Topology of the isometry group of the Urysohn space

Volume 207 / 2010

Julien Melleray Fundamenta Mathematicae 207 (2010), 273-287 MSC: 22F99, 54H11, 51F99. DOI: 10.4064/fm207-3-4

Abstract

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.

Authors

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

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