HOW TO FORCE A COUNTABLY TIGHT, INITIALLY ω_{1}-COMPACT AND NON-COMPACT SPACE?

I. JUHÁSZ AND L. SOUKUP

Abstract. Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional, Frechet-Urysohn, initially ω_{1}-compact and non-compact space X of size ω_{2} having the following property: for every open (or closed) set A in X we have |A| ≤ ω_{1} or |X \ A| ≤ ω_{1}.

1. Introduction

E. van Douwen and, independently, A. Dow [4] have observed that under CH an initially ω_{1}-compact T_{3} space of countable tightness is compact. (A space X is initially κ-compact if any open cover of X of size ≤ κ has a finite subcover, or equivalently any subset of X of size ≤ κ has a complete accumulation point). Naturally, the question arose whether CH is needed here, i.e. whether the same is provable just in ZFC. The question became even more intriguing when in [2] D. Fremlin and P. Nyikos proved the same result from PFA. Quite recently, A. V. Arhangel’skiĭhas devoted the paper [1] to this problem, in which he has raised many related problems as well.

2  How to force…  April 6, 1996

In [7] M. Rabus has answered the question of van Douwen and Dow in the negative. He constructed by forcing a Boolean algebra B such that the Stone space St(B) includes a counterexample X of size ω_{2} to the van Douwen–Dow question, in fact St(B) is the one point compactification of X, hence X is also locally compact. The forcing used by Rabus is closely related to the one due to J. Baumgartner and S. Shelah in [3], which had been used to construct a thin very tall superatomic Boolean algebra. In particular, Rabus makes use of a so-called Δ-function f (which was also used and introduced in [3]) with some extra properties that are satisfied if f is obtained by the original, rather sophisticated forcing argument of Shelah from [3].

1991 Mathematics Subject Classification. 54A25,03E35.

Key words and phrases. initially ω_{1}-compact, countable tightness, Frechet-Urysohn, right separated space, forcing, Δ-function, normal, locally compact.

The preparation of this paper was supported by the Hungarian National Foundation for Scientific Research grant no. 16391.

In this paper we give an alternative forcing construction of counterexamples to the van Douwen–Dow question, which we think is simpler, more direct and more intuitive than the one in [7]. First of all, we directly force a topology τ_{f} on ω_{2} that yields an example from a Δfunction (with no extra properties) in the ground model which also satisfies CH. There is a wide variety of such ground models since they are easily obtained when one forces a Δ-function or because _{ω_{1}} implies the existence of a Δ-function (cf. [3]).

Let us recall the definition of the Δ-functions from [3].

Definition 1.1. Let f : [ω_{2}]^{2} ⟶ [ω_{2}]^{≤ω} be a function with f{α, β} ⊂

α ∩ β for {α, β} ∈ [ω_{2}]^{2}. (1) We say that two finite subsets x and y of ω_{2} are good for f provided that for α ∈ x ∩ y, β ∈ x \ y and γ ∈ y \ x we always have

(a) α < β, γ ⟹ α ∈ f{β, γ},

(b) α < β ⟹ f{α, γ} ⊂ f{β, γ},

(c) α < γ ⟹ f{α, β} ⊂ f{γ, β}.

(2) We say that f is a Δ-function if every uncountable family of finite subsets of ω_{2} contains two sets x and y which are good for f.

Both in [3] and [7] the main use of the Δ-function f is to suitably restrict the partial order of finite approximations to a structure on ω_{2} so as to become c.c.c. This we do as well, but in the proof of the countable compactness of τ_{f} we also need the following simple result that yields an additional property of Δ-functions provided CH also holds. In fact, only property 1.1.(a) is needed for this.

Lemma 1.2. Assume that CH holds, f is a Δ-function, {c_{α} : α < ω_{2}} are pairwise disjoint finite subsets of ω_{2} and B ∈ [ω_{2}]^{ω}. Then for each n ∈ ω there are distinct ordinals α_{0}, α_{1}, …, α_{n−1} ∈ ω_{2} such that

B ⊂ ⋂{f(ξ, η) : ξ ∈ c_{α_{i}}, η ∈ c_{α_{j}}, i < j < n}.

Proof. We can assume that sup B < min c_{α} for each α < ω_{2}. Denote by S(n) the statement of the lemma for n. We prove S(n) by induction on n. The first non-trivial case is n=2. Assume indirectly that S(2) fails. Then for each α < β < ω_{2} there is b_{α,β} ∈ B such that b_{α,β} ∉ f(ξ, η) for some ξ ∈ c_{α} and η ∈ c_{β}. By CH the Erdős- Rado partition theorem [6] has the form ω_{2} ⟶ (ω_{1})_{ω}^{2}, thus there are I ∈ [ω_{2}]^{ω_{1}} and b ∈ B such that for each α ≠ β ∈ I we have b ∉ f(ξ, η) for some ξ ∈ c_{α} and η ∈ c_{β}.

April 6, 1996  How to force…  3

Let d_{α} = c_{α} ∪ {b} for α ∈ I. Since f is a Δ-function there are α ≠ β ∈ I such that d_{α} and d_{β} are good for f. But d_{α} ∩ d_{β} = {b} and b < min c_{α}, min c_{β}, so by 1.1.(a) we have b ∈ f(ξ, η) for each ξ ∈ c_{α} and η ∈ c_{β} contradicting the choice of b = b_{α,β}. Thus S(2) holds.

Assume now that S(n) holds for some n ≥ 2 and we prove S(2n). Applying S(n) ω_{2}-many times for B and suitable final segments of {c_{α} : α < ω_{2}} we can obtain ω_{2}-many pairwise disjoint n-element sets {α_{0}^{ν}, α_{1}^{ν}, …, α_{n−1}^{ν}} ⊂ ω_{2} such that for each ν < ω_{2}

B ⊂ ⋂{f(ξ, η) : ξ ∈ c_{α_{i}^{ν}}, η ∈ c_{α_{j}^{ν}}, i < j < n}.

Let d_{ν} = ⋃{c_{α_{i}^{ν}} : i < n} for ν < ω_{2}. Applying S(2) for B and the sequence {d_{ν} : ν < ω_{2}} we get ordinals ν < μ < ω_{2} such that B ⊂ f(ξ, η) for all ξ ∈ d_{ν} and η ∈ d_{μ}. In other words, if i, j < n, ξ ∈ c_{α_{i}^{ν}} and η ∈ c_{α_{j}^{μ}} then B ⊂ f(ξ, η). Therefore the set {α_{0}^{ν}, α_{1}^{ν}, …, α_{n−1}^{ν}, α_{0}^{μ}, α_{1}^{μ}, …, α_{n−1}^{μ}} witnesses S(2n).

The following, even simpler, result about arbitrary functions f : [ω_{2}]^{2} ⟶ [ω_{2}]^{≤ω} with f{α, β} ⊂ α ∩ β for {α, β} ∈ [ω_{2}]^{2} will also be needed.

Lemma 1.3. If f is a function as above then for each K, K′ ∈ [ω_{2}]^{≤ω} there is a countable set cl_{f}(K, K′) ⊂ ω_{2} such that

(a) K ⊂ cl_{f}(K, K′), sup K = sup cl_{f}(K, K′),

(b) ∀ξ ∈ cl_{f}(K, K′) ∀η ∈ cl_{f}(K, K′) ∪ K′ (f{ξ, η} ⊂ cl_{f}(K, K′)).

Proof. Let K(0) = K, K(n + 1) = K(n) ∪ ⋃{f{ξ, η} : ξ ∈ K(n), η ∈ K(n) ∪ K′} and cl_{f}(K, K′) = ⋃_{n<ω} K(n).

The topology τ_{f} that we will construct on ω_{2} is right separated (in the natural order of ω_{2}) and is also locally compact and 0-dimensional. Thus for each α ∈ ω_{2} one can fix a compact (hence closed) and open neighbourhood H(α) of α such that max H(α) = α. Conversely, if we can fix for each α ∈ ω_{2} such a right-separating compact open neighbourhood H(α) then the family {H(α) : α < ω_{2}} determines the whole topology τ on ω_{2}. In fact, using the notation U(α, b) = H(α) \ ⋃{H(β) : β ∈ b}, it is easy to check that for each α ∈ ω_{2} the family B_{α} = {U(α, b) : b ∈ [α]^{<ω}} is a τ-neighbourhood base of α. Therefore, our notion of forcing consists of finite approximations to a family H = {H(α) : α < ω_{2}} like above.

Now, if H = {H(α) : α < ω_{2}} is as required and β < α < ω_{2} then either (i) β ∈ H(α) or (ii) β ∉ H(α). If (i) holds then H(β) \ H(α), if (ii) holds then H(β) ∩ H(α) is a compact open subset of β, hence there is a finite subset of β, call it i{α, β}, such that this set is covered by H[i{α, β}] = ⋃{H(γ) : γ ∈ i{α, β}} It may come as a surprise, but the existence of such a function i is also sufficient to insure that the collection H be as required. More precisely, we have the following result.

Definition 1.4. If H = {H(α) : α ∈ ω_{2}} is a family of subsets of ω_{2} such that max H(α) = α for each α ∈ ω_{2} then we denote by τ_{H} the topology on ω_{2} generated by H ∪ {ω_{2} \ H : H ∈ H} as a subbase.

Clearly, τ_{H} is a 0-dimensional, Hausdorff and right separated topology in which the elements of H are clopen.

Theorem 1.5. Assume that H is as in definition 1.4 above and there is a function i : [ω_{2}]^{2} ⟶ [ω_{2}]^{<ω} satisfying i{α, β} ⊂ α ∩ β for each

{α, β} ∈ [ω_{2}]^{2} such that if β < α then β ∈ H(α) implies H(β)\H(α) ⊂ H[i{α, β}] and β ∉ H(α) implies H(β) ∩ H(α) ⊂ H[i{α, β}]. Then each H(α) is compact in the topology τ_{H}, hence τ_{H} is locally compact.

Proof. We do induction on α ∈ ω_{2}. Assume that for each β ∈ α we know H(β) is compact in τ_{H}. By Alexander’s subbase lemma it suffices to show that any cover K of H(α) by members of H and their complements has a finite subcover. Let K ∈ K be such that α ∈ K. If K = H(γ) then α ≤ γ. The case α = γ is trivial so assume α < γ. But then H(α)\K ⊂ H[i{α, γ}] and by our inductive hypothesis H(β) is compact for each β ∈ i{α, γ} hence so is H(β) ∩ H(α) \ K being closed in H(β). Therefore H(α) \ K is compact and so some finite K_{0} ⊂ K covers it. But then K_{0}∪{K} covers H(α), hence we are done. A similar argument works if K = ω_{2} \ H(γ), then using H(α) ∩ H(γ) ⊂ H[i{α, γ}] if α < γ (or the compactness of H(γ) if γ < α).

It is now very natural to try to force a generic 0-dimensional, locally compact and right separated topology on ω_{2} by finite approximations (or pieces of information) of H and i. As was already mentioned, the Δfunction f comes into the picture when one wants to make this forcing c.c.c. The technical details of this are done in section 2.

We call the family H coherent if β ∈ H(α) implies H(β) ⊂ H(α). Clearly, this makes things easier because then H(β) \ H(α) = ∅, hence there is no problem covering it, the requirement on i is only that if β ∉ H(α) and β < α then H(β) ∩ H(α) ⊂ H[i{α, β}]. The original forcing of Baumgartner and Shelah from [3] (when translated to scattered, i.e. right separated, locally compact spaces rather than superatomic Boolean algebras) actually produced such a coherent family H. This is interesting because if H is coherent and τ_{H} is separable, which we have almost automatically if H is obtained generically, then τ_{H} is also countably tight!

