Let $A$ and $B$ be non-empty open and disjoint sets on an $S^2$. Does there exist (at least) one closed $C^0$ curve in $S^2\setminus(A\cup B)$.
If it is not true, what other condition has to be required to make it true.
Comment: It seems clear that any continuous path $\alpha(\tau)$ defined for $\tau\in[0,1]$ with $\alpha(0)\in A$ and $\alpha(1)\in B$ has to intersect $S^2\setminus(A\cup B)$ in at least one point.
Interpreting the $S^2$ as the Riemannsphere this can also be interpreted as a question about open sets on the plane.