Analytic groups and pushing small sets apart

van Mill, Jan

doi:10.1090/S0002-9947-09-04665-0

Analytic groups and pushing small sets apart
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Trans. Amer. Math. Soc. 361 (2009), 5417-5434 Request permission

Abstract:

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 $f\colon X\to X$ such that $f(A)\cap B=\emptyset$. 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.

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