ON MAD FAMILIES AND SEQUENTIAL ORDER

ALAN DOW

1. Introduction

We investigate the question of the existence of a compact sequential space with sequential order more than 2. We establish a connection with an interesting problem about maximal families of almost disjoint subsets of ω. We also obtain a partial result by showing that it is consistent that every countably tight compact space of weight b is Frechet-Urysohn. The main task is to work with Shelah’s forcing for obtaining a model of b < s = a.

The sequential order of a space is the supremum of the number of times that one must iterate the taking limits of convergent sequences in order to get the closure; that is, for A ⊂ X, A^{(0)} = A, A^{(1)} = {x ∈ X : (∃{x_{n} : n ∈ ω} ⊂ A) {x_{n} : n ∈ ω} converges to x}, and A^{(α)} = (∪_{β<α}A^{(β)})^{(1)}. A space is sequential if and only if it has a sequential order (≤ ω_{1}), and a space is Fréchet-Urysohn if its sequential order is one. A simple example of a compact space with sequential order two is the one point compactification of a Ψ-space, i.e. a space defined from a maximal almost disjoint family of a subsets of ω.

It was shown, by Bashkirov [Bas74], to follow from CH that there are compact spaces of any sequential order up to ω_{1} and that MA implies the existence of a compact space of sequential order three.

2. Totally MAD families

Let us consider a simple construction of a compact space with sequential order three from the assumption b = c. We will then introduce an interesting property and question about mad families arising from this construction (what we will call totally mad families).

Proposition 2.1 (b = c). There is a compact space with sequential order 3.

Proof. We define two families of subsets of ω, {a_{α} : α < c} and {b_{α} : α < c}, by induction on α. The inductive hypotheses are

(1) {a_{β} : β < α} are almost disjoint,

(2) for each β, γ < α, a_{β} is either almost contained in b_{β} or is almost disjoint from it,

(3) for β < γ < α, there is a finite subset (possibly empty) F_{β,γ} of γ + 1 such that b_{β} ∩ b_{γ} is almost equal to ∪{a_{ξ} : ξ ∈ F_{β,γ}}.

1980 Mathematics Subject Classification (1985 Revision). Primary 54D55, 03E05, Secondary 03E35, 54A20.

Key words and phrases. compact, sequential, mad family, forcing.

The goal is to have that {a_{α} : α < c} is a maximal almost disjoint family and for each infinite A ⊂ c there is an infinite A′ ⊂ A and a β < c so that a_{α} ⊂^{∗} b_{β} for each α ∈ A′. Therefore we simply fix enumerations, {x_{α} : α < c} and {y_{α} : ω ≤ α < c} of P(ω) and [c]^{ω} respectively, where y_{α} ⊂ α for all α ≥ ω.

So we add two more conditions to our inductive hyptheses:

(4) for each β < α, x_{β} meets some member of {a_{γ} : γ ≤ β} in an infinite set,

(5) for each γ < α, there is β ≤ γ so that a_{ξ} ⊂^{∗} b_{β} for infinitely many ξ in y_{γ}. Let us show that we can find suitable a_{α} and b_{α}. Consider x_{α}; if x_{α} ∩ a_{β} is

infinite for some β < α then let a_{α} = ∅. So assume that x_{α} does not meet a_{β} in an infinite set for any β < α. If there is a β < α such that x_{α} meets b_{β} in an infinite set, then let a_{α} = x_{α} ∩ b_{β} (note that, by inductive hypothesis 3, this β is unique). If there is no such β, then we can let a_{α} = x_{α}.

Next consider the sequence {a_{ξ} : ξ ∈ y_{α}}. If there is a β < α such that a_{ξ} ∩ b_{β} is infinite (hence cofinite) for infinitely many ξ ∈ y_{α}, then let b_{α} = ∅. Otherwise note that since b = c, we can find a function h ∈ ^{y_{α}}ω such that for each β < α, there is a finite F_{α,β} ⊂ y_{α} such that a_{ξ} ⊂ b_{β} for each ξ ∈ F_{α,β}, a_{ξ} ∩ b_{β} is finite for each ξ ∈ y_{α} − F_{α,β} and (a_{ξ} − h(ξ)) ∩ b_{β} is empty for all but finitely many ξ ∈ y_{α}. Furthermore, a_{β} ∩ ⋃_{ξ∈y_{α}} a_{ξ} − h(ξ) is finite for all β ≤ α. So we let b_{α} = ⋃_{ξ∈y_{α}} a_{ξ} − h(ξ). Note that it would suffice if we could do this for an infinite subset of y_{α}.

The Stone space of the Boolean subalgebra of P(ω) generated by the union of the two families {a_{α} : α < c} and {b_{α} : α < c} together with the finite sets is our desired space X. The usual Ψ-space constructed from the maximal almost disjoint family {a_{α} : α < c} is a subspace of X. The two scattering levels of Ψ constitute the first two scattering levels of X. The next scattering level of X consists of ultrafilters corresponding to each b_{α}, i.e. {b_{α} − (∪F ∪ n) : n ∈ ω and F is a finite subset of {a_{γ} : γ < c}} generates an ultrafilter of B. There is only one other point in the space, i.e. we have the one-point compactification of the points described so far. It is therefore trivial that X is compact and sequential. The inductive hypotheses 4 and 5 guarantee that the sequential order of X is at least 3, in fact it is equal to 3.

In the above construction, consider the step in which we chose b_{α}. We were given the family {a_{ξ} : ξ ∈ y_{α}} (an arbitrary countable subfamily of {a_{ξ} : ξ < c}). We can fix an enumeration {ξ_{n} : n ∈ ω} of y_{α} and, by removing a finite set from each, we may assume that the a_{ξ_{n}} are pairwise disjoint. Now, for each γ, there is a partial function f_{γ} from a subset of ω to ω such that a_{γ} ∩ a_{ξ_{n}} is not empty iff n is in the domain of f_{γ} and, for n in the domain of f_{γ}, f_{γ}(n) is the largest element of a_{γ} ∩ a_{ξ_{n}}. Similarly, we may assume that b_{γ} ∩ a_{ξ_{n}} is either empty or infinite for each n not in the domain of f′_{γ} and, for other n, f′_{γ}(n) is the largest value of a_{ξ_{n}} ∩ b_{γ}. In the above proof, we found a function h which bounded all of the f_{γ} and f′_{γ} for γ < α. However it was not necessary to have a total function, it would have sufficed to pass to an infinite subset of {ξ_{n} : n ∈ ω}. We would say that {a_{γ} : γ < α} is not totally unbounded with respect to {a_{ξ_{n}} : n ∈ ω} as per the following definition.

Definition 2.2. (1) Given B and A, infinite families of subsets of ω, say that A is \underline{totally unbounded} with respect to B, if for each infinite B′ ⊂ B and each h ∈ ^{B′}ω, there is an a ∈ A such that a ∩ ⋃{b − h(b) : b ∈ B′} is infinite.

(2) A maximal almost disjoint family A is \underline{totally} mad if for each infinite B ⊂ A no subset of A of cardinality less than c is totally unbounded with respect to B.

Question 1. (1) Is there a totally mad family?

(2) Does b = ω_{1} imply there is a totally mad family?

Remark 2.1. A totally mad family has a refinement which is a completely separable mad family. A mad family is said to be completely separable (introduced by Hechler and Shelah [Hec71, ErSh72]) if each X ⊂ ω which meets infinitely many in an infinite set will actually contain one of them. The existence of a completely separable mad family has been established under various cardinal hypotheses ([BS89]) but it is not known to follow from b = ω_{1} < s. Our hope is that that there is a modification of Shelah’s model of b < s (discussed later in the paper) in which every compact separable sequential space will have sequential order at most 2 and that in all liklihood there will be no totally mad family.

The next result will be an integral tool in our construction of a totally mad family (Proposition 2.6).

