A pointfree theory of Pervin spaces

TL;DR

This paper develops a pointfree framework for Pervin spaces using Frith frames, establishing dualities with topological spaces and analyzing their categorical properties, including completeness and completion.

Contribution

It introduces Frith frames as a pointfree analogue of Pervin spaces and extends duality results, providing new categorical insights into their structure.

Findings

Dual adjunction between Pervin spaces and Frith frames

Frith frames form a full coreflective subcategory of quasi-uniform frames

Characterization and construction of complete Frith frames

Abstract

We lay down the foundations for a pointfree theory of Pervin spaces. A Pervin space is a set equipped with a bounded sublattice of its powerset, and it is known that these objects characterize those quasi-uniform spaces that are transitive and totally bounded. The pointfree notion of a Pervin space, which we call Frith frame, consists of a frame equipped with a generating bounded sublattice. In this paper we introduce and study the category of Frith frames and show that the classical dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, we do not have an equivalence between the categories of transitive and totally bounded quasi-uniform frames and of Frith frames, but we show that the latter is a full coreflective subcategory of the former. We also explore the notion of completeness of Frith frames inherited from quasi-uniform frames, providing a characterization of those Frith frames that are complete and a description of the completion of an arbitrary Frith frame.