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 $f(-x)=f(x)$? 



Borsuk–Ulam theorem on the sphere with expluded poles
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.


