Some geometric and dynamical properties of the Urysohn space (Q935257)

This is a survey article about the geometry and dynamical properties of the Urysohn space. Most of the results presented are part of the author's Ph.D. thesis and were published in the articles [Fundam. Math. 189, No.~1, 53--60 (2006; Zbl 1089.22019)] and [Topology Appl. 154, 384--403 (2007; Zbl 1113.54017)]. Amongst others, the following topics are dealt with in the paper: One-point metric extensions (Katětov maps); Construction of the Urysohn space; Simple geometric properties of \(\mathbb U\); Algebraic and topological properties of \(\text{Iso}(\mathbb U)\); Action of \(\text{Iso}(\mathbb U)\) on \(\mathcal F(\mathbb U)\); Conjugacy in \(\text{Iso}(\mathbb U)\) and fixed points of isometries. Some of the results presented are new, including the construction of translations and the proof that \(\text{Iso}(\mathbb U)\) is not a divisible group.