Metrization of Quasi-Uniformities, Powerset Monads, and Qualitative Robustness Analysis

2026-09-02

AI Summary

研究通过建立拟一致空间与量值度量空间之间的等价关系，解决了它们在拓扑学和定量鲁棒性分析中的关联问题。

Abstract

We study the relationship between quasi-uniform spaces, topological spaces, and quantale-valued metric spaces. Our main result is a metrization theorem establishing an equivalence between the category of quasi-uniform spaces and a category of quantale-valued metric spaces. We also obtain a quantale-based metrization theorem for arbitrary topological spaces that refines existing constructions. These results identify quasi-uniformities as the appropriate qualitative counterpart of quantale-valued metrics. Building on this correspondence, we show that the Hausdorff-Smyth monad on quantale-valued metric spaces, which is used in quantitative robustness analysis, arises as a lifting of a corresponding monad on quasi-uniform spaces along the equivalence. This provides a unified categorical framework connecting topology, quasi-uniformity, and quantitative robustness analysis.

Problem

• quasi-uniform spaces
• topological spaces
• quantale-valued metric spaces
• metrization theorem
• robustness analysis

Innovation

• metrization theorem
• quasi-uniform spaces
• quantale-valued metric spaces
• Hausdorff-Smyth monad
• categorical framework