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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 GERMANYE-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 λλ, 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 λλ that are definable from elements of λOrd 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 λλ that are definable from elements of λOrd 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 λλ and consequences of resurrection axioms for definable subsets of λλ.

Keywords

03E3503E6003E1503E15generalized descriptive set theoryperfect set propertyBanach–Mazur gamelong games
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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