There was an MO question that recently got resolved, asking for counterexamples in the general theory of length spaces. However, a reasonable follow-up question remains, so I thought I'd ask it here.
Especially, there was one theorem given and one counterexample given:
If $X$ is connected, locally path-connected, locally compact, separable and metric, there is a path metric on $X$.
There is a connected, locally path-connected, separable metric space with no equivalent path metric.
The author of the question just wanted some connected, locally path-connected metric space with no equivalent path metric, but the thread leaves open the following question:
Is there a connected, locally path-connected, locally compact metric space with no equivalent path metric?
In particular the space can't be separable (equivalently, 2nd countable). I figured I'd throw it on here and let people root through the standard examples of such spaces, if they find it interesting. Carrying no path metric is equivalent to there being no equivalent convex metric. "A Course in Metric Geometry" covers the basic definitions and properties of length spaces if someone who's unfamiliar wants to take a dive; I'll expand the question with definitions and add a bounty if it goes unresolved for a while.
Best Answer
No such space exists, since every connected, locally compact, metrizable space is separable.
Non separable locally compact connected metric space