The Baire Space Ordered by Eventual Domination: Spectra

These are notes of the author’s talk on various types of spectra associated naturally with the eventually domination ordering on the Baire space ωω , given at the General Topology Symposium at Kobe University in December 2002. The report comes in two parts: in the first half, we present an outline of the lecture, giving ideas of some of the arguments without going too deeply into details. The second part presents the technical niceties of some proofs. This part was circulated previously under the title Chubu Marginalia [2]. 1. Outline of the lecture The Baire space ω is the set of all functions from the natural numbers ω to ω, equipped with the product topology of the discrete topology. Given f, g ∈ ω say that g eventually dominates f (f ≤∗ g in symbols) if f(n) ≤ g(n) holds for all but finitely many n ∈ ω. A family F ⊆ ω is called unbounded if there is no g ∈ ω with f ≤∗ g for all f ∈ F . F ⊆ ω is said to be dominating if for all g ∈ ω there is f ∈ F with g ≤∗ f . It is easy to see that a dominating family is also unbounded. We let b := min{|F|; F ⊆ ω unbounded}, the (un)bounding number. d := min{|F|; F ⊆ ω dominating} is the dominating number. The cardinal invariants b and d characterize the combinatorial structure of (ωω,≤∗). Fact 1.1. א1 ≤ b ≤ cf(d) ≤ d ≤ c and b is regular. (Here, cf means cofinality, and c = |2ω| = |R| stands for the size of the continuum.) As a leitmotiv for this talk we address: What other notions can be used to describe the combinatorial structure of (ωω,≤∗)? ♣♣♣ For a given preorder (P,≤) (that is, ≤ is reflexive and transitive, but not necessarily antisymmetric), Fuchino and Soukup [4] defined the following four spectra. (i) the unbounded chain spectrum S↑(P ), the set of all regular cardinals κ such that there is an unbounded increasing chain of length κ in P ; (ii) the hereditarily unbounded set spectrum S(P ), the set of all cardinals κ such that there is A ⊆ P of size κ such that all subsets of A of size κ are unbounded in P while all subsets of A of size less than κ are bounded in P ; ∗The author is supported by the Kobe Technical Club KTC (神戸大学工学振興会)