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The Other Closure and Complete Sublocales

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A Correction to this article was published on 29 June 2018

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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.

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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.

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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.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Maria Manuel Clementino & Jorge Picado

  2. Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské nám. 24, 11800, Praha 1, Czech Republic

    Aleš Pultr

Authors
  1. Maria Manuel Clementino
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  2. Jorge Picado
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  3. Aleš Pultr
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Correspondence to Jorge Picado.

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Communicated by Bernhard Banaschewski.

Dedicated with thanks to our friend Bob Lowen.

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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

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