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 of functions from into -*bounded* if there is a countable sequence of functions 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 . We show that most of the usual definability results about the structure of countable subsets of have corresponding versions which hold about -bounded subsets of . For example, we show that every -bounded subset of has a ``bound'' and also that for any there are largest -bounded and sets. We need here the axiom of projective determinacy if . In order to study the notion of -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 whose special cases include countability, being of the first category and -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