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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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} ${\mathcal {P}}$ 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 Failed to parse (unknown function "\grp"): {\displaystyle \mathcal{T}_{\grp}} -the category of topological groupoids and topological groupoid homomorphisms.