Theorem 1.6. If there is a coherent family H of right separating compact open sets for a separable topology τ on ω_{2} then t(⟨ω_{2}, τ⟩) = ω.

Proof. Let X = ⟨ω_{2}, τ⟩. Then for each α ∈ ω_{2} we have t(α, X) = t(α, H(α)), hence it suffices to prove t(α, H(α)) = ω. If we had t(α, H(α)) = ω_{1} then we could find A ⊂ H(α) such that α ∈ \overline{A} but α ∉ \overline{B} for every countable B ⊂ A. Next we can find a strictly increasing sequence S = {x_{ν} : ν < ω_{1}} in A such that x_{ν} > sup F_{ν}, where F_{ν} = \overline{\{x_{μ} : μ < ν\}}. This construction can be carried out as sup F_{ν} < α because α ∉ F_{ν}, F_{ν} is compact and every initial segment of H(α) is open and so one of them should cover F_{ν}. Now for each ν < ω_{1} there is a finite subset b_{ν} of F_{ν} such that F_{ν} ⊂ H[b_{ν}]. If μ < ν then b_{μ} ⊂ F_{μ} ⊂ F_{ν} ⊂ H[b_{ν}] implies that for each β ∈ b_{μ} we have H(β) ⊂ H[b_{ν}] by the coherence of H, hence H[b_{μ}] ⊂ H[b_{ν}]. But x_{μ} ∈ H[b_{ν}] \ H[b_{μ}] for each μ < ν ∈ ω_{1}, hence the H[b_{ν}]’s yield a strictly increasing ω_{1}- sequence of clopen sets in a separable space, which is a contradiction completing the proof.

Ironically, this general result that gives countable tightness so easily cannot be used in our construction because we had to abandon the coherency of H in our effort to insure countable compactness (implied by the initial ω_{1}-compactness) of τ_{H}.

We mentioned above that our examples, by genericity, are separable. But this is not a coincidence. It is well-known and very easy to prove that if X is an initially ω_{1}-compact space then t(X) ≤ ω implies that X has no uncountable free sequence. (Moreover, if X is T_{3} the converse of this is also true.) Hence the following easy, but perhaps not widely known, result immediately implies that any non-compact, initially ω_{1}- compact space of countable tightness contains a countable subset whose closure is not compact. Thus if there is a counterexample to the van Douwen–Dow question then there is also a separable one.

Lemma 1.7. If Y is a non-compact topological space, then for some ordinal μ the space Y contains a free sequence {y_{ξ} : ξ < μ} ⊂ Y with non-compact closure.

Proof. If Y is non-compact, then Y has an strictly increasing open cover {U_{α} : α < κ} for some regular cardinal κ. We pick points y_{ξ} ∈ X and ordinals α_{ξ} < κ by recursion as follows. If the closure of the set Y_{ξ} = \{y_{η} : η < ξ\} in Y is compact, then pick α_{ξ} ∈ κ such that \overline{Y_{ξ}} ⊂ U_{α_{ξ}} and let y_{ξ} ∈ Y \ U_{α_{ξ}}.

The sequence α_{ξ} is strictly increasing because y_{η} ∈ U_{α_{ξ}}\U_{α_{η}} for η < ξ. So for some ξ ≤ κ the closure of Y_{ξ} is non-compact. But Y_{ξ} is also free because for each η < ξ we have \overline{Y_{η}} ⊂ U_{α_{η}} and (Y_{ξ} \ Y_{η}) ∩ U_{α_{η}} = ∅. So we are done.

Note that under CH the weight of a separable T_{3} space is ≤ ω_{1}, and an initially ω_{1}-compact space of weight ≤ ω_{1} is compact, hence the CH result of van Douwen and Dow is a trivial consequence of 1.7. Arhangel’skiĭraised the question, [1, problem 3], whether in this CH can be weakened to 2^{ω} < 2^{ω_{1}}? We shall answer this question in the negative: theorem 3.9 implies that the existence of a counterexample to the van Douwen–Dow question is consistent with practically any cardinal arithmetic that violates CH.

In [1, problem 17] Arhangel’skiĭasked if it is provable in ZFC that an initially ω_{1}-compact subspace of a T_{3} space of countable tightness is always closed. (Clearly this is so under CH or PFA, or in general if the answer to the van Douwen–Dow question is “yes”.) In view of our next result both Rabus’ and our spaces give a negative answer to this question. More generally we have the following result.

Theorem 1.8. If X is a locally compact counterexample to van Douwen– Dow then the one-point compactification αX = X ∪ {p} of X also has countable tightness. On the other hand, X is an initially ω_{1}-compact non-closed subset of αX.

Proof. Let A ⊂ X be such that p ∈ \overline{A} ( i.e. \overline{A}^{X} is not compact). By lemma 1.7 and our preceding remark then there is a countable set S ⊂ \overline{A}^{X} such that \overline{S}^{X} is not compact. But by t(X) = ω then there is a countable T ⊂ A for which S ⊂ \overline{T}^{X}, hence \overline{T}^{X} is non-compact as well, so p ∈ \overline{T}. Consequently we have t(p, αX) = ω and so t(αX) = ω.

2. The forcing construction

The following notation will be used in the definition of the poset P_{f}. Given a function h and a ⊂ dom(h) we write h[a] = ∪{h(ξ) : ξ ∈ a}. Given non-empty sets x and y of ordinals with sup x ≠ sup y let

x ∗ y = \{ \begin{matrix}x ∩ y & if sup x ∉ y and sup y ∉ x, \\ x \ y & if sup x ∈ y, \\ y \ x & if sup y ∈ x. \end{matrix}

Definition 2.1. For each function f : [ω_{2}]^{2} ⟶ [ω_{2}]^{≤ω} satisfying

f(α, β) ⊂ α ∩ β for any {α, β} ∈ [ω_{2}]^{2} we define a poset P_{f} = ⟨P_{f}, ≤⟩ as follows. The underlying set of P_{f} is the family of triples p = ⟨a, h, i⟩ for which

(i) a ∈ [ω_{2}]^{<ω}, h : a ⟶ P(a) and i : [a]^{2} ⟶ P(a) are functions, (ii) max h(ξ) = ξ for each ξ ∈ a , (iii) i(ξ, η) ⊂ f(ξ, η) for each {ξ, η} ∈ [a]^{2},

(iv) h(ξ) ∗ h(η) ⊂ h[i(ξ, η)] for each {ξ, η} ∈ [a]^{2}.

We will often write p = ⟨a^{p}, h^{p}, i^{p}⟩ for p ∈ P_{f}. For p, q ∈ P_{f} let p ≤ q if a^{p} ⊃ a^{q}, h^{p}(ξ) ∩ a^{q} = h^{q}(ξ) for ξ ∈ a^{q}, and i^{p} ⊃ i^{q}. If p ∈ P_{f}, α ∈ a^{p}, b ⊂ a^{p}∩α, let us write u^{p}(α, b) = h^{p}(α)\h^{p}[b].

Lemma 2.2. For each α < ω_{2} the set D_{α} = {p ∈ P_{f} : α ∈ a^{p}} is dense in P_{f}.

Proof. Let q ∈ P_{f} with α ∉ a^{q}. Define the condition p ≤ q by the stipulations a^{p} = a^{q} ∪ {α}, h^{p}(α) = {α}, h^{p}(ξ) = h^{q}(ξ) and i^{p}(α, ξ) = ∅ for ξ ∈ a^{q}. Then clearly p ∈ D_{α}.

Definition 2.3. If G is a P_{f}-generic filter over V, in V [G] we can define the topological space X_{f}[G] = X_{f} = ⟨ω_{2}, τ_{f}⟩ as follows. For α ∈ ω_{2} put H(α) = ⋃{h^{p}(α) : p ∈ G ∧ α ∈ a^{p}}, let H = {H(α) : α < ω_{2}} and let τ_{f} = τ_{H} as defined in 1.4, that is, τ_{f} is the topology on ω_{2} generated by H ∪ {ω_{2} \ H : H ∈ H} as a subbase.

If G is a P_{f}-generic filter over V then by lemma 2.2 we have ⋃{a^{p} : p ∈ G} = ω_{2}, and for each α < ω_{2} max H(α) = α and H(α) is clopen in X_{f}. Thus X_{f} is 0-dimensional and right separated. Of course, neither f nor i is needed for this. As was explained in section 1, we need f to be a Δ-function in order to make P_{f} c.c.c (which insures that no cardinal is collapsed), and the function i is used to make X_{f} also locally compact.

Theorem 2.4. If CH holds and f is a Δ-function, then P_{f} satisfies the c.c.c and V^{P_{f}} ⊧ “X_{f} = ⟨ω_{2}, τ_{f}⟩ is a 0-dimensional, right separated, locally compact space having the following properties:

(i) t(X_{f}) = ω, (ii) ∀A ∈ [ω_{2}]^{ω_{1}} ∃α ∈ ω_{2} |A ∩ H(α)| = ω_{1} (iii) ∀A ∈ [ω_{2}]^{ω} (\overline{A} is compact or |ω_{2} \ \overline{A}| ≤ ω_{1}).” Consequently, in V^{P_{f}}, X_{f} is a locally compact, normal, countably tight, initially ω_{1}-compact but non-compact space.

Proof of theorem 2.4. To show that P_{f} satisfies c.c.c we will proceed in the following way. We first formulate when two conditions p and p′ from P_{f} are called good twins (definition 2.5), then we construct the amalgamation r = p + p′ of p and p′ (definition 2.6) and show that r is a common extension of p and p′ in P_{f}. Finally we prove in lemma 2.8 that every uncountable family of conditions contains a couple of elements which are good twins.

Definition 2.5. Let p = ⟨a, h, i⟩ and p′ = ⟨a′, h′, i′⟩ be from P_{f}. We say that p and p′ are good twins provided

(1) p and p′ are twins, i.e., |a| = |a′| and the natural order-preserving bijection e = e_{p,p′} between a and a′ is an isomorphism between p and p′:

(i) h′(e(ξ)) = e′′h(ξ) for each ξ ∈ a, (ii) i′(e(ξ), e(η)) = e′′i(ξ, η) for each {ξ, η} ∈ [a]^{2} , (iii) e(ξ) = ξ for each ξ ∈ a ∩ a′,

(2) i(ξ, η) = i′(ξ, η) for each {ξ, η} ∈ [a ∩ a′]^{2},

(3) a and a′ are good for f.

Let us remark that, in view of (ii) and (iii), condition (2) can be replaced by “i(ξ, η) ⊂ a ∩ a′ for each {ξ, η} ∈ [a ∩ a′]^{2}”.

Definition 2.6. If p = ⟨a, h, i⟩ and p′ = ⟨a′, h′, i′⟩ are good twins we define the amalgamation r = ⟨b, g, j⟩ of p and p′ as follows:

Let b = a ∪ a′. For ξ ∈ h[a ∩ a′] ∪ h′[a ∩ a′] define

δ_{ξ} = min{δ ∈ a ∩ a′ : ξ ∈ h(δ) ∪ h′(δ)}.

Now, for any ξ ∈ b let

g(ξ) = \{ \begin{matrix}h(ξ) ∪ h′(ξ) & if ξ ∈ a ∩ a′, \\ h(ξ) ∪ \{η ∈ a′ \ a : δ_{η} ∈ h(ξ)\} & if ξ ∈ a \ a′, \\ h′(ξ) ∪ \{η ∈ a \ a′ : δ_{η} ∈ h′(ξ)\} & if ξ ∈ a′ \ a. \end{matrix} (•)

Finally for {ξ, η} ∈ [b]^{2} let

(••) j(ξ, η) = \{ \begin{matrix}i(ξ, η) & if ξ, η ∈ a, \\ i′(ξ, η) & if ξ, η ∈ a′, \\ f(ξ, η) ∩ b & otherwise. \end{matrix}

(Observe that j is well-defined because 2.5.(2) holds.) We will write r = p + p′ for the amalgamation of p and p′.

