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 equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As so, they have an underlying bitopological structure and inherit a natural notion of completion. In this talk, we will start
by exploring the bitopological nature of Pervin spaces and of Frith frames, proving a couple of categorical equivalences involving zero-dimensional structures. We will then see that the categories of T₀ complete Pervin spaces and of complete Frith frames are dually equivalent. This allows us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings.

This is based on joint work with Anna Laura Suarez.