Definable Functions in Urysohns Metric Space

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.

💡 Research Summary

The paper investigates definable functions on the Urysohn sphere U when U is regarded as a metric structure in the empty continuous signature—that is, a language containing no function or relation symbols other than the metric itself. The authors prove a striking dichotomy: any definable map f : Uⁿ → U is either a coordinate projection (i.e., f(x₁,…,xₙ)=x_i for some i) or its image is relatively compact, meaning that the closure of f(Uⁿ) is a compact subset of U.

The proof relies on basic tools from continuous model theory. Because the language is empty, the type space S₁(T) consists of a single orbit under the automorphism group of U, and every 1‑type is determined solely by distances to parameters. This extreme homogeneity forces any definable function to be governed by distance patterns alone. The authors first show that if the image of f is not relatively compact, then one can extract an ε‑separated infinite set in the range. Such a set would induce infinitely many distinct distance‑type realizations, contradicting the fact that, in the empty signature, only countably many distance‑types can be realized over a finite parameter set. Consequently, a non‑compact definable function must be indistinguishable from a projection onto one of the input coordinates.

If the image is relatively compact, the authors use the ultrahomogeneity and completeness of U to argue that the closure of the image is itself a compact metric subspace that embeds isometrically into U. Moreover, they derive additional topological constraints: the image must be connected, complete, and its topological dimension cannot exceed that of U. In particular, the only constant definable functions are those whose constant value is a fixed point of all automorphisms, i.e., a point definable without parameters, which does not exist in the empty signature.

From the dichotomy, several natural operations on U are shown to be non‑definable. For example, the average distance function, the metric midpoint of a pair, the barycenter of a finite tuple, and various group‑action‑derived maps (such as the displacement function of an isometry) all produce images that are not relatively compact and are not projections, hence they cannot be definable. This demonstrates that many intuitively simple geometric constructions lie outside the expressive power of continuous logic in the empty language.

The paper concludes with a discussion of the compact‑range case. The authors prove that if f(Uⁿ) is compact, then its closure is a compact, connected subspace of U that inherits the universal property of U in a restricted sense. They also pose open questions about how the picture changes when the language is enriched with a few additional symbols, or when one studies analogous phenomena in other universal homogeneous structures such as the random graph or the rational Urysohn space.

Overall, the work provides a clear and elegant characterization of definable functions on the Urysohn sphere, revealing a severe limitation: apart from trivial projections, any definable function must have a tightly bounded (compact) range. This contributes to our understanding of the interplay between homogeneity, metric geometry, and definability in continuous model theory.