Karol Borsuk

Karol Borsuk was a mathematician whose work helped develop topology, particularly the study of retracts and the relationship between geometric and topological structure. His collaboration with Stanisław Ulam produced results about continuous mappings and topological invariants.^[ref] The Borsuk-Ulam theorem states that every continuous map from the sphere $S^n$ to ${\mathbb{R}}^n$ takes the same value at some pair of antipodal points. The equal-value pair is forced by global topology, even though the map may have no explicit symmetry. This provides a tool for existence proofs rather than a recipe for locating the pair.