Topology of the isometry group of the Urysohn space

Julien Melleray. "Topology of the isometry group of the Urysohn space." Fundamenta Mathematicae 207.3 (2010): 273-287. <http://eudml.org/doc/282711>.

@article{JulienMelleray2010,
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 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.},
author = {Julien Melleray},
journal = {Fundamenta Mathematicae},
keywords = {Urysohn metric space; Polish metric space; Polish group; Katětov map},
language = {eng},
number = {3},
pages = {273-287},
title = {Topology of the isometry group of the Urysohn space},
url = {http://eudml.org/doc/282711},
volume = {207},
year = {2010},
}

TY - JOUR
AU - Julien Melleray
TI - Topology of the isometry group of the Urysohn space
JO - Fundamenta Mathematicae
PY - 2010
VL - 207
IS - 3
SP - 273
EP - 287
AB - 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.
LA - eng
KW - Urysohn metric space; Polish metric space; Polish group; Katětov map
UR - http://eudml.org/doc/282711
ER -