# Strongly Complete Profinite Groups.

I've been reading about profinite groups and have encountered the notion of strong completeness. I.e. that a profinite group $G$ is strongly complete if it is isomorphic to it's profinite completion or equivalently, if every subgroup of finite index is open. My problem is that I am not understanding why these conditions are equivalent. I cannot find a reference for this fact; every mention of it I find states that this equivalence is "obvious." I believe the equivalence stems from the fact that if all subgroups of finite index of $G$ are open then the set of subgroups of finite index forms a fundamental system of open neighborhoods of $1$ in $G$ which allows one to reconstruct the topology. I would appreciate any help understanding this, or any references on this fact.

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