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Geometry in Urysohn's universal metric space

Julien Melleray
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Abstract

Recently, 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.

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Julien Melleray. 2005-06-22. Geometry in Urysohn's universal metric space. https://arxiv.org/abs/math/0505508

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