Proposition 2.3. Let R = {R_{n} : n ∈ ω} be a partition of ω into infinite parts. Then there is a family D such that

(1) R ∪ D is a mad family on ω;

(2) every D ∈ D is a transversal of R, i.e., |D ∩ R_{n}| ≤ 1 for every n ∈ ω;

(3) whenever B ∈ [R]^{ω} and D_{0} ∈ [D]^{<𝔠}, then there is an infinite B′ ⊆ B and a function g : B′ → ω such that the set D ∩ ⋃{B \ g(B) : B ∈ B′} is finite for every D ∈ D_{0}.

Therefore the family R∪D is a mad family in which no subset of D with cardinality less than 𝔠 is totally unbounded with respect to R.

Proof. By [BPS80], we may fix T , a tree π-base for [ω]^{ω} of height at most 𝔟 (i.e. T ⊂ [ω]^{ω} forms a tree under the ordering of reverse inclusion mod finite and T is dense in [ω]^{ω} with respect to this ordering). It is easily arranged that T is dense in [ω]^{ω} with respect to the usual reverse inclusion ordering since we can just ensure that for each T ∈ T and each n ∈ ω, there is a T′ ∈ T with T′ ⊂ T − n. For X ⊆ ω, let dom X denote the set {n ∈ ω : X ∩ R_{n} ≠ ∅}. For every X ⊆ ω with dom X infinite, choose some T (X) ∈ T so that T (X) ⊆ dom X and T (X) ≠ T (X′), whenever X ≠ X′. This is clearly possible since for every infinite M ⊆ ω, the set of all T ∈ T with T ⊆ M is of size 𝔠.

Enumerate [ω]^{ω} as {X_{α} : α < 𝔠} and proceed by transfinite induction. Suppose we have found {D_{β} : β < α} for some α < 𝔠. If X_{α} is almost disjoint with all R_{n}, n ∈ ω, and all D_{β}, β < α, then let D_{α} be a subset of X_{α}, which is a transversal of R and such that dom D_{α} = T (X_{α}), otherwise let D_{α} = ∅. Let D = {D_{α} : α < 𝔠, D_{α} ≠ ∅}.

The conditions (1) and (2) immediately follow from the inductive construction. In order to verify (3), let D_{0} ∈ [D]^{<𝔠} and B ∈ [R]^{ω}. There is some T_{0} ∈ T such that for every n ∈ T_{0}, R_{n} ∈ B. There are 𝔠 almost disjoint members of T , all contained in T_{0} and |D_{0}| < 𝔠, hence there is some T ⊆ T_{0}, T ∈ T such that the inclusion dom D ⊆^{∗} T holds for no D ∈ D_{0}. Since T is a tree under ^{∗}⊇ and since dom D ∈ T for every D ∈ D, we have, for D ∈ D_{0}, that either T ∩ dom D is finite or T ⊆^{∗} dom D. For D ∈ D_{0}, let f_{D} ∈ ^{T} ω be defined by f_{D}(n) = 0, if n ∉ dom D, and by f_{D}(n) = k, if n ∈ T and {k} = D ∩R_{n}. Since the height of T is less or equal to 𝔟, there is some f ∈ ^{T} ω with f ^{∗}> f_{D} for all D ∈ D_{0} such that dom D ^{∗}⊇ T. Now, it remains to put B′ = {R_{n} : n ∈ T }, define g(R_{n}) = f(n) and (3) holds with this choice of B′ and g.

Corollary 2.4. Let α ∈ ω_{1} be a limit ordinal, let C = {C_{β} : β < α} be a decreasing mod finite chain of subsets of ω. Then there is a family D(C) ⊆ [ω]^{ω} such that

(1) for every D ∈ D(C) and every β < α, D ⊆^{∗} C_{β};

(2) D(C) is an almost disjoint family;

(3) D(C) is maximal with respect to (1) and (2);

(4) if B is an almost disjoint family of sets such that for every B ∈ B there is some β < α with B ⊆^{∗} (C_{β} \ C_{β+1}) and if D_{0} ∈ [D(C)]^{<𝔠}, then there is some infinite B′ ⊆ B and a function g : B′ ⟶ ω such that the set

D ∩ ⋃{B \ g(B) : B ∈ B′}

is finite for every D ∈ D_{0}.

Proof. Choose an increasing sequence ⟨α_{n} : n ∈ ω⟩ cofinal in α. Let R_{0} = ω − C_{α_{0}} and for n > 0 define R_{n} = C_{α_{n}} \ (C_{α_{n+1}} ∪ ⋃_{k<n} R_{k}). Apply Proposition 2.3 to get D = D(C). (1),(2) and (3) of the corollary follow from (1) and (2) of Lemma 2.3. If there is some β < α such that the set B′ = {B ∈ B : B ∩ C_{β} is finite} is infinite, put g to be the constant 0 and (4) follows. Otherwise there is some infinite set B′ such that the set {β < α : B ⊆^{∗} C_{β}, B ∈ B′} is cofinal in α and moreover, for every n ∈ ω, there is at most one B ∈ B′ with B ⊆^{∗} R_{n}. Now, Lemma 2.3 easily applies.

Lemma 2.5. Let R = {R_{n} : n ∈ ω} be a partition of ω with each R_{n} infinite, let Y ∈ [ω]^{ω} be almost disjoint with all R_{n}. Then there is a family H = {H_{α} : 0 < α < 𝔟} such that:

(1) H_{α+1} (for each α < 𝔟) is an infinite set, which is almost disjoint with R_{n} for each n ∈ ω;

(2) H_{α} (for each limit α < 𝔟) is a decreasing mod finite chain, H_{α} = {C_{α,β} : β < α} with each C_{α,β} almost disjoint with R_{n} for each n < ω, ;

(3) if α < β < 𝔟, then H_{α+1} ∩ H_{β+1} is finite;

(4) if α < β ≤ γ < δ < 𝔟, then C_{β,α} ∩ C_{δ,γ} is finite;

(5) if α < β < γ < 𝔟 or β < γ ≤ α < 𝔟, then H_{α+1} ∩ C_{γ,β} is finite;

(6) whenever X ∈ [ω]^{ω} is almost disjoint with each R_{n} (n ∈ ω), then either X ∩ H_{α+1} is infinite for some α < 𝔟 or there is some limit α < 𝔟 such that X ∩ C_{α,β} is infinite for all β < α;

(7) |Y ∩ H_{1}| = ω.

Proof. Enumerate R_{n} = {r(n, k) : k ∈ ω} and fix a family of strictly increasing functions {f_{α} : α < 𝔟} ⊂ ^{ω}ω which is unbounded with respect to <^{∗} and, moreover, f_{α} <^{∗} f_{β} for α < β < 𝔟. In addition, assume that {n : Y ∩{r(n, k) : k < f_{1}(n)} ≠ ∅} is infinite. Let H_{1} = {r(n, k) : n ∈ ω, k ∈ ω, k < f_{1}(n)}; H_{α+1} = {r(n, k) : n ∈ ω, k ∈ ω, f_{α}(n) ≤ k < f_{α+1}(n)} for 0 < α < 𝔟, and finally let C_{α,β} = {r(n, k) : n ∈ ω, k ∈ ω, f_{β}(n) ≤ k < f_{α}(n)} for β < α < 𝔟, α limit.

It is straightforward to verify that, since for all α < β < 𝔟, f_{α}(n) < f_{β}(n) holds for almost all n ∈ ω, (1) up to (5) are satisfied. (6) holds because the family {f_{α} : α < 𝔟} is unbounded and all the functions f_{α} are strictly increasing. By the choice of f_{1}, (7) holds.

Proposition 2.6 ( 𝔟 = 𝔰 = ω_{1} or 𝔟 = 𝔠). There is a totally mad family on ω.

Proof. Of course the statement follows trivially from 𝔟 = 𝔠, so we assume 𝔟 = 𝔰 = ω_{1}. Fix a splitting family {S_{ξ} : ξ < ω_{1}} and an unbounded family of functions {f_{α} : α < 𝔟} ⊆ ^{ω}ω satisfying: for α < β < 𝔟, f_{α} <^{∗} f_{β}, moreover, lim f_{β}(n) − f_{α}(n) = ∞ and f_{0} ≥ id.

