Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 2 / a / i / Solution

A subset $S\subseteq G$ topologically generates $G$ exactly whenfor every $j$. In the usual presentation by surjective finite quotients this reads $⟨p_j(S)⟩=G_j$. Indeed, a subgroup is dense exactly when its image in every finite discrete quotient is the whole quotient. This is the finite-quotient criterion for topological generation.