ℕ be the projection on the first coordinate. A compact space is called an orthogonal image of ℕ^{∗} if there is a continuous map f : W^{∗} → X such that the diagonal map βπ △ f : W^{∗} → ℕ^{∗} × X is onto. Theorem 2.5 of [4] states that products of 𝔠 (or fewer) orthogonal images of ℕ^{∗} are again orthogonal images of ℕ^{∗}. Thus the following proposition more than shows that 𝕂 is a continuous image of ℕ^{∗}.

Proposition 3.8. The space 𝔻 is an orthogonal image of ℕ^{∗}.

Proof. Let {q_{l} : l ∈ ℕ} be a countable dense subset of ℂ and define f : W → ℂ by f(k, l) = q_{l}; observe that βf maps W^{∗} onto ℂ and that βf(u) = q_{l} for all u in {⟨k, l⟩ : 2^{k} ⩾ l}^{∗}. This readily implies that βπ △ βf maps W^{∗} onto ℂ.

A minor modification of the usual argument that nonempty G_{δ}-subsets of W^{∗} have nonempty interior lets us associate with every x ∈ ℂ a subset A_{x} of W that meets all but finitely many of the vertical lines V_{k} = {⟨k, l⟩ : l ⩽ 2^{k}} and such that βf[A_{x}^{∗}] = {x}. Now define g : W → W by g(k + 1, 2l) = g(k + 1, 2l + 1) = ⟨k, l⟩ (and g(0, 0) = ⟨0, 0⟩); observe that B_{x} = g^{←}[A_{x}] meets all but finitely many V_{k} in at least two points, so that we may split it into two parts, C_{x} and D_{x}, each of which meets all but finitely many V_{k}.

Now we turn the map βf ∘ βg : W^{∗} → ℂ into a map from W^{∗} to 𝔻: every point of D_{x}^{∗} will be mapped to ⟨x, 1⟩ and the points u of W^{∗} \ ⋃_{x} D_{x}^{∗} will be mapped to ⟨(βf ∘ βg)(u), 0⟩. It is straightforward to verify that the map h thus obtained witnesses that 𝔻 is an orthogonal image of ℕ^{∗}. □

Theorem 3.9 (2^{ℵ_{1}} = 𝔠). The space 𝕂 has a compact subspace X that is not an image of ℕ^{∗}.

Proof. We obtain X by removing a (suitably chosen) set of isolated points from 𝕂. We enumerate ℂ as {r_{α} : α < 𝔠} and we use our assumption 2^{ℵ_{1}} = 𝔠 to enumerate the family [ω_{1}]^{ℵ_{1}} as {A_{α} : α ∈ 𝔠} with cofinal repetitions. The set of isolated points that we keep is {⟨⟨r_{α}, 1⟩, ⟨r_{β}, 1⟩⟩ : α ∉ A_{β} or β ∉ A_{α}}. Furthermore, we let U_{α} be intersection of X with the clopen ‘cross’

𝔻 × {⟨r_{α}, 1⟩} ∪ {⟨r_{α}, 1⟩} × 𝔻.

Note that then for all α one has A_{α} = {ξ ∈ ω_{1} : U_{ξ} ∩ U_{α} = ∅}.

Now suppose that f is a mapping of ℕ^{∗} onto X and for each α ∈ 𝔠 fix a representative a_{α} ⊆ ℕ for f^{←}[U_{α}]. Observe that thus A_{α} = {ξ ∈ ω_{1} : a_{ξ} ∩ a_{α} =^{∗} ∅}. Fix an ℵ_{1}-sized ℵ_{0}-ideal subalgebra B of P(ℕ)/fin that contains {a_{ξ} : ξ ∈ ω_{1}}.

For each b ∈ B let S_{b} = {ξ : a_{ξ} < b} and pick α ∈ 𝔠 such that both S_{b} \ A_{α} and S_{b} ∩ A_{α} have cardinality ℵ_{1} whenever S_{b} has cardinality ℵ_{1}. Now B ∩ a_{α}^{⊥} is countably generated and it contains the uncountable set {a_{ξ} : ξ ∈ A_{α}}; it follows that there is a b < a_{α}^{⊥} such that S_{b} is uncountable. But now pick any ξ ∈ S_{b} \ A_{α}. Then a_{ξ} ⊆^{∗} b ⊆^{∗} a_{α}^{⊥} yet a_{ξ} ∩ a_{α} ≠^{∗} ∅ — a clear contradiction. □

4. Other cardinal invariants

In this section we relate 𝔪_{c}(B) to other known cardinal invariants of Boolean Algebras; we have already connected 𝔪_{c} to the idea of reaping. We formalize this idea in the following definition, which is analogous to the cardinal 𝔯 in P(ℕ)/fin (see [5] or [1]).

Definition 4.1. A subset A of a Boolean algebra B is reaped by the element b ∈ B, if b and its complement meet every non-zero element of A. The cardinal 𝔯(B) is defined as the minimum cardinality of a subset A of B that is not reaped by any element of B.

Our discussion after Definition 2.9 therefore establishes the inequality 𝔯(B) ⩾ 𝔪_{c}(B). The other half, so to speak, of 𝔪_{c} is provided by the proper analogue, for arbitrary Boolean algebras, of the cardinal number 𝔡.

In [6] Van Douwen showed that 𝔡 is equal to the number 𝔡_{2} from the following definition.

Definition 4.2. If D ⊆ ω^{ω} and A ⊆ [ω]^{ℵ_{0}} then D is said to dominate on A if for each g ∈ ω^{ω} there are d ∈ D and a ∈ A such that g(n) < d(n) for each n ∈ a. The cardinal 𝔡_{2} is defined as

𝔡_{2} = min{|A| + |D| : A ⊆ [ω]^{ℵ_{0}}, D ⊆ ω^{ω} and D dominates on A}.

To find a natural analogue of the cardinal invariant 𝔡_{2} in a general Boolean algebra we proceed along the lines of Rothberger’s work on the cardinals 𝔟 and 𝔡. For this we say that an ideal I in a Boolean algebra B is co-generated by a set S if I = S^{⊥}. We will say that I is countably co-generated if there is a countably infinite set that co-generates it but, in order to avoid cumbersome consideration of cases, no finite set co-generates it. The cardinal invariant 𝔡 is naturally equal to the minimum cardinal of a cofinal subset of any countably co-generated nonprincipal ideal in P(ℕ)/fin and likewise 𝔟 is the cardinal of an unbounded subset of a countably co-generated non-principal ideal in P(ℕ)/fin — this is so because any countably co-generated ideal in P(ℕ)/fin is naturally isomorphic to the ideal in P(ω × ω)/fin that is generated by the set of the form

L_{f} = {⟨m, n⟩ : m ∈ ω, n ⩽ f(m)},

where f ∈ ω^{ω}.