We shall construct simultaneously, for ξ < ω_{1}, a family 𝔐_{ξ} ⊂ [ω]^{ω} consisting of decreasing mod finite chains of countable limit length, a mad family A_{ξ} on ω and an almost disjoint family D_{ξ}, with the aim to get ⋃_{ξ<ω_{1}} D_{ξ} totally mad.

The starting point is quite simple: let D_{0} = {D_{n} : n ∈ ω} be an arbitrary partition of ω with each D_{n} infinite, let 𝔐_{0} consist of just one chain C, where C = {ω \ ⋃_{k<n} D_{k} : n ∈ ω} and let A_{0} be the union of D_{0} ∪ D(C), where D(C) is the family described in Corollary 2.4.

Let ξ < ω_{1} and suppose that all D_{η}, 𝔐_{η} and A_{η} are known for η < ξ. The induction assumptions are as follows:

(a) for every η < ζ < ξ, A_{ζ} refines A_{η} and ⋃_{ξ≤ζ} D_{ξ} ⊂ A_{ζ};

(b) for every η < ζ < ξ and for every chain C ∈ 𝔐_{η} there is some C′ ∈ 𝔐_{ζ} with C ⊆ C′ and C′ is an end-extension of C;

(c) for every η < ξ, for every D ∈ ⋃_{ζ≤η} D_{ζ} and for every chain C ∈ 𝔐_{η} there is some C ∈ C with D ∩ C finite, similarly, for two distinct chains C, C′ ∈ 𝔐_{η} there is C ∈ C and C′ ∈ C′ with C ∩ C′ finite;

(d) for every η < ξ and for every chain C ∈ 𝔐_{η} there is a family D(C) chosen as in Corollary 2.4, such that for each A ∈ A_{η} either there is a C ∈ C such that A ∩ C is finite, or there is a D ∈ D(C) such that A ⊂ D;

(e) for every η < ξ and for every infinite X ⊆ ω, either |X ∩ D| = ω for some D ∈ ⋃_{ζ≤η} D_{ζ} or there is a chain C ∈ 𝔐_{η} such that |X ∩ C| = ω for all C ∈ C;

(f) if η + 1 < ξ, then for every C ∈ 𝔐_{η+1} there is some C ∈ C with C ⊆ S_{η} or C ∩ S_{η} = ∅;

(g) if η + 1 < ξ and C ∈ 𝔐_{η}, then for every infinite B ⊂ D_{η+1} such that, for each D ∈ B, D ⊂^{∗} C for all C ∈ C, there is a C′ ∈ 𝔐_{η+1} and a descending mod finite sequence {C_{n} : n ∈ ω} ⊂ C′ such that, for each n, there is a B ∈ B which is contained mod finite in C_{n} − C_{n+1} while if η < ξ is a limit ordinal, then D_{η} is empty.

If ξ < ω_{1} is limit, let 𝔐_{ξ} consist of all chains C such that for every η < ξ there is some C_{η} ∈ 𝔐_{η} with C = ⋃_{η<ξ} C_{ξ}. For every C ∈ 𝔐_{ξ}, let D(C) be the almost disjoint family resulting from the application of Corollary 2.4 to C. Then B defined as ⋃_{η<ξ} D_{η} ∪ ⋃{D(C) : C ∈ 𝔐_{ξ}} is a maximal almost disjoint family on ω; let A_{ξ} be an arbitrary mad family on ω which refines B as well as all A_{η}, η < ξ. Since ξ < ω_{1}, such an A_{ξ} is easy to find. Finally, put D_{ξ} = ∅.

For a successor ordinal ξ = η + 1, we proceed more carefully. First, denote by Y the family of all Y ∈ [ω]^{ω} such that the set {C ∈ 𝔐_{η} : for every C ∈ C, |Y ∩C| = ω} is of size 𝔠. Then for every Y ∈ Y select one chain C(Y) ∈ 𝔐_{η} such that Y meets all C ∈ C(Y) in an infinite set and do it so that distinct Y ’s are paired to distinct C(Y)’s.

Consider C ∈ 𝔐_{η}. There is some α < ω_{1} such that C = {C_{β} : β < α} and in the previous induction step some cofinal sequence ⟨α_{n} : n ∈ ω⟩ was selected in order to find D(C) using Corollary 2.4. Let R_{n} = C_{α_{n}} \ (C_{α_{n+1}} ∪ ⋃_{k<n} R_{k}) and apply Lemma 2.5 to the family R = {R_{n} : n ∈ ω} to select the family {H_{α} : α < 𝔟}. Since all members of D(C) are transversals of R and since lim f_{α+1}(n)−f_{α}(n) = ∞, every H_{α+1} meets uncountably many D’s from D(C), hence also uncountably many A’s from A_{η}. For every α < 𝔟 (= ω_{1}) choose a countable family {A_{n}(C, α) : n ∈ ω} ⊆ A_{η} such that A_{n}(C, α) ∩ H_{α+1} is infinite for every n, and put D_{n}(C, α) = {A_{n}(C, α) ∩ H_{α+1} : n ∈ ω}. In the case that C = C(Y), just one modification is necessary: we shall guarantee that D_{0}(C, 1) meets Y in an infinite set. This is always possible with the aid of Lemma 2.5 (7), and due to the fact that A_{η} is a mad family. Let D_{ξ} be the set of all infinite members of {D_{n}(C, α)∩S_{η} : n ∈ ω, α ∈ ω_{1}, C ∈ 𝔐_{η}} ∪ {D_{n}(C, α) − S_{η} : n ∈ ω, α ∈ ω_{1}, C ∈ 𝔐_{η}}.

If C ∈ 𝔐_{η}, then 𝔐_{ξ}(C) will denote the family, which we now define, of all chains to which C extends. We now take care of condition (f). For α + 1 < ω_{1}, if all sets H_{α+1}∩S_{η} \⋃_{k<n} D_{k}(C, α) are infinite, then C∪{H_{α+1}∩S_{η} \⋃_{k<n} D_{k}(C, α) : n ∈ ω} belongs to 𝔐_{ξ}(C). Similarly, if all sets H_{α+1} ∩ (ω \ S_{η}) \ ⋃_{k<n} D_{k}(C, α) are infinite, then C ∪ {H_{α+1} ∩ (ω \ S_{η}) \ ⋃_{k<n} D_{k}(C, α) : n ∈ ω} belongs to 𝔐_{ξ}(C) as well. If α < ω_{1} is limit, then C ∪H_{α} is a chain, however, we again take the member S_{η} from the splitting family into account: C ∪{C_{α,β} ∩S_{η} : β < α} and C ∪{C_{α,β} \S_{η} : β < α} will be members of 𝔐_{ξ}(C), provided they do not contain finite sets. Clearly, for each 0 < α < ω_{1}, at least one of the chains defined above is put into 𝔐(C). Set 𝔐_{ξ} = ⋃{𝔐_{ξ}(C) : C ∈ 𝔐_{η}}.

It remains to define A_{ξ}. Similar to the limit step, let D(C) be the almost disjoint family resulting from the application of Corollary 2.4 to the chain C. Then B = ⋃_{η≤ξ} D_{η} ∪ ⋃{D(C) : C ∈ 𝔐_{ξ}} is a mad family and it remains to define A_{ξ} = {A ∩ B : A ∈ A_{η}, B ∈ B, |A ∩ B| = ω}.

This completes the inductive definitions. The verification of the induction assumptions is straightforward and will be left to the reader.

