Let $(X,d)$ be a metric space and $(K\_X , h\_d)$ be the associated metric space of nonempty compact subsets of $X$ with the Hausdorff metric. It is well known that $K\_X$ inherits certain topological (and analytic) properties from $X$. For example, if $X$ is compact, then so is $K\_X$; and if $X$ is complete, then so is $K\_X$.


Is there a reference which further explores the properties that $K\_X$ inherits from $X$? In particular, if $X$ is locally compact then is $K\_X$ also?



Topological properties inherited by the Hausdorff metric space

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> Michael, Ernest. “[Topologies on spaces of subsets.](https://www.ams.org/journals/tran/1951-071-01/S0002-9947-1951-0042109-4/S0002-9947-1951-0042109-4.pdf)” Transactions of the American Mathematical Society 71 (1951): 152-182.
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The above paper shows in section $4$ that many properties of $X$, including being locally compact, are inherited by $K(X)$ (the latter space is called $\mathcal C(X)$ in the paper, while the author uses $2^X$ to refer to the hyperspace of closed sets of $X$, for which a more modern notation is $F(X)$).


In my comment I said that being zero-dimensional is also preserved but not shown in the paper, I now noticed that it is in fact shown there, together with more connectedness properties. The relationship between $\dim X$ and $\dim K(X)$ is much harder in general and has been widely studied, there is a whole chapter concerning this topic in the book *Hyperspaces* by Illanes and Nadler.


