# PlanetPhysics/Categories of Polish Groups and Polish Spaces

### IntroductionEdit

Let us recall that a *Polish space* is a separable, completely metrizable topological space, and
that Polish groups are metrizable (topological) groups whose topology is Polish, and thus they admit a compatible metric which is left-invariant; (a topological group is *metrizable* iff is Hausdorff, and the identity of 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
**Failed to parse (syntax error): {\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 groupsEdit

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

is obviously a subcategory of the category of topological groups; moreover,
is a subcategory of **Failed to parse (unknown function "\grp"): {\displaystyle \mathcal{T}_{\grp}}**
-the category of topological groupoids and topological groupoid homomorphisms.