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Géométrie de l'espace d'Urysohn et théorie descriptive des ensembles

Abstract : The starting point of this thesis was the study of the geometric properties of a remarkable Polish metric space, built by Urysohn in 1925. This space was almost forgotten for 60 years but has been more and more studied ever since Katětov gave another construction of it in 1986. This new construction is based on the space E(X) of Katětov maps on a metric space X, and these maps are the main tool we use for our study. We provide a characterization of the Polish metric spaces X such that E(X) is separable, then use these maps to show (answering a question of A.S Kechris) that any compact metrizable group is isomorphic to the isometry group of a compact metric space. We use similar techniques to study different geometric properties of the Urysohn space and its isometry group. We also apply our methods to the study of some “definable” classification problems; most notably we compute the Borel complexity of the relation of (linear) isometry between separable Banach spaces.
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Contributor : Julien Melleray <>
Submitted on : Monday, February 27, 2006 - 6:34:55 PM
Last modification on : Wednesday, December 9, 2020 - 3:12:06 PM
Long-term archiving on: : Saturday, April 3, 2010 - 8:05:51 PM


  • HAL Id : tel-00011694, version 1


Julien Melleray. Géométrie de l'espace d'Urysohn et théorie descriptive des ensembles. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2005. Français. ⟨tel-00011694⟩



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