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：广义度量空间是通过弱化距离函数的要求（例如对称性）并允许其取值于比非负实数集更一般的结构（例如量词）而获得的。量词值度量空间因其在程序/系统的定量推理以及定义各种行为度量概念中的应用而日益受到重视。我们研究了当量词连续时，在量词值度量空间框架下的不精确性与鲁棒性。特别地，我们探讨了捕捉分析鲁棒性的鲁棒拓扑与豪斯多夫-史密斯半度量之间的关系。为此，我们定义了一个预序增强的单子$\mathsf{P}_S$，称为豪斯多夫-史密斯单子，并且当$Q$为连续量词且$X$为$Q$-度量空间时，我们将$\mathsf{P}_S(X)$上的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。