Borsuk–Ulam Theorem

NOTE

For every continuousmap $f:S^2\to {\mathbb{R}}^2$ there are antipodal points with $f(x)=f(-x)$

This can be used to prove that When ever $S^2$ is expressed as the union of three closed sets $A_1,A_2,A_3$ then at least one of these sets must contain a pair of antipodal points $\{x,-x\}$

Important

This theorem holds more generally for maps $S^n\to {\mathbb{R}}^n$