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  "body": "\n## Summary\n\nAn open subgroup of a [[topological group|92700QAB5Y]] is a [[subgroup|0X1JXT47GA]] that is an open set.\n[[Open subgroups are also closed|DKX3EAVQX6]], so they are unions of connected components.\nThey are central in the study of totally disconnected groups.\nIn a profinite group the open normal subgroups form a base of neighbourhoods of the identity.\nA connected topological group has no open subgroup other than itself.\n\n## Explanations\n\n### L1 human\n\nAn open subgroup is a subgroup that contains a whole neighbourhood of each of its points.\nBecause the group can be shifted around, it is enough that it contains a neighbourhood of the identity.\nFor example, in the $p$-adic integers the multiples of $p^n$ form an open subgroup for each $n$.\nIn the real line under addition, the only open subgroup is the whole line.\n",
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  "markdown": "https://tamnd.github.io/math2/node/QYCFH68RD1.md",
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  "name": "Open subgroup",
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