Finite-quotient criterion for topological generation
= Finite-quotient criterion for topological generation
For a <profinite group> $G=\varprojlim_jG_j$, a subset $S\subseteq G$ is a <topological generating set> exactly when $\langle p_j(S)\rangle=p_j(G)$ in every finite quotient. Indeed, a subgroup of a profinite group is dense exactly when its image in every finite continuous quotient is surjective.