Point-Free Geometries: Proximities and Quasi-Metrics

In the Euclidean geometry points are the primitive entities. Point-based spatial construction is dominant but apparently, in a constructive point of view and a na\"{\i}ve knowledge of space, the region-based spatial theory is more quoted, as recent and past literature strongly suggest. The point-free geometry refers directly to sets, the {\it spatial regions}, and {\it relations between regions} rather than referring to points and sets of points. One of the approach to point-free geometry proposes as primitives the concept of region and quasi-metric, a non-symmetric distance between regions, yielding a natural notion of diameter of a region that, under suitable conditions, allows to reconstruct the canonical model. The intended canonical model is the hyperspace of the non-empty regularly closed subsets of a metric space equipped with the Hausdorff excess. The canonical model can be enriched by adding more qualitative structure involving a distinguished countable subfamily of regions, {\it bounded regions}, and a group of {\it similitudes} preserving bounded regions, so eventually producing a metric geometry whose points, roughly speaking decreasing sequences of bounded regions with vanishing diameters, have some specific features preserved by similitudes and different metric geometries for distinct bounded regions.