  Generalized metric spaces are obtained by weakening the requirements (e.g.,
symmetry) on the distance function and by allowing it to take values in
structures (e.g., quantales) that are more general than the set of non-negative
real numbers. Quantale-valued metric spaces have gained prominence due to their
use in quantitative reasoning on programs/systems, and for defining various
notions of behavioral metrics.
  We investigate imprecision and robustness in the framework of quantale-valued
metric spaces, when the quantale is continuous. In particular, we study the
relation between the robust topology, which captures robustness of analyses,
and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched
monad $\mathsf{P}_S$, called the Hausdorff-Smyth monad, and when $Q$ is a
continuous quantale and $X$ is a $Q$-metric space, we relate the topology
induced by the metric on $\mathsf{P}_S(X)$ with the robust topology on the
powerset $\mathsf{P}(X)$ defined in terms of the metric on $X$.
