A pointfree theory of Pervin spaces

Abstract

We lay down the foundations for a pointfree theory of Pervin spaces. APervin space is a set equipped with a bounded sublattice of its powerset, and it isknown that these objects characterize those quasi-uniform spaces that are transitiveand totally bounded. The pointfree notion of a Pervin space, which we call Frithframe, consists of a frame equipped with a generating bounded sublattice. In thispaper we introduce and study the category of Frith frames and show that the classicaldual adjunction between topological spaces and frames extends to a dual adjunctionbetween Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, wedo not have an equivalence between the categories of transitive and totally boundedquasi-uniform frames and of Frith frames, but we show that the latter is a fullcoreflective subcategory of the former. We also explore the notion of completenessof Frith frames inherited from quasi-uniform frames, providing a characterizationof those Frith frames that are complete and a description of the completion of anarbitrary Frith frame.