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Is there a theorem which states: "Every infinite metric space that is complete, connected and locally connected, is arc-wise connected"?

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  • $\begingroup$ According to the MathWorld article on arc-wise connectedness, "every locally compact, connected, locally connected metrizable topological space is arcwise-connected". So if you were willing to trade completeness for local compactness ... $\endgroup$ Commented Feb 24, 2016 at 21:24
  • $\begingroup$ It's even locally arcwise connected, and thus arc-wise connected as well (as a connected locally arcwise connected space is arcwise connected as well). $\endgroup$ Commented Feb 24, 2016 at 21:31
  • $\begingroup$ @HagenvonEitzen we can generalise to completely metrisable as well. $\endgroup$ Commented Feb 28, 2016 at 18:28

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Yes, as I already stated in this answer. A proof is in Hocking and Young. It's a classical result from the 1920's.

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  • $\begingroup$ Many thanks for this information. I found out that it was possible for a metric space to be connected and locally co $\endgroup$ Commented Feb 25, 2016 at 18:50
  • $\begingroup$ Many thanks for this information. I found out that it was possible for an infinite metric space to be connected and locally connected without containing a single arc. But apparently this was proved only fairly recently. $\endgroup$ Commented Feb 25, 2016 at 19:03
  • $\begingroup$ @GarabedGulbenkian the references in the linked answer are pretty old already. The positive result for complete spaces is older. $\endgroup$ Commented Feb 25, 2016 at 19:42

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