What is borsuk ulam theorem?

The Borsuk-Ulam theorem is a famous result in topology and geometry that relates to the properties of continuous functions. Specifically, it states that for any continuous function that maps a sphere of any dimension to Euclidean space of one less dimension, there must be at least two antipodal points that get mapped to the same point in space. In other words, there is no way to consistently distinguish points on a sphere that are diametrically opposite from one another based purely on their image under a continuous function.

The theorem was first conjectured by Karol Borsuk in 1932 and then proved by him and his collaborator Stanisław Ulam in 1933. Since then, it has been applied in many areas of mathematics, including algebraic topology, differential geometry, and combinatorics. The Borsuk-Ulam theorem has also been used to derive other important results, such as the Hairy Ball Theorem, which states that there is no continuous vector field on a sphere that has no singularities. Overall, the theorem has had a profound influence on the study of topology and its relationship to other mathematical disciplines.