Point-free geometries: proximities and quasi-metrics (Q1948559)

In [Math. Struct. Comput. Sci. 16, No. 1, 115--137 (2006; Zbl 1112.54012)], \textit{A. Di Concilio} and \textit{G. Gerla} proposed a `point-free' approach to metric spaces substantially different from what is usually understood by that term (see the review of the paper in [Zbl 1112.54012] for more information), assuming as only primitives the concepts of region and quasi-metric. The canonical model is the hyperspace of the nonempty regular closed subsets of a metric space equipped with the Hausdorff excess measure. In the present paper, the author discusses the (non-symmetric) nearness and proximity aspects of the construction of those point-free metric geometries via quasi-metrics.