Lemma 2.7. If p and p′ are good twins then their amalgamation, r = p + p′, is a common extension of p and p′ in P_{f}.

Proof. First we prove two claims.

Claim 2.7.1. Let η ∈ a and δ ∈ a ∩ a′. Then η ∈ h(δ) if and only if δ_{η} is defined and δ_{η} ∈ h(δ). (Clearly, we also have a symmetric version of this statement for η ∈ a′.)

Proof of claim 2.7.1. Assume first η ∈ h(δ). Then δ_{η} is defined and clearly δ_{η} ∈ h(δ) if δ_{η} = δ. So assume δ_{η} ≠ δ. Since i(δ_{η}, δ) ⊂ a ∩ a′ and max i(δ_{η}, δ) < δ_{η} we have η ∉ h[i(δ_{η}, δ)] by the choice of δ_{η}. Thus from p ∈ P_{f} we have

η ∉ h(δ_{η}) ∗ h(δ). (†)

Then h(δ_{η})∗h(δ) ≠ h(δ_{η})∩h(δ) by (†). Since η ∈ h(δ) implies δ_{η} < δ, we actually have h(δ_{η}) ∗ h(δ) = h(δ_{η}) \ h(δ). Thus δ_{η} ∈ h(δ) by the definition of the operation ∗.

On the other hand, if δ_{η} ∈ h(δ), then either δ_{η} = δ or h(δ_{η}) ∗ h(δ) = h(δ_{η}) \ h(δ). Thus η ∈ h(δ) because in the latter case again η ∉ h[i(δ_{η}, δ)], hence (†) holds.

Claim 2.7.2. If ξ ∈ a ∩ a′ then

g(ξ) = h(ξ) ∪ {η ∈ a′ \ a : δ_{η} ∈ h(ξ)} = h′(ξ) ∪ {η ∈ a \ a′ : δ_{η} ∈ h′(ξ)}.

Proof of claim 2.7.2. Conditions 2.5.1(i) and (iii) imply h(ξ) ∩ a ∩ a′ = h′(ξ) ∩ a ∩ a′ and so

g(ξ) = h(ξ) ∪ h′(ξ) = h(ξ) ∪ ((a′ \ a) ∩ h′(ξ)).

By claim 2.7.1 we have

(a′ \ a) ∩ h′(ξ) = {η ∈ a′ \ a : δ_{η} ∈ h′(ξ)}.

But by 2.5.(1) we have δ_{η} ∈ h(ξ) iff δ_{η} ∈ h′(ξ), hence it follows that

g(ξ) = h(ξ) ∪ {η ∈ a′ \ a : δ_{η} ∈ h(ξ)}.

The second equality follows analogously.

Next we check r ∈ P_{f}. Conditions 2.1.(i)–(iii) for r are clear by the construction. So we should verify 2.1.(iv).

Let ξ ≠ η ∈ b and α ∈ g(ξ) ∗ g(η). We need to show that α ∈ g[j(ξ, η)]. We will distinguish several cases.

Case 1. ξ, η ∈ a ( or ξ, η ∈ a′ ).

Since g(ξ) ∩ a = h(ξ) and g(η) ∩ a = h(η) we have (g(ξ) ∗ g(η)) ∩ a = h(ξ) ∗ h(η) by the definition of operation ∗. Thus (g(ξ) ∗ g(η)) ∩ a ⊂ h[i(ξ, η)] = g[j(ξ, η))]∩a ⊂ g[j(ξ, η)]. So we can assume that α ∈ a′ \a. We know that δ_{α} is defined because α ∈ g(ξ) ∪ g(η) is also satisfied. Since α ∈ g(ξ) iff δ_{α} ∈ h(ξ) and α ∈ g(η) iff δ_{α} ∈ h(η) by (•) it follows that δ_{α} ∈ h(ξ) ∗ h(η). Thus there is ν ∈ i(ξ, η) such that δ_{α} ∈ h(ν). But i(ξ, η) = j(ξ, η) and α ∈ g(ν) by (•). Thus α ∈ g[j(ξ, η)].

Case 2. ξ ∈ a \ a′ and η ∈ a′ \ a.

We can assume that α ∈ a, since the α ∈ a′ case is done symmetrically.

Subcase 2.1. g(ξ) ∗ g(η) = g(η) \ g(ξ).

Then α ∈ g(η) and η ∈ g(ξ) so δ_{α} and δ_{η} are both defined and δ_{α} ∈ h′(η), δ_{η} ∈ h(ξ) hold, hence α ≤ δ_{α} < η < δ_{η} < ξ. But a and a′ are good for f, so by 1.1(a) we have δ_{α} ∈ f(η, ξ) ∩ b = j(ξ, η). Thus α ∈ h(δ_{α}) ⊂ g(δ_{α}) ⊂ g[j(ξ, η)] which was to be proved.

Subcase 2.2. g(ξ) ∗ g(η) = g(ξ) ∩ g(η) or g(ξ) ∗ g(η) = g(ξ) \ g(η).

Since now α ∈ g(ξ) ∗ g(η) ⊂ g(ξ), by the definition of the operation ∗ we have

|{α, ξ} ∩ g(η)| = 1. (1)

Thus, by the definition of g(η), δ^{∗} = min{δ ∈ a∩a′ : α ∈ h(δ)∨ξ ∈ h(δ)} is well-defined and δ^{∗} < η. If δ^{∗} < ξ then α ∈ h(δ^{∗}) and by 1.1(a) we have δ^{∗} ∈ f(ξ, η) ∩ b = j(ξ, η) for a and a′ are good for f, and so α ∈ g[j(ξ, η)].

Thus we can assume ξ < δ^{∗}. We know that δ^{∗} = δ_{α} or δ^{∗} = δ_{ξ} by the choice of δ^{∗}, but δ_{α} = δ_{ξ} is impossible by (1). Thus

|{α, ξ} ∩ h(δ^{∗})| = 1. (2)

Since α ∈ g(ξ) implies α ∈ h(ξ) and we have ξ < δ^{∗}, (2) implies α ∈ h(ξ) ∗ h(δ^{∗}) and so α ∈ h[i(ξ, δ^{∗})]. But i(ξ, δ^{∗}) ⊂ f(ξ, δ^{∗}) ⊂ f(ξ, η) because a and a′ are good for f, so 1.1(b) or (c) may be applied. Consequently, we have i(ξ, δ^{∗}) ⊂ j(ξ, η) by (••). Hence α ∈ g[j(ξ, η)] which was to be proved.

Since we investigated all the cases it follows that r satisfies 2.1.(iv), that is, r ∈ P_{f}. Since r ≤ p, q are clear from the construction, the lemma is proved.

Lemma 2.8. Every uncountable family F of conditions in P_{f} contains a couple of elements which are good twins. Consequently, P_{f} satisfies c.c.c.

Proof. By standard counting arguments F contains an uncountable subfamily F′ such that every pair p ≠ p′ ∈ F′ satisfies 2.5.(1)-(2). But f is a Δ-function, so there are p ≠ p′ ∈ F′ such that a^{p} and a^{p′} are good for f, i.e. p and p′ satisfies 2.5.(3), too. In other words, p and p′ are good twins and so r = p + p′ is a common extension of p and p′ in P_{f}.

Let G be a P_{f}-generic filter over V. As in definition 2.3, let H(α) = ⋃{h^{p}(α) : p ∈ G ∧ α ∈ a^{p}} for α ∈ ω_{2}, and let τ_{f} be the topology on ω_{2} generated by {H(α) : α ∈ ω_{2}} ∪ {ω_{2} \ H(α) : α ∈ ω_{2}} as a subbase. Put i = ⋃{i^{p} : p ∈ G}.

Since X_{f} is generated by a clopen subbase and max(H(α)) = α for each α ∈ ω_{2} by 2.1(ii), it follows that X_{f} is 0-dimensional and right separated in its natural well-order.

The following proposition is clear by 2.1.(iv) and by the definition of H and i.

Proposition . H(α) ∗ H(β) ⊂ H[i(α, β)] for α < β < ω_{2}. So by 1.5 every H(α) is a compact open set in X_{f}.

Definition 2.9. For α ∈ ω_{2} and b ∈ [α]^{<ω} let

U(α, b) = H(α) \ H[b]

and let

B_{α} = {U(α, b) : b ∈ [α]^{<ω}}.

By theorem 1.5 every H(α) is compact and B_{α} is a neighborhood base of α in X_{f}. Thus X_{f} is locally compact and the neighbourhood base B_{α} of α consists of compact open sets.

Unfortunately, the family H = {H(α) : α < ω_{2}} is not coherent, so we can’t apply theorem 1.6 to prove that X_{f} is countably tight. It will however follow from the following result.

Lemma 2.10. In V^{P_{f}}, if a sequence {z_{ζ} : ζ < ω_{1}} ⊂ H(β) converges to β, then there is some ξ < ω_{1} such that β ∈ \overline{\{z_{ζ} : ζ < ξ\}}.

Proof. Assume on the contrary that for each ξ < ω_{1} we can find a finite subset b_{ξ} ⊂ β such that {z_{ζ} : ζ < ξ} ∩ U(β, b_{ξ}) = ∅, that is, {z_{ζ} : ζ < ξ} ⊂ H[b_{ξ}].

Fix now a condition p ∈ P_{f} which forces the above described situation and decides the value of β. Then, for each ξ < ω_{1} we can choose a condition p_{ξ} ≤ p which decides the value of z_{ξ} and b_{ξ}. We can assume that {a^{p_{ξ}} : ξ < ω_{1}} forms a Δ-system with kernel D, z_{ξ} ∈ a^{p_{ξ}} \ D and that z_{ξ} < z_{η} for ξ < η < ω_{1}.

Claim . Assume that ξ < η < ω_{1}, p_{ξ} and p_{η} are good twins and r = p_{ξ} + p_{η}. Then r^{∥}—“z_{ξ} ∈ H[D ∩ β]”.

Proof of the claim. Indeed, z_{ξ} ∈ h^{r}[b_{η}] because r ≤ p_{ξ}, p_{η}. Since z_{ξ} ∈ a^{p_{ξ}} \ a^{p_{η}}, (•) and claim 2.7.2 imply that z_{ξ} ∈ h^{r}[b_{η}] holds if and only if δ_{z_{ξ}} ∈ D = a^{p_{ξ}} ∩ a^{p_{η}} is defined and δ_{z_{ξ}} ∈ h^{p_{η}}[b_{η}]. Since b_{η} ⊂ β, we also have δ_{z_{ξ}} < β and so z_{ξ} ∈ h^{r}[D ∩ β].

Applying lemma 2.8 to appropriate final segments of {p_{ξ} : ξ < ω_{1}} we can choose, by induction on μ < ω_{1}, pairwise different ordinals μ < ξ_{μ} < η_{μ} < ω_{1} with η_{μ} < ξ_{ν} if μ < ν such that p_{ξ_{μ}} and p_{η_{μ}} are good twins. Let r_{μ} = p_{ξ_{μ}} + p_{η_{μ}}. Since P_{f} satisfies c.c.c there is a condition q ≤ p such that q^{∥}—“ |{μ ∈ ω_{1} : r_{μ} ∈ G}| = ω_{1}”. Thus, by the claim q^{∥}—“|{z_{ξ} : ξ < ω_{1}}∩H[D∩β]| = ω_{1}”, i.e. the neighbourhood U(β, D∩β) of β misses uncountably many of the points z_{ξ} which contradicts that β is the limit of this sequence.

