# categories of Polish groups and Polish spaces

## 0.1 Introduction

###### Definition 0.1.

Let us recall that a Polish space is a separable, completely metrizable topological space, and that Polish groups $G_{P}$ are metrizable (topological) groups whose topology is Polish, and thus they admit a compatible metric $d$ which is left-invariant; (a topological group $G_{T}$ is metrizable iff $G_{T}$ is Hausdorff, and the identity $e$ of $G_{T}$ has a countable neighborhood basis).

###### Remark 0.1.

Polish spaces can be classified up to a (Borel) isomorphism according to the following provable results (http://planetmath.org/PolishSpacesUpToBorelIsomorphism):

• All uncountable Polish spaces are Borel isomorphic to $\mathbb{R}$ equipped with the standard topology;

This also implies that all uncountable Polish space have the cardinality of the continuum.

• Two Polish spaces are Borel isomorphic if and only if they have the same cardinality.

Furthermore, the subcategory of Polish spaces that are Borel isomorphic is, in fact, a Borel groupoid.

## 0.2 Category of Polish groups

###### Definition 0.2.

The category of Polish groups $\mathcal{P}$ has, as its objects, all Polish groups $G_{P}$ and, as its morphisms the group homomorphisms $g_{P}$ between Polish groups, compatible with the Polish topology $\Pi$ on $G_{P}$.

###### Remark 0.2.

$\mathcal{P}$ is obviously a subcategory of $\mathcal{T}_{grp}$ the category of topological groups; moreover, $\mathcal{T}_{grp}$ is a subcategory of $\mathcal{T}_{{\mathbb{G}}}$ -the category of topological groupoids and topological groupoid homomorphisms.

 Title categories of Polish groups and Polish spaces Canonical name CategoriesOfPolishGroupsAndPolishSpaces Date of creation 2013-03-22 18:25:04 Last modified on 2013-03-22 18:25:04 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 19 Author bci1 (20947) Entry type Topic Classification msc 54H05 Classification msc 28A05 Classification msc 28A12 Classification msc 28C15 Synonym subcategory of topological groupoid category Related topic Category Related topic Metrizable Related topic CategoryOfBorelSpaces Related topic PolishSpacesUpToBorelIsomorphism Related topic TopologicalGroup2 Related topic MeasureSpace Related topic PolishSpace Defines Polish group homomorphism Defines metrizable topological groups