POINT-COFINITE COVERS IN THE LAVER MODEL

ARNOLD W. MILLER AND BOAZ TSABAN

Abstract. Let S_{1}(Γ, Γ) be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. 𝔟 is the minimal cardinality of a set of reals not satisfying S_{1}(Γ, Γ). We prove the following assertions:

(1) If there is an unbounded tower, then there are sets of reals of cardinality 𝔟, satisfying S_{1}(Γ, Γ).

(2) It is consistent that all sets of reals satisfying S_{1}(Γ, Γ) have cardinality smaller than 𝔟.

These results can also be formulated as dealing with Arhangel’skiĭ’s property α_{2} for spaces of continuous real-valued functions.

The main technical result is that in Laver’s model, each set of reals of cardinality 𝔟 has an unbounded Borel image in the Baire space ω^{ω}.

1. Background

Let P be a nontrivial property of sets of reals. The critical cardinality of P, denoted non(P), is the minimal cardinality of a set of reals not satisfying P. A natural question is whether there is a set of reals of cardinality at least non(P), which satisfies P, i.e., a nontrivial example.

We consider the following property. Let X be a set of reals. U is a point-cofinite cover of X if U is infinite, and for each x ∈ X, {U ∈ U : x ∈ U} is a cofinite subset of U.^{1} Having X fixed in the background, let Γ be the family of all point-cofinite open covers of X. The following properties were introduced by Hurewicz [8], Tsaban [19], and Scheepers [15], respectively.

U_{fin}(Γ, Γ): For all U_{0}, U_{1}, ⋯ ∈ Γ, none containing a finite subcover, there are finite F_{0} ⊆ U_{0}, F_{1} ⊆ U_{1}, … such that {⋃ F_{n} : n ∈ ω} ∈ Γ.

U_{2}(Γ, Γ): For all U_{0}, U_{1}, ⋯ ∈ Γ, there are F_{0} ⊆ U_{0}, F_{1} ⊆ U_{1}, … such that |F_{n}| = 2 for all n, and {⋃ F_{n} : n ∈ ω} ∈ Γ.

1Historically, point-cofinite covers were named γ-covers, since they are related to a property numbered γ in a list from α to ϵ in the seminal paper [7] of Gerlits and Nagy.

S_{1}(Γ, Γ): For all U_{0}, U_{1}, ⋯ ∈ Γ, there are U_{0} ∈ U_{0}, U_{1} ∈ U_{1}, …

such that {U_{n} : n ∈ ω} ∈ Γ.

Clearly, S_{1}(Γ, Γ) implies U_{2}(Γ, Γ), which in turn implies U_{fin}(Γ, Γ). None of these implications is reversible in ZFC [19]. The critical cardinality of all three properties is 𝔟 [9].^{2}

Bartoszyński and Shelah [1] proved that there are, provably in ZFC, totally imperfect sets of reals of cardinality 𝔟 satisfying the Hurewicz property U_{fin}(Γ, Γ). Tsaban proved the same assertion for U_{2}(Γ, Γ) [19]. These sets satisfy U_{fin}(Γ, Γ) in all finite powers [2].

We show that in order to obtain similar results for S_{1}(Γ, Γ), hypotheses beyond ZFC are necessary.

2. Constructions

We show that certain weak (but not provable in ZFC) hypotheses suffice to have nontrivial S_{1}(Γ, Γ) sets, even ones which possess this property in all finite powers.

Definition 2.1. A tower of cardinality κ is a set T ⊆ [ω]^{ω} which can be enumerated bijectively as {x_{α} : α < κ}, such that for all α < β < κ, x_{β} ⊆^{∗} x_{α}.

A set T ⊆ [ω]^{ω} is unbounded if the set of its enumeration functions are unbounded, i.e., for any g ∈ ω^{ω} there is an x ∈ T such that for infinitely many n, g(n) is less than the n-th element of x.

