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 such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of T 0 complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr's characterizations of sober and T D topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions.

We revisit results concerning the connection between subspaces of a space and sublocales of its locale of open sets. The approach we present is based on the notion of sublocale as a concrete subcollection of a locale. We characterize the frames L such that the spatial sublocales of S(L) perfectly represent the subspaces of pt(L). We prove choice-free, weak versions of the results by Niefield and Rosenthal characterizing those frames such that all their sublocales are spatial. We do so by using a notion of essential prime which does not rely on the existence of enough minimal primes above every element. We will re-prove Simmons' result that spaces such that the sublocales of Ω(X) perfectly represent their subspaces are exactly the scattered spaces. We will characterize scattered spaces in terms of a strong form of essentiality for primes. We apply these characterizations to show that, when L is a spatial frame and a coframe, pt(L) is scattered if and only if it is T D , and this holds if and only if all the primes of L are completely prime.

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