= Open subgroups of the p-adic integers
{title2=$H\text{ open}\ \Longleftrightarrow\ [\mathbb Z_p:H]<\infty$}
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.
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