Urysohn space; bilipschitz homeomorphism; modulus of continuity; reconstruction theorem; extension theorem
We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. As an application, we prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.
 Fonf, V. P., Rubin, M.: Reconstruction theorem for homeomorphism groups without small sets and non-shrinking functions of a normed space
. Preprint in Math Arxiv. Available at http://arxiv.org/abs/math/0510120
 Huhunaišvili, G. E.: On a property of Uryson's universal metric space
. Dokl. Akad. Nauk SSSR (N.S.) 101 (1955), 607-610 Russian. MR 0072454
 Urysohn, P. S.: Sur un espace métrique universel I, II. Bull. Sci. Math. 51 (1927), 43-64, 74-90.