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arXiv · 0907.0413

Topology of the isometry group of the Urysohn space

Julien Melleray
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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. 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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Julien Melleray. 2009-11-21. Topology of the isometry group of the Urysohn space. https://arxiv.org/abs/0907.0413

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