Scheepers [16] proved that if 𝔱 = 𝔟, then there is a set of reals of cardinality 𝔟, satisfying S_{1}(Γ, Γ). If 𝔱 = 𝔟, then there is an unbounded tower of cardinality 𝔟, but the latter assumption is weaker.

Lemma 2.2 (folklore). If 𝔟 < 𝔡, then there is an unbounded tower of cardinality 𝔟.

Proof. Let B = {b_{α} : α < 𝔟} ⊆ ω^{ω} be a 𝔟-scale, that is, each b_{α} is increasing, b_{α} ≤^{∗} b_{β} for all α < β < 𝔟, and B is unbounded.

As |B| < 𝔡, B is not dominating. Let g ∈ ω^{ω} exemplify that. For each α < 𝔟, let x_{α} = {n : b_{α}(n) ≤ g(n)}. Then T = {x_{α} : α < 𝔟} is an unbounded tower: Clearly, x_{β} ⊆^{∗} x_{α} for α < β. Assume that T is bounded, and let f ∈ ω^{ω} exemplify that. For each α, writing x_{α}(n) for the n-th element of x_{α}:

b_{α}(n) ≤ b_{α}(x_{α}(n)) ≤ g(x_{α}(n)) ≤ g(f(n))

for all but finitely many n. Thus, g ∘ f shows that B is bounded. A contradiction. □

2Blass’s survey [4] is a good reference for the definitions and details about the special cardinals mentioned in this paper.

Theorem 2.3. If there is an unbounded tower (of any cardinality), then there is a set of reals X of cardinality 𝔟, which satisfies S_{1}(Γ, Γ).

Theorem 2.3 follows from the following two propositions.

Proposition 2.4. If there is an unbounded tower, then there is one of cardinality 𝔟.

Proof. By Lemma 2.2, it remains to consider the case 𝔟 = 𝔡. Let T be an unbounded tower of cardinality κ. Let {f_{α} : α < 𝔟} ⊆ ω^{ω} be dominating. For each α < 𝔟, pick x_{α} ∈ T which is not bounded by f_{α}. {x_{α} : α < 𝔟} is unbounded, being unbounded in a dominating family. □

Scheepers’s mentioned proof actually establishes the following result, to which we give an alternative proof.

Proposition 2.5 (essentially, Scheepers [16]). For each unbounded tower T of cardinality 𝔟, T ∪ [ω]^{<ω} satisfies S_{1}(Γ, Γ).

Proof. Let T = {x_{α} : α < 𝔟} be an unbounded tower of cardinality 𝔟. For each α, let X_{α} = {x_{β} : β < α} ∪ [ω]^{<ω}. Let U_{0}, U_{1}, … be point-cofinite covers of X_{𝔟} = T ∪ [ω]^{<ω}. We may assume that each U_{n} is countable and that U_{i} ∩ U_{j} = ∅ whenever i ≠ j.

By the proof of Lemma 1.2 of [6], for each k there are distinct U_{0}^{k}, U_{1}^{k}, ⋯ ∈ U_{k}, and an increasing sequence m_{0}^{k} < m_{1}^{k} < …, such for each n and k,

{x ⊆ ω : x ∩ (m_{n}^{k}, m_{n+1}^{k}) = ∅} ⊆ U_{n}^{k}.

As T is unbounded, there is α < 𝔟 such that for each k, I_{k} = {n : x_{α} ∩ (m_{n}^{k}, m_{n+1}^{k}) = ∅} is infinite.

For each k, {U_{n}^{k} : n ∈ ω} is an infinite subset of U_{k}, and thus a point-cofinite cover of X_{α}. As |X_{α}| < 𝔟, there is f ∈ ω^{ω} such that

∀x ∈ X_{α} ∃k_{0} ∀k ≥ k_{0} ∀n > f(k) x ∈ U_{n}^{k}.

For each k, pick n_{k} ∈ I_{k} such that n_{k} > f(k) and U_{n_{k}}^{k} ≠ U_{n_{l}}^{l} for l < k.

