# Borsuk–Ulam theorem

In mathematics, the Borsuk–Ulam theorem, formulated by Stanislaw Ulam and proved by Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.

According to (Matoušek 2003, p. 25), the first historical mention of the statement of this theorem appears in (Lyusternik 1930). The first proof was given by (Borsuk 1933), where the formulation of the problem was attributed to Ulam. Since then, many alternative proofs have been found by various authors, as collected in (Steinlein 1985).

## Proof

We use the stronger statement that every odd (antipodes-preserving) mapping h : Sn−1 → Sn−1 has odd degree.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f : Sn → Rn that does not equalize on any antipodes then we can construct a map g : Sn → Sn−1 by the formula

$g(x)=\frac{f(x)-f(-x)}{|f(x)-f(-x)|}$

since f never equalizes antipodes the denominator never vanishes. Note that g is an antipode preserving map. Now let h : Sn−1 → Sn−1 be the restriction of g to the equator. By construction, h is antipode-preserving, and thus has non-zero degree. By construction, h extends to the whole upper hemisphere of Sn, and as such is null-homotopic. A null-homotopic map has degree zero, contradicting our only assumption, namely that f exists.

## Corollaries

• No subset of Rn is homeomorphic to Sn.
• The Lusternik–Schnirelmann theorem: If the sphere Sn is covered by n + 1 open sets, then one of these sets contains a pair (x, −x) of antipodal points. (this is equivalent to the Borsuk–Ulam theorem)
• The Ham sandwich theorem: For any compact sets A1, ..., An in Rn we can always find a hyperplane dividing each of them into two subsets of equal measure.
• The Brouwer fixed-point theorem (Matoušek 2003, p. 25; Su 1997).
• 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 case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, but this can be shown to be true much more easily using the intermediate value theorem.
• Sperner's Lemma