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On a notion of smallness for subsets of the Baire space

Kechris, Alexander S. (1977) On a notion of smallness for subsets of the Baire space. Transactions of the American Mathematical Society, 229 . pp. 191-207. ISSN 0002-9947.

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Let us call a set A ⊆ ω^ω of functions from ω into ω σ-bounded if there is a countable sequence of functions (α_n: n Є ω)⊆ ω^ω 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. We show that most of the usual definability results about the structure of countable subsets of ω^ω have corresponding versions which hold about σ-bounded subsets of ω^ω. For example, we show that every Σ_(2n+1^1 σ-bounded subset of ω^ω has a Δ_(2n+1)^1 "bound" {α_m: m Є ω} and also that for any n ≥ 0 there are largest σ-bounded Π_(2n+1)^1 and Σ_(2n+2)^1 sets. We need here the axiom of projective determinacy if n ≥ 1. In order to study the notion of σ-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 ω^ω whose special cases include countability, being of the first category and σ-boundedness and for which one can generalize all the main results of the present paper.

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Additional Information:© 1977 American Mathematical Society. Received by the editors December 10, 1975.
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Other Numbering System NameOther Numbering System ID
MathSciNet ReviewMR0450070
Classification Code:AMS (MOS) subject classifications (1970). Primary 04A15, 02K30, 28A05, 54H05; Secondary 02F35, 02K05, 02K25, 02K35, 04A30
Record Number:CaltechAUTHORS:20130522-135105483
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Official Citation:On a notion of smallness for subsets of the Baire space Alexander S. Kechris. Trans. Amer. Math. Soc. 229 (1977), 191-207
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38638
Deposited By: Ruth Sustaita
Deposited On:22 May 2013 22:22
Last Modified:03 Oct 2019 04:59

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