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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]].\nAn *open subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $H$ of $G$ that is an [[open set|02BH3MCBQB]] of $G$.\n\n## Remarks\n\nExample: in $\\R \\times \\Z$ with the product topology, where $\\Z$ is discrete, the subgroup $\\R \\times \\set{0}$ is open, because $\\set{0}$ is open in $\\Z$.\nNon-example: $\\Z$ is a subgroup of $(\\R, +)$ but it is not open, since no interval around $0$ lies in $\\Z$.\nA subgroup $H$ is open if and only if it is a [[neighbourhood|0ZCY0Z80B6]] of the identity $e$.\nIndeed, if $U \\subseteq H$ is a neighbourhood of $e$, then $hU \\subseteq H$ is a neighbourhood of each $h \\in H$.\n",
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  "name": "Open subgroup",
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