gn.general topology - 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?