How many miles to Erice? — Cofinal types of sets of metrics

This is a draft of the proof of the main theorem in the author’s lecture entitled Approximations to βX versus cofinal types of sets of metrics, which will be presented in Advances in Set-theoretic Topology (Conference in honour of Tsugunori Nogura on his 60th birthday) held in Erice, Sicily, Italy. 1 Tukey relations between directed sets Let (D,≤) and (E,≤) directed ordered sets. A mapping ϕ from D to E is called a Tukey mapping if the image of an unbounded subset of D by ϕ is an unbounded subset of E, or equivalently, if the inverse image of a bounded subset of E is a bounded subset of D. We write (D,≤) ≤T (E,≤) (and often say D is Tukey below E, or E is cofinally finer than D) if there is a Tukey mapping from D to E. A mapping ψ from E to D is called a convergent mapping if the image of a cofinal subset of E by ψ is a cofinal subset of D. It is easily checked that (D,≤) ≤T (E,≤) if and only if there is a convergent mapping from E to D. We write (D,≤) ≡T (E,≤) (and often say D is Tukey equivalent to E, D is cofinally similar to E, or D and E have the same cofinal type) if both (D,≤) ≤T (E,≤) and (E,≤) ≤T (D,≤) hold. In particular, if there is a mapping from D to E which is both Tukey and convergent, then (D,≤) ≡T (E,≤) holds. 2 The main theorem Throughout this section, we assume that X is a locally compact separable metrizable space and the subspace of nonisolated points of X (the first 1 Cantor–Bendixson derivative of X, often denoted by X(1)) is not compact. Since X is embedded into the Hilbert cube H = [0, 1]ω as an open subspace, we fix such an embedding and regard X as an open subspace of H. Let µ be a “canonical ” metric function on H. Let M(X) denote the set of all compatible metrics on X. For d1, d2 ∈ M(X), we write d1 ≼ d2 if the identity function on X is uniformly continuous as a function from (X, d2) to (X, d1). The following equivalent conditions for d1 ≼ d2 are known. Lemma 2.1. Each of the following conditions are equivalent to d1 ≼ d2. 1. ud1X ≤ ud2X