In hyperspace theory, if $(X,d)$ is a metric space, then the Hausdorff distance between nonempty subsets $A$ and $B$ of $X$ can be defined as $$H(A,B)=\text{max}\{\text{sup}\{d(a,B): a\in A\},\text{sup}\{d(b,A): b\in B\} \}$$ This is "the farthest distance any point of $B$ is from the set $A$, or the farthest any point of $A$ is from $B$, whichever is greater." In general, $H$ is not a metric, but a pseudometric (for $H(A,B)=0$ if and only if $\overline{A}=\overline{B}$, where overline denotes closure.)
Every article I've read about the Hausdorff distance only studies it in the hyperspace of closed sets, because it is a metric there, however I am interested in its properties as a (possibly extended) pseudometric in the power set of $X$ minus the empty set. I find the property I mentioned above incredibly useful and interesting, so I was wondering if there are any texts that address this topic any further. Pseudometric spaces seem more exotic and attractive to me lately.
Does anyone know more about Hausdorff distance as a pseudometric?