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

$\begingroup$ No, there is an injection from $U^2$ to $\mathbb R^2$. $\endgroup$– WojowuJan 2, 2018 at 13:52
1 Answer
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.