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Stabilizers of closed sets in the Urysohn space

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Melleray J.

Fundamenta Mathematicae

2006

tom 189

nr 1

53-60

Building on earlier work of Katětov, Uspenskij proved in [8] that the group of isometries of Urysohn's universal metric space 𝕌, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of 𝕌 such that G is topologically isomorphic to the group of isometries of 𝕌 which map F onto itself.

Topology of the isometry group of the Urysohn space

100%

Melleray J.

Fundamenta Mathematicae

2010

tom 207

nr 3

273-287

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.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Melleray J.

1 2010

1 2006

Wyniki wyszukiwania

Stabilizers of closed sets in the Urysohn space

Topology of the isometry group of the Urysohn space