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

Louveau, A.; Saint-Raymond, J.

doi:10.1090/S0002-9947-1987-0911079-0

Borel classes and closed games: Wadge-type and Hurewicz-type results
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by A. Louveau and J. Saint-Raymond PDF

Trans. Amer. Math. Soc. 304 (1987), 431-467 Request permission

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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