Typical families and T-compactness in proximity spaces

Abstract

In this work we introduce a family of subsets of a proximity space, called the typical family, defined as the complement of a cluster family, and we study its main properties and its relation to the cluster concept. We show explicitly that the typical family is, up to the language of proximity spaces, the ideal that is dual to the associated cluster in the classical sense of Choquet’s grill ideal duality, and we compare it in detail with ideals, filters, grills, and bornologies, establishing a necessary and sufficient condition under which it is a bornology. Building on this family we define a generalized compactness notion, called T-compactness, relate it to the classical notion of compactness with respect to an ideal, and establish preservation results for finite unions, closed subsets, and via a pushforward construction introduced here images under proximity isomorphisms. We illustrate the framework with a fully worked finite numeric example. We close with a discussion of the broader implications of these results for the relationship between cluster theory and generalized compactness.