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  "body": "\n## Definition\n\nLet $G$ be a [[Hausdorff topological group|7GKCGQ7215]].\nThe *two-sided uniformity* of $G$ is the coarsest uniform structure finer than both the left and the right [[uniform structures|P89SJ7JWJE]] of $G$.\n$G$ is *complete* if every [[Cauchy filter|WZC4J62B1G]] for the two-sided uniformity converges.\nA *completion* of $G$ is a pair $(\\hat G, \\iota)$ such that:\n\n1. $\\hat G$ is a complete Hausdorff topological group;\n2. $\\iota \\colon G \\to \\hat G$ is an [[isomorphism of topological groups|T3A3JTW6Q4]] from $G$ onto the subgroup $\\iota(G)$ with the subspace topology;\n3. $\\iota(G)$ is dense in $\\hat G$.\n\n## Remarks\n\nEvery Hausdorff topological group has a completion, and any two completions are isomorphic by a unique isomorphism compatible with the maps $\\iota$.\nExample: $(\\R, +)$ with the inclusion of $(\\Q, +)$.\nThe group $\\R$ is complete, $\\Q$ carries the subspace topology, and $\\Q$ is dense in $\\R$.\nNon-example: $(\\R, +)$ with the inclusion of $(\\Z, +)$ fails clause 3, since $\\Z$ is closed in $\\R$ and not dense.\nFor an abelian group the left, right and two-sided uniformities agree.\nIn general the completion for the left uniformity alone need not be a group.\n",
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