How wild can the topology of a length space be? That is,
Let $X$ be a metric space where the distance between two points $x,y \in X$ is the infinum of lengths of rectifiable paths from $x$ to $y$. What can be said about the topology of $X$? Can spaces of this form be completely characterized?
What if we additionally assume that $X$ is separable, compact, connected, and locally path-connected? In this case, $X$ is a Peano continuum -- does every Peano continuum admit a length metric?
What if we relax some of the conditions in (2)?
In particular, I might guess that every length space is locally path-connected, but it's not obvious that this is true. And I think the Hawaiian earring would be an example of a length space which is not locally contractible (or even semilocally simply connected).
For a bonus question:
- Is the topology of an Alexandrov space (in the curvature sense) significantly more controlled than the topology of a general length space?