Topological generating set

A subset $S$ of a topological group topologically generates it when the ordinary subgroup $⟨S⟩$ is dense.

An element of a topological generating set of a topological group. A finite tuple consists of topological generators when the abstract subgroup it generates is dense in the group; it need not equal that abstract subgroup.

A topological group is topologically finitely generated when it has a finite topological generating set.

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.