Topometric characterization of type spaces in continuous logic

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 $\{p:d(p,U)<\epsilon \}$ is open for every open $U$ and $\epsilon >0$). Furthermore, we show that this can always be accomplished with a stable theory.