Proposition 4.3. If I ⊆ P(ℕ)/fin is a countably co-generated ideal then 𝔡_{2} is equal to the minimum cardinality of a subset J of I such that no member of I meets every member of J.

Proof. Given A and D let J be the set of graphs g ↾ a, where g ∈ D and a ∈ A. Observe that g ↾ a is disjoint from L_{f} iff f(n) < g(n) for all n ∈ a.

Conversely, given J construct for each J ∈ J a function g_{J} with domain a_{J} = {m : (∃n)(⟨m, n⟩ ∈ J)} whose graph is contained in J. Clearly if L_{f} ∩ J = ∅ then g_{J}(n) > f(n) for n ∈ a_{J}. □

It is clear that this proof uses a lot of the underlying structure of P(ℕ)/fin; one cannot hope to do something similar for arbitrary Boolean algebras.

Definition 4.4. For a Boolean algebra B, let 𝔡_{2}(B) be the minimum cardinal κ such that whenever I is a countably co-generated ideal of B and A is a subset of I of cardinality less than κ, there is a b ∈ I that meets each member of A.

We get the following characterization of 𝔪_{c}(B), with the immediate corollary that 𝔪_{c} = min{𝔯, 𝔡}.

Theorem 4.5. If B is a Boolean algebra with 𝔪_{c}(B) > ℵ_{0} then 𝔪_{c}(B) is the minimum of 𝔯(B) and 𝔡_{2}(B).

Proof. We have already seen that 𝔯(B) ⩾ 𝔪_{c}(B).

Let I be co-generated by the countable set S and let J be a subset of I of cardinality less than 𝔪_{c}(B). By our assumption on 𝔪_{c}(B) the set A = S ∪ J has cardinality less than 𝔪_{c}(B) as well so that it is ℵ_{0}-ideal complete. There is therefore an element b of B such that s < b for all s ∈ S and such that b reaps J; then b′ is an element of I that meets all elements of J. Thus 𝔡_{2}(B) ⩾ 𝔪_{c}(B).

Next let A be a subset of B, of size less than both 𝔯(B) and 𝔡_{2}(B). Let S and T be countable subsets of A with S ⊥ T and divide A into three subsets: A_{S}, the set of elements a for which there is a finite subset F of S such that a ⩽ ⋁ F; the set A_{T}, defined similarly, and A_{r}, the rest of A. Applying 𝔪_{c}(B) > ℵ_{0} we find for each a ∈ A_{r} a nonzero element b_{a} below a that is in (A_{S} ∪ A_{T})^{⊥} and we find b ∈ B such that S ⊆ b^{↓} and T ⊆ b^{⊥}. Because |A| < 𝔡_{2}(B) we can find b_{1} < b and b_{2} < b′ such that

• b_{1} ⊥ S and for all a ∈ A_{r} if b ∩ b_{a} ≠ 0 then b_{1} ∩ b_{a} ≠ 0, and

• b_{2} ⊥ T and for all a ∈ A_{r} if b′ ∩ b_{a} ≠ 0 then b_{2} ∩ b_{a} ≠ 0

Because |A| < 𝔯(B) we can find c_{1} < b_{1} that reaps all possible b_{1} ∩ b_{a} and likewise we can find c_{2} < b_{2}. Finally then d = b ∩ c′_{1} ∩ c_{2} is as required in the definition of ℵ_{0}-ideal completeness. □

Remark 4.6. We can now see that 𝔪_{c} = 𝔠 does not imply MA_{countable}; indeed, it is well-known that in the Laver model MA_{countable} fails but that also 𝔡 = 𝔯 = 𝔠.

5. General structure of (ℵ_{1}, ℵ_{0})-ideal algebras

In this section we explore the general structure of (ℵ_{1}, ℵ_{0})-ideal algebras. It is straightforward to check that “being an ℵ_{0}-ideal subalgebra of” is a transitive relation.

Proposition 5.1. If A is an ℵ_{0}-ideal subalgebra of B and B is an ℵ_{0}-ideal subalgebra of C, then A is an ℵ_{0}-ideal subalgebra of C. □

We have already mentioned Parovičenko’s theorem that under CH the algebra P(ℕ)/fin is the unique (ℵ_{1}, ℵ_{0})-ideal algebra with 𝔪_{c} = 𝔠. This leads us to the following definition.

Definition 5.2. A Boolean algebra B is Cohen-Parovičenko if B is (∗, ℵ_{0})-ideal and 𝔪_{c}(B) = 𝔠.

In the special case when 𝔠 = ℵ_{2} we have a convenient characterization in terms of well-orderings at our disposal.

Proposition 5.3. If 𝔠 = ℵ_{2} then an algebra B of cardinality 𝔠 is Cohen-Parovi-čenko if and only if for each enumeration B = {b_{α} : α ∈ ω_{2}} the set of λ ∈ ω_{2} for which B_{λ} = {b_{α} : α ∈ λ} is both an ℵ_{0}-ideal and an ℵ_{0}-ideal complete subalgebra is closed and unbounded in ω_{2}. □

The reason for adding the prefix ‘Cohen’ is contained in the following proposition, which together with Theorem 5.5 gives a ‘factorization’ of Steprāns’ characterization of P(ℕ)/fin (Theorem 2.13). The proposition itself combines Theorem 2.7 and Proposition 2.12.

Proposition 5.4. In the Cohen model P(ℕ)/fin is Cohen-Parovičenko. □

The following theorem combines Parivičenko’s and Steprāns’ theorems into one. We have been unable to find any sort of similar result in the case that 𝔠 > ℵ_{2}.

Theorem 5.5. If 𝔠 ⩽ ℵ_{2}, then all Cohen-Parovičenko Boolean algebras of cardinality 𝔠 are pairwise isomorphic. □

To show that this theorem is never vacuous we now construct (∗, ℵ_{0})-ideal algebras with prescribed 𝔪_{c}-numbers, including 𝔠. Thus we see that the idealness of an algebra has no bearing on its 𝔪_{c}-number. By contrast, a careful inspection of [13, Proposition 2.3 and Corollary 2.4] will reveal that if P(ℕ)/fin is (κ, ℵ_{0})-ideal and cf[κ]^{ℵ_{0}} = κ then 𝔪_{c} > κ, thus showing that 𝔪_{c} = 𝔠 in case P(ℕ)/fin is (∗, ℵ_{0})-ideal and there are cofinally many κ below 𝔠 with cf[κ]^{ℵ_{0}} = κ.

Theorem 5.6. For each regular cardinal κ ⩽ 𝔠 such that cf[κ]^{ℵ_{0}} = κ, there is an algebra of cardinality 𝔠 that is (∗, ℵ_{0})-ideal and has κ as its 𝔪_{c}-number. Thus, if 𝔠 is regular then there is Cohen-Parovičenko algebra of cardinality 𝔠.

We split the construction into two propositions.

