SOLVED
Does every connected set in $\mathbb{R}^n$ contain a connected subset which is not a point and not homeomorphic to the original set?
If $n\geq 2$ does every connected set in $\mathbb{R}^n$ contain more than $2^{\aleph_0}$ many connected subsets?
Asked by Erdős in the 1940s, who thought the answer to both questions is yes. The answer to both is in fact no, as shown by Rudin
[Ru58] (conditional on the continuum hypothesis).