Let $K$ be a compact subset of $\mathbb R^2$ such that $\mathbb R^2\setminus K$ is not connected. Is it true that $K$ contains a simple closed curve?
No. One counterexample is the closure of topologist's sine curve $y=\sin (1/x)$, $0<x<1$, plus a curve connecting its "good" end to the line segment at the other end. The complement is open and clearly not path-connected, hence not connected.