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