If I'm not mistaken, the space $\mathcal{T}$ of all complete theories of a countable language is a compact Hausdorff space, and moreover it is second-countable, since it has as its base the sets of the form $\langle \phi \rangle = \{T \in \mathcal{T} \; | \; T \models \phi\}$, which is countable provided that the language is countable. If I'm not mistaken (and I may very well be!), any space that satisfies these three conditions is metrizable.
Question: Is there any interesting metric on this space?