# What does the property that path-connectedness implies arc-connectedness imply?

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?

• 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? – Benoit Jubin Oct 21 '09 at 1:09

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).
• 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. – KP Hart May 25 '10 at 9:27