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categories of Polish groups and Polish spaces (Topic)

Introduction

Definition 0.1   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).
Remark 0.1  

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

Definition 0.2   The category 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 $G_P$.
Remark 0.2   $\mathcal{P}$ is obviously a subcategory of $\mathcal{T}_{grp}$ the category of topological groups; moreover, $\mathcal{T}_{grp}$ is a subcategory of $\mathcal{T}_{{\mathbb{G}}}$ -the category of topological groupoids and topological groupoid homomorphisms.



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Keywords:  categories of Polish groups, Polish spaces

Cross-references: topological groupoids, topological groups, Polish topology, homomorphisms, morphisms, objects, category, Borel groupoid, isomorphism, identity, topological group, metric, groups, Polish groups, topological

This is version 1 of categories of Polish groups and Polish spaces, born on 2009-02-04.
Object id is 493, canonical name is CategoriesOfPolishGroupsAndPolishSpaces.
Accessed 292 times total.

Classification:
Physics Classification: 00. (GENERAL)
Pending Errata and Addenda
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