Stabilizers of closed sets in the Urysohn space

Volume 189 / 2006

Julien Melleray Fundamenta Mathematicae 189 (2006), 53-60 MSC: Primary 22F50; Secondary 51F99, 22A05. DOI: 10.4064/fm189-1-4


Building on earlier work of Katětov, Uspenskij proved in \cite{Uspenskij2} that the group of isometries of Urysohn's universal metric space $\mathbb U $, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group $G$, there exists a closed subset $F$ of $\mathbb U$ such that $G$ is topologically isomorphic to the group of isometries of $\mathbb U$ which map $F$ onto itself.


  • Julien MellerayÉquipe d'Analyse Fonctionnelle
    Université Paris 6
    Boîte 186, 4 Place Jussieu
    75252 Paris Cedex 05, France

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