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categories of Polish groups and Polish spaces∗ bci1† 2013-03-22 1:46:11 0.1 Introduction Definition 0.1. Let us recall that a Polish space is a separable, completely metrizable topological space, 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 group GT is metrizable iff GT is Hausdorff, and the identity e of GT 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 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 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 groupoids and topological groupoid homomorphisms. ∗ hCategoriesOfPolishGroupsAndPolishSpacesi created: h2013-03-2i by: hbci1i version: h41068i Privacy setting: h1i hTopici h54H05i h28A05i h28A12i h28C15i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1