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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:

  1. Is the topology of an Alexandrov space (in the curvature sense) significantly more controlled than the topology of a general length space?
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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

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  • $\begingroup$ Thanks! That Bing reference is just what I was hoping for! I'm a bit confused, though -- are you saying that length spaces are locally path connected? Or are you saying rather that it must be taken as an additional assumption? I still can't see why length spaces should be locally path connected, so if that's the claim, could you give a hint? $\endgroup$
    – Tim Campion
    Commented Jun 15, 2019 at 0:26
  • $\begingroup$ They are: Take $p$ in a length space and an (open) ball $B$ of radius $r$ around $p$. Pick $q \in B$, then $d(p,q) < r$ and there is a curve of length less than $r$ joining $p$ and $q$. By the triangle inequality, this curve lies entirely in $B$, so $B$ is path connected. $\endgroup$
    – Sergio Zamora
    Commented Jun 15, 2019 at 1:00
  • $\begingroup$ Thanks, I see that is indeed almost immediate! $\endgroup$
    – Tim Campion
    Commented Jun 15, 2019 at 1:02
  • $\begingroup$ Upon closer inspection, Bing only shows that a compact, locally connected continuum has a convex metric if additionally it is finite-dimensional. I'd be curious if the finite-dimensionality hypothesis can be lifted. $\endgroup$
    – Tim Campion
    Commented Jun 15, 2019 at 15:56
  • $\begingroup$ My bad, wrong reference projecteuclid.org/euclid.bams/1183514375 Theorem 8 $\endgroup$
    – Sergio Zamora
    Commented Jun 19, 2019 at 4:07

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