# PlanetPhysics/Categories of Polish Groups and Polish Spaces

### Introduction

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

Polish spaces can be classified up to a (Borel) isomorphism according to the following provable results:

• All uncountable Polish spaces are Borel isomorphic to $\displaystyle \mathbb{R$ 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.

### Category of Polish groups

The \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} of Polish groups} ${\displaystyle {\mathcal {P}}}$  has, as its objects, all Polish groups ${\displaystyle G_{P}}$  and, as its morphisms the group homomorphisms ${\displaystyle g_{P}}$  between Polish groups, compatible with the Polish topology ${\displaystyle \Pi }$  on ${\displaystyle G_{P}}$ .

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