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PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

Published online by Cambridge University Press:  09 January 2018

PHILIPP SCHLICHT
PHILIPP SCHLICHT*
Affiliation:
MATHEMATISCHES INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN GERMANY E-mail: schlicht@math.uni-bonn.de
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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 = 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.

Keywords

03E3503E6003E1503E15generalized descriptive set theoryperfect set propertyBanach–Mazur gamelong games

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1317 - 1355
Copyright
Copyright © The Association for Symbolic Logic 2017 

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