Proposition 5.7. There is a (∗, ℵ_{0})-ideal algebra B of size 𝔠 with 𝔪_{c}(B) ⩾ κ.

Proof. We obtain B as the direct limit of a sequence ⟨B_{ξ} : ξ < μ⟩, where μ is the ordinal 𝔠 · κ if κ < 𝔠 and μ = 𝔠 otherwise.

We begin by letting B_{0} be the two-element algebra. At limit stages ξ set B_{ξ} = \underrightarrow{lim}_{η<ξ} B_{η}. Carry an enumeration, \{⟨S_{ξ}, T_{ξ}⟩ : ξ ∈ μ\} with cofinal repetitions, of pairs of countably infinite subsets of B so that S_{ξ} ∪ T_{ξ} ⊆ B_{ξ} and S_{ξ} ⊥ T_{ξ} for all ξ. For simplicity, assume that S_{ξ} and T_{ξ} are strictly increasing sequences (if infinite) or singletons (if finite, where an empty set may always be replaced by {0}).

To construct B_{ξ+1} from B_{ξ} first take the completion B̃_{ξ} of B_{ξ} and in it we define s_{ξ} and t_{ξ} by s_{ξ} = ⋁ S_{ξ} and t′_{ξ} = ⋁ T_{ξ}; note that s_{ξ} ⩽ t_{ξ}. There are two cases to consider.

If s_{ξ} < t_{ξ} then we let B_{ξ+1} be the subalgebra of B̃_{ξ}^{2} generated by the diagonal {⟨b, b⟩ : b ∈ B_{ξ}} and the element b_{ξ} = ⟨s_{ξ}, t_{ξ}⟩. Observe that ⟨s_{ξ}, t_{ξ}⟩ does exactly what is required in Definition 2.9 — with A = B_{ξ}, S = S_{ξ} and T = T_{ξ}. Also observe that B_{ξ} is an ℵ_{0}-ideal subalgebra of B_{ξ+1}: a typical element b of B_{ξ+1} looks like (b_{ξ} ∧ a_{0}) ∨ (b′_{ξ} ∧ a_{1}) and from this it is easily seen that the countable set A_{b} = {(s ∧ a_{0}) ∨ (t ∧ a_{1}) ∨ (a_{0} ∧ a_{1}) : s ∈ S_{ξ}, t ∈ T_{ξ}} generates b^{↓} ∩ B_{ξ}: if a ⩽ b then a ⩽ a_{0} ∨ a_{1} and so we have to cover a ∧ a′_{1} (which is below b_{ξ} ∧ a_{0}), a ∧ a′_{0} (which is below b′_{ξ} ∧ a_{1}) and a ∧ a_{0} ∧ a_{1}.

If s_{ξ} = t_{ξ} then this still works if S_{ξ} and T_{ξ} are both infinite or both finite but not if, say, S_{ξ} is infinite and T_{ξ} is finite, for then we seek an element b_{ξ} such that s < b_{ξ} < t_{ξ} for all s ∈ S_{ξ} — note that in this case T_{ξ} = {t′_{ξ}} and that t_{ξ} belongs to B_{ξ}. To remedy this we take the Stone space X_{ξ} of B_{ξ} and consider the closed set C_{ξ} = s_{ξ} \ ⋃ S_{ξ}. We let B_{ξ+1} be the clopen algebra of the subspace Y_{ξ} = (X_{ξ} × {0}) ∪ (C_{ξ} × {1}) of X_{ξ} × {0, 1}. Observe that b_{ξ} = s_{ξ} × {0} does what we want: for every s ∈ S_{ξ} we have s < b_{ξ} < t_{ξ}, because t_{ξ} now corresponds to b_{ξ}∪(C_{ξ}×{1}). A typical element b of B_{ξ+1} now looks like (b_{ξ}∧a_{0})∨(c_{ξ}∧a_{1})∨(t′_{ξ}∧a_{2}), where c_{ξ} = C_{ξ} × {1} — because C_{ξ} ⊆ s_{ξ} we have b ∈ B_{ξ} iff we can take a_{0} = a_{1}. As above, we can verify that A_{b} = {(s∧a_{0})∨(t_{ξ} ∧a_{1})∨(t′_{ξ} ∧a_{2}) : s ∈ S_{ξ}} generates b^{↓} ∩ B_{ξ}. If a ⩽ b then t′_{ξ} ∧ a ⩽ t_{ξ} ∧ a_{2}, so we concentrate on a^{†} = a ∧ t_{ξ} = (a ∧ b_{ξ}) ∨ (a ∧ c_{ξ}). Now a^{†} ∧ a′_{1} ⩽ b_{ξ}, so there is an s ∈ S_{ξ} above a^{†} ∧ a′_{1}. For this s we have a ⩽ (s ∧ a_{0}) ∨ (t_{ξ} ∧ a_{1}) ∨ (t′_{ξ} ∧ a_{2}).

We shall refer to ξ as of type 0 if we simply adjoin ⟨s_{ξ}, t_{ξ}⟩; the other ξ will be of type 1.

It is straightforward to check that 𝔪_{c}(B) ⩾ κ: if |A| < κ then A ⊆ B_{η} for some η and if S and T are countable subsets of A with S ⊥ T then there is a ξ above η with ⟨S, T ⟩ = ⟨S_{ξ}, T_{ξ}⟩; the element b_{ξ} is as required for A, S and T.

We show that B is (∗, ℵ_{0})-ideal by showing that M ∩B is ℵ_{0}-ideal in B whenever M is an elementary substructure of H(𝔠^{+}) with ⟨⟨S_{ξ}, T_{ξ}⟩ : ξ < μ⟩ and ⟨B_{ξ} : ξ < μ⟩ both in M, and with |M| less than 𝔠 and regular.

Let b ∈ B \ M, take e ∈ b^{↓} ∩ M, and fix the δ and ξ for which b ∈ B_{δ+1} \ B_{δ} and e ∈ B_{ξ+1} \ B_{ξ} respectively.

Claim 1. If ξ ≠ δ then there is an a ∈ A_{b} with e ⩽ a. Proof. If ξ < δ then e ∈ B_{δ} and we are done.

If ξ > δ we consider two cases. If ξ is of type 0 and e = (b_{ξ} ∧ e_{0}) ∨ (b′_{ξ} ∧ e_{1}) then e_{0} ∧ b′ ∈ b_{ξ}^{⊥}, hence e_{0} ∧ b′ ⩽ t for some t in T_{ξ}; likewise e_{1} ∧ b′ ⩽ s for some s in S_{ξ}. But then e ⩽ (e_{0} ∧ t′) ∨ (e_{1} ∧ s′) ⩽ b, where the middle element belongs to B_{ξ}; it follows that there is an a ∈ A_{b} with e ⩽ a.

