Erdős Problems

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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?

#910: [Er82e]

topology

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).