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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f \\colon S^n \\to \\R^n$ be a [[continuous map|0VGFHTM75Y]] on the [[$n$-sphere|T62369RFH6]] $S^n \\subset \\R^{n+1}$.\nThen there is a point $x \\in S^n$ with $f(x) = f(-x)$.\n\n## Remarks\n\nAn equivalent form says that there is no continuous odd map $g \\colon S^n \\to S^{n-1}$, that is, no map with $g(-x) = -g(x)$ for all $x$.\nIndeed, if $f(x) \\neq f(-x)$ for all $x$, then $g(x) = (f(x) - f(-x)) / \\norm{f(x) - f(-x)}$ is such a map.\nOne proof uses the fact that an odd map $S^{n-1} \\to S^{n-1}$ has odd [[degree|TMFG2P912R]].\nThe restriction of an odd map $S^n \\to S^{n-1}$ to the equator is an odd map that extends over a hemisphere.\nSo it is null-homotopic and has degree $0$, a contradiction.\n\nExample: for $n = 1$, let $h(x) = f(x) - f(-x)$ on the circle.\nThen $h(-x) = -h(x)$, so $h$ takes values of both signs or vanishes, and by the intermediate value theorem it has a zero.\n\nExample of use: at any moment there are two antipodal points on the Earth with the same temperature and the same air pressure, if both are continuous functions on $S^2$.\nThe theorem fails for maps $S^n \\to \\R^{n+1}$: the inclusion $S^n \\subset \\R^{n+1}$ takes different values at $x$ and $-x$.\n\n## Explanations\n\n### L1 human\n\nIf you squash a sphere flat into a space of one dimension less, two opposite points of the sphere always land on the same spot.\nFor a balloon pressed flat onto a table, some pair of opposite points ends up one on top of the other.\n\n### L2 human\n\nThe theorem is the starting point of the use of topology in combinatorics and geometry.\nIt implies the [[ham sandwich theorem|GDXX60SNG5]], the Lusternik–Schnirelmann covering theorem, and Lovász's computation of the chromatic number of Kneser graphs.\nIts combinatorial counterpart is [[Tucker's lemma|PADZ7D9KBT]], from which it can also be proved.\n",
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  "name": "Borsuk–Ulam theorem",
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