Topology of the isometry group of the Urysohn space

Details Topology of the isometry group of the Urysohn space Topology of the isometry group of the Urysohn space

2009-07-02

arxiv.org/abs/0907.0413v2

Mathematics

Metric Geometry

Scientific paper

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.

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