If ξ is of type 1 and e = (b_{ξ} ∧ e_{0}) ∨ (c_{ξ} ∧ e_{1}) ∨ (t′_{ξ} ∧ e_{2}) then t′_{ξ} ∧ e_{2} belongs to B_{ξ}, so we concentrate on the other parts of e — and we assume e_{0}, e_{1} ⩽ t_{ξ}. Observe that b_{ξ} ∧ e_{0} ⩽ t_{ξ} ∧ e_{0} ⩽ b: use the fact that b ∈ B_{ξ}. Next e_{1} ∧ b′ ⩽ b_{ξ}, so that there is s ∈ S_{ξ} with e_{1} ∧ b′ ⩽ s and hence c_{ξ} ∧ e_{1} ⩽ s′ ∧ e_{1} ⩽ b. We see that e ⩽ (t_{ξ} ∧ e_{0}) ∨ (s′ ∧ e_{1}) ∨ (t′_{ξ} ∧ e_{2}) ⩽ b, where the middle element belongs to B_{ξ}; again we can find our a ∈ A_{b} with e ⩽ a. △

This claim essentially takes of the case δ ∉ M: by our obvious inductive assumption we have for every a ∈ A_{b} a countable generating set C_{a} for a^{↓} ∩ M. By the claim the countable set C_{b} = ⋃_{a∈A_{b}} C_{a} generates b^{↓} ∩ M (note that ξ ∈ M, so ξ ≠ δ).

To fully finish the proof we must show what to do if δ ∈ M. The set C_{b} still takes care of the e with ξ ≠ δ. The following two claims show what to add to C_{b} in order to take care of the e with ξ = δ.

Claim 2. If δ is of type 0, e = (b_{δ} ∧ e_{0}) ∨ (b′_{δ} ∧ e_{1}) and b = (b_{δ} ∧ a_{0}) ∨ (b′_{δ} ∧ a_{1}) then there are c_{0} ∈ C_{a_{0}} and c_{1} ∈ C_{a_{1}} such that e ⩽ (b_{δ} ∧ c_{0}) ∨ (b′_{δ} ∧ c_{1}) ⩽ b. Proof. Simply observe that b_{δ} ∧ e_{i} ∈ a_{i}^{↓} ∩ M for i = 0, 1. △

We see that we must add {(b_{δ} ∧ c_{0}) ∨ (b′_{δ} ∧ c_{1}) : c_{0} ∈ C_{a_{0}}, c_{1} ∈ C_{a_{1}}} to C_{b}.

Claim 3. If δ is of type 1 and e = (b_{δ} ∧ e_{0}) ∨ (c_{δ} ∧ e_{1}) ∨ (t′_{δ} ∧ e_{2}) and b = (b_{δ} ∧ a_{0}) ∨ (c_{δ} ∧ a_{1}) ∨ (t′_{δ} ∧ a_{2}) Then there are c_{0} ∈ C_{a_{0}}, c_{1} ∈ C_{a_{1}} and c_{2} ∈ C_{a_{2}} such that e ⩽ (b_{δ} ∧ c_{0}) ∨ (c_{δ} ∧ c_{1}) ∨ (t′_{δ} ∧ c_{2}) ⩽ b. Proof. Simply observe that b_{δ}∧e_{0} ∈ a_{0}^{↓}∩M, c_{δ}∧e_{1} ∈ a_{1}^{↓}∩M and t′_{δ}∧e_{2} ∈ a_{2}^{↓}∩M. △

Now add {(b_{δ} ∧c_{0})∨(c_{δ} ∧c_{1})∨(t′_{δ} ∧c_{2}) : c_{0} ∈ C_{a_{0}}, c_{1} ∈ C_{a_{1}}, c_{2} ∈ C_{a_{2}}} to C_{b}. □

Note that if κ = 𝔠 we are done: the algebra B is (∗, ℵ_{0})-ideal with 𝔪_{c}(B) = 𝔠. In the case where κ < 𝔠 we use the cofinality assumption to find a subalgebra of B with the right properties.

Proposition 5.8. If κ < 𝔠 then the algebra B constructed in the proof of Proposition 5.7 contains an algebra A of cardinality 𝔠 with 𝔪_{c}(A) = κ.

Proof. We fix a cofinal subfamily {Y_{α} : α < κ} of [κ]^{ℵ_{0}} with Y_{α} ⊆ α for all α. We also assume that, for every α < κ, all ordered pairs ⟨S, T ⟩ with S, T ⊆ B_{𝔠·α} occur in the list {⟨S_{ξ}, T_{ξ}⟩ : 𝔠 · α ⩽ ξ < 𝔠 · (α + 1)}. This enables us to choose, recursively, λ_{α} ∈ [𝔠 · α, 𝔠 · (α + 1)) such that S_{λ_{α}} = {b_{λ_{β}} : β ∈ Y_{α}} and T_{λ_{α}} = {0}. Note that then b_{λ_{β}} < b_{λ_{α}} whenever β ∈ Y_{α}. In what follows we abbreviate b_{λ_{α}} by p_{α}.

Because the Y_{α} form a cofinal family in [κ]^{ℵ_{0}}, the family {p_{α} : α < κ} is ℵ_{0}- directed, i.e., if F ⊆ κ is countable then there is an α such that p_{β} < p_{α} for all β ∈ F. It follows that I = {b : (∃α)(b ⩽ p_{α})} is a P -ideal. We set F = {b′ : b ∈ I} and consider the subalgebra A = I ∪F of B. It is clear that 𝔪_{c}(A) ⩽ κ: no element of A reaps the family {p_{α} : α < κ}.

To show 𝔪_{c}(A) ⩾ κ we take a subalgebra D of A of size less than κ and countable subsets S and T of D with S ⊥ T; we assume S and T are increasing sequences. Also, fix α < κ such that D ⊆ B_{𝔠·α} and for every d ∈ D there is β < α with d ⩽ p_{β} or d′ ⩽ p_{β}. If some member of S or T belongs to F then any b ∈ B that witnesses this instance of ℵ_{0}-ideal completeness of D in B automatically belongs to A.

In the other case, when S ∪ T ⊆ I, we can assume that Y_{α} contains, for every a ∈ S ∪ T, a β such that a ⩽ p_{β}; but then S ∪ T ⊆ p_{α}^{↓}. Also note that p_{α} meets every nonzero element of B_{λ_{α}} and hence of D. Now choose ζ ∈ [𝔠·α, 𝔠·(α+1)) such that S_{ζ} = S ∪ {p′_{α}} and T_{ζ} = T. Let d ∈ D ∩ b_{ζ}^{↓}; there is an s ∈ S with d ⩽ s ∨ p′_{α}, then d ∧ s′ ⩽ p′_{α} and so d ∧ s′ = 0 whence d ⩽ s. We see that S generates b_{ζ}^{↓} ∩ D and, similarly, that T generates b_{ζ}^{⊥} ∩ D.

