The Urysohn universal metric space is homeomorphic to a Hilbert space
General Topology
2021-08-27 v1
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).
Cite
@article{arxiv.math/0309101,
title = {The Urysohn universal metric space is homeomorphic to a Hilbert space},
author = {Vladimir Uspenskij},
journal= {arXiv preprint arXiv:math/0309101},
year = {2021}
}
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5 pages