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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[compact|0QPHW703PV]] [[metric space|063W5JMY85]] and let $(U_i)_{i \\in I}$ be an [[open cover|0NVPNC12JY]] of $X$.\nThen there is a real $\\delta \u003e 0$ such that each subset $A \\subseteq X$ with [[diameter|0692CGQF00]] $\\operatorname{diam}(A) \u003c \\delta$ is contained in $U_i$ for some $i \\in I$.\n\n## Remarks\n\nSuch a number $\\delta$ is a Lebesgue number of the cover.\nEquivalently, for each $x \\in X$, the ball $B(x, \\delta / 2)$ lies in one of the sets $U_i$.\nCompactness is needed: the cover of $(0, 1)$ by the sets $(1/n, 1)$ has no Lebesgue number.\n\n## Explanations\n\n### L1 human\n\nIn a compact metric space, each open cover has a uniform size: each small enough set fits completely into one set of the cover.\nYou can think of $\\delta$ as a safe size for a window that you move around the space.\n",
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