We finish by showing that A is (∗, ℵ_{0})-ideal. The notation b^{↓} will always mean the set computed in B. Let M be any elementary substructure of H(𝔠^{+}) of regular size less than 𝔠 such that ⟨B_{ξ} : ξ < μ⟩ and ⟨λ_{α} : α ∈ κ⟩ are members of M; this ensures that M ∩ B is an ℵ_{0}-ideal subalgebra of B. We shall show that for any b in B, the ideal b^{↓} ∩ (M ∩ A) is countably generated; we denote the countable generating set, when found, by b^{M,A}.

Fix δ < μ so that b ∈ B_{δ+1} \ B_{δ} and assume we have found a^{M,A} for all a ∈ B_{δ}. By Claim 1 in the proof of Proposition 5.7 the set ⋃_{a∈A_{b}} a^{M,A} takes care of all e ∈ b^{↓} ∩ (M ∩ A), except possibly those in B_{δ+1} \ B_{δ}. In particular we can set b^{M,A} = ⋃_{a∈A_{b}} a^{M,A} when δ ∉ M.

Thus we are left with the case where δ ∈ M. If there is an e ∈ b^{↓} ∩ M ∩ F then C_{b} ∩ F generates b^{↓} ∩ M ∩ A, where C_{b} is as defined in the proof of Proposition 5.7. In the other case, where b^{↓} ∩ M ∩ F = ∅, we add

{(b_{δ} ∧ c_{0}) ∨ (b′_{δ} ∧ c_{1}) : c_{0} ∈ a_{0}^{M,A}, c_{1} ∈ a_{1}^{M,A}}

to b^{M,A} if δ is of type 0 and we add

{(b_{δ} ∧ c_{0}) ∨ (c_{δ} ∧ c_{1}) ∨ (t′_{δ} ∧ c_{2}) : c_{0} ∈ a_{0}^{M,A}, c_{1} ∈ a_{1}^{M,A}, c_{2} ∈ a_{2}^{M,A}}

if δ is of type 1.

Indeed, if e ∈ b^{↓} ∩ (M ∩ A) then e belongs to I ∩ M and hence so do e ∧ b_{δ} and e ∧ b′_{δ}. Note that e ∧ b_{δ} ⩽ a_{0} and e ∧ b′_{δ} ⩽ a_{1} so that there are c_{0} ∈ a_{0}^{M,A} and c_{1} ∈ a_{1}^{M,A} with e ∧ b_{δ} ⩽ c_{0} ⩽ a_{0} and e ∧ b′_{δ} ⩽ c_{1} ⩽ a_{1} respectively.

If δ is of type 1 then we observe that e ∧ b_{δ}, e ∧ c_{δ} and e ∧ t′_{δ} all belong to I ∩ M and are below a_{0}, a_{1} and a_{2} respectively. □

Mapping F -spaces onto βℕ. Every compact F -space contains a copy of βℕ: it follows straight from the definition of F -space that the closure of a countably infinite relatively discrete subset is homeomorphic to βℕ. Thus, in a manner of speaking, βℕ is a minimal F -space. Bell has asked whether βℕ is also minimal in the mapping-onto sense: does every infinite compact zero-dimensional F -space map onto βℕ? The ease with which βℕ can be embedded into such a space belies the dual difficulty in constructing an embedding of P(ℕ) into its algebra of clopen sets. Indeed, we show by means of a Cohen-Parovičenko algebra that such an embedding does not always exist. Before that we prove that Bell’s question has a positive answer if the Continuum Hypothesis is assumed.

Proposition 5.9 (CH). Every infinite compact zero-dimensional F -space maps onto βℕ.

Proof. It suffices to prove that P(ℕ) will embed into B where B is infinite and has no (ω, ω)-gaps. Fix any sequence {b_{n} : n ∈ ω} of pairwise disjoint non-zero elements of B. Let {a_{α} : α ∈ ω_{1}} be an enumeration of P(ℕ) so that a_{n} = {n} for each n ∈ ω. Inductively choose elements b_{α} ∈ B so that the mapping a_{α} → b_{α} lifts to an isomorphism from the algebra generated by {a_{β} : β ⩽ α}. If a_{α} is in the algebra generated by its predecessors then there is nothing to do. Otherwise, by the inductive hypothesis, the ideal I generated by {b_{β} : a_{β} < a_{α}} is disjoint from the ideal J generated by {b_{β} : a_{β} ∧ a_{α} = 0}. Since B has no (ω, ω)-gaps, there is a b_{α} ∈ B such that I ⊆ b_{α}^{↓} and J ⊆ b_{α}^{⊥}. □

Theorem 5.10. It is consistent that there is an infinite compact zero-dimensional F -space that does not map onto βℕ.

Proof. It is consistent with 𝔠 = ℵ_{2} that P(ℕ)/fin contains an ω_{2}-chain, this happens, e.g., if MA holds. But now let B be the Cohen-Parovičenko algebra from Theorem 5.6. Clearly S(B) is a compact zero-dimensional F -space. Assume that f maps S(B) onto βℕ and let b_{n} = f^{←}(n) for each n. Now let I be the ideal in B generated by {b_{n} : n ∈ ω}. By the forthcoming Corollary 5.12 B/I is still (ℵ_{1}, ℵ_{0})- ideal and so, by Proposition 2.5, does not contain an ω_{2}-chain. However B/I is isomorphic to the algebra of clopen subsets of the closed set X \ ⋃_{n} b_{n} = f^{←}(ℕ^{∗}) and certainly does contain ω_{2}-chains. □

This proof does not work in the ℵ_{2}-Cohen model, where P(ℕ)/fin is the Cohen-Parovičenko algebra. We therefore ask, also in the hope of establishing the consistency with ¬CH of a yes answer to Bell’s question, the following.

Question 1. Is it true in the ℵ_{2}-Cohen model that every compact zero-dimensional F -space does map onto βℕ?

Quotient algebras. Under CH one can use Parovičenko’s theorem to find many copies of ℕ^{∗} inside of ℕ^{∗}: the proof usually boils down to showing that a quotient of P(ℕ)/fin by some ideal is isomorphic to P(ℕ)/fin. The same can be done in the Cohen model because many quotients of Cohen-Parovičenko algebras are again Cohen-Parovičenko. First we consider quotients by small ideals.

Lemma 5.11. If B is a Boolean algebra, A is an ℵ_{0}-ideal subalgebra, and I is an ideal which is generated by I ∩ A, then A/I is an ℵ_{0}-ideal subalgebra of B/I.

Proof. Fix any b ∈ B and fix a cofinal sequence {a_{n} : n ∈ ω} ⊆ b^{↓} ∩ A. Let c ∈ A be such that c/I < b/I, which means that c \ b is covered by some member d of I ∩ A. It follows then that c \ d < b. Hence there is n such that c \ d < a_{n} < b. But now it follows that c/I < a_{n}/I. □

Corollary 5.12. If I is an ℵ_{1}-generated ideal in a (κ, ℵ_{0})-ideal Boolean algebra B, then B/I is also a (κ, ℵ_{0})-ideal Boolean algebra. □

Another interesting consequence is that ω_{1}^{∗} is not the image of the Stone space of an (ℵ_{1}, ℵ_{0})-ideal algebra.

