TITLE: What does the property that path-connectedness implies arc-connectedness imply?
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: 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).