Galois theory tells us that when we have a finite field extension L/K satisfying certain conditions, we have an automorphism group Gal(L/K) and a natural bijection between its subgroups and the intermediate fields of L/K.
In this talk, I will extend the above construction to infinite extensions. The naive way of doing so, saying that subgroups correspond to intermediate fields, doesn't quite work, because, as we will see, there are many more subgroups. Instead, we will impose a topology on Gal(L/K), defined roughly by considering L/K together with all its finite intermediate extensions. This allows us to construct our bijection.
I will assume familiarity with finite Galois theory and basic point-set topology, but nothing more.