2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75302Recently, much interest was devoted to the Urysohn universal metric space U and its isometries; this paper is a contribution to this field of research. In particular, we study some properties of isometries of U, and prove the following result, which answers a question of Urysohn: if X is a subset of U is such that, for any X' which is isometric to X there exists an isometry of U which maps X to X', then X is compact (the converse was already well-known). We also answer in the negative a question of Clemens, proving that any Polish metric space is isometric to the set of fixed points of some isometry of U.v2: corrected some grammatical errors and typos, and added a part dealing with properties of the sets of fixed points of isometriesMetric Geometry51F (primary) 99 (secondary)text