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A Pervin space is a set equipped with a bounded sublattice of its powerset,while its pointfree version, called Frith frame, consists of a frame equippedwith a generating bounded sublattice. It is known that the dual adjunctionbetween topological spaces and frames extends to a dual adjunction betweenPervin spaces and Frith frames, and that the latter may be seen asrepresentatives of certain quasi-uniform structures. As such, they have anunderlying bitopological structure and inherit a natural notion of completion.In this paper we start by exploring the bitopological nature of Pervin spacesand of Frith frames, proving some categorical equivalences involvingzero-dimensional structures. We then provide a conceptual proof of a dualitybetween the categories of $T_0$ complete Pervin spaces and of complete Frithframes. This enables us to interpret several Stone-type dualities as arestriction of the dual adjunction between Pervin spaces and Frith frames alongfull subcategory embeddings. Finally, we provide analogues of Banaschewski andPultr's characterizations of sober and $T_D$ topological spaces in the settingof Pervin spaces and of Frith frames, highlighting the parallelism between thetwo notions.

Pervin spaces and Frith frames: bitopological aspects and completion