Applications of topology to logic?

Question

At Ieke Moerdijk's homepage, one can read that his research interests include "applications of topology to mathematical logic".

I know very few such applications (essentially I only know topological semantics for intuitionistic logic, and sheaves, but I wouldn't call the last one "applications of topology to logic", rather "topos theory" or something in the likes of it), and am surprised that someone would list this as a research interest: I must therefore be mistaken and missing a lot of things.

Does anyone have any idea what this could mean, what some applications of topology to mathematical logic are ? (not necessarily specifically those that Moerdijk studies)

Answer 1

One example, is Stone space. https://en.wikipedia.org/wiki/Type_(model_theory)#Stone_spaces

You can consider the set of $n$ types as a topological space. You can then use theorems from topology to study these types. Futhermore, I can't find a reference but I believe (correct me if wrong) that you can put a group structure on this space, and resulting Haar measure and bunch of other stuff.

The omitting types theorem is a result that is stated in terms of Stone space.

Answer 2

This is more or less exactly what I'm doing.

The typical place where topology occurs in model theory are the type spaces mentioned by Zachary, and omitting types is probably a result in pure model theory that is proved directly using topology, and more precisely, the Baire category theorem.

Most connections between topology and model theory are tied, directly or indirectly, to the topology on type spaces. They are not always Stone spaces: in continuous logic, they are not totally disconnected, and (for metric structures) they have a natural metric.

A slightly different flavour of topology occurring in model theory is the definable topology (of which there are several inequivalent notions, but nevermind that for now). The simplest example of that can be found in the real closed and p-adically closed fields, where we have a definable "metric" (albeit with possibly nonstandard values). This can be used to show, for example, that a "definably compact" group in a real closed field is "definably amenable" (i.e. admits a finitely additive invariant measure on the set of all definable subsets). Roughly speaking, we obtain this measure by passing to a canonical compact Hausdorff quotient and pulling back the Haar measure which can be found there.

In general, with each definable group $G$, we can associate certain compact topological groups $G/G^0$, $G/G^{00}$ and $G/G^{000}$ (there are some nuances related to the parameters we allow, but it's not important here), of which the first is profinite and the second is Hausdorff (one of the results of my thesis is expressing the third as the quotient of a compact Hausdorff group), which can then be used to understand the properties of $G$ itself.

Furthermore, if we interpret formulas as $0-1$ valued functions (or just as functions, in case of continuous logic), then the standard notions (like stability) translate to topological properties of families of continuous functions (like weak compactness). The reason for this is that the underlying combinatorics are essentially the same. Nonetheless, the topological point of view can provide new insight into the logical phenomena.