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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $K \\subseteq X$.\nThe set $K$ is compact when $K$, with the [[subspace topology|0D5YHQGDJY]], is a [[compact space|0QPHW703PV]].\n\n## Remarks\n\nBy [[compactness via open covers|0P6PMJB4MA]], $K$ is compact exactly when each [[cover|0NVPNC12JY]] of $K$ by open sets of $X$ has a finite subcover.\nEach finite subset of $X$ is compact. The empty set is compact.\nIn $\\mathbb{R}$, the interval $[0, 1]$ is compact, but $(0, 1)$ and $[0, \\infty)$ are not. The [[Heine–Borel theorem|07PFYV65MB]] describes the compact subsets of $\\mathbb{R}^n$.\n\n## Explanations\n\n### L1 human\n\nA compact set behaves in many ways like a finite set.\nFor example, a continuous real function on a compact set is bounded and attains its maximum, as a function on a finite set does.\n",
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