Borel games and the Baire property

Abstract

The Borel game operations are a natural generalization of the operation (A). It is shown that these operations preserve the property of Baire in all topological spaces. Applications are given to invariant descriptive set theory and the model theory of infinitary logic. Introduction. It is a classical theorem that the operation (A) preserves the Baire property in an arbitrary topological space. There are two proofs-e.g., in Kuratow- ski's Topology (6). One (? 11, VII) is valid in general; while the other (? 19, I, 4) works only if some countability assumption is made for the space (e.g., for a separable metric, but not for an arbitrary metric space).' Corresponding to each Borel subset of VP(X) is an operation GL which can be construed as a generalization of the operation (A). It is called a Borel game operation. D. A. Martin (7) established the fundamental fact that all Borel games are determinate. Using Martin's result, R. Solovay showed that the Borel game opera- tions preserve the Baire property in all spaces which are Polish (separable, complete metrizable). In this paper, we show (Theorem A) that the result holds in arbitrary topological spaces. At the same time, using a recently discovered method of Kechris (5), we shall (in Theorem B) extend to all spaces a result of Burgess, which locates in a useful way the position of the outcome set when GL is applied 'mod meager', in terms of the input sets 'mod meager'. We also will show that these results have a K-version in which games are played on an arbitrary infinite cardinal K, and meager, Baire property, etc., are replaced by their usual K-versions. In ?1, a detailed statement of the results is given. 1. Detailed statement of results. Let R be any nonempty set and let r (usually subscripted) range over R. Suppose Q is a subset of R' (the set of all functions on the set co of natural numbers to R). We write