Borel and projective sets from the point of view of compact sets

Abstract

In this paper we prove several results concerning the complexity of a set relative to compact sets. We prove that for any Polish space X and Borel set B ⊆ X, if B is not , then there exists a compact zero-dimensional P ⊆ X such that p ∩ X is not . We also show that it is consistent with ZFC that, for any A ⊆ ωω, if for all compact K ⊆ ωωA ∩ K is , then A is . This generalizes to in place of assuming the consistency of some hypotheses involving determinacy. We give an alternative proof of the following theorem of Saint-Raymond. Suppose X and Y are compact metric spaces and f is a continuous surjection of X onto Y. Then, for any A ⊆ Y, A is in Y iff f−1(A) is in X. The non-trivial part of this result is to show that taking pre-images cannot reduce the Borel complexity of a set. The techniques we use are the definability of forcing and Wadge games.