Given that $K$ is a connected subset of $\mathbb{R}^2$ such that $\forall x\in K, K\setminus\{x\}$ is not connected, then
- K must be homeomorphic to an interval of $\mathbb{R}$
- K must have empty interior.
Well, I feel that 1 is correct but I'm not able to make it formal, and I'm not sure about 2.
Thank you for help.