  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. The proof is basedon
a lemma about extensions of metric spaces by finite metric spaces, which wealso
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 the intersection of A and B pointwise.
