I have seen infinite Galois theory. I have observed that most sources prove that the Galois group (say, of an infinite extension) is compact by using the homeomorphic embedding of the Galois group inside the product of Galois groups of finite Galois sub-extensions and invoking Tychonoff's theorem.
Since the topology on the (infinite) Galois group is already known explicitly, could one come up with a proof that does not use Tychonoff's theorem? Or the embedding of the Galois group inside the product? A proof that it more intrinsic?
One can prove that Krull topology is precisely the induced subspace topology of the product space of finite Galois groups where we equip all the latter with the discrete topology. This subspace is the space of coherent sequences, also called the inverse limit (or rather a direct construction thereof).
This is the most natural way of studying the infinite Galois groups as they carry the structure of an inverse limit anyway. One can, of course, invoke Tychonoff to prove that the direct product of finite discrete spaces is compact. But this is not necessary and there is an elementary argument.
The main idea is to combine that the product topology has a open sets products of open sets with finite support, meaning that only finitely many factors are proper subspaces, with the fact that all the factors of the direct product are finite spaces. Hence any open set in an arbitrary open cover can be completed to a finite subcover quite trivially. I leave the details to you.
EDIT: This works not quite as I describe but the changes necessary are minor: One can restrict to checking compactness on a basis for the topology (per this post; although the idea is extremely simple) and this is then done as above.
I will leave you with this PDF. Although it features the same mantra of using Tychonoff, the rest of it develops infinite Galois theory directly from a simple minded definition of the Krull topology. This is in stark contrast to more abstract-algebraic treatments which often heavily rely on inverse limits and such.