Topology of the isometry group of the Urysohn space

Julien Melleray¹

¹ Université de Lyon CNRS Université Lyon 1 Institut Camille Jordan 43 blvd du 11 novembre 1918 F-69622 Villeurbanne Cedex, France

Fundamenta Mathematicae, Tome 207 (2010) no. 3, pp. 273-287

Résumé

Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space $\ell^2({\mathbb N})$. The proof is based on a lemma about extensions of metric spaces by finite metric spaces, which we also use to investigate (answering a question of I. Goldbring) the relationship, when $A,B$ are finite subsets of the Urysohn space, between the group of isometries fixing $A$ pointwise, the group of isometries fixing $B$ pointwise, and the group of isometries fixing $A \cap B$ pointwise.

Détail

DOI : 10.4064/fm207-3-4

Keywords: using classical results infinite dimensional geometry isometry group urysohn space endowed its usual polish group topology homeomorphic separable hilbert space ell mathbb proof based lemma about extensions metric spaces finite metric spaces which investigate answering question goldbring relationship finite subsets urysohn space between group isometries fixing pointwise group isometries fixing pointwise group isometries fixing cap pointwise