Corollary 2.11. t(X_{f}) = ω.

Proof. Assume on the contrary that α ∈ ω_{2} and t(α, X_{f}) = t(α, H(α)) = ω_{1}. Then there is an ω-closed set Y ⊂ H(α) that is not closed. Since the subspace H(α) is compact and right separated and so it is pseudoradial , for some regular cardinal κ there is a sequence {z_{ξ} : ξ < κ} ⊂ Y which converges to some point β ∈ H(α) \ Y. Since Y is ω-closed and |Y | ≤ | H(α)| = ω_{1} we have κ = ω_{1}. By lemma 2.10 there is some ξ < ω_{1} with β ∈ \overline{\{z_{ζ} : ζ < ξ\}} ⊂ Y contradicting β ∉ Y.

Lemma 2.12. In V^{P_{f}}, for each uncountable A ⊂ X_{f} there is β ∈ ω_{2} such that |A ∩ H(β)| = ω_{1}.

Proof. Assume that p^{∥}— “Ȧ = {α̇_{ξ} : ξ < ω_{1}} ∈ [ω_{2}]^{ω_{1}}”. For each ξ < ω_{1} pick p_{ξ} ≤ p and α_{ξ} ∈ ω_{2} such that p_{ξ}^{∥}—α̇_{ξ} = α̂_{ξ}. Since P_{f} satisfies c.c.c we can assume that the α_{ξ} are pairwise different. Let sup{α_{ξ} : ξ < ω_{1}} < β < ω_{2}. Now for each ξ < ω_{1} define the condition q_{ξ} ≤ p_{ξ} by the stipulations a^{q_{ξ}} = a^{p_{ξ}}∪{β}, h^{q_{ξ}}(β) = a^{q_{ξ}} and i^{q_{ξ}}(ν, β) = ∅ for ν ∈ a^{p_{ξ}}. Then q_{ξ} ∈ P_{f} and q_{ξ}^{∥}—α̇_{ξ} ∈ H(β). But P_{f} satisfies c.c.c, so there is q ≤ p such that q^{∥}— “|{ξ ∈ ω_{1} : q_{ξ} ∈ G}| = ω_{1}”. Thus q^{∥}— “|A ∩ H(β)| = ω_{1}.”

Since every H(β) is compact, lemma 2.12 above clearly implies that X_{f} is ω_{1}-compact, i.e. every subset S ⊂ X_{f} of size ω_{1} has a complete accumulation point.

Now we start to work on (iii): in V^{P_{f}} the closure of any countable subset Y of X_{f} is either compact or it contains a final segment of ω_{2}. If Y is also in the ground model, then actually the second alternative occurs and this follows easily from the next lemma.

Lemma 2.13. If p ∈ P_{f}, β ∈ a^{p}, b ⊂ a^{p} ∩ β, α ∈ β \ a^{p}, then there is a condition q ≤ p such that α ∈ u^{q}(β, b).

Proof. Define the condition q ≤ p by the following stipulations: a^{q} = a^{p} ∪ {α}, h^{q}(α) = {α},

h^{q}(ν) = \{ \begin{matrix}h^{p}(ν) ∪ \{α\} & if β ∈ h^{p}(ν) \\ h^{p}(ν) & if β ∉ h^{p}(ν) \end{matrix}

for ν ∈ a^{p}, and let i^{q} ⊃ i^{p} and i^{q}(α, ν) = ∅ for ν ∈ a^{p}.

To show q ∈ P_{f} we need to check only 2.1(iv). Assume that α ∈ h^{q}(ν) ∗ h^{q}(μ). Then by the construction of q we have β ∈ h^{p}(ν) ∗ h^{p}(μ).

Thus there is ξ ∈ i^{p}(ν, μ) with β ∈ h^{p}(ξ). But then α ∈ h^{q}(ξ), so by i^{q}(ν, μ) = i^{p}(ν, μ) we have α ∈ h^{q}[i^{q}(ν, μ)]. In view of h^{p}(ν) ∗ h^{p}(μ) ⊂ h^{p}[i^{p}(ν, μ)] we are done. Thus q ∈ P_{f}, q ≤ p and clearly α ∈ u^{q}(β, b), so we are done.

This lemma yields the following corollary.

Corollary 2.14. If Z ∈ [ω_{2}]^{ω} ∩ V and β ∈ ω_{2} \ sup Z then β ∈ \overline{Z}.

Proof. Let U(β, b) be a neighbourhood of β, b ∈ [β]^{<ω}. Since p^{∥}—“U(β, b) ⊃ u^{p}(β, b)” for each p ∈ P_{f} and the set

D_{β,b,Z} = {q ∈ P_{f} : u^{q}(β, b) ∩ Z ≠ ∅}

is dense in P_{f} by the previous lemma, it follows that U(β, b) intersects Z. Consequently β ∈ \overline{Z}.

The space X_{f} is right separated, i.e. scattered, so we can consider its Cantor-Bendixon hierarchy. According to corollary 2.14 for each α < ω_{2} the set A_{α} = [ωα, ωα + ω) is a dense set of isolated points in X_{f}⌈(ω_{2} \ ωα). Thus the α^{th} Cantor-Bendixon level of X_{f} is just A_{α}. Therefore X_{f} is a thin very tall, locally compact scattered space in the sense of [8]. Let us emphasize that CH was not needed to get this result, hence we have also given an alternative proof of the main result of [3].

Now we continue to work on proving property 2.4(iii) of X_{f}.

Given p ∈ P_{f} and b ⊂ a^{p} let p⌈b = ⟨b, h, i^{p}⌈[b]^{2}⟩ where h is the

function with dom(h) = b and h(ξ) = h^{p}(ξ) ∩ b for ξ ∈ b. Let us remark that p⌈b in not necessarily in P_{f}. In fact, p⌈b ∈ P_{f} if and only if i^{p}(ξ, η) ⊂ b for each ξ ≠ η ∈ b. Especially, if b is an initial segment of a^{p}, then p⌈b ∈ P_{f}. The order ≤ of P_{f} can be extended in a natural way to the restrictions of conditions: if p and q are in P_{f}, b ⊂ a^{p}, c ⊂ a^{q}, define p⌈b ≤ q⌈c iff b ⊃ c, h^{p}(ξ) ∩ c = h^{q}(ξ) ∩ c for each ξ ∈ c, and i^{p}⌈[c]^{2} = i^{q}⌈[c]^{2}. Clearly if p⌈b ∈ P_{f} and q⌈c ∈ P_{f} then the two definitions of ≤ coincide.

Definition 2.15. Let p, p′ ∈ P_{f} with a^{p} = a^{p′}. We write p ≺ p′ if for each α ∈ a^{p} and b ⊂ a^{p} ∩ α we have u^{p}(α, b) ⊂ u^{p′}(α, b).

The following technical result will play a crucial role in the proof of 2.4(iii). Part (c) in it will enable us to “insert” certain things in H(γ_{0}) in a non-trivial way. But there is a price we have to pay for this: this is the point where the coherency of the H(α) has to be abandoned. Part (d) will be needed in section 3.

Lemma 2.16. Assume that s = ⟨a^{s}, h^{s}, i^{s}⟩ ∈ P_{f}, a^{s} = S ∪ E ∪ F, Q ⊂ S, S <_{On} E <_{On} F, E = {γ_{i} : i < k}, γ_{0} < γ_{1} < ⋯ < γ_{k−1}, F = {γ_{i,0}, γ_{i,1} : i < k}, moreover

(i) ∀i < k h^{s}(γ_{i,0}) ∩ h^{s}(γ_{i,1}) = h^{s}[Q ∪ E], (ii) ∀i < k ∀ξ ∈ S f(ξ, γ_{i}) = f(ξ, γ_{i,0}) = f(ξ, γ_{i,1}). Then there is a condition r = ⟨a^{r}, h^{r}, i^{r}⟩ with a^{r} = S ∪ E such that (a) r ≤ s⌈S,

(b) r ≤ s⌈(Q ∪ E),

(c) S \ h^{s}[Q ∪ E] ⊂ h^{r}(γ_{0}),

(d) s⌈(S ∪ E) ≺ r.

Proof. Let a^{r} = S ∪ E and write C = S \ h^{s}[Q ∪ E]. For ξ ∈ a^{r} we set

h^{r}(ξ) = \{ \begin{matrix}h^{s}(ξ) ∪ C & if ξ = γ_{i} and γ_{0} ∈ h^{s}(γ_{i}), \\ h^{s}(ξ) & otherwise. \end{matrix}

For ξ ≠ η ∈ a^{r} we let

i^{r}(ξ, η) = \{ \begin{matrix}i^{s}(ξ, η) & if ξ, η ∈ Q ∪ E or ξ, η ∈ S, \\ f(ξ, η) ∩ a^{r} & otherwise. \end{matrix}

Finally let r = ⟨a^{r}, h^{r}, i^{r}⟩.

We claim that r satisfies the requirements of the lemma. (a), (b) and (c) are clear from the definition of r, once we establish that r ∈ P_{f}. To see that it suffices to check only 2.1.(iv) because the other requirements are clear from the construction of r. So let ξ < η ∈ a^{r}. We have to show h^{r}(ξ) ∗ h^{r}(η) ⊂ h^{r}[i^{r}(ξ, γ_{i})].

If ξ, η ∈ S, then h^{r}(ξ) ∗ h^{r}(η) ⊂ h^{r}[i^{r}(ξ, η)] holds because r⌈S = s⌈S ∈ P_{f}.

So we can assume that η = γ_{i} for some i < k.

Case 1. ξ ∈ S.

Subcase 1.1. ξ ∉ h^{r}(γ_{i}), hence h^{r}(ξ) ∗ h^{r}(γ_{i}) = h^{r}(ξ) ∩ h^{r}(γ_{i}).

In this case we also have h^{s}(ξ) ∗ h^{s}(γ_{i}) = h^{s}(ξ) ∩ h^{s}(γ_{i} and so

h^{s}(ξ) ∩ h^{s}(γ_{i}) ⊂ h^{s}[i^{s}(ξ, γ_{i})] ⊂ h^{r}[i^{r}(ξ, γ_{i})] (3)

for i^{r}(ξ, γ_{i}) ⊃ i^{s}(ξ, γ_{i},). If γ_{0} ∉ h^{s}(γ_{i}) then h^{r}(γ_{i}) = h^{s}(γ_{i}) and since h^{r}(ξ) = h^{s}(ξ) we have h^{r}(ξ) ∗ h^{r}(γ_{i}) = h^{s}(ξ) ∩ h^{s}(γ_{i}) ⊂ h^{r}[i^{r}(ξ, γ_{i})] by (3). Assume now that γ_{0} ∈ h^{s}(γ_{i}). Thus h^{r}(γ_{i}) = h^{s}(γ_{i}) ∪ C, and so ξ ∉ C, that is ξ ∈ h^{s}[Q ∪ E]. Then h^{r}(ξ) ∗ h^{r}(γ_{i}) = h^{r}(ξ) ∩ h^{r}(γ_{i}) = (h^{s}(ξ) ∩ h^{s}(γ_{i})) ∪ (h^{s}(ξ) ∩ C). By (3) above it is enough to show that h^{s}(ξ) ∩ C ⊂ h^{r}[i^{r}(ξ, γ_{i})]. Since h^{s}(ξ) ∩ C = ∅ for ξ ∈ Q we can assume that ξ ∉ Q. By (i) h^{s}[Q ∪ E] = h^{s}(γ_{i,0}) ∩ h^{s}(γ_{i,1}) and so

h^{s}(ξ) ∩ C = h^{s}(ξ) \ h^{s}[Q ∪ E] = (h^{s}(ξ) \ h^{s}(γ_{i,0})) ∪ (h^{s}(ξ) \ h^{s}(γ_{i,1})). (4)

Since ξ ∈ h^{s}[Q ∪ E] ⊂ h^{s}(γ_{i,j}) for j = 0, 1 we have

h^{s}(ξ) \ h^{s}(γ_{i,j}) = h^{s}(ξ) ∗ h^{s}(γ_{i,j}) ⊂ h^{s}[i^{s}(ξ, γ_{i,j})]. (5)

By (ii), i^{s}(ξ, γ_{i,j}) ⊂ f(ξ, γ_{i,j}) ∩ a^{s} = f(ξ, γ_{i}) ∩ a^{s}. Since ξ ∉ Q it follows that f(ξ, γ_{i}) ∩ a^{s} = i^{r}(ξ, γ_{i}). Thus from (5) we obtain

h^{s}(ξ) \ h^{s}(γ_{i,j}) ⊂ h^{s}[i^{r}(ξ, γ_{i})] = h^{r}[i^{r}(ξ, γ_{i})]. (6)

Putting (4) and (6) together we get h^{s}(ξ) ∩ C ⊂ h^{r}[i^{r}(ξ, γ_{i})] which was to be proved.

