Question
(Fulton's Algebraic Topology, A First Course, Problem 4.40) Suppose the sphere $S^2=A\cup B$ where $A,B\subseteq S^2$ are two closed subsets of $S^2$. Is it true that either $A$ or $B$ must contain a closed connected set $X$ such that $X=X^*$, the set of antipoles of points of $X$?
I have no idea on how to proceed. It's an exercise immediately after Borsuk-Ulam theorem. The condition that $X$ is connected is somewhat tough. Otherwise, the existence is a direct (much weaker) result of Borsuk-Ulam.
Any idea? Thanks!
wspin's counterexample:
Identify the sphere with the inscribed cube and let $A$ be the front, bottom, and back faces. Let $B$ be the top, left, and right faces. Then each connected component of $A\cap A^\star$ is disjoint from its image under the antipodal map, and similarly for $B \cap B^\star$. Assume $X \subset A$. Then $X\subset A\cap A^\star$, so that $X$ is contained in one the connected components, a contradiction since $X^\star =X$. Similarly $X$ cannot be contained in $B$.