$\sigma$-idéaux engendrés par des ensembles fermés et théorèmes d’approximation

Abstract:

This paper is motivated by the study of *-games on $\omega$, and by a question of D. A. Martin on the strength of the hypothesis ${\text {AD}}_\omega ^{\ast }$ that every *-game on ${\omega ^\omega }$ is determined. A general study of the $\sigma$-ideals of subsets of ${\omega ^\omega }$ generated by closed sets encompasses the *-games and the “perfect set property". Using associated games, we extend for these ideals many properties known for countable sets, under various hypotheses of determinacy. Our methods thus apply also to other examples of regularity properties, such as those introduced by A. S. Kechris. Finally, a general theorem of approximation by analytic sets in Solovay’s model is proved which, together with the preceding results, gives the solution of Martin’s problem: ${\text {AD}}_\omega ^{\ast }$ is true in Solovay’s model.