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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.