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The Urysohn universal metric space is homeomorphic to a Hilbert space

Vladimir Uspenskij
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Abstract

The Urysohn universal metric space U is characterized up to isometry by the following properties: (1) U is complete and separable; (2) U contains an isometric copy of every separable metric space; (3) every isometry between two finite subsets of U can be extended to an isometry of U onto itself. We show that U is homeomorphic to the Hilbert space l_2 (or to the countable power of the real line).

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Vladimir Uspenskij. 2003-09-05. The Urysohn universal metric space is homeomorphic to a Hilbert space. https://arxiv.org/abs/math/0309101

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