Corollary 5.13. The algebra P (ω_{1})/fin cannot be embedded into an (ℵ_{1}, ℵ_{0})-ideal algebra.

Proof. This proceeds much as the proof of Theorem 5.10 since P (ω_{1})/ctble is a quotient of P (ω_{1})/fin by an ℵ_{1}-generated ideal and contains ω_{2}-chains. □

Corollary 5.14 (¬CH). If B is Cohen-Parovičenko and I is an ℵ_{1}-generated ideal then B/I is again Cohen-Parovičenko.

Proof. It remains only to prove that 𝔪_{c}(B/I) = 𝔠. To do so, fix countable subsets S and T of B so that s ∧ t ∈ I for each s ∈ S and t ∈ T. Since S and T are countable it is routine to inductively remove from each member of S and T some member of I so as to ensure that s ∧ t = 0 for each s ∈ S and t ∈ T. Now suppose that A is a subalgebra of B that contains S and T and has cardinality less than 𝔠. We may assume that A contains a generating set for I. Since B is Cohen-Parovičenko there is a b ∈ B such that S generates b^{↓} ∩ A and T generates b^{⊥} ∩ A. Now suppose a/I is below b/I, i.e., a \ b ∈ I. Since A ∩ I generates I, there is a c ∈ A ∩ I such that a \ b < c. Thus a \ c < b and so there is a finite join, s, of members of S such that a \ c < s. It follows that {s/I : s ∈ S} generates (b/I)^{↓} ∩ A/I. Similarly (b/I)^{⊥} ∩ A/I is generated by {t/I : t ∈ T }. □

Another situation that occurs frequently is that one has a lifting for the ideal I, this is a Boolean homomorphism l : B/I → B with the property that l(b/I)/I = b/I. In dual terms this means that the closed set F = S(B) \ ⋃{i^{∗} : i ∈ I} is a retract of S(B). The retraction r and the lifting l are connected by the formula l(b/I) = r^{←}[b^{∗} ∩ F] for each b ∈ B.

Theorem 5.15. If I is an ideal on B for which there is a lifting l : B/I → B then for each ℵ_{0}-ideal subalgebra A of B such that l[A/I] ⊆ A the quotient A/I is an ℵ_{0}-ideal subalgebra of B/I. Therefore, if B is an (κ, ℵ_{0})-ideal Boolean algebra, then so is B/I.

Proof. Let A be an ℵ_{0}-ideal subalgebra of B such that l[A/I] ⊆ A. Fix any b ∈ B. We will show that (b/I)^{↓} ∩ A/I is countably generated. In fact, suppose that {a_{n} : n ∈ ω} generates l(b/I)^{↓} ∩ A. Fix any x ∈ A such that x/I < b/I. By assumption, x^{†} = l(x/I) is also a member of A. Furthermore l(x/I) ⩽ l(b/I), hence there is an n such that x^{†} ⩽ a_{n}. Clearly then x/I = x^{†}/I ⩽ a_{n}/I. □

6. Other remainders and applications to ℕ^{∗}

We say that a zero-dimensional space K is Cohen-Parovičenko if its algebra of clopen sets is Cohen-Parovičenko. In this section we are interested in identifying which remainders of σ-compact locally compact spaces can be Cohen-Parovičenko; by ‘remainder’ we mean the Čech-Stone remainder βX \ X — commonly denoted by X^{∗}. We then apply this information and the results of the previous section to the study of ℕ^{∗} under the assumption that it is Cohen-Parovičenko. We are motivated by the somewhat classical results about ℕ^{∗} that are known to follow from CH (see [21]). The predisposition of this section is to assume that ℕ^{∗} is Cohen-Parovičenko and to determine how this affects the structure of ℕ^{∗} and of other remainders.

In what follows, whenever X is a zero-dimensional compact space, we write 𝔪_{c}(X), 𝔯(X) and 𝔡_{2}(X) for the values that these functions have on the Boolean algebra CO(X) of clopen subsets of X. We first prove a lemma concerning the behaviour of 𝔡_{2} and 𝔯 under continuous mappings.

Lemma 6.1. If f : X → Y is an open continuous surjection then 𝔡_{2}(Y) ⩾ 𝔡_{2}(X) and 𝔯(Y) ⩽ 𝔯(X).

Proof. Let I be an ideal of CO(Y), co-generated by the family {c_{n} : n ∈ ω}, and let A be a subfamily of I of size less than 𝔡_{2}(X). In CO(X) we can find an element b such that b ∩ f^{←}[c_{n}] = ∅ for all n and b ∩ f^{←}[a] ≠ ∅ for all a ∈ A. Because the map f is open the set f[b] is clopen, it also belongs to I and it meets every element of A.

Next let C be a family of clopen subsets of X, of size less than 𝔯(Y). Because f is open the family {f[c] : c ∈ C} consists of clopen sets and so we can find b ∈ CO(Y) that reaps it. Then f^{←}[b] reaps the family C. □

Our first result is somewhat surprising. It implies that if CH fails then most remainders are not Cohen-Parovičenko. Recall that a space is basically disconnected if each cozero-set has clopen closure — dually: the algebra of clopen subsets is countably complete. Unless stated otherwise the spaces we are considering are all zero-dimensional.

Proposition 6.2. Let X be the topological sum of countably many compact spaces that are not basically disconnected. If its remainder X^{∗} is (ℵ_{1}, ℵ_{0})-ideal then 𝔡 = ℵ_{1}.

Proof. Write X = ⨁_{n∈ω} X_{n} and fix for each n an infinite family {a(n, m) : m ∈ ω} of pairwise disjoint clopen sets of X_{n} so that D_{n}, the closure of their union, is not open.

Assume that M is an ℵ_{0}-covering elementary substructure of some H(θ) of size ℵ_{1} that contains X and the family {a(n, m) : n, m ∈ ω}. We show that M ∩ ℕ^{ℕ} is cofinal in ℕ^{ℕ}. Let f : ℕ → ℕ be a strictly increasing function not in M. We find g ∈ M such that f ⩽ g.

Let b = ⋃{a(n, m) : m ⩽ f(n)}; observe that b is also not in M. We take a countable subfamily C of M ∩ CO(X) that is cofinal in b^{⊥} ∩ M — this means that c ∩ b is compact for all c ∈ C and that whenever d ∈ M and d ∩ b is compact there is c ∈ C such that the difference d \ c is compact.

