Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$ ?
Also ; does every connected metric space $X$ contains a connected subset which is homeomorphic with $X$ ?
UPDATE : So as noticed by @orangeskid ; the answer to the 2nd question is "no" by considering $X=S^1$ . The first question still remains unanswered