Abstract:The aim of the paper is to prove that in the unbounded Urysohn universal space $\mathbb U$ there is a functor of extension of $\Lambda $-isometric maps (i.e.\ dilations) between central subsets of $\mathbb U$ to $\Lambda $-isometric maps acting on the whole space. Special properties of the functor are established. It is also shown that the multiplicative group $\mathbb R \setminus \{0\}$ acts continuously on $\mathbb U$ by $\Lambda $-isometries.
Keywords: Urysohn's universal space, ultrahomogeneous spaces, functor, extensions of isometries
AMS Subject Classification: 54C20 54E40 54E50