There are two cases to consider. If there is a c ∈ C such that the set I_{c} = {n : (∃m)(c ∩ a(n, m) ≠ ∅)} is infinite then we are done. Indeed, define h ∈ M by h(n) = min{m : c ∩ a(n^{+}, m) ≠ ∅}, where n^{+} = min{l ⩾ n : l ∈ I_{c}}. Because c ∩ b is compact there is an l such that c ∩ a(n, m) = ∅ whenever n ⩾ l and m ⩽ f(n). It follows that for n ⩾ l we have h(n) = h(n^{+}) > f(n^{+}) > f(n). Now define g by g(n) = max{h(n), f(n)}; this g belongs to M because it is a finite modification of h and it is as required.

In the other case, where I_{c} is finite for all c, we may assume C ∈ M: indeed, take a countable D ∈ M with C ⊆ D and replace C by the set of elements d of D for which I_{d} is finite. By subtracting a compact part from each c we can also assume that every c is disjoint from every a(n, m).

But now from an enumeration {c_{n} : n ∈ ω} of C (that is in M) we define the clopen set

c = ⋃{X_{n} ∩ (c_{0} ∪ ⋯ ∪ c_{n}) : n ∈ ω}.

It follows that c is in M ∩ b^{⊥}. Now for each n, D_{n} ∩ c is empty, but D_{n} is not equal to X_{n} \c since D_{n} is not open. Therefore, there is some d ∈ M ∩b^{⊥} such that c ⊆ d and, for each n, d ∩ X_{n} \ c is not empty. It follows that d \ c_{k} is not compact for any k and so {c_{k} : k ∈ ω} is not a generating set for b^{⊥} ∩ M. Therefore this case does not occur. □

Theorem 6.3 (¬CH). If X = ⨁_{n∈ω} X_{n} is the topological sum of countably many compact spaces that are not basically disconnected then the remainder of X is not Cohen-Parovičenko.

Proof. Choose x_{n} ∈ X_{n} for all n and observe that D = cl{x_{n} : n ∈ ω} ∩ X^{∗} is homeomorphic to ℕ^{∗}. The map that send X_{n} to the point x_{n} induces an open retraction from X^{∗} onto D. It follows that 𝔡 ⩾ 𝔡_{2}(X^{∗}), so if CO(X^{∗}) is (ℵ_{1}, ℵ_{0})- ideal then, by Proposition 6.2, we get 𝔡_{2}(X^{∗}) = ℵ_{1} < 𝔠. □

Remark 6.4. Clearly it follows from the previous result that if ℕ^{∗} is Cohen-Parovi-čenko and CH fails then (ω × (ω + 1))^{∗} is not homeomorphic to ω^{∗}. Using this fact and tracking the location of both clopen and nowhere dense P -set copies of ℕ^{∗} in their remainders, one can easily show that, in addition, ω ×(ω +1) and ω ×(ω^{2} +1) do not have homeomorphic remainders either.

It is also worth mentioning the following result since it has already found applications in Functional Analysis, see [10].

Corollary 6.5 (¬CH). If P(ℕ)/fin is Cohen-Parovičenko and C is a non-compact cozero set in ℕ^{∗}, then the closure of C is not a retract of ℕ^{∗}.

Proof. Let C be a non-compact cozero subset of ℕ^{∗}. It follows that C is a countable union of compact open subsets of ℕ^{∗} and, as is well-known, that the closure of C is just its Čech-Stone compactification. Now if the closure were a retract of ℕ^{∗}, then its clopen algebra would be an (ℵ_{1}, ℵ_{0})-algebra. The boundary of C, which is homeomorphic to βC \ C, is a G_{δ}-set in the closure of C; hence its clopen algebra is also an (ℵ_{1}, ℵ_{0})-algebra by Lemma 5.11.

However, no clopen subset of ℕ^{∗} is basically disconnected so by Proposition 6.2 we have 𝔡 = ℵ_{1}. But we assumed that 𝔪_{c} and hence 𝔡 was equal to 𝔠. □

With the previous results in mind it is tempting to hope that for σ-compact locally compact X and Y, if X^{∗} and Y^{∗} were homeomorphic then X and Y would be homeomorphic-modulo-compact-sets in some sense. For example, we do not know if (ω × 2^{ω})^{∗} and (ω × 2^{ω_{1}})^{∗} are homeomorphic in the Cohen model.

We do however know of other spaces whose remainder is Cohen-Parovičenko. The proof of this fact is a rather interesting use of the basic results we have developed about Cohen-Parovičenko algebras. Recall that the Gleason cover or absolute of a compact space X is denoted by E(X) and that E(X) is just the Stone space of the complete Boolean algebra of regular open subsets of X. We write E_{κ} for ω ×E(2^{κ}).

Lemma 6.6. Let X = ⨁_{n∈ω} X_{n} be the topological sum of basically disconnected compact spaces. Then 𝔡_{2}(X^{∗}) = 𝔡.

Proof. Lemma 6.1 gives us 𝔡 ⩾ 𝔡_{2}(X^{∗}). To prove the other inequality we take an ideal in CO(X^{∗}) that is co-generated by a strictly increasing sequence. Translating this into CO(X) we get an increasing sequence ⟨C_{k} : k ∈ ω⟩ of clopen sets in X such that for all k the difference C_{k+1} \ C_{k} is not compact (for convenience we assume C_{0} = ∅), and the ideal I in CO(X) consisting of those sets D for which every intersection D ∩ C_{k} is compact.

For every n, k ∈ ω put a(n, k) = X_{n} ∩ (C_{k+1} \ C_{k}). This transforms the C_{k} into an increasing sequence ⟨c_{k} : k ∈ ω⟩ of subsets of the countable set A = {a(n, k) : n, k ∈ ω}, where c_{k} = {a(n, l) : n ∈ ω, l < k}. To every D ∈ I corresponds the set x_{D} = {a ∈ A : D ∩ a ≠ ∅}; observe that x_{D} ∩ c_{k} is finite for each k. Because each X_{n} is basically disconnected the sets D_{n} = cl ⋃_{k} a(n, k) are clopen (maybe empty); we put Y = X \ ⋃_{n} D_{n}.

Let J be a subfamily of I of size less than 𝔡 and consisting of non-compact sets. Fix an infinite subset d of A such that d ∩ c_{k} is finite for all k and such that d ∩ x_{D} is infinite whenever D ∈ J and x_{D} is infinite. Finally put C = Y ∪ ⋃ d. Clearly C ∩C_{k} is compact for every k. If D ∈ J and x_{D} is finite then D∩Y is not compact; if x_{D} is infinite then D ∩ ⋃ d is not compact. □

Remark 6.7. A similar result does not hold for 𝔯. Indeed, consider the space E_{κ}; a clopen set in its remainder E_{κ}^{∗} is determined by a clopen set of E_{κ} itself. In turn a clopen subset of E_{κ} is determined by a regular open subset of ω × 2^{κ} and it is well-known that such a regular open set depends on at most countably many coordinates. Thus, if C is a family of fewer than κ many clopen sets in E_{κ}^{∗} then we can find an α ∈ κ such that no element of C depends on α. But this means that the clopen set ω × π\overleftarrow{α}(0) (or rather the clopen subset of E_{κ}^{∗} determined by it) reaps the family C. We deduce that 𝔯(E_{κ}^{∗}) ⩾ κ and hence that, for example, 𝔯(E_{𝔠}^{∗}) > 𝔯 in models where 𝔠 > 𝔯.

