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  "body": "\n## Statement\n\nLet $G$ be a Hausdorff [[locally compact group|AHV05ZBCYW]] that is [[totally disconnected|AW3J9AQSA0]].\nThen every [[neighbourhood|0ZCY0Z80B6]] of the identity $e$ contains a [[compact|0R6EWFKRCF]] [[open subgroup|QYCFH68RD1]] of $G$.\n\n## Remarks\n\nThe proof has two steps.\nA locally compact Hausdorff totally disconnected space has a basis of compact open sets, so the given neighbourhood contains a compact open set $V \\ni e$.\nBy compactness of $V$ there is a symmetric open neighbourhood $W$ of $e$ with $V W \\subseteq V$.\nThen all products of elements of $W$ lie in $V$, so the subgroup generated by $W$ is open and lies in $V$.\nAn open subgroup is closed, so this subgroup is compact.\n\nExample: in the $p$-adic numbers $\\Q_p$, the subgroups $p^k \\Z_p$ with $k \\ge 0$ are compact and open, and they shrink to $0$.\nIn $\\mathrm{GL}_n(\\Q_p)$, the congruence subgroups $I + p^k \\mathrm{M}_n(\\Z_p)$ with $k \\ge 1$ play the same role.\n\nLocal compactness is necessary.\nThe group $\\Q$ with the topology of $\\R$ is totally disconnected and not locally compact.\nAn open subgroup of it contains an interval around $0$, so it is all of $\\Q$, which is not compact.\n\n## Explanations\n\n### L1 human\n\nIn a totally disconnected group, points cannot be joined by any connected piece.\nVan Dantzig's theorem says that, when the group is also locally compact, there are arbitrarily small compact subgroups that are also open.\nThey play the role of small balls around the identity.\nFor example, in the $p$-adic numbers the balls $p^k \\Z_p$ are themselves subgroups.\n\n### L2 human\n\nThe theorem is the starting point of the structure theory of totally disconnected locally compact groups.\nIt shows that such a group is a union of cosets of a compact open subgroup $U$, with a discrete coset space $G/U$.\nA totally disconnected compact group is profinite, so these groups are locally profinite.\nCompact open subgroups are used to define [[Cayley–Abels graphs|QWG66JE1JE]] and the [[Hecke algebras|6H3WCCHZB5]] of $p$-adic groups.\n",
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