# Pervin spaces and Frith frames: bitopological aspects and completion

## TL;DR

This paper explores the bitopological and categorical structures of Pervin spaces and Frith frames, establishing dualities and characterizations that extend classical topological dualities into a pointfree and quasi-uniform context.

## Contribution

It provides a categorical equivalence between zero-dimensional complete Pervin spaces and complete Frith frames, extending Stone-type dualities and characterizations to this setting.

## Key findings

- Categorical equivalences involving zero-dimensional structures
- Duality between $T_0$ complete Pervin spaces and complete Frith frames
- Analogues of Banaschewski and Pultr's characterizations in this context

## Abstract

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.