Robust Topology and the Hausdorff-Smyth Monad on Metric Spaces over Continuous Quantales

We define a (preorder-enriched) category Met of quantale-valued metric spaces and uniformly continuous maps, with the essential requirement that the quantales are continuous. For each object $(X,d,Q)$ in this category, where X is the carrier set, Q is a continuous quantale, and d: X × X → Q is the metric, we consider a topology τd on X, which generalizes the open ball topology, and a topology τd,R on the powerset P(X), called the robust topology, which captures robustness with respect to small perturbations of parameters. We define a (preorder-enriched) monad PS on Met, called the Hausdorff-Smyth monad, which captures the robust topology, in the sense that the open ball topology of the object PS(X,d,Q) coincides with the robust topology τd,R for the object $(X,d,Q)$. We prove that every topology arises from a quantale-valued metric. As such, our framework provides a foundation for quantitative reasoning about imprecision and robustness in a wide range of computational and physical systems.