# Borsuk-Ulam Theorem

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## 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 $\map f x = \map f {-x}$.

## Proof

## Source of Name

This entry was named for Karol Borsuk and StanisÅ‚aw Marcin Ulam.