SCHWEITZER 2020 Problem 5

Prove that for a nowhere dense, compact set $K\subset \mathbb{R}^2$ the following are equivalent:

(i) $K=\bigcup_{i=1}^{\infty}K_n$ where $K_n$ is a compact set with connected complement for all $n$ .

(ii) $K$ does not have a nonempty closed subset $S\subseteq K$ such that any neighborhood of any point in $S$ contains a connected component of $\mathbb{R}^2 \setminus S$ .