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  "body": "\n## Definition\n\nA [[metric space|063W5JMY85]] $(X, d)$ is complete when each [[Cauchy sequence|0MCK8K2D7Z]] in $X$ [[converges|06JYPJ4WHH]] to a point of $X$.\n\n## Remarks\n\nThe real line $\\mathbb{R}$ with the usual metric is complete. So is $\\mathbb{R}^n$ with the Euclidean metric.\nThe rationals $\\mathbb{Q}$ and the open interval $(0, 1)$ are not complete. In $(0, 1)$, the sequence $1/n$ is Cauchy but has no limit in $(0, 1)$.\nCompleteness depends on the metric, not only on the topology: $(0, 1)$ is homeomorphic to $\\mathbb{R}$, but only $\\mathbb{R}$ is complete with the usual metric.\nEach metric space has a [[completion|0G4J7Z7WPJ]].\n\n## Explanations\n\n### L1 human\n\nA complete space has no holes: each sequence that looks convergent really has a limit in the space.\nThe rationals have holes, for example at $\\sqrt{2}$, and the reals fill them.\n",
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