We show that every abelian Polish group is the topological factor-group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the unitary group.

The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book Becker-Kechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same. Locally compact Polish groups, i.e., second countable locally compact groups, are the traditional obje...

In the presence of suitable power spaces, compactness ofX can be characterized as the singleton {∅} being open in the space A(X). Equivalently, this means that universal quantification over a compact space preserves open predicates. Using the language of represented spaces, one can make sense of notions such as a Σ 2 subset of the space of Σ 2 -subsets of a given space. This suggests higher-ord...

For a continuous action of a countable discrete group G on a Polish space X, a countable Borel partition P of X is called a generator if GP ∶= {gP ∶ g ∈ G,P ∈ P} generates the Borel σ-algebra of X. For G = Z, the Kolmogorov–Sinai theorem gives a measuretheoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite...

We identify four countable topological spaces S2, S1, SD, and S0 which serve as canonical examples of topological spaces which fail to be quasi-Polish. These four spaces respectively correspond to the T2, T1, TD, and T0-separation axioms. S2 is the space of rationals, S1 is the natural numbers with the cofinite topology, SD is an infinite chain without a top element, and S0 is the set of finite...

In this article we give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture on Borel functions from an analytic subset of a Polish space into a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to prove that several prominent results in recursion t...

Building on earlier work of Katětov, Uspenskij proved in [9] that the group of isometries of Urysohn’s universal metric space U, endowed with the product topology, is a universal Polish group (i.e it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F o...

Non-abelian Polish groups arising as countable products of countable groups can be tame in arbitrarily complicated ways. This contrasts with some results of Solecki who revealed a very different picture in the abelian case. 0. Group trees. The class of all Polish (completely metrizable, separable) groups may be naturally divided into two classes. 0.1. Definition. A Polish group G is tame if whe...

(A) A Polish metric space is a complete separable metric space (X, d). Our first goal in this paper is to determine the exact complexity of the classification problem of Polish metric spaces up to isometry. Our work was motivated by a recent paper of Vershik [1998], where the author remarks (in the beginning of Section 2): “The classification of Polish spaces up to isometry is an enormous task....

We construct a homogeneous connected Polish space X on which no א0-bounded topological group acts transitively. In fact, X is homeomorphic to a convex subset of Hilbert space `.