Let be an algebraic (infinite) Galois extension with group .

We can define the Krull topology on the group .

We wish to show this topology is totally disconnected.

Let some subset of .

Then there exists a subgroup of the form where is a finite Galois extension with (this is just a property on how this topology is defined).

Then, .

Thus, is a union of two non-empty open sets.

My question is why are those sets open?