$p$-adic Galois reduction map is open? Let $K$ be a finite extension of $\mathbb{Q}_p$, and $k$ be the residue field. Then we have the continuous surjective map $G_K\rightarrow G_k\cong \hat{\mathbb{Z}}$.
Question: Is this map a open map between the two topological groups?
The motivation is that if this map is open, then I can deduce that the Weil group $W$ (as the inverse image of $\mathbb{Z}$) is dense in $G_K$ if we consider the subspace topology of $W$ in $G_K$. Because the inverse image of a dense subset in a continous open map between topological groups is also dense!
Thanks!

Hints:
Galois groups are profinite groups. Profinite groups are compact Hausdorff; every continuous map between compact Hausdorff spaces is closed.
Further, a closed normal subgroup of a profinite group is open if and only if it has finite index in the group. And, a profinite group has a basis of neighbourhoods of $1$ consisting of open normal subgroups.
