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

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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.