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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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 ... – Hagen von Eitzen Feb 24 '16 at 21:24
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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). – Henno Brandsma Feb 24 '16 at 21:31
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@HagenvonEitzen we can generalise to completely metrisable as well. – Henno Brandsma Feb 28 '16 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.
Henno Brandsma
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Many thanks for this information. I found out that it was possible for a metric space to be connected and locally co – Garabed Gulbenkian Feb 25 '16 at 18:50
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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. – Garabed Gulbenkian Feb 25 '16 at 19:03
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@GarabedGulbenkian the references in the linked answer are pretty old already. The positive result for complete spaces is older. – Henno Brandsma Feb 25 '16 at 19:42