Let M be a Metric space which is connected, locally connected, and contains more than one point. Do these conditions on M imply that M is always arc-wise connected? I have been unable to find a precise statement-to this effect-in the literature. But I have not been able to cook up any counter-examples.
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For completely metrisable spaces this holds. This is classical, see Engelking 6.3.11. We then have locally arcwise connected (so even stronger) and so arcwise connected as well, from standard arguments. – Henno Brandsma Feb 23 '16 at 18:55
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No. Based on chapter 3-2 of Hocking and Young: There is an example due to R.L. Moore: "A connected and regular point set which contains no arc", Bull. AMS 32, 331-332 (1926), has an example of a subset of the plane which is locally connected, and such that every 2 points lie in a continuum, but which contains no arc. Also, Knaster and Kuratowski ("A connected and connected im kleinen point set which contains no perfect set", Bull. AMS 33, 106-109 (1927)) has a connected and locally connected subset of the plane that contains no continuum at all.
In Hocking and Young they claim the example after theorem 3-6 is such an example as well, without proof.
They also prove the classical theorem that for complete(ly metrisable) such spaces we do have local arcwise connectedness (and thus arc-wise connectedness as well), in theorem 3-17, and the example above show that completeness is essential. Some intuition is presented at the start of chapter 3-2.
Henno Brandsma
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