On a notion of smallness for subsets of the Baire space

Abstract

Let us call a set A ⊆ ω ω A \subseteq {\omega ^\omega } of functions from ω \omega into ω σ \omega \;\sigma -bounded if there is a countable sequence of functions { α n : n ∈ ω } ⊆ ω ω \{ {\alpha _n}:n \in \omega \} \subseteq {\omega ^\omega } 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 {\omega ^\omega } . We show that most of the usual definability results about the structure of countable subsets of ω ω {\omega ^\omega } have corresponding versions which hold about σ \sigma -bounded subsets of ω ω {\omega ^\omega } . For example, we show that every Σ 2 n + 1 1 σ \Sigma _{2n + 1}^1\;\sigma -bounded subset of ω ω {\omega ^\omega } has a Δ 2 n + 1 1 \Delta _{2n + 1}^1 “bound” { α m : m ∈ ω } \{ {\alpha _m}:m \in \omega \} and also that for any n ⩾ 0 n \geqslant 0 there are largest σ \sigma -bounded Π 2 n + 1 1 \Pi _{2n + 1}^1 and Σ 2 n + 2 1 \Sigma _{2n + 2}^1 sets. We need here the axiom of projective determinacy if n ⩾ 1 n \geqslant 1 . In order to study the notion of σ \sigma -boundedness a simple game is devised which plays here a role similar to that of the standard ∗ ^\ast -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 ∗ ^\ast - and ∗ ∗ ^{ \ast \ast } - (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 {\omega ^\omega } whose special cases include countability, being of the first category and σ \sigma -boundedness and for which one can generalize all the main results of the present paper.