In mathematics, the Borsuk–Ulam theorem, named after Stanislaw Ulam and 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 (), the first historical mention of the statement of this theorem appears in (). The first proof was given by (), where the formulation of the problem was attributed to Ulam. Since then, many alternate proofs have been found out by various auth...