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 $X$.

翻译：广义度量空间通过放宽距离函数的性质（例如对称性）并允许其取值于比非负实数集更一般的结构（如Quantale）而得到。Quantale值度量空间因其在程序/系统定量推理以及定义各种行为度量概念中的应用而日益重要。我们研究了当Quantale为连续时，在Quantale值度量空间框架下的不精确性与鲁棒性。特别地，我们分析了捕捉分析鲁棒性的鲁棒拓扑与Hausdorff-Smyth半度量之间的关系。为此，我们定义了一个预序强化单子$\mathsf{P}_S$，称为Hausdorff-Smyth单子，并且当$Q$为连续Quantale且$X$为$Q$-度量空间时，我们将$\mathsf{P}_S(X)$上由该度量诱导的拓扑与基于$X$上度量定义的幂集$\mathsf{P}(X)$上的鲁棒拓扑联系起来。