Originally Posted by

**ThePerfectHacker** Let $\displaystyle K/F$ be an algebraic (infinite) Galois extension with group $\displaystyle G$.

We can define the Krull topology on the group $\displaystyle G$.

We wish to show this topology is totally disconnected.

Let $\displaystyle \phi_1,\phi_2 \in X$ some subset of $\displaystyle G$.

Then there exists a subgroup $\displaystyle N\subseteq G$ of the form $\displaystyle \text{Gal}(K/E)$ where $\displaystyle E/F$ is a finite Galois extension with $\displaystyle \phi_2\not \in \phi_1 N$ (this is just a property on how this topology is defined).

Then, $\displaystyle X = (\phi_1 N \cap X) \cup ((G-\phi_1 N) \cap X)$.

Thus, $\displaystyle X$ is a union of two non-empty open sets.

My question is why are those sets open?