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 ℓ²(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 ∩ B pointwise.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-3-4,
author = {Julien Melleray},
title = {Topology of the isometry group of the Urysohn space},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {273-287},
zbl = {1202.22001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-3-4}
}
Julien Melleray. Topology of the isometry group of the Urysohn space. Fundamenta Mathematicae, Tome 209 (2010) pp. 273-287. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-3-4/