TITLE: Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$?
QUESTION [8 upvotes]: 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

REPLY [4 votes]: Take $X$ a $1$-dimensional circle. $X$ does not contain any proper subspaces homeomorphic to a circle, since any connected proper subspace is a segment. 
${\bf Added:}$ 
The answer to the first question is yes for spaces that contain a segment. It seems a lot of connected metric spaces contain a segment.