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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $A \\subseteq X$.\nAn open cover of $A$ is a family $(U_i)_{i \\in I}$ of [[open subsets|01ZQQDCERP]] of $X$ with $A \\subseteq \\bigcup_{i \\in I} U_i$.\nA subcover is a subfamily $(U_j)_{j \\in J}$, with $J \\subseteq I$, that is still an open cover of $A$.\nA subcover is finite when $J$ is a finite set.\n\n## Remarks\n\nThe intervals $(-n, n)$ for $n \\ge 1$ form an open cover of $\\mathbb{R}$. It has no finite subcover.\nThe intervals $(1/n, 1)$ for $n \\ge 2$ form an open cover of $(0, 1)$ with no finite subcover.\nA set $K$ is [[compact|00R0X1WGPX]] when each open cover of $K$ has a finite subcover.\n\n## Explanations\n\n### L1 human\n\nAn open cover is a collection of open sets that together cover the whole set.\nYou can think of them as patches, and a finite subcover uses only finitely many patches to cover the set.\n",
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