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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On a notion of smallness for subsets of the Baire space
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by Alexander S. Kechris PDF
Trans. Amer. Math. Soc. 229 (1977), 191-207 Request permission

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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • 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