Uniformization and anti-uniformization properties of ladder systems

Abstract

Let S denote a stationary subset of limit ordinals of ω1. A ladder system on S is a sequence {Lα : α ∈ S} such that each Lα is an unbounded subset of α of order type ω. A ladder system is uniformizable if for each sequence 〈fα : α ∈ S〉 of functions fα : Lα → ω there is an F : ω1 → ω such that F 1 Lα =∗ fα for each α ∈ S. I.e., for each α ∈ S, {β ∈ Lα : F (β) 6= fα(β)} is finite. We now formulate natural weakenings of uniformizable denoted, for each n ∈ ω, by Pn: A ladder system is said to satisfy Pn if for each f : S → ω there is an F : ω1 → [ω] such that for each α ∈ S (a) F 1 Lα is eventually constant with value sα, and (b) f(α) ∈ sα. Note that P0 is equivalent to the version of uniformizable obtained by considering only sequences of constant functions fα. We will say that a ladder system satisfies P<ω if for each f : S → ω there is an F : ω1 → [ω] satisfying (a) and (b) above. If we drop the requirement that the restrictions F 1 Lα are eventually constant we obtain uniformization properties that we denote Mn and M<ω. E.g., a ladder system is said to satisfy M<ω if for each f : S → ω there is an F : ω1 → [ω] such that for each α ∈ S, f(α) ∈ F (β) for all but finitely many β ∈ Lα. Most of these uniformization properties can be characterized in terms of properties of a certain topological space naturally associated to any ladder system. If L is a ladder system, let XL denote the topology space ω1 × {0} ∪ S × {1} where every point (α, 0) is isolated and for each α ∈ S, a basic neighborhood of (α, 1) consists of {(α, 1)} along with a cofinite subset of Lα×{0}. Such a space is always first countable and locally compact. The stationarity of S implies that it is not collectionwise Hausdorff. Spaces XL have been considered by many to construct examples of normal not collectionwise Hausdorff spaces (see [11] and [2]). It is folklore that a ladder system L satisfies P0 if and only if XL is normal. The property M<ω is characterized by XL being countably metacompact. For this reason, we will say that a ladder system L is countably metacompact in the case that it satisfies M<ω.