We claim that {U_{n_{k}}^{k} : k ∈ ω} is a point-cofinite cover of X_{𝔟}: If x ∈ X_{α}, then x ∈ U_{n_{k}}^{k} for all but finitely many k, because n_{k} > f(k) for all k. If x = x_{β}, β ≥ α, then x ⊆^{∗} x_{α}. For each large enough k, m_{n_{k}}^{k} is large enough, so that x∩(m_{n_{k}}^{k}, m_{n_{k}+1}^{k}) ⊆ x_{α} ∩(m_{n_{k}}^{k}, m_{n_{k}+1}^{k}) = ∅, and thus x ∈ U_{n_{k}}^{k}. □

Remark 2.6. Zdomskyy points out that for the proof to go through, it suffices that {x_{α} : α < 𝔟} is such that there is an unbounded {y_{α} : α < 𝔟} ⊆ [ω]^{ω} such that for each α, x_{α} is a pseudointersection of {y_{β} : β < α}. We do not know whether the assertion mentioned here is weaker than the existence of an unbounded tower.

We now turn to nontrivial examples of sets satisfying S_{1}(Γ, Γ) in all finite powers. In general, S_{1}(Γ, Γ) is not preserved by taking finite powers [9], and we use a slightly stronger hypothesis in our construction.

Definition 2.7. Let 𝔟_{0} be the additivity number of S_{1}(Γ, Γ), that is, the minimum cardinality of a family F of sets of reals, each satisfying S_{1}(Γ, Γ), such that the union of all members of F does not satisfy S_{1}(Γ, Γ).

𝔱 ≤ 𝔥, and Scheepers proved that 𝔥 ≤ 𝔟_{0} ≤ 𝔟 [17]. It is open whether 𝔟_{0} = 𝔟 . If 𝔱 = 𝔟 or 𝔥 = 𝔟 < 𝔡, then there is an unbounded tower of cardinality 𝔟_{0}.

Theorem 2.8. For each unbounded tower T of cardinality 𝔟_{0}, all finite powers of T ∪ [ω]^{<ω} satisfy S_{1}(Γ, Γ).

Proof. We say that U is an ω-cover of X if no member of U contains X as a subset, but each finite subset of X is contained in some member of U. We need a multidimensional version of Lemma 1.2 of [6].

Lemma 2.9. Assume that [ω]^{<ω} ⊆ X ⊆ P (ω), and let e ∈ ω. For each open ω-cover U of X^{e}, there are m_{0} < m_{1} < … and distinct U_{0}, U_{1}, ⋯ ∈ U, such that for all x_{0}, …, x_{e−1} ⊆ ω, (x_{0}, …, x_{e−1}) ∈ U_{n} whenever x_{i} ∩ (m_{n}, m_{n+1}) = ∅ for all i < e.

Proof. As U is an open ω-cover of X^{e}, there is an open ω-cover V of X such that {V^{e} : V ∈ V} refines U [9].

Let m_{0} = 0. For each n ≥ 0: Assume that V_{0}, …, V_{n−1} ∈ V are given, and U_{0}, …, U_{n−1} ∈ U are such that V_{i}^{e} ⊆ U_{i} for all i < n. Fix a finite F ⊆ X such that F^{e} is not contained in any of the sets U_{0}, …, U_{n−1}. As V is an ω-cover of X, there is V_{n} ∈ V such that F ∪ P ({0, …, m_{n}}) ⊆ V_{n}. Take U_{n} ∈ U such that V_{n}^{e} ⊆ U_{n}. Then U_{n} ∉ {U_{0}, …, U_{n−1}}. As V_{n} is open, for each s ⊆ {0, …, m_{n}} there is k_{s} such that for each x ∈ P (ω) with x ∩ {0, …, k_{s} − 1} = s, x ∈ V_{n}. Let m_{n+1} = max{k_{s} : s ⊆ {0, …, m_{n}}}.

