Analytic groups and pushing small sets apart

We say that a space X has the separation property provided that if A and B are subsets of X with A countable and B first category, then there is a homeomorphism f: X → X such that f(A) ∩ B = Ø. We prove that a Borel space with this property is Polish. Our main result is that if the homeomorphisms needed in the separation property for the space X come from the homeomorphisms given by an action of an analytic group, then X is Polish. Several examples are also presented. © 2009 American Mathematical Society