Stone duality associates to each (finitary classical propositional) theory a topological space whose points are the models. The theory can be fully recovered from the space—this is a form of completeness: everything we need to know about the theory is encoded in the topology, and we can replace the bureaucracy of syntax by suitable topological notions.

Notably, ultraproducts of models can also be read on the space of models: the ultraproduct of a family of models corresponds to the limit of an ultrafilter on points. Through the Stone duality, this shows that ultraproducts encode all we need to know about the theory and explains the central role of the ultraproduct construction.

Of course, finitary classical propositional logic is very limited—we can go a lot further: geometric logic is a very powerful logic (encompassing classical first-order logic) that has a topological interpretation (actually a toposical interpretation).

In this talk I will explain how the ultraproduct operation induces a spaced-like structure on the models of a geometric theory, opening the way of a new understanding of model-theoretic results, where the ultraproduct of models become the ultraconvergence of points—in a formal way.