Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
The following result is due to R.L. Moore.
Let $K\subseteq\mathbb C$ be compact. Suppose that $K$ is connected, and that $\mathbb C\setminus K$ is connected. Then $\partial K$ is connected.
Does anyone know the original source? Thank you.