A. Di Concilio

Point-Free Geometries: Proximities and Quasi-Metrics

In the Euclidean geometry points are the primitive entities. Point-based spatial construction is dominant but apparently, from a constructive point of view and a naive 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 spatial regions, and relations between regions rather than referring to points and sets of points. One of the approaches to point-free geometry proposes as primitives the concepts of region and quasi-metric, a non-symmetric distance between regions, yielding a natural notion of diameter of a region that, under suitable conditions, makes it possible 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 family of bounded regions and a group of similitudes preserving bounded regions, so producing a metric geometry in which shape is relevant. The main purpose of this article is to highlight the role of nearness and emphasize the proximity aspects taking part in the construction by quasi-metrics of point-free geometries.In the Euclidean geometry points are the primitive entities. Point-based spatial construction is dominant but apparently, from a constructive point of view and a naïve 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 spatial regions, and relations between regions rather than referring to points and sets of points. One of the approaches to point-free geometry proposes as primitives the concepts of region and quasi-metric, a non-symmetric distance between regions, yielding a natural notion of diameter of a region that, under suitable conditions, makes it possible 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 family of bounded regions and a group of similitudes preserving bounded regions, so producing a metric geometry in which shape is relevant. The main purpose of this article is to highlight the role of nearness and emphasize the proximity aspects taking part in the construction by quasi-metrics of point-free geometries.

Descriptive Proximities. Properties and Interplay Between Classical Proximities and Overlap

The theory of descriptive nearness is usually adopted when dealing with subsets that share some common properties, even when the subsets are not spatially close. Set description arises from the use of probe functions to define feature vectors that describe a set; nearness is given by proximities. A probe on a nonempty set X is an n-dimensional, real-valued function that maps each member of X to its description. We establish a connection between relations on an object space X and relations on the corresponding feature space. In this paper, the starting point is what is known as

On local versions of metric uc-ness, hypertopologies and function space topologies

\mathcal {P}_\Phi

Topologizing Homeomorphism Groups

PΦ proximity (two sets are

ℵ-completeness as E-compactness via the Uniform Dimensiona

\mathcal {P}_\Phi

Topologizing homeomorphism groups of rim-compact spaces

PΦ-near or