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A space X is path-connected if any two points are the endpoints of a path, that is, the image of a map [0,1] \to X. A space is arc-connected if any two points are the endpoints of a path, that, the image of a map [0,1] \to X which is a homeomorphism on its image. If X is Hausdorff, then path-connected implies arc-connected.

I was wondering about the converse: What properties must X have if path-connected implies arc-connected? In particular, what are equivalent properties?

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  • $\begingroup$ More precisely: what properties must have a space where two points that can be joined by a path can be joined by an arc. And more vaguely: why is it needed (or simply useful) to know that two points can be joined by an arc, and not simply a path? $\endgroup$
    – Benoit Jubin
    Commented Oct 21, 2009 at 1:09

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I don't have an answer, but here is an example to show it's not a local property that decides it. Consider the real line with two inseparable zeros, 0 and 0'. Clearly there is a path from 0 to 0' but not an arc. On the other hand, if you adjoin a point at infinity, making a circle with a double point on it, you can make such an arc going through infinity, and so the space is arc-connected.

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    $\begingroup$ That's not an embedding, though - the image is the whole space, and the circle with a double point isn't Hausdorff. $\endgroup$
    – Harry Altman
    Commented May 30, 2010 at 1:55
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It suffices that $X$ be Hausdorff: the path is then a compact metric image of [0,1] and as such arc-wise connected (do Problem 6.3.11 of Engelking's General Topology).

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  • $\begingroup$ But it is not necessary. The real line with a double origin is a counterexample. $\endgroup$
    – skupers
    Commented May 24, 2010 at 3:13
  • $\begingroup$ The best I can come up with then is an artificial ``any two points are connected by a locally connected metric continuum''. Any arc is such a continuum and the exercise mentioned above establishes that such continua are arcwise connected. It looks to me like that exercise, a step towards the Hahn-Mazurkiewicz theorem, will have to be a main part of any argument and any iff-condition that should set things up for a construction of an arc will be as artificial as what I wrote above. $\endgroup$
    – KP Hart
    Commented May 25, 2010 at 9:27

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