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

Abstract

For each countable ordinal ξ \xi and pair ( A 0 , A 1 ) ({A_0},\,{A_1}) of disjoint analytic subsets of 2 ω {2^\omega } , we define a closed game J ξ ( A 0 , A 1 ) {J_\xi }({A_0},\,{A_1}) and a complete Π ξ 0 \Pi _\xi ^0 subset H ξ {H_\xi } of 2 ω {2^\omega } such that (i) a winning strategy for player I constructs a ∑ ξ 0 \sum _\xi ^0 set separating A 0 {A_0} from A 1 {A_1} ; and (ii) a winning strategy for player II constructs a continuous map φ : 2 ω → A 0 ∪ A 1 \varphi :{2^\omega } \to {A_0} \cup {A_1} with φ − 1 ( A 0 ) = H ξ {\varphi ^{ - 1}}({A_0}) = {H_\xi } . Applications of this construction include: A proof in second order arithmetics of the statement "every Π ξ 0 \Pi _\xi ^0 non ∑ ξ 0 \sum _\xi ^0 set is Π ξ 0 \Pi _\xi ^0 -complete"; an extension to all levels of a theorem of Hurewicz about ∑ 2 0 \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.