On a notion of smallness for subsets of the Baire space

Abstract

Let us call a set A ⊆ ω^ω of functions from ω into ω σ-bounded if there is a countable sequence of functions (α_n: n Є ω)⊆ ω^ω such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of ω^ω. We show that most of the usual definability results about the structure of countable subsets of ω^ω have corresponding versions which hold about σ-bounded subsets of ω^ω. For example, we show that every Σ_(2n+1^1 σ-bounded subset of ω^ω has a Δ_(2n+1)^1 "bound" {α_m: m Є ω} and also that for any n ≥ 0 there are largest σ-bounded Π_(2n+1)^1 and Σ_(2n+2)^1 sets. We need here the axiom of projective determinacy if n ≥ 1. In order to study the notion of σ-boundedness a simple game is devised which plays here a role similar to that of the standard ^*-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the ^*- and ^(**)-(or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of ω^ω whose special cases include countability, being of the first category and σ-boundedness and for which one can generalize all the main results of the present paper.