Stabilizers of Closed Sets in the Urysohn Space

Building on earlier work of Katětov, Uspenskij proved in [9] that the group of isometries of Urysohn’s universal metric space U, endowed with the product 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 U such that G is topologically isomorphic to the group of isometries of U which map F onto itself.