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  "body": "\n## Summary\n\nThe completion of a [[Hausdorff topological group|7GKCGQ7215]] $G$ is a complete Hausdorff topological group that contains $G$ as a dense subgroup.\nCompleteness refers to the [[uniform structure|P89SJ7JWJE]] of the group, so it makes sense without a metric.\nEvery Hausdorff topological group has a completion for its two-sided uniformity, unique up to isomorphism; this is the Raikov completion.\nFor abelian groups this is the familiar completion with respect to Cauchy nets.\n\n## Explanations\n\n### L1 human\n\nThe rational numbers have holes: a sequence of rationals can get closer and closer together without having a rational limit.\nFilling in all such holes in the additive group of rationals gives the real numbers.\nThe completion of a topological group does the same thing in general: it adds limits for all the Cauchy nets, and the group law extends to the new points.\nFor example, the completion of the integers with the $p$-adic topology is the group of $p$-adic integers.\n",
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