Antipodal points of sphere Whenever $S^2$ is the union of three closed subsets $A_1$, $A_2$, and $A_3$, then at least one of these sets must contain a pair of antipodal points {${x,-x}$} in $S^{2}$ This is homework from topology (undergraduate) Can you help me ?

Construct a function $f \colon S^2 \to \mathbb{R}^2$ by setting $f(x) = (d(x,A_1), d(x,A_2))$ where $d$ is the standard metric on $S^2$. Apply the Borsuk-Ulam theorem to this function.

Details of the general case (the Lusternik–Schnirelmann theorem) can be found in Armstrong's book.