Subcase 1.2. ξ ∈ h^{r}(γ_{i}), hence h^{r}(ξ) ∗ h^{r}(γ_{i}) = h^{r}(ξ) \ h^{r}(γ_{i}).

If ξ ∈ h^{s}(γ_{i}) then since h^{s}(ξ) = h^{r}(ξ) we have h^{r}(ξ) ∗ h^{r}(γ_{i}) = h^{r}(ξ) \ h^{r}(γ_{i}) ⊂ h^{s}(ξ) \ h^{s}(γ_{i}) ⊂ h^{s}[i^{s}(ξ, γ_{i})] ⊂ h^{r}[i^{r}(ξ, γ_{i})] and we are done.

So we can assume that ξ ∉ h^{s}(γ_{i}) and so h^{r}(γ_{i}) ≠ h^{s}(γ_{i}). By the construction of r, we have γ_{0} ∈ h^{s}(γ_{i}), h^{r}(γ_{i}) = h^{s}(γ_{i}) ∪ C and so ξ ∈ C, i.e ξ ∉ h^{s}[Q ∪ E]. By (i) we can assume that ξ ∉ h^{s}(γ_{i,0}). So s ∈ P_{f} implies

h^{s}(ξ) ∩ h^{s}(γ_{i,0}) = h^{s}(ξ) ∗ h^{s}(γ_{i,0}) ⊂ h^{s}[i^{s}(ξ, γ_{i,0})] (7)

We have

h^{r}(ξ) ∗ h^{r}(γ_{i}) = h^{s}(ξ) \ (h^{s}(γ_{i}) ∪ C) ⊂ h^{s}(ξ) \ C (8)

and applying (i) again

h^{s}(ξ) \ C = h^{s}(ξ) ∩ h^{s}[Q ∪ E] ⊂ h^{s}(ξ) ∩ h^{s}(γ_{i,0}). (9)

By (ii), i^{s}(ξ, γ_{i,0}) ⊂ f(ξ, γ_{i,0}) ∩ a^{s} = f(ξ, γ_{i}) ∩ a^{s}. Since ξ ∉ Q ⊂ h^{s}[Q ∪ E] it follows that f(ξ, γ_{i}) ∩ a^{s} = i^{r}(ξ, γ_{i}) and so (7)–(9) together yield

h^{r}(ξ) ∗ h^{r}(γ_{i}) ⊂ h^{r}[i^{r}(ξ, γ_{i})] (10)

which was to be proved.

Case 2. ξ = γ_{j} for some j < i.

Since i^{s}(γ_{j}, γ_{i}) = i^{r}(γ_{j}, γ_{i}) we have

h^{s}(γ_{j}) ∗ h^{s}(γ_{i}) ⊂ h^{r}[i^{r}(γ_{i}, γ_{j})]. (11)

It is easy to check that

h^{r}(γ_{j}) ∗ h^{r}(γ_{i}) = \{ \begin{matrix}h^{s}(γ_{j}) ∗ h^{s}(γ_{i}) & if γ_{0} ∉ h^{s}(γ_{j}) ∗ h^{s}(γ_{i}), \\ h^{s}(γ_{j}) ∗ h^{s}(γ_{i}) ∪ C & if γ_{0} ∈ h^{s}(γ_{j}) ∗ h^{s}(γ_{i}). \end{matrix} (12)

So we are done if γ_{0} ∉ h^{s}(γ_{j}) ∗ h^{s}(γ_{i}). Assume γ_{0} ∈ h^{s}(γ_{j}) ∗ h^{s}(γ_{i}). Then there is γ_{l} ∈ i^{s}(γ_{j}, γ_{i}) with γ_{0} ∈ h^{s}(γ_{l}). Thus, by the construction of r we have

C ⊂ h^{r}(γ_{l}) ⊂ h^{r}[i^{r}(γ_{j}, γ_{i})] (13)

But (12) and (13) together imply what we wanted.

Thus we proved r ∈ P_{f}.

Clearly r satisfies 2.16.(a)–(c). To check 2.16.(d) write s′ = s⌈(S∪E) and let α ∈ S ∪ E and b ⊂ (S ∪ E) ∩ α. We need to show that u^{s′}(α, b) ⊂ u^{r}(α, b). Since S ∪ E is an initial segment of a^{s}, we have u^{s′}(α, b) = u^{s}(α, b). If α ∈ S, then also u^{s′}(α, b) = u^{r}(α, b), so we can assume α ∈ E.

Let ξ ∈ u^{s′}(α, b) = u^{s}(α, b). Then ξ ∈ h^{s}(α) ⊂ h^{s}[E], and hence ξ ∉ C. But h^{r}[b] \ h^{s}[b] ⊂ C, more precisely, it is empty or just C. Since ξ ∈ u^{s}(α, b), it follows that ξ ∉ h^{s}[b] and so ξ ∉ h^{r}[b] because ξ ∉ C. Thus ξ ∈ u^{r}(α, b). Hence r satisfies (d).

The lemma is proved.

Lemma 2.17. In V^{P_{f}}, if Y ⊂ ω_{2} is countable, then either \overline{Y} is compact or |ω_{2} \ \overline{Y} | ≤ ω_{1}.

Proof. Assume that 1_{P_{f}} ^{∥}—“Ẏ = {ẏ_{n} : n ∈ ω} ⊂ ω_{2}”. For each n ∈ ω fix a maximal antichain C_{n} ⊂ P_{f} such that for each p ∈ C_{n} there is α ∈ a^{p} with p^{∥}—“ẏ_{n} = α̂”. Let A = ⋃{a^{p} : p ∈ ⋃_{n<ω} C_{n}}. Since every

C_{n} is countable by c.c.c we have |A| = ω.

Assume also that 1_{P_{f}} ^{∥}— “\overline{Ẏ} is not compact”, that is, Y can not be covered by finitely many H(δ) in V^{P_{f}}.

Let

I = \{δ < ω_{2} : ∃p ∈ P_{f} p^{∥}—“δ ∉ \overline{Ẏ} ”\}.

Clearly 1_{P_{f}} ^{∥}— ω_{2} \ I ⊂ \overline{Ẏ}. Since ω_{2} \ I is in the ground model, by corollary 2.14 it is enough to show that ω_{2} \ I is infinite. Actually we will prove much more:

Claim . I is not stationary in ω_{2}.

Assume on the contrary that I is stationary. Let us fix, for each δ ∈ I, a condition p_{δ} ∈ P_{f} and a finite set D_{δ} ∈ [δ]^{<ω} such that p_{δ}^{∥}— “Ẏ ∩ U(δ, D_{δ}) = ∅”. For each δ ∈ I let Q_{δ} = a^{p_{δ}} ∩ δ and E_{δ} = a^{p_{δ}} \ δ. We can assume that D_{δ} ⊂ Q_{δ} and sup A < δ for each δ ∈ I.

Let B_{δ} = cl_{f}(A ∪ Q_{δ}, E_{δ}) for δ ∈ I (see 1.3). For each δ ∈ I the set B_{δ} is countable with sup(B_{δ}) = sup(A ∪ Q_{δ}), so we can apply Fodor’s pressing down lemma and CH to get a stationary set J ⊂ I and a countable set B ⊂ ω_{2} such that B_{δ} = B for each δ ∈ J.

By thinning out J and with a further use of CH we can assume that for a fixed k ∈ ω we have

(1) E^{δ} = {γ_{i}^{δ} : i < k} for δ ∈ J, γ_{0}^{δ} < γ_{1}^{δ} < ⋯ < γ_{k−1}^{δ},

(2) f(ξ, γ_{i}^{δ}) = f(ξ, γ_{i}^{δ′}) for each ξ ∈ B, δ, δ′ ∈ J and i < k.

Let δ = min J, D = D_{δ}, E = E_{δ}, p = p_{δ}, Q = Q_{δ}. By lemma 1.2 there are ordinals δ_{j} ∈ J with δ < δ_{0} < δ_{1} < ⋯ < δ_{2k−1} such that

B ∪ E ⊂ ⋂{f(ξ, η) : ξ ∈ E_{δ_{i}}, η ∈ E_{δ_{j}}, i < j < 2k}. (⋆)

For i < k and j < 2 let γ_{i} = γ_{i}^{δ} and γ_{i,j} = γ_{i}^{δ_{2i+j}}. Let F = {γ_{i,j} : i < k, j < 2}.

We know that a^{p} = Q ∪ E. Define the condition q ∈ P_{f} by the following stipulations:

(i) a^{q} = a^{p} ∪ F and q ≤ p, (ii) h^{q}(γ_{i,j}) = {γ_{i,j}} ∪ a^{p} for ⟨i, j⟩ ∈ k × 2 (iii) i^{q}(γ_{i_{0},j_{0}}, γ_{i_{1},j_{1}}) = a^{p} for ⟨i_{0}, j_{0}⟩ ≠ ⟨i_{1}, j_{1}⟩ ∈ k × 2, (iv) i^{q}(ξ, γ_{i,j}) = ∅ for ξ ∈ a^{p} and ⟨i, j⟩ ∈ k × 2. Since a^{p} ⊂ B ∪ E, (⋆) implies that q ∈ P_{f}.

Since 1_{P_{f}} ^{∥}— “H[Q ∪ E] ⊅ Ẏ ”, there is a condition t ≤ q, a natural number n and an ordinal α such that t^{∥}— “α = ẏ_{n}” but α ∈ a^{t}\h^{t}[Q∪E]. Since C_{n} is a maximal antichain we can assume that t ≤ v for some v ∈ C_{n}.

Let s = t⌈(B∪E∪F). Then s ∈ P_{f} because for each pair ξ < η ∈ a^{s} if ξ ∈ B then i^{t}(ξ, η) ⊂ f(ξ, η) ⊂ B and so i^{t}(ξ, η) ⊂ a^{s}, and if ξ, η ∈ E∪F then i^{t}(ξ, η) = i^{q}(ξ, η) ⊂ Q ∪ E ⊂ a^{s}. Moreover s ≤ v because a^{v} ⊂ B. Thus s^{∥}— “α = ẏ_{n}” and α ∉ h^{s}[Q ∪ E]. Let S = a^{s} ∩ B.

