I was looking into particular cases for the Poincaré-Bendixson theorem and I came across a topological problem about simply connectivity.

If $\gamma$ is a Jordan curve in ${\Bbb S}^2$ then using Jordan-Schoenflies, we have that ${\Bbb S}^2\setminus \gamma = U\sqcup V$ with $U$ and $V$ being simply connected (s.c.). Moreover, as the sphere minus $\gamma$ is homeomorphic to the sphere minus the equator, we also get that $\overline{U}$ and $\overline{V}$ are s.c.

Let $U$ be a s.c. open subset in ${\Bbb S}^2$. We note that $\overline{U}$ need not be s.c. (take an open annulus where you make a transversal cut). On the other hand, the complement $F=S^2\setminus U$ appears to me to be s.c.

${\rm Int} F$ need not be connected, but a connected component of ${\rm Int} F$ appears to me again to be s.c.

Do the above two claims hold in general? Or have I missed some obvious counter-examples?