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On \(\Game \mathbf{\Gamma } \)-complete sets. (English) Zbl 1453.03049

Let \(\Game\) denote the game quantifier on poinsets and pointclasses. A pointclass \(\boldsymbol\Gamma\) is called a Moschovakis class if for any set \(B \subset 2^{\omega} \times \omega^{\omega}\) in \(\boldsymbol\Gamma\) there exists a \(\Game\boldsymbol\Gamma\)-measurable function on \(\Game B\) which assigns to any \(x \in \Game B\) a winning strategy for Player I in the game on \(\omega\) with payoff set \(B_x\).
The main result of this paper is the following:
Theorem. Let \(\boldsymbol\Gamma\) be a Moschovakis class of Borel sets such that the class \(\Game \boldsymbol\Gamma\) has the (generalized) \(\omega\)-reduction property and is closed under preimages by total \(\Game\boldsymbol\Gamma\)-measurable functions. Then any set which is \(\Game\boldsymbol\Gamma\)-hard set for \(\Game\boldsymbol\Gamma\)-measurable reductions is \(\Game\boldsymbol\Gamma\)-hard (i.e., by a continuous reduction).
The hypotheses are satisfied by \(\boldsymbol\Gamma=\boldsymbol\Sigma_\alpha^0, \boldsymbol\Pi_\alpha^0\) (boldface), and in particular the main result of [A. Kechris, Proc. Am. Math. Soc. 125, No. 6, 1811–1814 (1997; Zbl 0864.03034)] follows as a corollary.
The main theorem is proved without using the effective theory. Among the ingredients of the proof, the authors prove (using a refined version of a result by A. Kechris [Lect. Notes Math. 669, 277–302 (1978, Zbl 0391.03024)] about an unfolded version of the Banach-Mazur game) that given some \(A\in\Game\boldsymbol\Gamma( 2^\omega \times X )\), there is a \(\Game\boldsymbol\Gamma\)-measurable map assigning to each \(\varepsilon\in 2^\omega\) such that the section \(A_\varepsilon\) is nonmeager, an injective continuous function \(2^\omega \to X\) with range included in the latter.

MSC:

03E15 Descriptive set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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