If x_{i} ∩ (m_{n}, m_{n+1}) = ∅ for all i < e, then (x_{0}, …, x_{e−1}) ∈ V_{n}^{e} ⊆ U_{n}. □

The assumption in the theorem implies that 𝔟_{0} = 𝔟. The proof is by induction on the power e of T ∪ [ω]^{<ω}. The case e = 1 follows from Theorem 2.5.

Let U_{0}, U_{1}, ⋯ ∈ Γ((T ∪ [ω]^{<ω})^{e}). We may assume that these covers are countable. As in the proof of Theorem 2.5 (this time using Lemma 2.9), there are for each k m_{0}^{k} < m_{1}^{k} < … and distinct U_{0}^{k}, U_{1}^{k}, ⋯ ∈ U_{k} (so that {U_{n}^{k} : n ∈ ω} ∈ Γ((T ∪[ω]^{<ω})^{e})), such that all y_{0}, …, y_{e−1} ⊆ ω, (y_{0}, …, y_{e−1}) ∈ U_{n}^{k} whenever y_{i} ∩ (m_{n}^{k}, m_{n+1}^{k}) = ∅ for all i < e.

Let α_{0} be such that X_{α_{0}} is not contained in any member of ⋃_{n} U_{n}. As T is unbounded, there is α such that α_{0} ≤ α < 𝔟, and for each k, I_{k} = {n : x_{α} ∩ (m_{n}^{k}, m_{n+1}^{k}) = ∅} is infinite.

Let Y = {x_{β} : β ≥ α}. (T ∪ [ω]^{<ω})^{e} \ Y^{e} is a union of fewer than 𝔟_{0} homeomorphic copies of (T ∪ [ω]^{<ω})^{e−1}. By the induction hypothesis, (T ∪[ω]^{<ω})^{e−1} satisfies S_{1}(Γ, Γ), and therefore so does (T ∪[ω]^{<ω})^{e} \Y^{e}. For each k, {U_{n}^{k} : n ∈ I_{k}} is a point-cofinite cover of (T ∪ [ω]^{<ω})^{e} \ Y^{e}, and thus there are infinite J_{0} ⊆ I_{0}, J_{1} ⊆ I_{1}, …, such that {⋂_{n∈J_{k}} U_{n}^{k} : k ∈ ω} is a point-cofinite cover of (T ∪ [ω]^{<ω})^{e} \ Y^{e}.^{3} For each k, pick n_{k} ∈ J_{k} such that: m_{n_{k}}^{k} > m_{n_{k−1}+1}^{k−1}, x_{α} ∩ (m_{n_{k}}^{k}, m_{n_{k}+1}^{k}) = ∅, and U_{n_{k}}^{k} ∉ {U_{n_{0}}^{0}, …, U_{n_{k−1}}^{k−1}}.

{U_{n_{k}}^{k} : k ∈ ω} ∈ Γ(T ∪ [ω]^{<ω}): If x ∈ (T ∪ [ω]^{<ω})^{e} \ Y^{e}, then x ∈ U_{n_{k}}^{k} for all but finitely many k. If x = (x_{β_{0}}, …, x_{β_{e−1}}) ∈ Y, then β_{0}, …, β_{e−1} ≥ α, and thus x_{β_{0}}, …, x_{β_{e−1}} ⊆^{∗} x_{α}. For each large enough k, m_{n_{k}}^{k} is large enough, so that x_{β_{i}}∩(m_{n_{k}}^{k}, m_{n_{k}+1}^{k}) ⊆ x_{α}∩(m_{n_{k}}^{k}, m_{n_{k}+1}^{k}) = ∅ for all i < e, and thus x ∈ U_{n_{k}}^{k}. □

There is an additional way to obtain nontrivial S_{1}(Γ, Γ) sets: The hypothesis 𝔟 = cov(N) = cof(N) provides 𝔟-Sierpiński sets, and 𝔟-Sierpiński sets satisfy S_{1}(Γ, Γ), even for Borel point-cofinite covers. Details are available in [18].

We record the following consequence of Theorem 2.3 for later use.

