Point-cofinite covers in the Laver model

arXiv:0910.4063v2 [math.GN] 4 Nov 2010 POINT-COFINITE COVERS IN THE LAVER MODEL ARNOLD W. MILLER AND BOAZ TSABAN Abstract. Let S1(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 S1(Γ, Γ). We prove the following assertions: (1) If there is an unbounded tower, then there are sets of reals of cardinality b, satisfying S1(Γ, Γ). (2) It is consistent that all sets of reals satisfying S1(Γ, Γ) have cardinality smaller than b. These results can also be formulated as dealing with Arhangel’ski˘ı’s property α2 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 ωω. 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. Ufin(Γ, Γ): For all U0, U1, · · · ∈Γ, none containing a finite sub- cover, there are finite F0 ⊆U0, F1 ⊆U1, . . . such that {S Fn : n ∈ω} ∈Γ. U2(Γ, Γ): For all U0, U1, · · · ∈Γ, there are F0 ⊆U0, F1 ⊆U1, . . . such that |Fn| = 2 for all n, and {S Fn : 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 ǫ in the seminal paper [7] of Gerlits and Nagy. 1 2 ARNOLD MILLER AND BOAZ TSABAN S1(Γ, Γ): For all U0, U1, · · · ∈Γ, there are U0 ∈U0, U1 ∈U1, . . . such that {Un : n ∈ω} ∈Γ. Clearly, S1(Γ, Γ) implies U2(Γ, Γ), which in turn implies Ufin(Γ, Γ). None of these implications is reversible in ZFC [19]. The critical car- dinality of all three properties is b [9].2 Bartoszy´nski and Shelah [1] proved that there are, provably in ZFC, totally imperfect sets of reals of cardinality b satisfying the Hurewicz property Ufin(Γ, Γ). Tsaban proved the same assertion for U2(Γ, Γ) [19]. These sets satisfy Ufin(Γ, Γ) in all finite powers [2]. We show that in order to obtain similar results for S1(Γ, Γ), hypothe- ses beyond ZFC are necessary. 2. Constructions We show that certain weak (but not provable in ZFC) hypotheses suffice to have nontrivial S1(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 {xα : α < κ}, such that for all α < β < κ, xβ ⊆∗xα. A set T ⊆[ω]ω is unbounded if the set of its enumeration functions are unbounded, i.e., for any g ∈ωω 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 S1(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 = {bα : α < b} ⊆ωω be a b-scale, that is, each bα is increasing, bα ≤∗bβ for all α < β < b, and B is unbounded. As |B| < d, B is not dominating. Let g ∈ωω exemplify that. For each α < b, let xα = {n : bα(n) ≤g(n)}. Then T = {xα : α < b} is an unbounded tower: Clearly, xβ ⊆∗xα for α < β. Assume that T is bounded, and let f ∈ωω exemplify that. For each α, writing xα(n) for the n-th element of xα: bα(n) ≤bα(xα(n)) ≤g(xα(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. POINT-COFINITE COVERS IN THE LAVER MODEL 3 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 S1(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 {fα : α < b} ⊆ωω be dominating. For each α < b, pick xα ∈T which is not bounded by fα. {xα : α < b} is unbounded, being unbounded in a dominating family. □ Define a topology on P(ω) by identifying P(ω) with the Cantor space 2ω, via characteristic functions. 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

…(Full text truncated)…

This content is AI-processed based on ArXiv data.