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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

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    $\begingroup$ See the Knuster-Kuratowski fan $\endgroup$
    – Teri
    Commented Dec 21, 2015 at 15:44
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    $\begingroup$ @Teri : How does that help ? $\endgroup$
    – user228168
    Commented Jan 3, 2016 at 8:31
  • $\begingroup$ interesting question. I can show it is true for compact Hausdorff spaces. my intuition says it should be true. I am working on a proof by cases. $\endgroup$ Commented Jan 10, 2016 at 5:24
  • $\begingroup$ @ForeverMozart : You can show it for compact connected Hausdorff spaces ? Could you please take the trouble to write that out in comment or as an answer ; It would be very very helpful . Thanks in advance $\endgroup$
    – user228168
    Commented Jan 10, 2016 at 5:38
  • $\begingroup$ @SaunDev See my question (and answer) here: math.stackexchange.com/questions/1572687/… . Every cpt Haus. space is the union of two disjoint connected subsets. At least one must be non-compact, thus not homeomorphic to the entire space. $\endgroup$ Commented Jan 10, 2016 at 5:59

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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.

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  • $\begingroup$ Okay , so that answers the 2nd question ; what about the first one ? $\endgroup$
    – user228168
    Commented Dec 21, 2015 at 13:12
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    $\begingroup$ @Saun Dev: This post math.stackexchange.com/questions/796635/… seems to be related to your question $\endgroup$
    – orangeskid
    Commented Dec 24, 2015 at 15:46

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