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  "body": "\n## Statement\n\nLet $G$ be a [[topological group|92700QAB5Y]] and let $H$ be an [[open subgroup|QYCFH68RD1]] of $G$.\nThen $H$ is a [[closed set|0ZCK8DQYN1]] in $G$.\n\n## Remarks\n\nThe [[left cosets|03DEH43F4H]] $gH$ are open, because left translation by $g$ is a homeomorphism.\nThe complement of $H$ is the union of the cosets $gH$ with $g \\notin H$, so it is open, and $H$ is closed.\nConversely, a closed subgroup of finite [[index|0DTXBQPA2S]] is open, since its complement is a finite union of closed cosets.\nExample: in the $p$-adic integers $\\Z_p$, each subgroup $p^n \\Z_p$ is open, hence also closed.\nThe converse of the statement fails: $\\Z$ is a closed subgroup of $\\R$ that is not open.\n\n## Explanations\n\n### L1 human\n\nIf a subgroup of a topological group is open, it is automatically closed as well.\nThe reason is that the group splits into copies of the subgroup, all shifted, and each copy is open.\nThe part outside the subgroup is a union of these open copies, so it is open too.\n\n### L2 human\n\nA first consequence: a connected topological group has no open subgroup except itself, since an open subgroup is a nonempty set that is both open and closed.\nFor instance, a connected group is generated by any neighbourhood of the identity, because the subgroup generated by an open set is open.\nIn profinite groups the open subgroups are exactly the closed subgroups of finite index, which links the topology with finite quotients.\n",
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