categories of Polish groups and Polish spaces

0.1 Introduction

Definition 0.1.

Let us recall that a Polish spaceMathworldPlanetmath is a separablePlanetmathPlanetmath, completely metrizable topological spaceMathworldPlanetmath, and that Polish groups GP are metrizable (topological) groups whose topology is Polish, and thus they admit a compatible metric d which is left-invariant; (a topological groupMathworldPlanetmath GT is metrizable iff GT is HausdorffPlanetmathPlanetmath, and the identityPlanetmathPlanetmathPlanetmathPlanetmath e of GT has a countableMathworldPlanetmath neighborhood basis).

Remark 0.1.

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

  • All uncountable Polish spaces are Borel isomorphic to R equipped with the standard topology;

    This also implies that all uncountable Polish space have the cardinality of the continuumMathworldPlanetmathPlanetmath.

  • Two Polish spaces are Borel isomorphic if and only if they have the same cardinality.

Furthermore, the subcategoryMathworldPlanetmath of Polish spaces that are Borel isomorphic is, in fact, a Borel groupoid.

0.2 Category of Polish groups

Definition 0.2.

The category of Polish groups 𝒫 has, as its objects, all Polish groups GP and, as its morphisms the group homomorphisms gP between Polish groups, compatible with the Polish topology Π on GP.

Remark 0.2.

𝒫 is obviously a subcategory of 𝒯grp the category of topological groups; moreover, 𝒯grp is a subcategory of 𝒯𝔾 -the category of topological groupoidsPlanetmathPlanetmathPlanetmathPlanetmath and topological groupoid homomorphismsMathworldPlanetmathPlanetmathPlanetmath.

Title categories of Polish groups and Polish spaces
Canonical name CategoriesOfPolishGroupsAndPolishSpaces
Date of creation 2013-03-22 18:25:04
Last modified on 2013-03-22 18:25:04
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 19
Author bci1 (20947)
Entry type Topic
Classification msc 54H05
Classification msc 28A05
Classification msc 28A12
Classification msc 28C15
Synonym subcategory of topological groupoid category
Related topic Category
Related topic Metrizable
Related topic CategoryOfBorelSpaces
Related topic PolishSpacesUpToBorelIsomorphism
Related topic TopologicalGroup2
Related topic MeasureSpace
Related topic PolishSpace
Defines Polish group homomorphism
Defines metrizable topological groups