Definable functions in Urysohn’s metric space

Abstract

Let $\mathfrak{U}$ denote the Urysohn sphere and consider $\mathfrak{U}$ as a metric structure in the empty continuous signature. We prove that every definable function $\mathfrak{U}^{n}\to\mathfrak{U}$ is either a projection function or else has relatively compact range. As a consequence, we prove that many functions natural to the study of the Urysohn sphere are not definable. We end with further topological information on the range of the definable function in case it is compact.