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  "body": "\n## Statement\n\nLet $n \\ge 0$ and let $A_1, \\ldots, A_{n+1}$ be closed subsets of the [[$n$-sphere|T62369RFH6]] $S^n \\subset \\R^{n+1}$ with $A_1 \\cup \\cdots \\cup A_{n+1} = S^n$.\nThen some $A_i$ contains a pair of antipodal points $x$ and $-x$.\n\n## Remarks\n\nThe theorem follows from the [[Borsuk–Ulam theorem|QHJ0GBQE2E]].\nWe may assume that every $A_i$ is nonempty, by replacing an empty one with a copy of a nonempty one.\nLet $f(x) = (d(x, A_1), \\ldots, d(x, A_n)) \\in \\R^n$, where $d(x, A)$ is the [[distance from $x$ to $A$|0WQ19TJV24]], and choose $x$ with $f(x) = f(-x)$.\nIf some coordinate of $f(x)$ is $0$, then $x$ and $-x$ both lie in the closed set $A_i$; otherwise neither lies in $A_1 \\cup \\cdots \\cup A_n$, so both lie in $A_{n+1}$.\nThe same conclusion holds when all the $A_i$ are open.\n\nExample: for $n = 1$, if the circle is the union of two closed sets, one of them contains two opposite points.\n\nThe number $n + 1$ cannot be raised to $n + 2$.\nProject the facets of a regular $(n+1)$-simplex centred at the origin radially onto $S^n$.\nThis covers $S^n$ by $n + 2$ closed sets, none of which contains an antipodal pair.\n\n## Explanations\n\n### L1 human\n\nIf you paint a sphere with $n + 1$ colours, where each colour region is closed, then some colour appears at two opposite points.\nOn an ordinary globe, painted with three colours, some colour is used at two antipodal places.\n\n### L2 human\n\nThe theorem is one of several equivalent forms of the Borsuk–Ulam theorem, and it can be proved from it and conversely.\nIt gives a short proof of Kneser's conjecture on the chromatic number of Kneser graphs.\nMore generally, covering and coincidence statements of this kind are a main entry point of topological methods in combinatorics.\n",
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