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Does every connected complete metric space with more than one point contain a non-trivial path? The pseudo-arc is an example of a connected metrizable space without a path.

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    $\begingroup$ This may be a bit extreme, but if you can work through the proof that the pseudoarc does not contain a nontrivial path, see if the lack of completeness is necessary. If not, then just take the completion of the pseudoarc for your example. $\endgroup$ Jun 26, 2014 at 2:01

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Such spaces do exist, since there are connected Polish spaces without non degenerate connected compact subsets. See the 1st answer to https://mathoverflow.net/questions/25171/how-thinly-connected-can-a-closed-subset-of-hilbert-space-be.

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  • $\begingroup$ Yes. But not all Polish spaces are complete. $\endgroup$
    – user156619
    Jun 26, 2014 at 2:23
  • $\begingroup$ That's irrelevant. By definition, each Polish space is completely metrizable. $\endgroup$ Jun 26, 2014 at 2:25
  • $\begingroup$ I get it now. Thanks $\endgroup$
    – user156619
    Jun 26, 2014 at 2:26

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