 Topological generator

A single topological generator of a topological group $G$ is an element $g$ whose cyclic subgroup is dense:More generally, a tuple $g_1,\dots ,g_d$ consists of topological generators when the subgroup they jointly generate is dense. Membership in such a tuple does not imply that one element alone generates densely.

In the additive group ${\mathbb{Z}}_p$ of p-adic integers, the element one is a topological generator: ordinary integers surject onto every finite quotient $\mathbb{Z}/p^n\mathbb{Z}$. The element $p$ generates only the proper closed subgroup $p{\mathbb{Z}}_p$. The profinite group ${\mathbb{Z}}_p^2$ has a two-element topological generating set but no single topological generator, since its quotient $(\mathbb{Z}/p\mathbb{Z}{)}^2$ is not cyclic. ^[ref]