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 homotopytheoretic 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 pathconnected 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$\begingroup$ arxiv.org/pdf/1008.2926.pdf $\endgroup$– Denis T.Nov 14 '21 at 2:50
In J. Andres, M. Väth, Calculation of Lefschetz and Nielsen Numbers in Hyperspaces for Fractals and Dynamical Systems, Proc. Amer. Math. Soc. 135 (2007), 479487, it was shown (esssentially, the result was already implicitly shown in D.W. Curtis, Hyperspaces of noncompact metric spaces, Compositio Math 40 (1980) (2), 139152, without explicitly noting it):
${\mathcal H}(X)$ is a union of disjoint open ARs (being its connected components) if and only if $X$ is locally continuumconnected.
${\mathcal H}(X)$ is an AR if and only if $X$ is locally continuumconnected and connected.
(Locally continuumconnected is a property between locally pathconnected and locally connected.)
In particular, ${\mathcal H}(X)$ is contractible in all cases of 2 while $X$, in general, is not.

$\begingroup$ Brilliant, this answers my question perfectly. Thanks! $\endgroup$ Nov 14 '21 at 21:18