Since i^{s}(γ_{i,0}, γ_{i,1}) = i^{q}(γ_{i,0}, γ_{i,1}) = Q ∪ E, and γ_{i,j} ∉ h^{s}(γ_{i,1−j}) we have

h^{s}(γ_{i,0}) ∩ h^{s}(γ_{i,1}) = h^{s}(γ_{i,0}) ∗ h^{s}(γ_{i,1}) ⊂ h^{s}[Q ∪ E]. (14)

Moreover, if ξ ∈ Q ∪ E and j < 2 then ξ ∈ h^{q}(γ_{i,j}) ⊂ h^{s}(γ_{i,j}) and i^{s}(ξ, γ_{i,j}) = ∅, consequently

h^{s}(ξ) \ h^{s}(γ_{i,j}) = h^{s}(ξ) ∗ h^{s}(γ_{i,j}) ⊂ h^{s}[i^{s}(ξ, γ_{i,j})] = ∅. (15)

Putting (14) and (15) together it follows that h^{s}(γ_{i,0})∩h^{s}(γ_{i,1}) = h^{s}[Q∪ E].

Thus we can apply 2.16 to get a condition r such that r ≤ s⌈(Q∪E) = p, r ≤ s⌈S ≤ v and α ∈ S \ h^{s}[Q ∪ E] ⊂ h^{s}(γ_{0}) = h^{s}(δ). Since D ⊂ Q, we have α ∈ h^{s}(δ) \ h^{s}[D].

Thus

r^{∥}—α ∈ Ẏ ∩ (H(δ) \ H[D]).

On the other hand r^{∥}—Ẏ ∩ (H(δ) \ H[D]) = ∅ because r ≤ p. With this contradiction the claim is proved and this completes the proof of the lemma.

Clearly, lemma 2.17 implies that X_{f} is countably compact.

Corollary 2.18. If F ⊂ X is closed (or open), then either |F | ≤ ω_{1} or |X \ F | ≤ ω_{1}.

Proof. If |F | = ω_{2} then F is not compact, so by lemma 1.7 F contains a free sequence Y with non-compact closure. But F is initially ω_{1}- compact and countably tight, so Y is countable. Consequently, we have |ω_{2} \ \overline{Y} | ≤ ω_{1} by lemma 2.17 and so |X \ F | ≤ ω_{1}.

Corollary 2.19. X_{f} is normal and z(X_{f}) = hd(X_{f}) ≤ ω_{1}.

Proof. To show that X_{f} is normal let F_{0} and F_{1} be disjoint closed subsets of X_{f}. Since at least one of them is compact by lemma 2.17 they can be separated by open subsets of X_{f} because X_{f} is T_{3}.

Concerning the hereditarily density of X_{f} it follows easily from corollary 2.18 that X_{f} does not contain a discrete subspace of size ω_{2}. But X_{f} is right separated, so all the left separated subspaces of X_{f} are of size ≤ ω_{1}, that is , z(X) ≤ ω_{1}.

Thus theorem 2.4 is proved.

We know that the space X_{f} is not automatically hereditarily separable, so the following question of Arhangel’skiĭ, [1, problem 5], remains unanswered: Is it true in ZFC that every hereditarily separable, initially ω_{1}-compact space is compact?

As we have seen our space X_{f} is normal. However, we don’t know whether X_{f} is or can be made hereditarily normal, i.e. T_{5}. This raises the following problem.

Problem 1. Is it provable in ZFC that every T_{5}, countably tight, initially ω_{1}-compact space is compact?

3. Making X_{f} Frechet-Urysohn

In [1, problem 12] Arhangel’skiĭasks if it is provable in ZFC that a normal, first countable initially ω_{1}-compact space is necessarily compact. We could not completely answer this question, but in this section we show that the Frechet-Urysohn property (which is sort of half-way between countable tightness and first countability) in not enough to get compactness.

To achieve that we want to find a further extension of the model V^{P_{f}} in which X_{f} becomes Frechet-Urysohn but its other properties are preserved, for example, X_{f} remains initially ω_{1}-compact and normal. Since X_{f} is countably tight and χ(X_{f}) ≤ ω_{1} it is a natural idea to make X_{f} Frechet-Urysohn by constructing a generic extension of V^{P_{f}} in which X_{f} remains countably tight and p > ω_{1}, i.e. MA_{ω_{1}}(σ-centered) holds (see [9, theorem 8]).

The standard c.c.c poset P which forces p > ω_{1} is obtained by a suitable finite support iteration of length 2^{ω_{1}}. During this iteration in the α^{th} step we choose a non-principal filter F ⊂ P(ω) generated by at most ω_{1} elements and we add a new subset A of ω to the α^{th} intermediate model so that A is almost contained in every element of F, i.e. A \ F is finite for each F ∈ F. It is well-known and easy to see that P has property K. Thus, by theorem 3.1 below, X_{f} remains countably tight in V^{P_{f} ∗R} and so indeed X_{f} becomes Frechet-Urysohn in that model. Moreover, theorem 3.2 implies that the ω_{1}-compactness of X_{f} is also preserved. Unfortunately, we could not prove that forcing with P preserves the countable compactness of X_{f}.

So, instead of aiming at p > ω_{1} we will consider only those filters during the iteration which are needed in proving the Frechet-Urysohn property of X_{f}. As we will see, we can handle these filters in such a way that our iterated forcing R preserves not only the countable compactness of X_{f} but also property (iii) from theorem 2.4: in V^{P_{f} ∗R} the closure of any countable subset of X_{f} is either compact or contains a final segment of ω_{2}. Of course, this will insure the preservation of the normality of X_{f} as well.

We start with the two easy theorems, promised above, about the preservation of countable tightness and ω_{1}-compactness of X_{f} under certain c.c.c forcings.

Theorem 3.1. If the topological space X is right separated, compact, countably tight and the poset R has property K then forcing with R preserves the countable tightness of X.

Proof. First we recall that X remains compact (and clearly right separated) in any extension of the ground model by [5, lemma 7]. Since F(X) = t(X) for compact spaces, assume indirectly that 1_{R}^{∥}— “{ż_{ξ} : ξ < ω_{1}} ⊂ X is a free sequence”. For every ξ < ω_{1} we have that 1_{R}^{∥}—“\overline{\{ż_{ζ} : ζ < ξ\}} and \overline{\{ż_{ζ} : ξ ≤ ζ < ω_{1}\}} are disjoint compact sets” and X is T_{3}, so we can fix a condition p_{ξ} ∈ P, open sets U_{ξ} and V_{ξ} from the ground model and a point z_{ξ} ∈ X such that \overline{U_{ξ}} ∩ \overline{V_{ξ}} = ∅ and

p_{ξ}^{∥}—“ {ż_{ζ} : ζ < ξ} ⊂ U_{ξ}, {ż_{ζ} : ξ ≤ ζ < ω_{1}} ⊂ V_{ξ} and ż_{ξ} = z_{ξ}.”

Since R has property K, there is an uncountable set I ⊂ ω_{1} such that the conditions {p_{ξ} : ξ ∈ I} are pairwise compatible.

We claim that the sequence {z_{ξ} : ξ ∈ I} is an uncountable free sequence in the ground model which contradicts F(X) = t(X) = ω. Indeed let ξ ∈ I. If ζ ∈ I ∩ ξ, then p_{ξ} and p_{ζ} has a common extension q in P and we have

q^{∥}—“ż_{ζ} = z_{ζ} and {ż_{η} : η < ξ} ⊂ U̇_{ξ}.”

Hence z_{ζ} ∈ U_{ξ}. Similarly for ζ ∈ I \ ξ we have z_{ζ} ∈ V_{ξ}. Therefore \overline{U_{ξ}} and \overline{V_{ξ}} separate \overline{\{z_{ζ} : ζ ∈ I ∩ ξ\}} and \overline{\{z_{ζ} : ζ ∈ I \ ξ\}} which implies that {z_{ξ} : ξ ∈ I} is really free.

Theorem 3.2. Forcing with a c.c.c poset R over V^{P_{f}} preserves property 2.4.(ii) of the space X_{f}, i.e. for each uncountable A ⊂ X_{f} there is β ∈ ω_{2} such that A ∩ H(β) is uncountable.

Proof. We work in V^{P_{f}}. Assume that r^{∥}—_{R} “Ȧ = {α̇_{ξ} : ξ < ω_{1}} ∈ [X_{f} ]^{ω_{1}}”. For each ξ < ω_{1} pick a condition r_{ξ} ≤ r from R which decides the value of α̇_{ξ}, r_{ξ}^{∥}—_{R} “α̇_{ξ} = α_{ξ}”. Since R satisfies c.c.c, {α_{ξ} : ξ ∈ ω_{1}} is uncountable, hence as X_{f} has property (ii) in V^{P_{f}}, for some β < ω_{2} the set I = H(β) ∩ {α_{ξ} : ξ < ω_{1}} is also uncountable. Since R satisfies c.c.c there is a condition q ≤ r in R such that q^{∥}—_{R} “|{ξ ∈ I : r_{ξ} ∈ G}| = ω_{1}”, where G is the R-generic filter over V^{P_{f}}. Thus q^{∥}—_{R} “|Ȧ ∩ H(β)| = ω_{1}” which was to be proved.

Of course, theorem 3.2 implies that forcing with any c.c.c poset R preserves the ω_{1}-compactness of X_{f}. It is much harder to find a property of a poset R which guarantees that forcing with R over V^{P_{f}} preserves the countable compactness of X_{f}. We will proceed in the following way. In definition 3.3 we formulate when a poset Ṙ is called nice (over P_{f}), and then in theorem 3.4 we show that forcing with a nice poset preserves not only the countable compactness of X_{f}, but also property 2.4(iii): the closure of any countable subset of X_{f} is either compact or contains a final segment of ω_{2}. Finally, in definitions 3.5 and 3.6 we describe a class of finite support iterated forcings, which by theorem 3.7 are nice and have property K , and then in theorem 3.9 we show that forcing with a suitable member of this class makes X_{f} Frechet-Urysohn.

Definition 3.3. Let Ṙ be a name for a poset in V^{P_{f}}. We say that Ṙ is nice (over P_{f}) if there is a dense subset D of the iteration P_{f} ∗ Ṙ with the following property: If ⟨p_{0}, r_{0}⟩ , ⟨p_{1}, r_{1}⟩ ∈ D, p, p′ ∈ P_{f} are such that p ≤ p_{0}, p_{1}, p′ ≤ p_{0}, p_{1}, and

p^{∥}—“r_{0} and r_{1} are compatible in Ṙ”,

moreover we have either p ≤ p′ or p ≺ p′ (see definition 2.15) then we also have

p′^{∥}—“r_{0} and r_{1} are compatible in Ṙ”.

Theorem 3.4. If CH holds in V, f is a Δ-function and Ṙ is a P_{f}- name for a c.c.c poset which is nice over P_{f}, then 2.4.(iii) is preserved by forcing with Ṙ, i.e.

V^{P_{f} ∗Ṙ} ⊧ ∀Y ∈ [X_{f} ]^{ω}( \overline{Y} is compact or |X_{f} \ \overline{Y} | ≤ ω_{1} ).

Proof. Let D ⊂ P_{f} ∗ Ṙ witness that Ṙ is nice.

Assume that 1_{P ∗Ṙ}^{∥}—“Ẏ = {ẏ_{n} : n ∈ ω} ⊂ ω_{2}”. For each n ∈ ω fix a maximal antichain C_{n} ⊂ D such that for each ⟨p, r⟩ ∈ C_{n} there is

α ∈ a^{p} with ⟨p, r⟩^{∥}—“ẏ_{n} = α̂”. Let A = ⋃{a^{p} : ⟨p, r⟩ ∈ ⋃_{n<ω} C_{n}}. Since

every C_{n} is countable by c.c.c we have |A| = ω.

Assume that 1_{P_{f} ∗Ṙ}^{∥}— “\overline{Ẏ} is not compact”, that is, Ẏ can not be covered by finitely many H(δ) in V^{P_{f} ∗Ṙ}.

