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PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

  • PHILIPP SCHLICHT (a1)

Abstract

We extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire space λ λ, where λ is an uncountable cardinal with λ = λ. In the first main theorem, we show that the perfect set property for all subsets of λ λ that are definable from elements of λ Ord is consistent relative to the existence of an inaccessible cardinal above λ. In the second main theorem, we introduce a Banach–Mazur type game of length λ and show that the determinacy of this game, for all subsets of λ λ that are definable from elements of λ Ord as winning conditions, is consistent relative to the existence of an inaccessible cardinal above λ. We further obtain some related results about definable functions on λ λ and consequences of resurrection axioms for definable subsets of λ λ.

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