Let $U,V$ be disjoint non-empty connected open subsets of the sphere $S^2$ such that $\partial U=\partial V$ and $\operatorname{cl}(U\cup V)=S^2$. Must $U$ and $V$ be simply connected?
This seems intuitively obvious, but I'm not sure how to best show it. I have a painstaking proof where one basically "rasterizes" the problem and reduces it to some discrete statement about any finite grid, but this feels like a poor way to go about it. Is there a better way to show this statement?
This follows from the fact that a connected open subset of $S^2$ is simply connected iff its complement is connected. This fact itself is proved in many complex analysis textbooks, see also Complement is connected iff Connected components are Simply Connected