Let

I = \{γ < ω_{2} : ∃ ⟨p, r⟩ ∈ P_{f} ∗ Ṙ ⟨p, r⟩^{∥}—“γ ∉ \overline{Ẏ} ”\}.

Since ω_{2} \ I is in the ground model and 1_{P_{f} ∗Ṙ}^{∥}— ω_{2} \ I ⊂ \overline{Ẏ}, it is enough to show that ω_{2} \ I is infinite. Indeed, in this case \overline{ω_{2} \ I} contains a final segment of ω_{2} by corollary 2.14. (Note that the closure of a ground model set does not change under any further forcing.) Thus the next claim completes the proof of this theorem.

Claim . I is not stationary.

Assume on the contrary that I is stationary. For each δ ∈ I fix a condition ⟨p_{δ}, r_{δ}⟩ ∈ P_{f} ∗ Ṙ and a finite set D_{δ} ∈ [δ]^{<ω} such that ⟨p_{δ}, r_{δ}⟩^{∥}— “Y ∩ U(δ, D_{δ}) = ∅”. For each δ ∈ I write Q_{δ} = a^{p_{δ}} ∩ δ and E_{δ} = a^{p_{δ}} \ δ. We can assume that D_{δ} ⊂ Q_{δ}, δ ∈ E_{δ} and sup A < δ for each δ ∈ I.

Let B_{δ} = cl_{f}(A ∪ Q_{δ}, E_{δ}) for δ ∈ I (see 1.3). Since B_{δ} is a countable set with sup(B_{δ}) = sup(A ∪ Q_{δ}) and so sup(B_{δ}) < δ, we can apply Fodor’s pressing down lemma and CH to get a stationary set J ⊂ I and a countable set B ⊂ ω_{2} such that B_{δ} = B for each δ ∈ J.

By thinning out J and with a another use of CH we can assume that for some fixed k ∈ ω we have (1) E^{δ} = {γ_{i}^{δ} : i < k} for δ ∈ J, γ_{0}^{δ} < γ_{1}^{δ} < ⋯ < γ_{k−1}^{δ},

(2) f(ξ, γ_{i}^{δ}) = f(ξ, γ_{i}^{δ′}) for each ξ ∈ B, δ, δ′ ∈ J and i < k.

Let δ = min J, D = D_{δ}, E = E_{δ}, p = p_{δ}, r = r_{δ}, Q = Q_{δ}. By lemma 1.2 there are 2k ordinals δ_{j} ∈ J with δ < δ_{0} < δ_{1} < ⋯ < δ_{2k−1} from J such that

B ∪ E ⊂ ⋂{f(ξ, η) : ξ ∈ E_{δ_{i}}, η ∈ E_{δ_{j}}, i < j < 2k}. (⋆)

For i < k and j < 2 let γ_{i} = γ_{i}^{δ} and γ_{i,j} = γ_{i}^{δ_{2i+j}}. Let F = {γ_{i,j} : i < k, j < 2}.

We know that a^{p} = Q ∪ E. Define the condition q ∈ P_{f} by the following stipulations:

(i) a^{q} = a^{p} ∪ F and q ≤ p, (ii) h^{q}(γ_{i,j}) = {γ_{i,j}} ∪ a^{p} for ⟨i, j⟩ ∈ k × 2 (iii) i^{q}(γ_{i_{0},j_{0}}, γ_{i_{1},j_{1}}) = a^{p} for ⟨i_{0}, j_{0}⟩ ≠ ⟨i_{1}, j_{1}⟩ ∈ k × 2, (iv) i^{q}(ξ, γ_{i,j}) = ∅ for ξ ∈ a^{p} and ⟨i, j⟩ ∈ k × 2. Since a^{p} ⊂ B ∪ E, (⋆) implies that q ∈ P_{f} and so ⟨q, r⟩ ∈ P ∗ Ṙ.

Since 1_{P_{f} ∗Ṙ}^{∥}— “H[Q ∪ E] does not cover Y ”, there is a condition ⟨t, u⟩ ≤ ⟨q, r⟩ in P_{f} ∗ Ṙ, a natural number n and an ordinal α such that ⟨t, u⟩^{∥}— “α = ẏ_{n}” but α ∈ a^{t} \ h^{t}[Q ∪ E]. Since C_{n} is a maximal antichain we can assume that ⟨t, u⟩ ≤ ⟨c, d⟩ for some ⟨c, d⟩ ∈ C_{n}.

Let s = t⌈(B ∪ E ∪ F), w = t⌈(S ∪ E) and S = a^{s} ∩ B = a^{w} ∩ B. Then s and w are in P_{f} because for each pair ξ < η ∈ a^{s} if ξ ∈ B then i^{t}(ξ, η) ⊂ f(ξ, η) ⊂ B and so i^{t}(ξ, η) ⊂ S, and if ξ, η ∈ E ∪ F then i^{t}(ξ, η) = i^{q}(ξ, η) ⊂ Q ∪ E ⊂ a^{s} ∩ a^{w}. Moreover s ≤ w ≤ p, c because a^{c} ⊂ a^{t} ∩ B = S and a^{p} = Q ∪ E ⊂ a^{t}∩ ⊂ B ∪ E.

Since ⟨t, u⟩ ≤ ⟨p, r⟩ and ⟨t, u⟩ ≤ ⟨c, d⟩, so

(*) t^{∥}—“ r and d are compatible in Ṙ”.

Since t ≤ w ≤ p, c and Ṙ is nice for P_{f} we have

(**) w^{∥}—“ r and d are compatible in Ṙ”.

We know s ≤ c and so s^{∥}—“α = ẏ_{n}” and α ∈ a^{s} \ h^{s}[Q ∪ E].

Since i^{s}(γ_{i,0}, γ_{i,1}) = i^{q}(γ_{i,0}, γ_{i,1}) = Q ∪ E, and γ_{i,j} ∉ h^{s}(γ_{i,1−j}) we have

h^{s}(γ_{i,0}) ∩ h^{s}(γ_{i,1}) = h^{s}(γ_{i,0}) ∗ h^{s}(γ_{i,1}) ⊂ h^{s}[Q ∪ E]. (16)

If ξ ∈ Q ∪ E, then ξ ∈ h^{q}(γ_{i,j}) ⊂ h^{s}(γ_{i,j}) and i^{s}(ξ, γ_{i,j}) = ∅ and so

h^{s}(ξ) \ h^{s}(γ_{i,j}) = h^{s}(ξ) ∗ h^{s}(γ_{i,j}) ⊂ h^{s}[i^{s}(ξ, γ_{i,j})] = ∅. (17)

Putting (16) and (17) together it follows that h^{s}(γ_{i,0})∩h^{s}(γ_{i,1}) = h^{s}[Q∪ E]. Thus we can apply 2.16 to get a condition p^{∗} ∈ P_{f} such that p^{∗} ≤ s⌈(Q∪E) = p, p^{∗} ≤ s⌈S ≤ c, w ≺ p^{∗} and α ∈ S \h^{s}[Q∪E] ⊂ h^{p^{∗}}(γ_{0}) = h^{p^{∗}}(δ). But D ⊂ Q, so α ∈ h^{p^{∗}}(δ) \ h^{p^{∗}}[D]. Hence p^{∗}^{∥}—“α ∈ U(δ, D)”.

Since p^{∗} ≤ p, c and w ≺ p^{∗} and Ṙ is nice, it follows from (**) that

(***) p^{∗}^{∥}—“ r and d are compatible in Ṙ”.

Thus ⟨p, r⟩ and ⟨c, d⟩ have a common extension ⟨p^{∗}, r^{∗}⟩ in P ∗ Ṙ.

But then

⟨p^{∗}, r^{∗}⟩^{∥}—ẏ_{n} = α ∈ Ẏ ∩ U(δ, D)

because ⟨p^{∗}, r^{∗}⟩ ≤ ⟨c, d⟩. On the other hand ⟨p^{∗}, r^{∗}⟩^{∥}— “Ẏ ∩ U(δ, D) = ∅” because ⟨p^{∗}, r^{∗}⟩ ≤ ⟨p, r⟩. Contradiction, the claim is proved, which completes the proof of the theorem.

Definition 3.5. Assume that A ∈ [X_{f} ]^{ω} and α ∈ \overline{A}. We define the

poset Q(A, α) as follows. Its underlying set is [A]^{<ω} × [α]^{<ω} . If ⟨s, C⟩ and ⟨s′C′⟩ are conditions, let ⟨s, C⟩ ≤ ⟨s′C′⟩ if and only if s ⊃ s′, C ⊃ C′ and s \ s′ ⊂ U(α, C′). For q ∈ Q(A, α) write q = ⟨s^{q}, C^{q}⟩ and supp(q) = s^{q} ∪ C^{q}.

It is well-known and easy to see that if G is a Q(A, α)-generic filter, then S = ⋃{s^{q} : q ∈ G} is a sequence from A which converges to α, i.e. every open neighbourhood of α contains all but finitely many points of S.

Clearly Q(A, α) is σ-centered and well-met. In fact, if p_{0} = ⟨s_{0}, C_{0}⟩ and p_{1} = ⟨s_{1}, C_{1}⟩ are compatible, then p_{0} ∧ p_{1} = ⟨s_{0} ∪ s_{1}, C_{0} ∪ C_{1}⟩.

Definition 3.6. A finite support iterated forcing ⟨R_{ξ} : ξ ≤ κ⟩ over V^{P_{f}} is called an FU-iteration if for each ξ < κ we have

1_{P_{f} ∗R_{ξ}}^{∥}—R_{ξ+1} = R_{ξ} ∗ Q(Ȧ, α̇) for some Ȧ ∈ [X_{f} ]^{ω} and α̇ ∈ \overline{Ȧ}.

Since every Q(A, α) is σ-centered, it is clear that any FU-iteration is c.c.c, in fact it even has property K. The really important, and much less trivial, property of them is given in our next result.

Theorem 3.7. Any FU-iteration is nice over P_{f}.

Proof. Assume that ⟨R_{ξ} : ξ ≤ κ⟩ is an FU-iteration, R_{ξ+1} = R_{ξ}∗Q(Ȧ_{ξ}, α̇_{ξ}). Write Q^{∗} = [ω_{2}]^{<ω} × [ω_{2}]^{<ω}. Clearly 1_{P_{f} ∗R_{ξ}}^{∥}— “Q(Ȧ_{ξ}, α̇_{ξ}) ⊂ Q^{∗}”.

We consider the elements of P_{f} ∗ Ṙ_{κ} as pairs ⟨p, ṙ⟩, where p ∈ P_{f} and p^{∥}— “ṙ is a finite function, dom(ṙ) ⊂ κ and ṙ⌈ξ^{∥}—Ṙ_{ξ} ṙ(ξ) ∈ Q(Ȧ_{ξ}, α̇_{ξ}) for each ξ ∈ dom(ṙ)”.

Definition 3.8. A condition ⟨p, r⟩ ∈ P_{f} ∗ R_{κ} is called determined if r ∈ Fn(κ, Q^{∗}, ω), (i.e r is a finite function, not only a P_{f}-name of a finite function) moreover for each ξ ∈ dom(r) we have supp(r(ξ)) ⊂ a^{p} and there is an ordinal α ∈ a^{p} such that ⟨p, r⌈ξ⟩^{∥}— “α̇_{ξ} = α̂”.

Clearly the family D of the determined conditions is dense in P_{f} ∗R_{κ}. We claim that D witnesses that R_{κ} is nice. Indeed let ⟨p_{0}, r_{0}⟩ , ⟨p_{1}, r_{1}⟩ ∈ D, p, p′ ∈ P_{f} be such that p ≤ p_{0}, p_{1}, p′ ≤ p_{0}, p_{1},

(+) p^{∥}—“r_{0} and r_{1} are compatible in Ṙ”

and either p ≤ p′ or p ≺ p′.

