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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $K \\subseteq X$.\nThen $K$ is [[compact|00R0X1WGPX]] if and only if $K$, with the [[subspace metric|0T2WYDTRP2]], is [[complete|08EWT7J0SC]] and $K$ is [[totally bounded|0WE5BBHGM6]].\n\n## Remarks\n\nProof idea of the converse: in a totally bounded set, a diagonal argument gives a Cauchy subsequence of each sequence. Completeness gives a limit. Thus $K$ is sequentially compact, and so [[compact|0K8SGQSV7G]].\nIn a complete space $X$, a subset is compact if and only if it is closed and totally bounded.\nThe [[Heine–Borel theorem|07PFYV65MB]] follows, because bounded subsets of $\\mathbb{R}^n$ are totally bounded.\n\n## Explanations\n\n### L1 human\n\nCompactness in a metric space splits into two parts: no holes, and finitely many small pieces for each size.\nThe interval $(0, 1)$ is totally bounded but has holes at its ends. The real line has no holes but is not totally bounded.\n",
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