We have to verify that D = ⋃_{ξ<ω_{1}} D_{ξ} is totally mad. The almost disjointness of D should be clear, so let us first prove its maximality. Let Y ∈ [ω]^{ω} be arbitrary, assume that ξ(0) < ω_{1} is the first ordinal such that both sets Y ∩S_{ξ(0)} and Y \S_{ξ(0)} are infinite. If Y does not meet any D ∈ ⋃_{η≤ξ(0)} D_{η} in an infinite set, then by (e), there is some chain C in 𝔐_{ξ} such that Y ∩ C is infinite for all C ∈ C, however, by (f), there must be two chains C_{0} and C_{1} in 𝔐_{ξ} with this property. Choose two infinite subsets Y_{0} and Y_{1} of Y such that Y_{0} ⊆^{∗} C for all C ∈ C_{0} and Y_{1} ⊆^{∗} C for all C ∈ C_{1}. Let ξ(1) be maximum of the first ζ with both S_{ζ} ∩ Y_{0} and Y_{0} \ S_{ζ} infinite and of the first η with both S_{η} ∩ Y_{1} and Y_{1} \ S_{η} infinite. If Y still meets no D ∈ ⋃_{η≤ξ(1)} D_{η} in an infinite set, then Y_{0} is compatible with two chains C_{00}, C_{01} ∈ 𝔐_{ξ(1)}, both extending C_{0}, and similarly for Y (1). Proceeding further, if the branching continues, we arrive finally to a limit ξ = sup{ξ(n) : n ∈ ω}. Since Y meets all members from 𝔠 chains from 𝔐_{η}, our construction ensures that Y ∈ Y at the ξ + 1 st step of the induction and so Y ∩ D is infinite for some D ∈ D_{ξ+1}. Since Y was arbitrary, we conclude that D is a mad family.

Let B ∈ [D]^{ω}, E ∈ [D]^{<𝔠} be two disjoint subsets of D. Then there is some infinite {B_{n} : n ∈ ω} ⊆ B, some ξ < ω_{1} and a chain C = {C_{β} : β < α} ∈ 𝔐_{ξ} such that for some increasing sequence of ordinals ⟨β_{n} : n ∈ ω⟩ cofinal in α, B_{n} ⊆^{∗} C_{β_{n}} \ C_{β_{n}+1}. To see this, let δ be minimal such that B∩ ⋃_{η<δ} D_{η} is infinite; hence we may assume that B ⊂ ⋃_{η<δ} D_{η}. If δ = 0 then it is clear that we may take C to be the unique member of 𝔐_{0}. Otherwise, recursively choose a sequence C_{η} ∈ 𝔐_{η} so that β < η implies C_{β} ⊂ C_{η} ∈ 𝔐_{η} and there are infinitely many B ∈ B which are contained mod finite in every member of C_{η}. Clearly there is some ξ ≤ δ for which we cannot choose C_{ξ}. If ξ = β + 1 is a successor then we can apply inductive hypothesis (g) to find the required C ∈ 𝔐_{ξ} and sequence ⟨β_{n} : n ∈ ω⟩. If on the other hand, ξ is a limit, then C = ⋃_{η<ξ} C_{η} ∈ 𝔐_{ξ} is the desired chain and it should be clear, by inductive hypotheses (a) and (d), that there is a sequence ⟨β_{n} : n ∈ ω⟩ as required (i.e. there is an infinite sequence ⟨B_{n} : n ∈ ω⟩ ⊂ B and an increasing sequence ⟨η_{n} : n ∈ ω⟩ ⊂ ξ such that B_{n} is almost contained in every member of C_{η_{n}} but is almost disjoint from some member of C_{η_{n}+1}).

Having found C, let α be the length of C and let ⟨α_{n} : n ∈ ω⟩ be the increasing sequence that was used in the application of Corollary 2.4 when defining D(C). Choose an infinite subset B_{0} of B so that for each n ∈ ω, there is at most one B ∈ B_{0} such that B ⊂^{∗} C_{α_{n}} − C_{α_{n+1}}. Now, by construction, we have that for every E ∈ E, either there is an n such that E ∩ C_{α_{n}} is finite or there is a D ∈ D(C) such that E ⊂^{∗} D (by hypotheses (a) and (d)). Therefore by Lemma 2.3, there is a suitable B′ ⊂ B and function h ∈ ^{B′}ω witnessing that E is not totally unbounded with respect to B. The proposition is proved.

3. Shelah’s b < a poset

For the reader’s convenience we will reproduce the results from [She84] that we will need in the next section.

Definition 3.1. 1) Let K_{n} be the family of pairs (s, h), s a hereditarily finite set, h a partial function from P(s) to n + 1 such that

(1) h(s) = n

(2) if h(t) = ℓ + 1 (t ⊆ s), t = t_{1} ∪ t_{2} then h(t_{1}) ≥ ℓ or h(t_{2}) ≥ ℓ.

2) Let K_{≥n} = ⋃_{k≥n} K_{k} and K = ⋃_{n} K_{n}.

One example of a member of (s, h) ∈ K_{n} is to take s to be the set of integers [0, 2^{n}] and to define h(t) = log_{2}(|t|) for t ⊆ s.

Definition 3.2. An ordering on K is defined in two steps:

(1) Suppose (s, h) and (s′, h′) are members of K, we say that (s, h) ≤^{d} (s′, h′)

(we think of (s, h) as refining (s′, h′)) if: (a) s = s′;

(b) dom(h) ⊂ dom(h′);

(c) for some integer i, h(u) = max{0, h′(u)} for all u ∈ dom(h).

(2) (s, h) ≤^{e} (s′, h′) if for some s̄ ∈ dom h′, (s, h) = (s̄, h′ ↾ P(s̄)) .

(3) (s, h) < (s′, h′) if for some (s̄, h̄),

(s, h) ≤^{d} (s̄, h̄) ≤^{e} (s′, h′) .

We next define a set L consisting of finite trees of members of K whose maximal elements will be integers. These will form the component pieces of the conditions of our poset and they represent a certain measure of how likely it is that the finite sets of integers will be contained in the generic filter.

Definition 3.3. L is the union of all the L_{n} where for each n, L_{n} is the family of pairs (S, H) such that:

(1) S is a finite tree with a root;

(2) int(S) is the set of maximal nodes of S and will be a subset of ω, the set of non-maximal nodes of S will be denoted in(S) and we will assume that in(S) is disjoint from ω;

(3) H is a function whose domain is in(S) and let H_{x} denote H(x) for x ∈ in(S);

(4) For x ∈ in(S), (Suc_{S}(x), H_{x}) ∈ K_{≥n} where Suc_{S}(x) is the set of immediate successors of x in S.

If (S, H) ∈ L, then lev(S, H) = max{n : (S, H) ∈ L_{n}}.

The ordering on K will lift naturally to an ordering on L as follows.

Definition 3.4. We say that (S^{0}, H^{0}) ≤ (S^{1}, H^{1}) if S^{0} ⊆ S^{1}, they have the same root, in(S^{0}) = S^{0} ∩ in(S^{1}) and, for every x ∈ in(S^{0}), (Suc_{S^{0}}(x), H_{x}^{0}) ≤ (Suc_{S^{1}}(x), H_{x}^{1}).

If t ∈ L, we use (S^{t}, H^{t}) to denote its components. For (S, H) ∈ L and x ∈ in(S), let (S, H)^{[x]} = (S^{[x]}, H ↾ S^{[x]}) where S^{[x]} = {y ∈ S : x ≤ y in S }. An important tool in the proofs is to define half(S, H) for (S, H) in L_{n}, where half(S, H) = (S′, H′) is defined by S′ = S, dom(H′_{x}) = {A ∈ dom(H_{x}) : H_{x}(A) ≥ n/2} and H′_{x}(A) = [H_{x}(A) − n/2] (the greatest integer function). The usefulness is in the following

Fact 1. If t ∈ L_{n} and t′ ≤ half(t), then t̄ (canonically obtained from t′ and t) is such that t̄ ≤ t, int(t′) = int(t̄), and t̄ ∈ L_{[n/2]}. Define t̄ as follows: St̄ = S^{t′} and Ht̄ = H^{t} ↾ S^{t′} in the sense that for each x ∈ St̄, Ht̄ ; x = H_{x}^{t} ↾ P(Suc_{S^{t′}} (x))

An important fact for getting the splitting number to go up, as well as in some of the proofs of the properties of the poset we will define is the following.

Fact 2. If (S, H) ∈ L_{n+1} and int(S) = A_{0} ∪ A_{1}, then there is (S′, H′) ≤ (S, H) such that int(S′) ⊆ A_{0} or int(S′) ⊆ A_{1} and (S′, H′) ∈ L_{n}.

