The Urysohn universal metric space is homeomorphic to a Hilbert space
By V. Uspenskij. 2003
53 citations recorded by the source corpus.
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 l2 (or to the countable power of the real line).
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/35fdfa7c172b2bb6e161c6e00cc5b76732c08252
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- On subgroups of minimal topological groups (2000, background): “For more on the Urysohn space, see [12, 23, 24, 27, 29, 31, 32, 50, 51 ], and papers in this volume.”
- Structural Ramsey theory of metric spaces and topological dynamics of isometry groups (2008, unknown): “here is ϕin iso(U) whose set of fixed points in Uis isometric to U. Next, we present the structures which are supported by U. We start with the topological characterization of U: Theorem 82 (Uspenskij [93]). Uis homeomorphic to ℓ2. Next, recall that a group is monothetic if it contains a dense subgro”