How wild can the topology of a length space be? That is,


**Questions:** 


1. 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?
2. 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?
3. 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:


4. Is the topology of an Alexandrov space (in the [curvature sense](https://en.wikipedia.org/wiki/Alexandrov_space)) significantly more controlled than the topology of a general length space?



Topology of length spaces
1 Look at 3.


3 It is almost obvious from the definition that length spaces are locally path connected, and if you assume local connectedness, 2 is true, but nontrivial:


Bing, R. H. A convex metric for a locally connected continuum. Bull. Amer. Math. Soc. 55 (1949), no. 8, 812--819. <https://projecteuclid.org/euclid.bams/1183514048>


4 For finite dimensional Alexandrov Spaces, it is not hard to show that there is an open dense subset which is a manifold. It is nontrivial that there is an even nicer stratification, and they are locally contractible.


<https://anton-petrunin.github.io/papers/alexandrov/perelmanASWCBFB2+.pdf>


The topological structure results are very well surveyed in the last chapter here:


Burago, Dmitri & Burago, Yuri & Ivanov, Sergei. (2001). A Course in Metric Geometry. Graduate Studies in Math.. 33. 
<https://www.math.psu.edu/petrunin/papers/akp-papers/bbi.pdf>