Definition 3.5. The forcing-notion Q is defined as follows:

(1) p ∈ Q if p = (W, T) where W is a finite subset of ω and T is a countably infinite set of pairwise disjoint members of L such that T − L_{n} is finite for each n. Let int(p) and int(T) denote the set ⋃{int(t) : t ∈ T }.

(2) Given t_{1} = (S_{1}, H_{1}), …, t_{k} = (S_{k}, H_{k}) all from L such that S_{i} ∩ S_{j} = ∅ for i ≠ j, t is \underline{built} from t_{1}, …, t_{k} if: There are incomparable nodes a_{1}, …, a_{k} of S such that every node of S is comparable with one of the a_{i} and such that (S, H)^{[a_{i}]} = (S_{i}, H_{i}) for i = 1, …, k.

(3) (W, T) ≤ (W′, T′) if W′ ⊂ W ⊂ W′ ∪ int(T′) and letting T′ = {t′_{0}, t′_{1}, … } and T = {t_{0}, t_{1}, … }, there are t̄_{i} ≤ t′_{i} for each i and pairwise disjoint finite subsets, B_{0}, B_{1}, …, of {t̄_{i} : i ∈ ω} such that, for each m ∈ ω, t_{m} is built from B_{m}. It follows that for each n only finitely many of the B_{m}’s will meet L − L_{n}.

(4) We also demand that for (W, T) ∈ Q and t, t′ ∈ T, max W < min(int(t)) and either max(int(t)) < min(int(t′)) or vice-versa.

It is natural to think of W as the root of the condition (W, T) ∈ Q and root preserving extensions will be important. The reader familiar with Axiom A forcings will observe that Q is an Axiom A poset and there is a natural choice for <_{n} similar to the <_{0} defined next.

Definition 3.6. For p ∈ Q we use (W^{p}, T^{p}) and p <_{0} q will mean that p < q and W^{p} = W^{q}. In addition, we will use {t_{n}^{p} : n ∈ ω} to denote T^{p} listed so that max(int(t_{n})) is increasing. If p ∈ Q, n ∈ ω, and W ⊆ 1 + max(int(t_{n}^{p})) then p_{W}^{n} denotes the condition (W, {t_{ℓ}^{p} : ℓ > n}).

Definition 3.7. For a proper ideal I ⊂ P(ω) (which includes all finite sets) let Q[I] denote the set of q ∈ Q such that for each A ∈ I, there are infinitely many t ∈ T^{q} such that int(t) ∩ A is empty.

Proposition 3.8. (1) If p ∈ Q and τ_{n} are Q-names of ordinals, then there is a q <_{0} p, such that for each k ≤ n < ω and each W ⊆ max[int(t_{n}^{q})] + 1, q_{W}^{n} forces a value on τ_{k} iff some <_{0} extension of q_{W}^{n} forces a value on τ_{k}.

(2) Q is proper.

(3) ⊩_{Q} “ {n : (∃p ∈ G_{Q})[n ∈ W^{p}]} is an infinite subset of ω which P(ω)V does not split”.

Proof. Parts (1) and (2) are proven below for Q[I], the more difficult case, so we will not prove them here. Item (3) follows from Fact 2 above.

Proposition 3.9. Let C_{ω_{1}} denote the poset for adding ω_{1} Cohen reals. In V^{C_{ω_{1}}} the following hold for an ideal I ∈ V with [ω]^{<ω} ⊂ I ⊂ P(ω):

(1) If p ∈ Q[I] and τ_{n} are Q[I]-names of ordinals then there is a <_{0} extension q of p such that: q ∈ Q[I], and for each k ≤ n < ω and each W ⊂ max[int(t_{n})] + 1, q_{W}^{n} forces a value on τ_{k} iff some <_{0} extension of q_{W}^{n} forces a value on τ_{k}.

(2) Q[I] is proper.

(3) ⊩_{Q[I]} “ {n : (∃p ∈ G_{Q[I]}) n ∈ W^{p}} is an infinite subset of ω which is almost disjoint from every A ∈ I”.

Proof. (1) Let λ be a large enough regular cardinal and let M be a countable elementary submodel of (H(λ), ∈, V ∩ H(λ)) to which all of I, the set of ω_{1} many Cohen reals given by C_{ω_{1}}, Q[I], p, and τ_{n}, for each n ∈ ω, all belong. Let δ = M∩ω_{1}. Define by induction on n < ω, q^{n} ∈ Q[I] ∩ M, t_{n} and k_{n} < ω such that:

(1) each q^{n} is a <_{0} extension of p,

(2) q^{n} ≤ q^{ℓ} for ℓ < n and if W ⊂ k_{n}, m < n + 1 and some <_{0} extension of

(W, T^{q^{n}}) forces a value on τ_{m}, then (W, T^{q^{n}}) does so.
(3) k_{n} > k_{ℓ} and k_{n} > max(int(t_{ℓ})) for ℓ < n.
(4) every ℓ ∈ int(q^{n}) is greater than k_{n}.
(5) t_{n} ∈ T^{q^{n}} and lev(t_{n}) > n and min(int(t_{n})) is greater than k_{n}.

To do so, first choose k_{n}, then q^{n}, and at last t_{n}. We want in the end to let T^{q} = {t_{n} : n < ω}. One point is missing. Why does q = (W^{p}, T^{q}) belong to Q[I] (not just to Q)? The answer is that it may not but we may choose a real r ∈ ^{ω}ω which is Cohen generic over V [M] and choose t_{n} to be the r(n)-th possible one according to some fixed well-ordering from V of the hereditarily finite sets (i.e. all possible t’s). Then for any A ∈ I there are infinitely many such t_{n} for which int(t_{n})∩A = ∅, hence there will be infinitely many n such that int(t_{n})∩A is empty.

Now the proof of (2) is quite standard. Let M etc. be as above. Recall that from [She82], it suffices to show that there is a condition q < p such that for each τ ∈ M which is a Q[I]-name of an ordinal, q forces that τ takes on a value from M. So in condition (1) we may take {τ_{n} : n ∈ ω} to be the list of all Q[I]-names of ordinals which are members of M. Let r be any extension of q which forces a value on some τ_{k}. Let n > k be chosen so that W^{r} ⊂ max[int(t_{n})] + 1. Clearly (W^{r}, {t_{ℓ} : ℓ > n}) has a <_{0}-extension (namely r) which forces a value on τ_{k}. Therefore q_{W^{r}}^{n} forces a value on τ_{k} and, since q_{W^{r}}^{n} is a member of M, this value is in M.

Statement (3) follows trivially from the fact that for each A ∈ I, the set {q ∈ Q[I] : int(T^{q}) ∩ A = ∅} is dense in Q[I].

We will also need the fact that Q is almost ^{ω}ω-bounding – this is certainly the property of Q which is the most difficult to obtain (and which does not hold for Q[I] for many choices of I). This proof we will need to reproduce carefully as we will need to generalize it in the next section. We refer the reader to [She82] for the basic properties of almost ^{ω}ω-bounding forcings: e.g. that a countable support iteration of almost ^{ω}ω-bounding proper posets is itself weakly bounding. We give the definitions below.

Lemma 3.10. Let (∅, T) ∈ Q and let W be a family of finite subsets of int(T) so that

(3.1) for every (∅, T′) < (∅, T), there is a W ⊂ int(T′) with W ∈ W .

Then for each k ∈ ω, there is a t ∈ L_{k} appearing in some (∅, T′) < (∅, T) such that, for every t′ ≤ t, there is a W ∈ W with W ⊆ int(t′).

Proof. Let T be enumerated as {t_{n} : n ∈ ω} and assume with no loss of generality that W is closed upwards. We first do the case k = 1. The desired tree t will be built from a set {t′_{n} : n ∈ u} where t′_{n} ≤ t_{n} for each n. For this we need a finite subset u of ω together with a partial function H: P(u) → ω such that H(u) = 1 and H(u_{1}) ≥ 0 or H(u_{2}) ≥ 0 whenever u = u_{1} ∪ u_{2} and whenever H(v) ≥ 0, then for every choice t̄_{n} ≤ t′_{n} (n ∈ v) the set ⋃_{ℓ∈v} int(t̄_{n}) belongs to W.

