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 $$G_P$$ 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).

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


 * 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.

Category of Polish groups
The \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} 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 $$G_P$$.

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