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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.

Apply part i toThe element $\alpha$ generates the additive cyclic group $\mathbb{Z}/p^n\mathbb{Z}$ exactly when it is coprime to $p$, which is equivalent to its reduction modulo $p$ being nonzero. Hence $\{\alpha \}$ is a topological generating set of the additive group ${\mathbb{Z}}_p$ if and only if $\alpha \not\equiv 0(\mathrm{mod}p)$.

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