This suggests defining a partial function H: [ω]^{<ω} → ω by the following conditions:

1. u ∈ dom(H) iff for every ℓ ∈ u the tree half(t_{ℓ}) belongs to L_{1} and for all possible choices t′_{ℓ} ≤ half(t_{ℓ}) (ℓ ∈ u) the set ⋃_{ℓ∈u} int(t′_{ℓ}) belongs to W, and

2. H(u) ≥ n + 1 iff whenever u = u_{1} ∪ u_{2} one has H(u_{1}) ≥ m or H(u_{2}) ≥ m. It is assumed tacitly that H(u) is always chosen as large as possible so that for example, H(u) = 0 implies there are v, w ⊂ u such that u = v ∪ w but v, w ∉ dom(H). The need to use half(t_{ℓ}) will become apparent shortly.

Once we have u with H(u) ≥ 1 we define t = (S^{t}, H^{t}) by taking u as the root of the tree S^{t} and putting the trees half(t_{ℓ}) as its immediate successors. The function H_{u}^{t} is taken to be H on the successors of u and the appropriate H^{half(t_{ℓ})} higher up. This defines an element of L_{1} that is built using trees that were below elements of T and as such it appears in a condition (∅, T′) below (∅, T).

Now if t′ < t then the successors of u that are in t′ form a set u′ that is in dom(H^{t′}) and hence in dom(H) so that we have H(u′) ≥ 0. But t′ is built using t′_{ℓ} ≤ half(t_{ℓ}) with ℓ ∈ u′, so int(t′) = ⋃_{ℓ∈u′} int(t′_{ℓ}) belongs to W.

We show that there is such a u in three stages.

\underline{Stage A:} There is n such that if t′_{ℓ} < half(t_{ℓ}) is chosen for each ℓ < n, then ⋃_{ℓ<n} int(t′_{ℓ}) ∈ W. This is because, otherwise, the family of all ⟨t′_{ℓ} : ℓ < n⟩ such that n < ω, t′_{ℓ} < half(t_{ℓ}) for ℓ < n and ⋃_{ℓ<n} int(t′_{ℓ}) ∉ W form an ω-tree with finite branching. By König’s lemma there is an infinite branch, say ⟨t′_{n} : n ∈ ω⟩. But now there is a T′ such that (∅, T′) < (∅, T) and int(T′) = ⋃_{n∈ω} int(t′_{n}). The only problem with taking T′ ⊂ {t′_{n} : n ∈ ω} is that {lev(t′_{n}) : n ∈ ω} may be bounded. However by Fact 1, we can replace each t′_{n} by some t̄_{n} where the level of t̄_{n} is at least one-half of that of t_{n}. It follows that T′ = {t̄_{n} : n ∈ ω} is a suitable choice. Of course, the hypothesis on W contradicts the assumption that ⟨t′_{n} : n ∈ ω⟩ is a branch through the above tree – hence we finish. Note that if A ⊂ ω is such that {n : int(t_{n}) ∩ A = ∅} is infinite, then this is true of T′ as well. Therefore this stage generalizes directly for Q[I].

\underline{Stage B:} Repeated applications of Stage A will give 0 = n(0) < n(1) < … such that for every i and choice of t′_{ℓ} < half(t_{ℓ}) (n(i) ≤ ℓ < n(i + 1)), the set ⋃{int(t′_{ℓ}) : n(i) ≤ ℓ < n(i + 1)} is an element of W. Note that if i is large enough then half(t_{ℓ}) belongs to L_{1} for all ℓ ≥ n(i), so that [n(i), n(i + 1)) ∈ dom(H) for all but finitely many i.

\underline{Stage C:} Similarly we can produce a sequence 0 = m(0) < m(1) < … such that for every j ∈ ω and function h: [m(j), m(j + 1)) → ω with h(i) ∈ [n(i), n(i + 1)) for all i ∈ dom(h) and for every choice t′_{h(i)} ≤ half(t_{h(i)}), the set ∪{int(t′_{h(i)} : m(j) ≤ i < m(j + 1)} belongs to W. Let us remark however that if A is some subset of ω for which there are infinitely many n such that int(t_{n}) is disjoint from A, then this may not be true for a subsequence of t_{n}’s. Therefore this stage does not easily generalize to Q[I].

Now, take j large enough so that [n(i), n(i + 1)) ∈ dom(H) for i ≥ m(j) and consider u = [n(m(j)), n(m(j + 1)) ); we claim that H(u) ≥ 1. Indeed, let v ⊆ u. If there is i such that [n(i), n(i + 1) ) ⊂ v, then v ∈ dom(H), otherwise we may define h: [m(j), m(j + 1) ) → u with h(i) ∈ [n(i), n(i + 1) ) \ v for all i. It then follows, by Stage C, that u \ v ∈ dom(H).

Now observe that the above arguments apply to all (∅, T′) ≤ (∅, T) and hence that the family W_{1} consisting of those W that contain sets of the form ⋃_{ℓ∈u} int(t′_{ℓ}), where H(u) ≥ 1 and t′_{ℓ} ≤ half(t_{ℓ}) for all ℓ, also satisfies (3.1). We may define its associated function H_{1}. We note that H_{1}(u) ≥ 0 implies H(u) ≥ 1 and so H_{1}(u) ≥ 1 implies H(u) ≥ 2. Using high enough t_{ℓ} we can then find our required t ∈ L_{2}. An obvious induction completes the proof.

We can now reformulate this lemma in terms of forcing.

Lemma 3.11. Fix p ∈ Q, a sequence {τ_{n} : n ∈ ω} of Q-names of ordinals, and let q = (W^{q}, {t_{n} : n ∈ ω}) be as in Proposition 3.8, then there is an r <_{0} q such that if T^{r} = {t̄_{n} : n ∈ ω} the following holds.

For every n < ω, W ⊆ [0, max(int(t̄_{n})) + 1] and t′′_{n+1} ≤ t̄_{n+1},
there is W′ ⊂ int(t′′_{n+1}) such that (W ∪ W′, {t̄_{ℓ} : ℓ > n + 1})
forces a value on τ_{m} (m ≤ n)

(3.2)

Proof. Choose, inductively t̄_{n} using Lemma 3.10 as follows. Having chosen t̄_{n−1}, let m be large enough so that max[int(t̄_{n−1})] < min[int(t_{m})]. Let W be the set of all W ⊂ [min[int(t_{m})], ω) such that for each W′ with W^{q} ⊂ W′ ⊂ max[int(t̄_{n−1})] + 1 there is an r ≤ (W′, {t_{ℓ} : m ≤ ℓ < ω}) such that W^{r} − W′ ⊆ W and r forces a value on each of τ_{k} for k ≤ n. Clearly W satisfies the conditions in Lemma 3.10. Apply Lemma 3.10 to {t_{ℓ} : ℓ ≥ m} to obtain a condition t̄_{n} ∈ L_{n}.

Recall that a poset Q is almost ^{ω}ω-bounding if for every Q-name of a function from ω to ω and p ∈ Q, there is some g ∈ ^{ω}ω such that for every infinite A ⊂ ω, there is a condition p′ < p such that

p′ ⊩_{Q} {n ∈ A : f(n) < g(n)} is infinite .

A poset is weakly bounding if every unbounded family of functions from ω to ω remains unbounded in the extension.

Proposition 3.12. The forcing Q is almost ^{ω}ω-bounding.

