Definable Functions in Urysohn's Metric Space

TL;DR

This paper characterizes definable functions in the Urysohn sphere, showing they are either projections or have relatively compact ranges, and explores implications for natural functions in this metric space.

Contribution

It establishes a complete classification of definable functions in the Urysohn sphere and demonstrates non-definability of many natural functions.

Findings

Definable functions are projections or have relatively compact range

Many natural functions in the Urysohn sphere are not definable

Provides topological properties of the range of definable functions

Abstract

Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function from U^n to 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.