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Streszczenie

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.