I want to start study infinite galois theory and I have a problem with a basic definition:
The definition of Krull topology. So $L/K$ is a Galoisextension, $F$ is a subfield $K\subseteq F\subseteq L$ and $\mathcal{N}:=\{Gal(L/F)\subseteq Gal(L/K)\text{ }|\text{ }[F:K]<\infty, F/K \text{ galois}\}$.
We define an open neighborhood basis of $\sigma$ as a set of $\sigma\mathcal{N}$. Now I want to verify that this defines a topology. We have $U\subseteq Gal(L/K)$ is open if and only if there exists for every $\sigma\in Gal(L/K)$ an Element $S$ in the neighborhood basis such that $S\subseteq U$.
So I don't understand why the empty set is open can somebody explain it please ?
This doesn't need to be true for every $\sigma$ in $\mathrm{Gal}(L/K)$, it needs to be true for every $\sigma$ in $U$. If $U$ is the empty set, this is true vacuously - every $\sigma$ in $U$ admits some $S$ such that blah blah blah since there are no elements in $U$ to begin with.