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

We define a (preorder-enriched) category $\mathsf{Met}$ 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 $d: X \times X \to Q$ is the metric, we consider a topology $\tau_d$ on $X$, which generalizes the open ball topology, and a topology $\tau_{d,R}$ on the powerset $\mathsf{P}(X)$, called the robust topology, which captures robustness with respect to small perturbations of parameters. We define a (preorder-enriched) monad $\mathsf{P}_S$ on $\mathsf{Met}$, called the Hausdorff-Smyth monad, which captures the robust topology, in the sense that the open ball topology of the object $\mathsf{P}_S(X,d,Q)$ coincides with the robust topology $\tau_{d,R}$ 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.