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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]].\nA completion of $X$ is a pair $(\\hat{X}, \\iota)$ of a [[complete metric space|08EWT7J0SC]] $\\hat{X}$ and an [[isometry|0878DN3WCS]] $\\iota \\colon X \\to \\hat{X}$ with [[dense|0D198GPMTJ]] image $\\iota(X)$.\n\n## Remarks\n\nEach metric space has a completion. One construction takes the Cauchy sequences of $X$ and identifies two of them when the distance between their terms tends to $0$.\nThe completion is unique: for two completions $(\\hat{X}, \\iota)$ and $(\\hat{X}', \\iota')$, there is a unique isometric isomorphism $\\varphi \\colon \\hat{X} \\to \\hat{X}'$ with $\\varphi \\circ \\iota = \\iota'$.\nThe completion of $\\mathbb{Q}$ with the usual metric is $\\mathbb{R}$. This is the [[construction of the reals from Cauchy sequences|0D8HE681SC]].\nIf $X$ is already complete, then $X$ with the identity map is a completion of $X$.\n\n## Explanations\n\n### L1 human\n\nThe completion fills the holes of a metric space with new points, and it adds no more points than it needs.\nEach new point is a limit of a Cauchy sequence of old points.\n",
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