Corollary 2.10. For each unbounded tower T of cardinality 𝔟, T ∪ [ω]^{<ω} satisfies S_{1}(Γ, Γ) for open covers, but not for Borel covers.

Proof. The latter property is hereditary for subsets [18]. By a theorem of Hurewicz, a set of reals satisfies U_{fin}(Γ, Γ) if, and only if, each continuous image of X in ω^{ω} is bounded. It follows that the set T ⊆ T ∪[ω]^{<ω} does not even satisfy U_{fin}(Γ, Γ). □

3. A consistency result

By the results of the previous section, we have the following.

Lemma 3.1. Assume that every set of reals with property S_{1}(Γ, Γ) has cardinality < 𝔟, and 𝔠 = ℵ_{2}. Then ℵ_{1} = 𝔱 = cov(N) < 𝔟 = ℵ_{2}.

3Choosing infinitely many elements from each cover, instead of one, can be done by adding to the given sequence of covers all cofinite subsets of the given covers.

Proof. As there is no unbounded tower, we have that 𝔱 < 𝔟 = 𝔡. As 𝔠 = ℵ_{2}, ℵ_{1} = 𝔱 < 𝔟 = ℵ_{2}. Since there are no 𝔟-Sirepiński sets and 𝔟 = cof(N) = 𝔠, cov(N) < 𝔟. □

In Laver’s model [11], ℵ_{1} = 𝔱 = cov(N) < 𝔟 = ℵ_{2}. We will show that indeed, S_{1}(Γ, Γ) is trivial there. Laver’s model was constructed to realize Borel’s Conjecture, asserting that “strong measure zero” is trivial. In some sense, S_{1}(Γ, Γ) is a dual of strong measure zero. For example, the canonical examples of S_{1}(Γ, Γ) sets are Sierpiński sets, a measure theoretic object, whereas the canonical examples of strong measure zero sets are Luzin sets, a Baire category theoretic object. More about that can be seen in [18].

The main technical result of this paper is the following.

Theorem 3.2. In the Laver model, if X ⊆ 2^{ω} has cardinality 𝔟, then there is a Borel map f : 2^{ω} → ω^{ω} such that f[X] is unbounded.

Proof. The notation in this proof is as in Laver [11]. We will use the following slightly simplified version of Lemma 14 of [11].

Lemma 3.3 (Laver). Let ℙ_{ω_{2}} be the countable support iteration of Laver forcing, p ∈ ℙ_{ω_{2}}, and å be a ℙ_{ω_{2}}-name such that

p ⊩å ∈ 2^{ω}.

Then there are a condition q stronger than p, and finite U_{s} ⊆ 2^{ω} for each s ∈ q(0) extending the root of q(0), such that for all such s and all n:

q(0)_{t} ˆ q ↾ [1, ω_{2}) ⊩ “ ∃u ∈ Ǔ_{s} u = å ↾ n ”

for all but finitely many immediate successors t of s in q(0).

Assume that X ⊆ 2^{ω} has no unbounded Borel image in M[G_{ω_{2}}], Laver’s model. For every code u ∈ 2^{ω} for a Borel function f : 2^{ω} → ω^{ω} there exists g ∈ ω^{ω} such that for every x ∈ X we have that f(x) ≤^{∗} g.

By a standard Löwenheim-Skolem argument, see Theorem 4.5 on page 281 of [3], or section 4 on page 580 of [12], we may find α < ω_{2} such that for every code u ∈ M[G_{α}] there is an upper bound g ∈ M[G_{α}]. By the arguments employed by Laver [11, Lemmata 10 and 11], we may assume that M[G_{α}] is the ground model M.

Since the continuum hypothesis holds in M and |X| = 𝔟 = ℵ_{2}, there are p ∈ G_{ω_{2}} and å such that

p ⊩å ∈ X̊ and å ∉ M.

Work in the ground model M.

