Mycielski ideal and the perfect set theorem

Abstract

We make several observations on the Mycielski ideal and prove a version of the perfect set theorem concerning this ideal for analytic sets: If A ⊆ ω 2 A\subseteq {}^\omega 2 is an analytic set all projections of which are uncountable, then there is a perfect set B ⊆ A B\subseteq A a projection of which is the whole space. We also prove that (a modification of) an infinite game of Mycielski is determined for analytic sets.