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On a notion of smallness for subsets of the Baire space


Author: Alexander S. Kechris
Journal: Trans. Amer. Math. Soc. 229 (1977), 191-207
MSC: Primary 04A15; Secondary 54H05
DOI: https://doi.org/10.1090/S0002-9947-1977-0450070-1
MathSciNet review: 0450070
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0450070-1
Article copyright: © Copyright 1977 American Mathematical Society

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