Geometry in Urysohn's universal metric space
Metric Geometry
2007-05-23 v2
Abstract
Recently, much interest was devoted to the Urysohn universal metric space U and its isometries; this paper is a contribution to this field of research. In particular, we study some properties of isometries of U, and prove the following result, which answers a question of Urysohn: if X is a subset of U is such that, for any X' which is isometric to X there exists an isometry of U which maps X to X', then X is compact (the converse was already well-known). We also answer in the negative a question of Clemens, proving that any Polish metric space is isometric to the set of fixed points of some isometry of U.
Cite
@article{arxiv.math/0505508,
title = {Geometry in Urysohn's universal metric space},
author = {Julien Melleray},
journal= {arXiv preprint arXiv:math/0505508},
year = {2007}
}
Comments
v2: corrected some grammatical errors and typos, and added a part dealing with properties of the sets of fixed points of isometries