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

For each countable ordinal ( and pair (Ao,Al) of disjoint analytic subsets of 2@, we define a closed game J(AoS A1) and a complete IIt subset Ht of 2@ such that (i) a winning strategy for player I constructs a 24 set separating Ao from A1; and (ii) a winning strategy for player II constructs a continuous map (p: 2@ A() U A1 with (p-l(Ao) = H. Applications of this construction include: A proof in second order arithmetics of the statement "every IIt non SS set is II)-complete"; an extension to all levels of a theorem of Hurewicz about 22 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. Introduction. Let E = 2@ be the usual Cantor space, and following the standard modern terminology of Addison, let E: and fIt denote, respectively, the Borel subsets of E of additive and multiplicative Borel class (, starting with E:° = Open and TI° = Closed. The aim of this paper is to characterize, given a set A c 2@ and one of the Borel classes, say r, whether A is in r or not. By a "characterization", we have in mind the finding of some mathematical object associated with A and which positively witnesses that A E r or that A 0 r. Let r be the class E:t. If the set A is in E:, a natural candidate for such a witness is simply one particular construction of A as a E: set (a list of sequences of open sets together with instructions on the operations to be performed to get A in t steps). The problem of how to find such constructions, which is deeply connected to the structural properties of sets in the plane with E: sections, has been extensively investigated in papers by Novikov [N] and Kunugui [Kun] for (= 1 (see also Dellacherie [De]), Saint Raymond [SR 1] for t = 2, Bourgain [B 1, B 2] and Louveau [Lo 1] for t = 3, Louveau [Lo 2] for t > 3, in case A is known to be Borel. In case A ranges over larger classes, the problem is studied in papers by Stern [St], Kechris [Ke] and Louveau [Lo 3]. Received by the editors December 26, 1985 and, in revised form, November 19, 1986. These results have been presented at the Annual Meeting of the American Mathematical Society, New Orleans, January 1986, in a special session on Determinacy and large cardinals. 1980 Mclthemcltics Subject Clclssificcltion (1985 Revision). Primary O2K30, 04A15, 28A05; Secondary 54H05, 26A21.