On a notion of smallness for subsets of the Baire space

Abstract:

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