Borsuk–Ulam theorem

In mathematics, the Borsuk–Ulam theorem 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. In mathematics, the Borsuk–Ulam theorem 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. Formally: if f : S n → R n {displaystyle f:S^{n} o mathbb {R} ^{n}} is continuous then there exists an x ∈ S n {displaystyle xin S^{n}} such that: f ( − x ) = f ( x ) {displaystyle f(-x)=f(x)} . The case n = 1 {displaystyle n=1} can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously. The case n = 2 {displaystyle 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. The Borsuk–Ulam theorem has several equivalent statements in terms of odd functions. Recall that S n {displaystyle S^{n}} is the n-sphere and B n {displaystyle B^{n}} is the n-ball: According to Jiří Matoušek (2003, p. 25), the first historical mention of the statement of the Borsuk–Ulam theorem appears in Lyusternik & Shnirel'man (1930). The first proof was given by Karol Borsuk (1933), where the formulation of the problem was attributed to Stanislaw Ulam. Since then, many alternative proofs have been found by various authors, as collected by Steinlein (1985). The following statements are equivalent to the Borsuk–Ulam theorem. A function g {displaystyle g} is called odd (aka antipodal or antipode-preserving) if for every x {displaystyle x} : g ( − x ) = − g ( x ) {displaystyle g(-x)=-g(x)} . The Borsuk–Ulam theorem is equivalent to the following statement: A continuous odd function from an n-sphere into Euclidean n-space has a zero. PROOF: