Open subgroups of the p-adic integers ($H\text{ open}⟺[{\mathbb{Z}}_p:H]<\infty$)

The cosets $a+p^n{\mathbb{Z}}_p$ form a neighborhood basis in the additive group of p-adic integers. An open subgroup contains $p^n{\mathbb{Z}}_p$ for some $n$, so has finite index. Conversely, if the index is $m$, the finite quotient is annihilated by $m$, giving $m{\mathbb{Z}}_p\subset H$. Multiplication by the unit part of $m$ is bijective, so $m{\mathbb{Z}}_p=p^{v_p(m)}{\mathbb{Z}}_p$ is open. This proof applies to abstract subgroups without assuming beforehand that they are closed.