Proof. Let ḟ be a Q-name of a function from ω to ω and let p ∈ Q. For each n, let τ_{n} = ḟ(n) and obtain r = (W^{r}, T^{r}) as in Lemma 3.11. Let g: ω → ω be the function where g(n) is the largest k such that for some W ⊂ max(int(t_{n+1}^{r})) + 1, the condition (W, {t_{ℓ}^{r} : ℓ > n + 1} forces that ḟ(n) = k. Given any infinite A ⊂ ω, let p_{A} = (W^{p}, {t_{n+1}^{r} : n ∈ A}). Let r̃ < p_{A} and let n ∈ A be such that some t̃≤ t_{n}^{r} is used in building some member of T r̃. Of course W r̃ is one of the subsets used in defining g(n) and by Lemma 3.8, it follows that the value r̃ forces on τ_{n} is the same as that forced by (W r̃, {t_{ℓ}^{r} : ℓ > n + 1}). This completes the proof.

4. The Modification of Q[I]

One thing that was not proven in the previous sections is that Q[I] is almost ωω-bounding. Indeed, in general, we essentially only expect C_{ω_{1}} ∗ Q[I] to be almost ωω-bounding and not even for all ideals I. In [She84], Shelah shows that this is the case when I is the ideal generated by an almost disjoint family of subsets of ω. We can describe this another way. Consider the usual Ψ-space (so-called) generated by an almost disjoint family A, i.e. the space is ω∪{x_{A} : A ∈ A} and a neighbourhood base for x_{A} is {{x_{A}} ∪ A − n : n ∈ ω}. Let X be the one point compactification of this space and let x denote the new point added. What we want to remark is that X is a compact sequential space and that I is equal to I_{x} = {A ⊂ ω : x ∉ cl_{X}(A)}. In general, if X is a sequential space which is a compactification of ω and x ∈ X, let I_{X,x} = {A ⊂ ω : x ∉ cl_{X}(A)}. We intend to show in this section that there is a forcing P which is a countable support iteration of proper almost ^{ω}ω-bounding forcings so that Q[I_{X,x}] is almost ^{ω}ω-bounding in V^{P} (and so can be used in an iteration which will be weakly bounding).

We begin with the following. As usual, if X is a compact space and P is any forcing notion, then X generates canonically a compact space in the extension V^{P} as follows: assume that X is a subset of [0, 1]^{κ} for κ = w(X). In the extension, we let X refer to the closure of X^{V} in [0, 1]^{κ}. In the case that X is zero-dimensional, we are simply refering to the Stone space of the algebra of clopen sets in both models.

Lemma 4.1 (CH). Let X be a compactification of ω and let Q_{ω_{1}} be the countable support ω_{1}-length iteration in which Q is the iterand for each α < ω_{1}. Let P = Q_{ω_{1}} ∗ C_{ω_{1}}. In V^{P}, if (∅, {t_{n} : n ∈ ω}) ∈ Q then, there is an infinite J ⊂ ω and a sequence t′_{n} ≤ t_{n} for n ∈ J so that (∅, {t′_{n} : n ∈ J}) ∈ Q and there is an x ∈ X so that ∪{int(t′_{n}) : n ∈ J} converges to x in the space X.

Proof. It should be clear that we may assume that δ < ω_{1} is large enough so that {t_{n} : n ∈ ω} is a member of the model V^{Q_{δ}∗C_{δ}} where, of course, Q_{δ} is the iteration of δ copies of Q and C_{δ} is the obvious countable sub-poset of C_{ω_{1}}. Working in the model V^{Q_{δ}} we can inductively choose t′_{n} in such a way that for each forcing condition p ∈ C_{δ} there are infinitely many n such that there is a p′ < p which forces t_{n} to equal t′_{n} and such that (∅, {t′_{n} : n ∈ ω}) is a member of Q (i.e. t′_{m} ∈ L − L_{n} for at most finitely many m). Now find a coordinate α > δ so that (∅, {t′_{n} : n ∈ ω}) is a member of G(α) (i.e. the generic subset of Q given by the α-th coordinate in Q_{ω_{1}}). It is easy to check that, by genericity, J_{1} = {n : (∃t̄_{n}) t̄_{n} ≤ t′_{n} and (∃q ∈ G(α)) int(t̄_{n}) ⊂ W^{q}} is infinite. In addition we may assume (by passing to a subset) that lev(t̄_{n}) diverges to infinity. For each n ∈ J_{1}, fix t̄_{n} ≤ t′_{n} such that there is a q ∈ G(α) with int(t̄_{n}) ⊂ W^{q}. Finally, by the inductive construction of the t′_{n}’s, there is an infinite set J ⊂ J_{1} in V^{P} such that t_{n} = t′_{n} for each n ∈ J. This is our desired set J. Note that (∅, {t̄_{n} : n ∈ J}) is a member of Q. The only thing remaining is to see that ∪{int(t̄_{n}) : n ∈ ω} converges to some member of X. This however follows from the fact (Lemma 3.8) that {m : (∃q ∈ G(α)) m ∈ W^{q}} is not split by any basic open subset of X.

The key property that seems to make this go through is the following.

Definition 4.2. A poset P = Q[I_{X,x}] will be said to have the madf property if for each p ∈ P with T^{p} = {t_{n} : n ∈ ω} there are pairwise disjoint subsets of ω, {J_{n} : n ∈ ω} and a sequence {t′_{n} : n ∈ ω} with t′_{n} ≤ t_{n} for each n, so that for each B ⊂ ω such that B ∩ J_{n} is infinite for each n, (∅, {t′_{n} : n ∈ B}) is a member of P.

Lemma 4.3. Suppose that X is a compactification of ω with countable tightness and suppose that for each p ∈ Q, there is a y ∈ X and p′ ≤ p such that int(T^{p′}) converges to y. Then Q[I_{X,x}] has the madf property for each x ∈ X.

Proof. Let S be the set of y ∈ X such that there is a J_{y} ⊂ ω such that there is a T_{y} ≤ T_{J_{y}} for which int(T_{y}) converges to y (where T_{J} = {t_{n} : n ∈ J}). Note that by the hypothesis on X, every infinite subset of ω contains some such J. Since p ∈ Q[I_{X,x}], x is in the closure of S. Since X has countable tightness, there is a countable subset {s_{n} : n ∈ ω} ⊂ S which has x as a limit point. Note that if x ∈ S, then we may stop since it would follow that, below (∅, T_{x}), the posets Q[I_{X,x}] and Q are the same.

For each n, fix J_{s_{n}} and {t_{m}^{n} : m ∈ J_{s_{n}}} so that t_{m}^{n} ≤ t_{m} for each m, the set ∪{int(t_{m}^{n}) : m ∈ J_{s_{n}}} converges to s_{n} and (∅, {t_{m}^{n} : m ∈ J_{s_{n}}}) is in Q. Since we may shrink J_{s_{n}}, we may assume that {J_{s_{n}} : n ∈ ω} are pairwise disjoint (countable families of infinite sets have disjoint refinements). With no loss of generality (and it isn’t important) we may assume that ∪_{n∈ω}J_{s_{n}} is all of ω. For each m ∈ ω, there is a unique n, such that m ∈ J_{s_{n}}, define t′_{m} to be t_{m}^{n}. Observe that (∅, {t′_{m} : m ∈ ω}) is a member of Q[I_{X,x}]. Now suppose B ⊂ ω is such that B ∩J_{s_{n}} is infinite for each n, then (∅, {t′_{m} : m ∈ B}) is a member of Q[I_{X,x}]. To see this, fix any A ∈ I_{X,x}. Since x ∉ cl_{X}(A), there is an n such that s_{n} ∉ cl_{X}(A). It follows that the intersection of A with ∪{int(t_{m}^{n}) : m ∈ J_{s_{n}}} is finite. Therefore there are infinitely many members, m, of B ∩ J_{s_{n}} such that int(t_{m}^{n}) ∩ A is empty.

Lemma 4.4. Suppose that X is a compactification of ω and suppose that x ∈ X is such that Q[I_{X,x}] has the madf property. Let p = (∅, T) ∈ Q[I_{X,x}] and suppose that W is a set of finite subsets of ω such that for each (∅, T′) < (∅, T), there is a W ∈ W with W ⊂ int(T′). Then, for every k ∈ ω, there is a t ∈ L_{k} appearing in some (∅, T′) < (∅, T) such that, for every t′ ≤ t, there is a W ∈ W with W ⊂ int(t′).

