Topology of the isometry group of the Urysohn space
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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2009
- DOI:
- 10.48550/arXiv.0907.0413
- arXiv:
- arXiv:0907.0413
- Bibcode:
- 2009arXiv0907.0413M
- Keywords:
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- Mathematics - Metric Geometry