<Reference List> | |
Type: | Preprint |
National /International: | International |
Title: | Pervin spaces and Frith frames: bitopological aspects and completion |
Publication Date: | 2023-03-01 |
Authors: |
- Célia Borlido
- Anna Laura Suarez |
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. |
Institution: | DMUC 23-07 |
Online version: | http://www.mat.uc.pt...prints/eng_2023.html |
Download: | Not available |