# Borsuk-Ulam Theorem

## Theorem

Let $n$ be a positive integer.

Let $f: \mathbb S^n \to \R^n$ be a continuous mapping from an $n$-sphere to $\R^n$.

Then there exists $x \in \mathbb S^n$ such that $f \left({x}\right) = f \left({-x}\right)$.

## Proof

## Source of Name

This entry was named for Karol Borsuk and Stanisław Marcin Ulam.