Let q ≤ p be as in Lemma 3.3. Define

Q = {s ∈ q(0) : root(q(0)) ⊆ s}

and let U_{s}, s ∈ Q, be the finite sets from the Lemma. Let U = ⋃_{s∈Q} U_{s}. Define a Borel map f : 2^{ω} → ω^{Q} so that for every x ∈ 2^{ω} \ U and for each s ∈ Q: If f(x)(s) = n, then x ↾ n ≠ u ↾ n for each u ∈ U_{s}. For x ∈ U, f(x) may be arbitrary. There must be a g ∈ ω^{Q} ∩ M and r ≤ q such that

r ⊩f(å) ≤^{∗} ǧ.

Since p forced that a is not in the ground model, it cannot be that a is in U. We may extend r(0) if necessary so that if s = root(r(0)), then

r ⊩f(å)(s) ≤ ǧ(s).

But this is a contradiction to Lemma 3.3, since for all but finitely many t ∈ r(0) which are immediate extensions of s:

r(0)_{t} ˆ q ↾ [1, ω_{2}) ⊩f(å)(s) > ǧ(s).
□

In [20], Tsaban and Zdomskyy prove that S_{1}(Γ, Γ) for Borel covers is equivalent to the Kočinac property S_{cof}(Γ, Γ) [10], asserting that for all U_{0}, U_{1}, ⋯ ∈ Γ, there are cofinite subsets V_{0} ⊆ U_{0}, V_{1} ⊆ U_{1}, … such that ⋃_{n} V_{n} ∈ Γ. The main result of [5] can be reformulated as follows.

Theorem 3.4 (Dow [5]). In Laver’s model, S_{1}(Γ, Γ) implies S_{cof}(Γ, Γ).

For the reader’s convenience, we give Dow’s proof, adopted to the present notation.

Proof. A family H ⊆ [ω]^{ω} is ω-splitting if for each countable A ⊆ [ω]^{ω}, there is H ∈ H which splits each element of A, i.e.,

|A ∩ H| = |A \ H| = ω for all A ∈ A.

The main technical result in [5] is the following.

Lemma 3.5 (Dow). In Laver’s model, each ω-splitting family contains an ω-splitting family of cardinality < 𝔟.

Assume that X satisfies S_{1}(Γ, Γ). Let U_{0}, U_{1}, … be open point-cofinite countable covers of X. We may assume^{4} that U_{i} ∩ U_{j} = ∅ whenever i ≠ j. Put U = ⋃_{n<ω} U_{n}. We identify U with ω, its cardinality.

Define H ⊆ [U]^{ω} as follows. For H ∈ [U]^{ω}, put H ∈ H if and only if there exists V ∈ [U]^{ω}, a point-cofinite cover of X, such that H ∩ U_{n} ⊆^{∗} V for all n. We claim that H is an ω-splitting family. As H is closed under taking infinite subsets, it suffices to show that it is ω-hitting, i.e., for any countable A ⊆ [U]^{ω} there exists H ∈ H which intersects each A ∈ A. (It is enough to intersect each A ∈ A, since we may assume that A is closed under taking cofinite subsets.)

4To see why, replace each U_{n} by U_{n} \ ⋃_{i<n} U_{i}, and discard the finite ones. It suffices to show that S_{cof}(Γ, Γ) applies to those that are left.

Let A ⊆ [U]^{ω} be countable. For each n, choose sets U_{n,m} ∈ [U_{n}]^{ω}, m ∈ ω, such that for each A ∈ A, if A ∩ U_{n} is infinite, then U_{n,m} ⊆ A for some m. Apply the S_{1}(Γ, Γ) to the family {U_{n,m} : n, m ∈ ω}, to obtain a point-cofinite V ⊆ U such that V ∩ U_{n,m} is nonempty for all n, m.

Next, choose finite subsets F_{n} ⊆ U_{n}, n ∈ ω, such that for each A ∈ A with A ∩ U_{n} finite for all n, then A ⊆^{∗} ⋃_{n} F_{n}. Take H = V ∪ ⋃_{n} F_{n}. Then H is in H and meets each A ∈ A. This shows that H is an ω-splitting family.

