Abstract
Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or \(T_1\). Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting.
Similar content being viewed by others
Change history
29 June 2018
In the original publication of the article, the formulation of the c-subfitness condition (c-sfit) in Subsection 5.2 is inaccurate, with effect in Theorem 5.3.
References
Aull, C.E., Thron, W.J.: Separation axioms between \(T_0\) and \(T_1\). Indag. Math. 24, 26–37 (1963)
Banaschewski, B., Pultr, A.: Variants of openness. Appl. Categ. Struct. 2, 331–350 (1994)
Dikranjan, D., Giuli, E.: Closure operators I. Topol. Appl. 27, 129–143 (1987)
Dowker, C.H., Strauss, D.P.: Separation axioms for frames. Colloq. Math. Soc. Janos Bolyai 8, 223–240 (1974)
Dowker, C.H., Strauss, D.: \(T_1\)- and \(T_2\)-axioms for frames. In: Aspects of Topology, London Math. Soc. Lecture Note Ser. 93, pp. 325–335. Cambridge Univ. Press, Cambridge (1985)
Dube, T.: A note on weakly pseudocompact locales. Appl. Gen. Topol. 18, 131–141 (2017)
Herrlich, H.: A concept of nearness. Gen. Topol. Appl. 5, 191–212 (1974)
Herrlich, H., Pultr, A.: Nearness, subfitness and sequential regularity. Appl. Categ. Struct. 8, 67–80 (2000)
Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)
Johnstone, P.T.: Stone Spaces. Cambridge Univ. Press, Cambridge (1982)
Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. Mem. Am. Math. Soc. 309 (1984)
Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971)
Picado, J., Pultr, A.: Frames and Locales: Topology without Points, Frontiers in Mathematics, vol. 28. Springer, Basel (2012)
Picado, J., Pultr, A.: More on subfitness and fitness. Appl. Categ. Struct. 23, 323–335 (2015)
Picado, J., Pultr, A.: New aspects of subfitness in frames and spaces. Appl. Categ. Struct. 24, 703–714 (2016)
Picado, J., Pultr, A., Tozzi, A.: Ideals in Heyting semilattices and open homomorphisms. Quaest. Math. 30, 391–405 (2007)
Plewe, T.: Quotient maps of locales. Appl. Categ. Struct. 8, 17–44 (2000)
Simmons, H.: The lattice theoretic part of topological separation properties. Proc. Edinb. Math. Soc. 21(2), 41–48 (1978)
Simmons, H.: Regularity, fitness, and the block structure of frames. Appl. Categ. Struct. 14, 1–34 (2006)
Simpson, Alex: Measure, randomness and sublocales. Ann. Pure Appl. Logic 163, 1642–1659 (2012)
Acknowledgements
This work was partially supported by the Centre for Mathematics of the University of Coimbra (UID/MAT/00324/ 2013 funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020), and by projects P202/12/G061 (Grant Agency of the Czech Republic) and MTM2015-63608-P (Ministry of Economy and Competitiveness of Spain). The first author also acknowledges a sabbatical grant from FCT (grant SFRH/BSAB/127925/2016). Thanks are due to the referee for comments and suggestions that have helped improve the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Bernhard Banaschewski.
Dedicated with thanks to our friend Bob Lowen.
Rights and permissions
About this article
Cite this article
Clementino, M.M., Picado, J. & Pultr, A. The Other Closure and Complete Sublocales. Appl Categor Struct 26, 891–906 (2018). https://doi.org/10.1007/s10485-018-9516-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-018-9516-4
Keywords
- Frame
- Locale
- Frame congruence
- Sublocale
- Subfit frame
- c-subfit frame
- Fit frame
- Regular frame
- Fitted sublocale
- Codense sublocale
- Complete sublocale
- Weakly complete sublocale