Borsuk–Ulam theorem

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The Borsuk–Ulam theorem states that any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)

The case n = 2 is often illustrated by saying that at any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures. This assumes that temperature and barometric pressure vary continuously.

The Borsuk–Ulam theorem was first conjectured by Stanislaw Ulam. It was proved by Karol Borsuk in 1933.

There is an elementary proof that the Borsuk–Ulam theorem implies the Brouwer fixed point theorem.

A stronger statement related to Borsuk–Ulam theorem is that every antipode-preserving map

$f:\mathbb{S}^n\to\mathbb{S}^n$

has odd degree.

[edit] Corollaries of Borsuk-Ulam theorem

No subset of $\mathbb{R}^{n}$ is homeomorphic to $\mathbb{S}^{n}$
If the sphere $\mathbb{S}^n$ is covered by n+1 open sets, then one of these sets contains a pair (x,-x) of antipodal points
The Ham sandwich theorem (stating that for any compact sets $A_1,\ldots, A_n$ in $\mathbb{R}^n$ we can always find a hyperplane dividing each of them into two subsets of equal measure)

[edit] References

K. Borsuk, "Drei Sätze über die n-dimensionale euklidische Sphäre", Fund. Math., 20 (1933), 177-190.
Jiří Matoušek, "Using the Borsuk–Ulam theorem", Springer Verlag, Berlin, 2003. ISBN 3-540-00362-2.
L. Lyusternik and S. Shnirel'man, "Topological Methods in Variational Problems". Issledowatelskii Institut Matematiki i Mechaniki pri O. M. G. U., Moscow, 1930.
Borsuk-Ulam theorem implies the Brouwer fixed point theorem
Allen Hatcher: Algebraic Topology (free download)