Theorem 6.8. For each cardinal κ ⩽ 𝔠, the remainder of E_{κ} is Cohen-Parovičenko iff ℕ^{∗} is Cohen-Parovičenko.

Proof. We start out by observing two partial equivalences. Claim 1. 𝔡 = 𝔠 iff 𝔡_{2}(E_{κ}^{∗}) = 𝔠. Proof. By Lemma 6.6 we know that 𝔡_{2}(E_{κ}^{∗}) = 𝔡 for all κ. △

Claim 2. The algebra CO(E_{κ}^{∗}) is (∗, ℵ_{0})-ideal iff P(ℕ)/fin is. Proof. This follows by applying Theorem 5.15 twice. First: ℕ^{∗} is easily seen to be a retract of E_{κ}^{∗}, so P(ℕ)/fin is (∗, ℵ_{0})-ideal if CO(E_{κ}^{∗}) is. Second: βE_{κ} is a separable extremally disconnected compact space and hence can be embedded as retract in ℕ^{∗}, so that CO(βE_{κ}^{∗}) is (∗, ℵ_{0})-ideal if P(ℕ)/fin is and, by Corollary 5.12, so is the clopen algebra of E_{κ}^{∗}. △

We would be done if we could also prove 𝔯(E_{κ}^{∗}) = 𝔯 but by Remark 6.7 we know that this cannot be done. We circumvent this difficulty by showing that 𝔯 ⩾ 𝔡 if P(ℕ)/fin is (∗, ℵ_{0})-ideal. This will follow from the following technical lemma, which is in the spirit of Proposition 2.3 of [13], whose content was explained just before Theorem 5.6. □

Lemma 6.9. If κ < 𝔡 and P(ℕ)/fin is (κ, ℵ_{0})-ideal then also κ < 𝔯.

Proof. It suffices to show that if M ≺ H(θ) is ℵ_{0}-covering, of size κ and such that M ∩ P(ℕ) is ℵ_{0}-ideal in P(ℕ) then there is an r ∈ P(ℕ) that reaps M ∩ [ℕ]^{ℵ_{0}}. By Van Douwen’s characterization of 𝔡 (see Definition 4.2) there is f ∈ ^{ℕ}ℕ such that for every x ∈ M ∩ [ℕ]^{ℵ_{0}} and every g ∈ M ∩^{ℕ}ℕ there is an n ∈ x such that f(n) ⩾ g(n). Fix a countable subset C of M ∩ ^{ℕ}ℕ such that for every subset a of ℕ × ℕ with a ⊆ L_{f} there is c ∈ C such that a ⊆ L_{c}. As we can assume C ∈ M and because M knows that C is countable we can find g ∈ M ∩^{ℕ}ℕ such that c <^{∗} g for all c ∈ C. We claim that r = {n : f(n) ⩾ g(n)} is as required.

Now let x ∈ M ∩ [ℕ]^{ℵ_{0}}; the choice of f implies that r ∩ x is infinite. To show that r′ ∩ x is infinite consider a = L_{g} ∩ (x × ω). Clearly there is no c ∈ C with a ⊆^{∗} L_{c}, hence a \ L_{f} is infinite; this gives infinitely many n with g(n) > f(n). □

Remark 6.10. Many of the foregoing consequences of ℕ^{∗} being Cohen-Parovičenko do need the assumption of ¬CH. For example, it is shown in [7] that a homeomorphism between nowhere dense P -set subsets of ℕ^{∗} can be lifted to a homeomorphism on ℕ^{∗}. In addition, Steprāns [23] proves that all P -points can be taken to one another by autohomeomorphisms of ℕ^{∗} in the Cohen model (and it appears that only the assumption that 𝔪_{c} = ℵ_{2} = 𝔠 is used). However we can provide the following elegant contrasting result.

Proposition 6.11. If 𝔠 = ℵ_{2} and if ℕ^{∗} is Cohen-Parovičenko then there are two P -sets in ℕ^{∗}, of character ℵ_{1} and ℵ_{2} respectively, that are both homeomorphic to ℕ^{∗}.

Proof. Using Theorem 6.8 we see that E_{𝔠}^{∗} is Cohen-Parovičenko. We may therefore apply Theorem 5.5 to deduce that ℕ^{∗} and E_{𝔠}^{∗} are homeomorphic. Now fix one point x in E(2^{𝔠}); the set (ω × {x})^{∗} is a P -set of character 𝔠 in E_{𝔠}^{∗} and clearly homeomorphic to ℕ^{∗}.

To get a P -set of character ℵ_{1} we take a strictly decreasing chain ⟨a_{α} : α < ω_{1}⟩ of clopen sets in ℕ^{∗} whose intersection A is nowhere dense in ℕ^{∗} — see Remark 2.11 for the construction. Clearly then A is a P -set of character ℵ_{1}. The ideal I generated by the family {a′_{α} : α < ω_{1}} is ℵ_{1}-generated, so by Corollary 5.14 the algebra (P(ℕ)/fin)/I is Cohen-Parovičenko and hence isomorphic to P(ℕ)/fin. Its Stone space is A, which consequently is homeomorphic to ℕ^{∗}. □

7. Problems

Other reals. The Cohen model is probably the most intensively investigated model of ¬CH of all; this may explain our success in extracting key features of P(ℕ)/fin and ℕ^{∗} in that model. It would be of great interest if a similar thing could be done for other familiar models of ¬CH.

The Laver and Sacks (also side-by-side) models are particular favourites of the authors but the Random real model seems the most likely candidate for a successful investigation.

Characterizing P(ℕ)/fin. Theorems 2.13 and 5.5 lead one to hope that there is a characterization of P(ℕ)/fin in any Cohen model. As first steps on the way to such a result we ask the following questions.

Question 2. Is, in the ℵ_{3}-Cohen model, P(ℕ)/fin the unique Cohen-Parovičenko algebra?

Or, more generally.

Question 3. If 𝔠 = ℵ_{3} are then all Cohen-Parovičenko algebras of cardinality 𝔠 isomorphic?

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Department of Mathematics and Statistics, York University, 4700 Keele street, Toronto, Ontario, Canada M3J 1P3

E-mail address: dowa@yorku.ca

Faculty ITS, TU Delft, Postbus 5031, 2600 GA Delft, the Netherlands

E-mail address: k.p.hart@its.tudelft.nl
URL: http://aw.twi.tudelft.nl/~hart