categories of Polish groups and Polish spaces

0.1 Introduction

Definition 0.1.

Let us recall that a Polish space is a separable, completely metrizable topological space, andthat 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 provableresults (http://planetmath.org/PolishSpacesUpToBorelIsomorphism):

•
“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.

0.2 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 morphismsthe 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.

单词	CategoriesOfPolishGroupsAndPolishSpaces
释义	categories of Polish groups and Polish spaces 0.1 Introduction Definition 0.1. Let us recall that a Polish space is a separable, completely metrizable topological space, andthat 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 provableresults (http://planetmath.org/PolishSpacesUpToBorelIsomorphism): • “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. 0.2 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 morphismsthe 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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