I am looking for a reference to the proof of the following result: Let $E\subseteq\mathbb C$ be a continuum. Then the following assertions are equivalent:
(1) $E$ is locally connected;
(2) (i) each of the connected components of $\mathbb C\setminus E$ has a locally connected boundary and (ii) for any $\epsilon>0$ there are at most finitely many components of $\mathbb C\setminus E$ with diameter bigger than $\epsilon$.
I did not find it in the topology monographs by Kuratowski, but may be I overlooked it? And who proved this for the first time? Thanks.