We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures. Introduction. The Borel reducibility theory of Pol...

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is Π2 in the Borel hierarchy. ...

We prove that for every homogeneous and strongly locally homogeneous Polish space X there is a Polish group admitting a transitive action on X. We also construct an example of a homogeneous Polish space which is not a coset space and on which no separable metrizable topological group acts transitively.

In this article we consider the possible sets of distances in Polish metric spaces. By a Polish metric space we mean a pair (X, d), where X is a Polish space (a separable, completely-metrizable space) and d is a complete, compatible metric for X. We will consider two aspects. First, we will characterize which sets of reals can be the set of distances in a Polish metric space. We will also obtai...

A topological space is a Borel space if it is homeomorphic to a Borel subset of a Polish space (Bertsekas and Shreve, 1978, Definition 7.7). Examples of Borel spaces include any Borel subset of a Euclidean space Rd and, more generally, any Borel subset of a Polish space (Bertsekas and Shreve, 1978, Proposition 7.12). If X and Y are Borel spaces, a function g : X × Y → R is upper semianalytic if...

We consider the equivalence relation of isometry of separable, complete metric spaces, and show that any equivalence relation induced by a Borel action of a Polish group on a Polish space is Borel reducible to this isometry relation. We also consider the isometry relation restricted to various classes of metric spaces, and produce lower bounds for the complexity in terms of the Borel reducibili...

This paper continues the work [6]. For a Polish group G the notions of G-continuous functions and whirly actions are further exploited to show that: (i) A G-action is whirly iff it admits no nontrivial spatial factors. (ii) Every action of a Polish Lévy group is whirly. (iii) There exists a Polish monothetic group which is not Lévy but admits a whirly action. (iv) In the Polish group Aut (X,X, ...

For any non-totally disconnected Polish space, there is a family of c = 2א0 many Wadge incomparable finite level Borel subsets. If the space is additionally locally compact or locally connected, there is a family of 2 many Wadge incomparable subsets. In this note, we study the Wadge order for Polish spaces which are not totally disconnected. For a fixed Polish space, the Wadge order for the spa...

Let X be a Polish space, Y a separable metrizable space, and f : X → Y a continuous surjection. We prove that if the image under f of every open set or every closed set is resolvable, then Y is Polish. This generalizes similar results by Sierpiński, Vainštain, and Ostrovsky.