Analytic groups and pushing small sets apart

Abstract

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 : X → X such that f (A) ∩ B = ∅. 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.

Effros, Baire, Steinhaus and non-separability

“…The arguments are based on the following lemma. We note a corollary, observed earlier by van Mill in the case of metric topological groups ([vMil2,Prop. 3.4]), which concerns a co-meagre set, but we need its re…nement to a localized version for a non-meagre set.…”

“…We argue as in [vMil2] Prop 3.1 (1). Suppose otherwise; then X contains a non-empty meagre open set.…”

“…For further commentary (connections between convexity and the Baire property, relation to van Mill's separation property in [vMil2], certain specializations) see the extended version of this paper on arXiv.…”

The topology of ultrafilters as subspaces of 2ω

“…This is easily seen to be impossible by choosing g to be the multiplication by any coinfinite x ∈ U. Actually, something much stronger holds by the following result of Van Mill (see Proposition 3.4 in [17]).…”

Strong measure zero in separable metric spaces and Polish groups

“…The properties strongly GMS and weakly GMS are related to recent work done by van Mill [36]. The result relevant for us is that an analytic group which is not Polish admits a meager set M and a countable set C such that for any g ∈ G M ∩ C · g = ∅.…”