Proof. Let T = {t_{n} : n ∈ ω} and proceed as in Lemma 3.10. As mentioned in the proof of Lemma 3.10, the proofs of Stage A and Stage B are unchanged. We generalize the idea from [She84], where this is shown for the case X is the one point compactification of a Ψ-space, to prove Stage C. The rest of the proof is as in Lemma 3.10.

Fix the sequences {t′_{n} : n ∈ ω} and {J_{n} : n ∈ ω} as in Definition 4.2. The rest of the proof is virtually the same as in [She84].

In Stage A, we may assume that n(i+1) is chosen large enough so that [n(i), n(i+ 1)) ∩ J_{k} is not empty for each k ≤ n(i). Now fix a function ϕ from ω to ω so that ϕ(i) ≤ n(i) for each i and so that ϕ^{−1}(n) is infinite for each n (ϕ can be defined by induction).

\underline{Stage C:} We wish to define the sequence m(ℓ) so that for each function h with domain [m(ℓ), m(ℓ + 1)) and with h(i) ∈ [n(i), n(i + 1)) ∩ J_{ϕ(i)} for each relevent i, and for each selection of t̄_{h(i)} ≤ half(t′_{h(i)}), the set ⋃ int(t̄_{h(i)}) is in W. As in the proof of Stage A in Lemma 3.10, the idea is to note that if m(ℓ + 1) cannot be chosen, then we obtain an infinite sequence {t̄_{h(i)} : m(ℓ) ≤ i < ω} such that h(i) ∈ [n(i), n(i + 1)) ∩ J_{ϕ(i)} for each i and so that t̄_{h(i)} ≤ half(t′_{h(i)}). The role of the half(t′_{n})’s is the same in that they guarantee that we can find a suitable (∅, T̄) ≤ (∅, T) in Q which does not contain a member of W. However we must ensure that this (∅, T̄) is a member of Q[I_{X,x}] to obtain a contradiction. We do obtain such a contradiction, since B, the range of h, meets each J_{n} in an infinite set. Therefore, just as above, there are, for each A ∈ I_{X,x} infinitely many i so that t̄_{h(i)} misses A.

Corollary 4.5 (CH). If X is a compactification of ω and is such that in V^{C_{ω_{1}}} , Q[I_{X,x}] has the madf property, then, in V^{C_{ω_{1}}} , Q[I_{X,x}] is almost ^{ω}ω-bounding.

Proof. We work in V^{C_{ω_{1}}} and let ḟ be a Q[I_{X,x}]-name of an increasing function from ω to ω. Fix δ < ω_{1} large enough so that ḟ ∈ V^{C_{δ}}. We may invoke Lemmas 4.1 and 4.4, to find a sequence {t̄_{n} : n ∈ ω} just as we did in Lemma 3.11 except that we may use a real which is Cohen generic over V^{C_{δ}} to choose the next value for t̄_{n} from among all those satisfying the requirements. This genericity will ensure that (∅, {t̄_{n} : n ∈ ω}) ∈ Q[I_{X,x}]. We define a g similar to that defined in Lemma 3.12.

First of all, since there is no harm in refining the sequence of t̄_{n}’s, we may assume that there are pairwise disjoint J_{n}’s so that for each B which meets each J_{n} in an infinite set, (∅, {t̄_{n} : n ∈ B}) is in Q[I_{X,x}]. Choose an increasing sequence n(0, 0) < n(0, 1) < n(1, 1) < n(0, 2) < … < n(0, ℓ) < … < n(ℓ, ℓ) … so that n(m, ℓ) ∈ J_{m} for each m ≤ ℓ < ω. Clearly (∅, {t(n(m, ℓ)) : m ≤ ℓ < ω}) is in Q[I_{X,x}]. For each ℓ, define g(ℓ) large enough so that for each W ⊂ min int(n(0, ℓ)) and each t ≤ t̄(m, ℓ) (m ≤ ℓ), there is a W′ ⊂ int(t) such that (W ∪W′, {t̄_{n} : n(ℓ, ℓ) < n}) forces a value on ḟ(n(m, ℓ)) which is less than g(ℓ). We may assume that g is increasing.

Fix any A ∈ V^{C_{ω_{1}}} , and use the condition p_{A} = (W^{p}, {t̄_{n(m,ℓ)} : m ≤ ℓ ∈ A}). To finish the proof, suppose that r is any condition below p_{A}. There is an m < ℓ ∈ A such that t̄_{(m,ℓ)} is used in building a condition t′ of T^{r}. We know that int(t′) contains a set W′ so that (W^{r} ∪ W′, {t̄_{n} : n > n(ℓ, ℓ)}) forces a value on ḟ(n(m, ℓ)) which is less than g(ℓ). Since ḟ is increasing, it follows that g(ℓ) is forced to be greater than ḟ(ℓ).

The payoff in this section is the following interesting observation.

Theorem 4.6. There is a model of b = ω_{1} < ω_{2} = s = a = c in which every compact space of countable tightness and weight ω_{1} is actually Fréchet-Urysohn.

Proof. Start with a model of CH, and force with the countable support iteration of posets of the form Q_{ω_{1}} ∗ C_{ω_{1}} ∗ Q[I_{X,x}] where a suitable diamond sequence is used to choose which X and x to use. More precisely, diamond will predict a space X containing ω and a point x ∈ X. If X has countable tightness after forcing with Q_{ω_{1}} ∗ C_{ω_{1}} then we will use Q[I_{X,x}] otherwise just use the trivial poset. By the Lemmas in this section it follows that this iteration is weakly bounding, hence b will be ω_{1}.

In the extension assume that X is a compact space of countable tightness and weight ω_{1}. Let x be a limit point of the countable set ω ⊂ X. We may assume that ω is dense in X. The diamond sequence will have chosen a space X′ with the same base as X and the point x at some stage in the iteration. If we did not use Q[I_{X,x}] it was because the reflected copy of X did not have countable tightness. However uncountable tightness in compact spaces is upwards absolute (if cardinals are preserved), hence we were able to force with Q[I_{X,x}]. This forcing introduces a subset of ω which converges to x. The fact that a = ω_{2} follows from the fact that the one-point compactification of a Ψ-space is sequential but not Frechet-Urysohn. Finally, s = ω_{2} since we forced with Q cofinally.

Question 2. Does Q_{ω_{1}} preserve countable tightness in compact spaces? It is easy to see that C_{ω_{1}} does but it is certainly known that there are proper posets which do not [Fre88].

Question 3. Are there any totally mad families in this model? What if we do not use Cohen reals at ω_{1} limits?

The only thing blocking a proof, for us, of an affirmative answer to the first part of the previous question is that we need a “no” answer to the following question. However it probably has a “yes” answer, but then we would ask about towers which are maximal chains in ⋃_{ξ<ω_{1}} 𝔐_{ξ} from Proposition 2.6.

Question 4. Can Q fill maximal towers?

Recall that the proper forcing axiom, PFA, implies that all compact spaces of countable tightness are sequential. However at present the most complicated (in terms of sequential order) compact sequential space that we know under this axiom has sequential order only three!

Question 5. Does PFA imply that all compact spaces of countable tightness have sequential order at most three?

References

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[Bas74] A. I. Bashkirov. The classification of quotient maps and sequential bicompacta. Soviet Math. Dokl., 15, pages 1104-1109, 1974. Russian Original: Dokl. Akad. Nauk. SSSR, 217, pages 745-748, 1974.

[ErSh72] P. Erdös and S. Shelah. Separability properties of almost disjoint families of sets. Israel J., 13, pages 207–214, 1972

[Fre88] David H. Fremlin. Perfect pre-images of ω_{1} and the PFA. Top. Appl., pages 151–166, 1988.

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[She82] S. Shelah. Proper Forcing. Springer Lecture Notes, 1982.

[She84] S. Shelah. On cardinal invariants of the continuum. In Axiomatic Set Theory. Contemporary Mathematics, AMS, 1984.

York University, North York, Ontario, M3J 1P3

E-mail: Alan.Dow@mathstat.yorku.ca