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单词 CategoriesOfPolishGroupsAndPolishSpaces
释义

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, andthat 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 provableresults (http://planetmath.org/PolishSpacesUpToBorelIsomorphism):

  • 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 morphismsthe 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.

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