PlanetPhysics/Categories of Polish Groups and Polish Spaces

Introduction

Let us recall that a Polish space is a separable, completely metrizable [[../CoIntersections/|topological]] space, and that [[../InvariantBorelSet/|Polish groups]] $G_P$ are metrizable (topological) [[../TrivialGroupoid/|groups]] whose topology is Polish, and thus they admit a compatible [[../MetricTensor/|metric]] $d$ which is left-invariant; (a [[../TrivialGroupoid/|topological group]] $G_T$ is metrizable iff $G_T$ is Hausdorff, and the [[../Cod/|identity]] $e$ of $G_T$ has a countable neighborhood basis).

Polish spaces can be classified up to a (Borel) [[../IsomorphicObjectsUnderAnIsomorphism/|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 [[../BorelGroupoid/|Borel groupoid]].

Category of Polish groups

The \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} of Polish groups} $𝒫$ has, as its [[../TrivialGroupoid/|objects]], all Polish groups $G_P$ and, as its [[../TrivialGroupoid/|morphisms]] the group [[../TrivialGroupoid/|homomorphisms]] $g_P$ between Polish groups, compatible with the [[../InvariantBorelSet/|Polish topology]] $\mathrm{\Pi }$ on $G_P$.

$𝒫$ is obviously a subcategory of ${𝒯}_{grp}$ the category of [[../PolishGroup/|topological groups]]; moreover, ${𝒯}_{grp}$ is a subcategory of Failed to parse (unknown function "\grp"): {\displaystyle \mathcal{T}_{\grp}} -the category of [[../GroupoidHomomorphism2/|topological groupoids]] and topological groupoid homomorphisms.