# Homotopy type of the Hausdorff metric

Recall that if we have a metric space $$X$$ then we can consider the set of its nonempty compact subsets and equip this with a metric called the Hausdorff distance. Denote the resulting metric space $$\mathcal{H}(X)$$. We of course get an inclusion $$X \hookrightarrow \mathcal{H}(X)$$. Are any of the homotopy-theoretic properties of this map known? For instance, if $$X$$ is connected, is $$\mathcal{H}(X)$$ connected? (I think the answer is yes). Note if $$X$$ consists of $$n$$ discrete points, then $$\mathcal{H}(X)$$ consists of $$2^n - 1$$ discrete points, so certainly the map is not a weak homotopy equivalence (perhaps it would be fruitful to consider only connected or path-connected compact subsets). Furthermore, I wonder if there are any insights to be had about more categorical structure of this map - $$\mathcal{H}$$ is clearly a functor; is it a monad? If so, is there something meaningful to be said about it?

1. $${\mathcal H}(X)$$ is a union of disjoint open ARs (being its connected components) if and only if $$X$$ is locally continuum-connected.
2. $${\mathcal H}(X)$$ is an AR if and only if $$X$$ is locally continuum-connected and connected.
In particular, $${\mathcal H}(X)$$ is contractible in all cases of 2 while $$X$$, in general, is not.