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.
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These notions are not consistent over all the literature. For instance, in [6] our notions of \(T_0\) and compact are named join \(T_0\) and join compact, respectively, while compact is named pairwise compact in [21]. Moreover, in [6, 20] our notion of zero-dimensional is named pairwise zero-dimensional.
Under the identification of Pervin spaces with transitive and totally bounded quasi-uniform spaces, this functor forgets the quasi-uniform structure.
For the interested reader, this is the underlying bitopological space of the quasi-uniform space represented by \((X, {{\mathcal {S}}})\).
As for spaces, this is the underlying biframe of the quasi-uniform frame defined by (L, S).
This example was borrowed from [7, Example 5.13] The interested reader may show that the fixpoints of the adjunction \({\textbf{Sk}}_{\textrm{Frith}}\dashv {\textbf{B}}_+\) are precisely the underlying biframes of the quasi-uniform frames representable by a Frith frame in the sense of [7, Proposition 5.6].
The result cited is stated for a regular domain, but every zero-dimensional frame is regular.
In [11] Cauchy complete and Cauchy completion are simply named complete and completion, respectively.
Note that pairwise Stone spaces are the same as \(T_0\), compact and zero-dimensional bitopological spaces only under the assumption of the Prime Ideal Theorem.
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Funding
Célia Borlido was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Anna Laura Suarez received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.670624).
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Borlido, C., Suarez, A.L. Pervin Spaces and Frith Frames: Bitopological Aspects and Completion. Appl Categor Struct 31, 43 (2023). https://doi.org/10.1007/s10485-023-09749-6
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DOI: https://doi.org/10.1007/s10485-023-09749-6