# Borsuk–Ulam theorem on the sphere with expluded poles [closed]

Consider a sphere without two poles $U^2$. Will Borsuk–Ulam theorem still work, i.e. $\forall$ continuous functions $f:U^2 \rightarrow \mathbb{R}^2 ~\exists x \in U^2$ such as $f(-x)=f(x)$?

• No, there is an injection from $U^2$ to $\mathbb R^2$. Jan 2, 2018 at 13:52

No, it fails as soon as you remove one point: the stereographic projection is a bijection between $\mathbb{R}^2$ and a sphere minus a point.