Show that Borsuk -Ulam Theorem for $n=2$ is equivalent to the following statement : For any cover $A_1, A_2, $ and $A_3$ of $S^2$ with each $A_i$ closed, there is at least one $A_i$ containing a pair of antipodal points.
For one direction, the function $f:S^2\rightarrow \mathbb R^2$ where $f(x)=(d(x,A_1),d(x,A_2))$ is enough. For another direction, I don't even know how to start the proof. Can you help me?
For any continuous function $ f: S^2 \to \mathbb R^2 $ consider the function $ g (x) = f (-x) -f (x) $. Note that $ g (x) $ is odd for reflection through the origin.
Now cover $\mathbb R^2$ by three closed sets $ B_i $ such the only pair of points given by reflection through the origin that is contained in each set is the origin (paired with itself). Then consider the inverse images of these closed sets sitting in $ S^2$.You should be able to get the rest from there.