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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:

  1. If $X$ is connected, locally path-connected, locally compact, separable and metric, there is a path metric on $X$.

  2. 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.

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    $\begingroup$ What about the amazon metric on the plane (aka river metric, but in my first year it had the name "Amazonemetriek")? It is connected, non-separable, locally path-connected. And locally compact. What is a "convex metric" here? $\endgroup$ Apr 6 at 8:51
  • $\begingroup$ Hey! I tried to google this space, but I just found books on Amazon, or performance metrics of Amazon haha A convex metric is one such that for every $x, y \in X$, if $d(x,y) = 2r$ then there is a point $z$ with $d(x,z)=d(z,y) = r$. $\endgroup$ Apr 6 at 12:48
  • $\begingroup$ The river metric might be more findable. $\endgroup$ Apr 6 at 12:52
  • $\begingroup$ Ok yeah, I think I found it. But apparently it's convexifiable, assuming a "hyperconvex" space is convex. I'm still wading through the definitions, but I guess you're not the only one who thought the 'convex behavior' of this space is interesting: hindawi.com/journals/jfs/2017/4901762 $\endgroup$ Apr 6 at 12:55
  • $\begingroup$ Interesting. I always thought of the "amazonemetriek" as an example to teach first year maths students to really consider the definitions of a metric, instead of some intuition (most metrics on the plane that are commonly introduced are equivalent to the standard one, but this one stands out). It served to illustrate that metric spaces could be non-separable ye connected and that a seemingly two-dimensional space could have a cut point etc. DIdn't know people actually write papers on such examples. $\endgroup$ Apr 7 at 7:14
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No such space exists, since every connected, locally compact, metrizable space is separable.

Non separable locally compact connected metric space

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  • $\begingroup$ This theorem surprised me in its strength. I wonder why it's not considered a fundamental result, especially since the proof isn't hard. $\endgroup$ Apr 11 at 18:26

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