 Finite-quotient criterion for topological generation

For a profinite group $G={\underleftarrow{\lim }}_j{G}_j$, a subset $S\subseteq G$ is a topological generating set exactly when $⟨p_j(S)⟩=p_j(G)$ in every finite quotient. Indeed, a subgroup of a profinite group is dense exactly when its image in every finite continuous quotient is surjective.