Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
 They are homotopic to each other.
 They are settheoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ is disconnected?
Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
Is $S\setminus(C_1 \cup C_2)$ is disconnected? 

