ArXiv.orgPub Date : 2025-08-15DOI: 10.48550/arxiv.2508.11623
Francesco Dagnino, Amin Farjudian, Eugenio Moggi
{"title":"Robust Topology and the Hausdorff-Smyth Monad on Metric Spaces over Continuous Quantales","authors":"Francesco Dagnino, Amin Farjudian, Eugenio Moggi","doi":"10.48550/arxiv.2508.11623","DOIUrl":"https://doi.org/10.48550/arxiv.2508.11623","url":null,"abstract":"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 $τ_d$ on $X$, which generalizes the open ball topology, and a topology $τ_{d,R}$ 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 $τ_{d,R}$ 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.","PeriodicalId":521416,"journal":{"name":"ArXiv.org","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2025-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://arxiv.org/pdf/2508.11623","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147877365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}