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 ...

We demonstrate that a reproducing kernel Hilbert space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

We prove that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S∞ is continuous. We also show that a universally measure homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous. The present work is motivate...

We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture. We establish the existence of universal actions for any given Polish ...

We give a classical proof of the generalization of the characterization of smoothness to quotients of Polish spaces by Borel equivalence relations. As an application, we describe the extent to which any given Borel equivalence relation on a Polish space is encoded by the corresponding σ-ideal generated by the family of Borel sets on which it is smooth.

Let H be a product of countably infinite number of copies of an uncountable Polish space X . Let Σξ (Σξ ) be the class of Borel sets of additive class ξ for the product of copies of the discrete topology on X (the Polish topology on X), and let B = ∪ξ<ω1Σξ . We prove in the Lévy–Solovay model that Σξ = Σξ ∩B

The aim of the paper is to present a preliminary analysis of Polish children's gesticulation in narrating (re-telling) a cartoon. The children are nine year old native speakers of Polish. The analysis presented was focused mainly on the semantic correlation between gestural and speech modalities, the viewpoint of the gesture, types of gestures, gesture space and structure of

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

We establish the generic inexistence of stationary Borel probability measures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no σ-compact realization.