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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Notes
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.
References
Ball, R.N., Picado, J., Pultr, A.: Notes on exact meets and joins. Appl. Categ. Struct. 22, 699–714 (2014)
Banaschewski, B.: Coherent frames. In Continuous Lattices (Berlin, Heidelberg), B. Banaschewski and R.-E. Hoffmann, Eds., Springer Berlin Heidelberg, pp. 1–11 (1981)
Banaschewski, B.: Universal, zero-dimensional compactifications. In Categorical topology and its relation to analysis, algebra and combinatorics (Prague,: World Sci. Publ. Teaneck, NJ 1989, 257–269 (1988)
Banaschewski, B., Brümmer, G.C., Hardie, K.A.: Biframes and bispaces. Quaest. Math. 6(1–3), 13–25 (1983)
Banaschewski, B., Pultr, A.: Pointfree aspects of the \(T_D\) axiom of classical topology. Quaest. Math. 33(3), 369–385 (2010)
Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A.: Bitopological duality for distributive lattices and Heyting algebras. Math. Struct. Comput. Sci. 20(3), 359–393 (2010)
Borlido, C., Suarez, A. L.: A pointfree theory of Pervin spaces. Quaestiones Mathematicae, 1–40 (2023)
Cornish, W.: On H. Priestley’s dual of the category of bounded distributive lattices. Matematički Vesnik 12(27), 329–332 (1975)
Davey, B.A., Priestley, H.A.: Introduction to lattices and order. Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge (1990)
Frith, J. (1986) Structured frames. PhD thesis, University of Cape Town
Gehrke, M., Grigorieff, S., Pin, J.-E.: A topological approach to recognition. In Automata, languages and programming. Part II, vol. 6199 of Lecture Notes in Computer Science Springer, Berlin, pp. 151–162 (2010)
Jech, T. J.: The axiom of choice. Studies in Logic and the Foundations of Mathematics, Vol. 75. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, (1973)
Johnstone, P. T.: Stone spaces, vol. 3 of Cambridge studies in advanced mathematics. Cambridge University Press, (1982)
Johnstone, P. T.: Vietoris locales and localic semilattices. In: Lecture Notes in Pure and Applied Mathematics 101 (1985). Proceedings of a conference held in the University of Bremen, (1982)
Moshier, M.A., Pultr, A., Suarez, A.L.: Exact and strongly exact filters. Appl. Categ. Struct. 28(6), 907–920 (2020)
Niefield, S., Rosenthal, K.: Spatial sublocales and essential primes. Topol. Appl. 26(3), 263–269 (1987)
Picado, J.: Join-continuous frames, Priestley’s duality and biframes. Appl. Categ. Struct. 2(3), 297–313 (1994)
Picado, J.: Frames and Locales: Topology without points. Springer, Basel (2011)
Pin, J.E.: Dual space of a lattice as the completion of a Pervin space. In: International Conference on Relational and Algebraic Methods in Computer Science, pp. 24–40 (2017)
Reilly, I.L.: Zero dimensional bitopological spaces. Indagationes Mathematicae (Proceedings) 76(2), 127–131 (1973)
Salbany, S.: Bitopological spaces, compactifications and completions. Mathematical Monographs of the University of Cape Town, No. 1. University of Cape Town, Department of Mathematics, Cape Town, (1974)
Schauerte, A.: Biframes. PhD thesis, McMaster University, Hamilton, Ontario, (1992)
Vickers, S.: Constructive points of powerlocales. Math. Proc. Cambridge Philos. Soc. 122(2), 207–222 (1997)
Wilson, J. T.: The Assembly Tower and Some Categorical and Algebraic Aspects of Frame Theory. PhD thesis, Carnegie Mellon University, (1994)
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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Communicated by Matias Menni.
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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