By Lemma 3.5, there is an ω-splitting H′ ⊆ H of cardinality < 𝔟. For each H ∈ H′, let V_{H} witness that H is in H, i.e., V_{H} ⊆ U is a point-cofinite cover of X and H ∩ U_{n} ⊆^{∗} V_{H} for all n.

By the definition of 𝔟, we may find finite F_{n} ⊆ U_{n}, n ∈ ω, such that for each H ∈ H′,

H ∩ U_{n} ⊆ V_{H} ∪ F_{n}

for all but finitely many n. We claim that W = ⋃_{n} U_{n} \ F_{n} is pointcofinite. Suppose it is not. Then there is x ∈ X such that for infinitely many n, there is U_{n} ∈ U_{n} \ F_{n} with x ∉ U_{n}. Let H ∈ H′ contain infinitely many of these U_{n}. By the above inclusion, all but finitely many of these U_{n} are in V_{H}. This contradicts the fact that V_{H} is point-cofinite. □

We therefore have the following.

Theorem 3.6. In Laver’s model, each set of reals X satisfying S_{1}(Γ, Γ) has cardinality less than 𝔟.

Proof. By Dow’s Theorem, S_{1}(Γ, Γ) implies S_{cof}(Γ, Γ), which in turn implies S_{1}(Γ, Γ) for Borel covers [20]. The latter property is equivalent to having all Borel images in ω^{ω} bounded [18]. Apply Theorem 3.2. □

Thus, it is consistent that strong measure zero and S_{1}(Γ, Γ) are both trivial.

The proof of Dow’s Theorem 3.4 becomes more natural after replacing, in Lemma 3.5 “ω-splitting” by “ω-hitting”. This is possible, due to the following fact (cf. Remark 4 of [5]).

Proposition 3.7. For each infinite cardinal κ, the following are equivalent:

(1) Each ω-splitting family contains an ω-splitting family of cardinality < κ.

(2) Each ω-hitting family contains an ω-hitting family of cardinality < κ.

Proof. (1 ⇒ 2) Suppose A is an ω-hitting family. Let B = ⋃_{A∈A}[A]^{ω}. Then B is ω-splitting. By (1) there exists C ⊆ B of size < κ which is ω-splitting. Choose D ⊆ A of size < κ such that for every C ∈ C there exists D ∈ D with C ⊆ D. Then D is ω-hitting.

(2 ⇒ 1) Suppose A is an ω-splitting family. For each A ⊆ ω define

A^{∗} = \{2n : n ∈ A\} ∪ \{2n + 1 : n ∈ \overline{A}\}.

Then the family A^{∗} = {A^{∗} : A ∈ A} is ω-hitting. To see this, suppose that B is countable. Without loss we may assume that B = B_{0} ∪ B_{1} where each element of B_{0} is a subset of the evens and each element of B_{1} is a subset of the odds. For B ∈ B_{0} let C_{B} = {n : 2n ∈ B} and for B ∈ B_{1} let C_{B} = {n : 2n + 1 ∈ B}. Now put

C = {C_{B} : B ∈ B}.

Since A is ω-splitting there is A ∈ A which splits C. If B ∈ B_{0}, then A ∩ C_{B} infinite implies B ∩ A^{∗} infinite. If B ∈ B_{1} then \overline{A} ∩ C_{B} infinite implies B ∩ A^{∗} infinite.

By (2) there exists A_{0} ⊆ A of cardinality < κ such that A_{0}^{∗} is ω-hitting. We claim that A_{0} is ω-splitting. Given any B ⊆ ω let B′ = {2n : n ∈ B} and let B′′ = {2n + 1 : n ∈ B}. Given B ⊆ [ω]^{ω} countable, there exists A ∈ A_{0} such that A^{∗} hits each B′ and B′′ for B ∈ B. But this implies that A splits B. □

4. Applications to Arhangel’skiĭ’s α_{i} spaces

Let Y be a general (not necessarily metrizable) topological space. We say that a countably infinite set A ⊆ Y converges to a point y ∈ Y if each (equivalently, some) bijective enumeration of A converges to y. The following concepts are due to Arhangel’skiĭ. Y is an α_{1} space if for each y ∈ Y and each sequence A_{0}, A_{1}, … of countably infinite sets, each converging to y, there are cofinite B_{0} ⊆ A_{0}, B_{1} ⊆ A_{1}, …, such that ⋃_{n} B_{n} converges to y. Replacing “cofinite” by “singletons” (or equivalently, by “infinite”), we obtain the definition of an α_{2} space.

