I am trying to determine whether the following is true:
Let $H$ be a closed subgroup of a finitely generated free pro-$p$ group $F$. Does $F$ induce on $H$ the full pro-$p$ topology?
Stated in another way, is it true that every $p$-power index normal subgroup $U \unlhd H$ is given by the intersection $U = V \cap H$ where $V$ is a $p$-power index normal subgroup of $F$? I know that closed subgroups of free pro-$p$ groups are free, however I'm not sure if this fact is sufficient to conclude something about the topology...
Any help and direction is appreciated!
