Let $L/K$ be a galois extension, we defined an open neighborhood basis for every $\sigma\in Gal(L/K)$ as $\sigma Gal(L/E)$ with $K\subset E\subset L$ and $E/K$ finit and galois. Now i want to verify, that this defines a topology.
So every $O\subset Gal(L/K)$ is open if and only if for all $\sigma\in O$ exist one Galoisgroup $Gal(L/E)$ with $K\subset E\subset L$ and $E/K$ finit, galois such that $\sigma Gal(L/E)\subset O$.
I had shown that the intersection of open sets is open with the Composite field. Now my question is how can i show that the infinit union of open sets is open.
Let $O_i$ some open sets then for $\sigma \in \cup_i O_i$ there exists $k$ such that $\sigma\in O_k$ and we can say $\sigma Gal(L/E_k)\subset O_k$ ($E_i$ field depending on $i$). But i need an extension for all $\sigma\in \cup_i O_i$ i think about $Gal(L/\cap E_i)$ is this correct and how can i verify that ?