We first consider countable spaces.

Definition 4.1. Let X be a set of reals, and let U_{0}, U_{1}, … be countable point-cofinite covers of X. For each n, enumerate bijectively U_{n} = {U_{m}^{n} : m ∈ ω}. We associate to X a (new) topology τ on the fan S_{ω} = ω × ω ∪ {∞} as follows: ∞ is the only nonisolated point of S_{ω}, and a neighborhood base at ∞ is given by the sets

[∞]_{F} = {(n, m) : F ⊆ U_{m}^{n}}

for each finite F ⊆ X.

Lemma 4.2. In the notation of Definition 4.1: A converges to ∞ in τ if, and only if, U(A) = {U_{m}^{n} : (n, m) ∈ A} is a point-cofinite cover of X. □

Assume that there is an unbounded tower. By Corollary 2.10, there is a set of reals X satisfying S_{1}(Γ, Γ) but not S_{cof}(Γ, Γ). Let U_{0}, U_{1}, … be countable open point-cofinite covers of X witnessing the failure of S_{cof}(Γ, Γ). Then, by Lemma 4.2, (S_{ω}, τ) is α_{2} but not α_{1}. In particular, we reproduce the following.

Corollary 4.3 (Nyikos [13]). If there is an unbounded tower of cardinality 𝔟, then there is a countable α_{2} space, which is not an α_{1} space. □

Recall that by Proposition 2.4, it suffices to assume in Corollary 4.3 the existence of any unbounded tower.

Next, we consider spaces of continuous functions. Consider C(X), the family of continuous real-valued functions, as a subspace of the Tychonoff product ℝ^{X}, i.e., with the topology of pointwise convergence. Sakai [14] proved that X satisfies S_{1}(Γ, Γ) for clopen covers if, and only if, C(X) is an α_{2} space. The main result of [20] is that C(X) is α_{1} if, and only if, X satisfies S_{1}(Γ, Γ) for Borel covers (equivalently, each Borel image of X in ω^{ω} is bounded).

The Scheepers Conjecture is that for subsets of ℝ \ ℚ, S_{1}(Γ, Γ) for clopen covers implies S_{1}(Γ, Γ) for open covers. Dow [5] proved that in Laver’s model, every α_{2} space is α_{1}. By Theorem 3.2, we can add the last item in the following list.

Corollary 4.4. In Laver’s model, the following are equivalent for sets of reals X:

(1) C(X) is an α_{2} space;

(2) C(X) is an α_{1} space;

(3) X satisfies S_{1}(Γ, Γ) for clopen covers;

(4) X satisfies S_{1}(Γ, Γ) for open covers;

(5) X satisfies S_{1}(Γ, Γ) for Borel covers;

(6) |X| < 𝔟. □

On the other hand, Corollary 2.10 implies the following.

Corollary 4.5. If there is an unbounded tower, then there is a set of reals X such that C(X) is α_{2} but not α_{1}. □

Essentially, Corollary 4.3 is a special case of Corollary 2.10, whereas Corollary 4.5 is equivalent to Corollary 2.10.

Acknowledgment. We thank Lyubomyr Zdomskyy for his useful comments.

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(Miller) Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall 480 Lincoln Drive, Madison, Wisconsin 53706-1388, USA

E-mail address: miller@math.wisc.edu
URL: http://www.math.wisc.edu/~miller/

(Tsaban) Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

E-mail address: tsaban@math.biu.ac.il
URL: http://www.cs.biu.ac.il/~tsaban