Definable functions in Urysohn's metric space (Q358931)

The paper studies definable operations on the Urysohn metric space of diameter one \(\mathfrak U\) in the framework of continuous model theory.NEWLINENEWLINE It is proved that any definable map \(f:\mathfrak{U}^n\rightarrow\mathfrak U\) is either a projection or else has a relatively compact range. In the second case, its range is included in the closure of the set of parameters used to define \(f\). As a consequence, most of the usual such maps are not definable or else reduce to trivial cases. For example, there is no definable group operation on \(\mathfrak U\). Also, if \(n=1\) and \(f\) is either surjective or open or an isometric embedding, it must be the identity map.NEWLINENEWLINE Several similar consequences are obtained in this respect.