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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Borel classes and closed games: Wadge-type and Hurewicz-type results


Authors: A. Louveau and J. Saint-Raymond
Journal: Trans. Amer. Math. Soc. 304 (1987), 431-467
MSC: Primary 03E15; Secondary 04A15, 28A05, 54H05
MathSciNet review: 911079
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Abstract: For each countable ordinal $ \xi $ and pair $ ({A_0},\,{A_1})$ of disjoint analytic subsets of $ {2^\omega }$, we define a closed game $ {J_\xi }({A_0},\,{A_1})$ and a complete $ \Pi _\xi ^0$ subset $ {H_\xi }$ of $ {2^\omega }$ such that (i) a winning strategy for player I constructs a $ \sum _\xi ^0$ set separating $ {A_0}$ from $ {A_1}$; and (ii) a winning strategy for player II constructs a continuous map $ \varphi :{2^\omega } \to {A_0} \cup {A_1}$ with $ {\varphi ^{ - 1}}({A_0}) = {H_\xi }$. Applications of this construction include: A proof in second order arithmetics of the statement "every $ \Pi _\xi ^0$ non $ \sum _\xi ^0$ set is $ \Pi _\xi ^0$-complete"; an extension to all levels of a theorem of Hurewicz about $ \sum _2^0$ sets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieff's and Wadge's hierarchies.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0911079-0
PII: S 0002-9947(1987)0911079-0
Keywords: Borel classes, closed games, Wadge games, determinacy, Hurewicz' theorem
Article copyright: © Copyright 1987 American Mathematical Society