Write r_{i}(ξ) = ⟨s_{i}(ξ), C_{i}(ξ)⟩ for ξ ∈ dom(r_{i}). Let D = dom(r_{0}) ∪ dom(r_{1}). For i < 2 and ξ ∈ D let

s_{i}^{∗}(ξ) = \{ \begin{matrix}s_{i}(ξ) & if ξ ∈ dom(r_{i}), \\ ∅ & otherwise, \end{matrix}

and

C_{i}^{∗}(ξ) = \{ \begin{matrix}C_{i}(ξ) & if ξ ∈ dom(r_{i}), \\ ∅ & otherwise. \end{matrix}

Consider the condition r^{∗} ∈ Fn(κ, Q^{∗}, ω) defined by the stipulations dom(r^{∗}) = D and r^{∗}(ξ) = ⟨s_{0}^{∗}(ξ) ∪ s_{1}^{∗}(ξ), C_{0}^{∗}(ξ) ∪ C_{1}^{∗}(ξ)⟩ for ξ ∈ D. We show that p′^{∥}— “r^{∗} is a common extension of r_{0} and r_{1} in R_{κ}.” We prove a bit more: we show, by a finite induction, that for each ξ ∈ D

p′^{∥}—“r^{∗}⌈ξ + 1 ∈ R_{ξ+1} and r^{∗}⌈ξ + 1 ≤ r_{0}⌈ξ + 1, r_{1}⌈ξ + 1.”

The non-trivial step is when ξ ∈ dom(r_{0}) ∩ dom(r_{1}).

Since ⟨p′, r^{∗}⌈ξ⟩ ≤ ⟨p′, r_{i}⌈ξ⟩ by the induction hypothesis, it follows that ⟨p′, r^{∗}⌈ξ⟩^{∥}— “s_{i}(ξ) ∈ [Ȧ_{ξ}]^{<ω}” and so ⟨p′, r^{∗}⌈ξ⟩^{∥}— “r^{∗}(ξ) ∈ Q(Ȧ_{ξ}, α̇_{ξ})”, that is, p′^{∥}— “r^{∗}⌈ξ + 1 ∈ R_{ξ+1}”.

By (+) for some P_{f}-name ṙ and P_{f}∗R_{ξ}-name q̇ we have ⟨p, ṙ⟩ ∈ P_{f}∗R_{ξ} and ⟨p, ṙ⟩^{∥}— “q̇ ∈ Q(Ȧ_{ξ}, α̇_{ξ}) is a common extension of r_{0}(ξ) and r_{1}(ξ)”. Hence ⟨p, ṙ⟩^{∥}— “sq̇ \ s_{0}(ξ) ⊂ U(α, C_{0}(ξ)) and sq̇ ⊃ s_{1}(ξ)”. Therefore ⟨p, ṙ⟩^{∥}— “s_{1}(ξ) \ s_{0}(ξ) ⊂ U(α, C_{0}(ξ))”. By supp(r_{i}(ξ)) ⊂ a^{p}, this can only happen if

s_{1}(ξ) \ s_{0}(ξ) ⊂ u^{p}(α, C_{0}(ξ)). (†)

But both p ≤ p′ and p ≺ p′ together with (†) imply

s_{1}(ξ) \ s_{0}(ξ) ⊂ u^{p′}(α, C_{0}(ξ)). (‡)

Thus p′^{∥}— “s_{1}(ξ) \ s_{0}(ξ) ⊂ U(α, C_{0}(ξ))”, i.e. ⟨p′, r^{∗}⌈ξ⟩ “^{∥}—r^{∗}(ξ) ≤ r_{0}(ξ)”. Similarly, it can be proved that ⟨p′, r^{∗}⌈ξ⟩^{∥}— “r^{∗}(ξ) ≤ r_{1}(ξ)”. Thus we have carried out the induction step and the theorem is proved.

Now we are ready to prove the main result of this section.

Theorem 3.9. Assume that CH holds in the ground model V, there is a Δ-function f and λ is a cardinal such that ω_{1} < λ = λ^{ω}. Then in V^{P_{f}} there is an FU-iteration Ṙ_{λ} of length λ such that in V^{P_{f} ∗Ṙ_{λ}} the space X_{f} is Frechet-Urysohn and satisfies 2.4(i)–(iii), moreover V^{P_{f} ∗Ṙ_{λ}} ⊧“2^{κ} = (λ^{κ})^{V} for each cardinal κ ≥ ω”.

Proof. Since |P_{f}| = ω_{2} and P_{f} satisfies c.c.c we have (λ^{ω})^{V^{P_{f}}} ≤ ((|P_{f}|λ)^{ω})^{V} = (λ^{ω})^{V} = λ. Therefore, using a suitable book-keeping procedure in V^{P_{f}} (see [6, Ch VIII. 6.3] for this technique) we can construct an FUiteration ⟨R_{ξ} : ξ ≤ λ⟩, R_{ξ+1} = R_{ξ}∗Q(Ȧ_{ξ}, α̇_{ξ}), having the following property: for every pair ⟨A, α⟩ if A is a countable subset of X_{f} in V^{P_{f} ∗R_{λ}} and α ∈ \overline{A} then for some ξ < λ we have V^{P_{f} ∗R_{ξ}} ⊧ “Q(A_{ξ}, α_{ξ}) = Q(A, α)”. Thus (*) V^{P_{f} ∗R_{λ}} ⊧ “if α ∈ ω_{2} is in the closure of a countable set A ⊂ X_{f}

then there is a sequence {s_{n} : n ∈ ω} ⊂ A which converges to α”. Since R_{λ} has property K in V^{P_{f}}, by theorem 3.1 the space X_{f} remains countably tight in V^{P_{f} ∗R_{λ}}. Putting together this observation with (*) it follows that V^{P_{f} ∗R_{λ}} ⊧ “X_{f} is Frechet-Urysohn”.

An FU-iteration is nice by theorem 3.7, and so X_{f} has property 2.4(iii) in V^{P_{f} ∗R_{λ}} by theorem 3.4. Since X_{f} is right separated it remains locally compact in any extension of V^{P_{f}}. Since R_{λ} satisfies c.c.c the space X_{f} has property 2.4(ii) in V^{P_{f} ∗R_{λ}} by theorem 3.2. All this implies that X_{f} remains initially ω_{1}-compact and normal.

Finally we investigate the cardinal exponents in V^{P_{f} ∗Ṙ_{λ}}. Since |P_{f}| = ω_{2} and 1_{P_{f}} ^{∥}—“|R_{λ}| = λ”, the iteration P_{f} ∗ R_{λ} contains a dense subset D of cardinality ≤ λ. Since P_{f} ∗ R_{λ} satisfies c.c.c it follows that for each κ ≥ ω we have (2^{κ})^{V^{P_{f} ∗R_{λ}}} ≤ ((|D|^{ω})^{κ})^{V} = (λ^{κ})^{V}.

On the other hand every successor step of an FU-iteration introduces a new subset of a countable set, and so (2^{ω})^{V^{P_{f} ∗R_{λ}}} ≥ λ. Consequently, V^{P_{f} ∗R_{λ}} ⊧“2^{κ} = (2^{ω)^{κ}} ≥ λ^{κ} ≥ (λ^{κ})^{V} ”, which proves what we wanted.

Theorem 3.9 answers a question raised by Arhangel’skiĭ, [1, problem 3], in the negative: CH can not be weakened to 2^{ω} < 2^{ω_{1}} in the theorem of van Douwen and Dow. In fact we proved much more: the existence of a Frechet-Urysohn, initially ω_{1}-compact and non-compact space is consistent with practically any cardinal arithmetic that violates CH. More precisely, if we have a ZFC model V in which CH holds and λ = λ^{ω} ≤ 2^{ω_{1}}, then we can find a cardinal preserving generic extension W of V which contains a Frechet-Urysohn and normal counterexample to the van Douwen–Dow question, (2^{ω})^{W} = λ, moreover (2^{κ})^{W} = (2^{κ})^{V} for each κ ≥ ω_{1}. We can obtain W as follows. First we force the σcomplete poset P of Shelah (see [3]) which introduces a Δ-function f in V^{P}. Since P is σ-complete and |P | = ω_{2}, forcing with P does not change 2^{κ} for any κ ≥ ω. Now forcing with P_{f} over V^{P} introduces the counterexample X_{f} to the van Douwen–Dow question. Since |P_{f}| = ω_{2}, the cardinal exponents are the same in V^{P} and in V^{P_{f}} for uncountable cardinals and (2^{ω})^{V^{P ∗P_{f}}} = ω_{2}. Finally we can apply theorem 3.9 to get the desired final model W = V^{P ∗P_{f} ∗R_{λ}}. Let us remark that we have (2^{κ})^{W} = (2^{κ})^{V} for κ ≥ ω_{1} because (2^{κ})^{V} = (λ^{κ})^{V} by λ ≤ 2^{ω_{1}}.

Let us remark that for any cardinal κ the poset Fn(κ, 2, ω), i.e. the forcing notion that adds κ many Cohen reals, is clearly nice over P_{f}, as is witnessed by the dense set of the determined conditions. Thus, by theorems 3.1, 3.2 and 3.4 , adding Cohen reals will preserve properties 2.4(i)–(iii) of X_{f}, especially X_{f} remains countably tight and initially ω_{1}- compact. It is worthwhile to mention that, in contrast with this, Alan Dow proved in [4] that if CH holds in the ground model V then adding Cohen reals can not introduce a countably tight, initially ω_{1}-compact and non-compact T_{3} space.

Let us finish by formulating the following higher cardinal version of the van Douwen–Dow problem:

Problem 2. Is it provable in ZFC that an initially ω_{2}-compact T_{3} space of countable tightness is compact ?

The main problem is trying to use out approach that worked for ω_{2} (instead of ω_{3}) here is that no Δ-function may exist for ω_{3}! This problem has come up already in the efforts trying to lift the result of [3] from ω_{2} to ω_{3}.

References

[1] A. V. Arhangel’skiĭ, On countably compact and initially ω_{1}-compact topological spaces and groups, Math. Japonica, 40, No 1(1994), 39–53.

[2] Z. Balog, A. Dow, D. Fremlin, P. Nyikos, Countable tightness and proper forcing, Bull. Amer. Math. Soc. 19 (1988) 295–298.

[3] J. E. Baumgartner, S. Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic, 33 (l987), no. 2, 119-129.

[4] A. Dow , On initially κ-compact spaces, in: Rings of continuous functions, ed. C.E. Aull, Lecture Notes in pure and applied Mathematics, 1985., v.95, Marcel Dekker, Inc, New York and Basel, pp. 103–108.

[5] I. Juhász, W. Weiss, Omitting the cardinality of the continuum in scattered spaces, Topology and its Applications, 31 (1989) 19–27.

[6] K. Kunen, Set Theory, North Holland, Amsterdam - New York, 1980.

[7] M Rabus, A ω_{2}-minimal Boolean algebra, Logic E-print.

[8] J. Roitman, Height and width of superatomic Boolean Algebras, Proc. Amer. Math. Soc., 94 (1985), no 1, 9–14.

[9] J. R. Shoenfield, Martin’s Axiom, Am. Math. Monthly, 82(6) (1975), 610–617.

Mathematical Institute of the Hungarian Academy of Sciences, P.O.B 127, H-1364 Budapest, Hungary

E-mail address: juhasz@math-inst.hu

Mathematical Institute of the Hungarian Academy of Sciences, P.O.B 127, H-1364 Budapest, Hungary

E-mail address: soukup@math-inst.hu