Topometric characterization of type spaces in continuous logic

TL;DR

This paper characterizes when a compact topometric space can be realized as a type space of a continuous first-order theory, establishing conditions involving openness of the metric and stability.

Contribution

It provides a precise topometric characterization of type spaces in continuous logic, including the existence of stable theories for such spaces.

Findings

A topometric space is a type space of a continuous theory iff it is compact with an open metric.

Any such topometric space can be associated with a stable theory.

The characterization involves the openness of the metric in the topometric space.

Abstract

We show that a topometric space $X$ is topometrically isomorphic to a type space of some continuous first-order theory if and only if $X$ is compact and has an open metric (i.e., satisfies that $\{p : d(p,U) < \varepsilon\}$ is open for every open $U$ and $\varepsilon > 0$). Furthermore, we show that this can always be accomplished with a stable theory.