Is the commutator subgroup of a profinite group closed? Let $G$ be a profinite group, $[G,G]=\{ghg^{-1}h^{-1}|g,h\in G\}$ is a subgroup of $G$. Is $[G,G]$ closed?
In the case we are interested, $G$ is the absolute galois group of a local field.

In general is not true that for a profinite group $G$ the derived subgroup $G'$ is closed; But if $G$ is a pro-$p$-group, with $p$ a prime, and if $G$ is finitely generated then it becomes true. For example this is true for the group of $p$-adic integer $\mathbb{Z}_{p}$.
