= Topological generator of the additive group of p-adic integers
{title2=$\overline{\mathbb Z\alpha}=\mathbb Z_p$}
An element $\alpha\in\mathbb Z_p$ topologically generates the <additive group> of <p-adic integers> exactly when $\alpha\not\equiv0\pmod p$, equivalently when it is a <p-adic unit>. Topological generation means that the <closure> of the <cyclic subgroup> $\mathbb Z\alpha$ is all of $\mathbb Z_p$.
For a <p-adic unit>, <multiplication> by $\alpha$ permutes every finite <quotient group> $\mathbb Z/p^n\mathbb Z$. Thus its integer multiples meet every <coset> $a+p^n\mathbb Z_p$. These <cosets> form a <neighbourhood basis>, so the multiples are <dense>.
If instead $p\mid\alpha$, every multiple lies in the proper <closed subgroup> $p\mathbb Z_p$, making density impossible. In particular, one topologically generates $\mathbb Z_p$, whereas $p$ does not. The generated algebraic subgroup can remain <countable>; it is its <closure> that fills the uncountable <topological group>.
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