Let $\Omega:=\{z\in\Bbb{C};\;0<\Im(z)<1\},$ it's convex so simply connected.
As we can see the two connected components of $\Bbb{C}\setminus\Omega$ are not bounded.
Any exemple I can imagine of open connected sets of $\Bbb{C}$ that are simply connected have this property. So I imagine the following result is true:
Let $\Omega$ an open connected set of $\Bbb{C}.$ If $\Omega$ is simply connected then the c of $\Bbb{C}\setminus\Omega$ are not bounded.
I am aware that a proof use, probably, algebraic topology. It's not a problem to read but much harder to find, I think.
Is the result true? If yes, where I can find a proof?(even in the form of exercise)