A bitopological point-free approach to compactifications
type: Item
instance of: scholarly article
title: A bitopological point-free approach to compactifications
zbMATH Open document ID: 1222.06006
author: Olaf Karl Klinke
author: Achim Jung
author: M. Andrew Moshier
published in: Topology and its Applications
publication date: 2011-08-04T00:00:00Z
review text: A d-frame consists of two frames \(L_{-}\) and \(L_{+}\) together with two relations \(\mathsf{tot}\subseteq L_-\times L_+\) and \(\mathsf{con}\subseteq L_+\times L_-\) satisfying certain axioms (imposing, in particular, that \(\mathsf{tot}\) is an upper set in \(L_-\times L_+\) and a sublattice of \(L_{-}^{\text{op}}\times L_+\) and that \(\mathsf{con}\) is a lower set in \(L_+\times L_-\), a sublattice of \(L_{+}\times L_{-}^{\text{op}}\) and closed under directed joins in \(L_+\times L_-\)). D-frames were introduced by the last two authors of the present paper in [On the bitopological nature of Stone duality. Technical Report CSR-06-13, School of Computer Science, The University of Birmingham (2006)] and are a kind of point-free analogue of bitopological spaces (different from the well-known biframes of \textit{B. Banaschewski, G. C. L. Brümmer} and \textit{K. A. Hardie} [``Biframes and bispaces'', Quaest. Math. 6, 13--25 (1983; Zbl 0513.06005)]. Indeed, the corresponding category is dually adjoint to the category of bitopological spaces and bicontinuous maps. This adjunction yields a general bitopological version of Stone duality, encompassing a number of classical Stone-type dualities (namely, those of Stone for Boolean algebras and bounded distributive lattices, those of the two authors for strong proximity lattices with negation and the well-known Ehresmann-Bénabou duality for ordinary frames). In the paper under review, the authors investigate d-frames and point-free compactifications of d-frames. They study a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames by their associated proximities. Further, they show that the category of d-frames has a normal coreflection (a somewhat surprising fact) taking completely regular d-frames to regular normal d-frames, and that the Stone-Čech compactification of d-frames factors through it. They end the paper with an adjunction between the category…
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 54E55
zbMATH DE Number: 5935899
zbMATH Keywords: frame
zbMATH Keywords: locale
zbMATH Keywords: d-frame
zbMATH Keywords: compactification
zbMATH Keywords: bitopology
zbMATH Keywords: ideal completion
zbMATH Keywords: strong inclusion
zbMATH Keywords: proximity
MaRDI profile type: Publication
full work available at URL: j.topol.2011.05.024
OpenAlex ID: W2156162038
cites work: Q4843177
cites work: BIFRAMES AND BISPACES
cites work: Compactification of Frames
cites work: A Hausdorff Topology for the Closed Subsets of a Locally Compact Non-Hausdorff Space
cites work: Neuaufbau der Endentheorie
cites work: Continuous Lattices and Domains
cites work: Q3965241
cites work: Asymmetry and duality in topology
cites work: Q3864573
cites work: Stable Compactification I
cites work: Q3994458
cites work: Information systems for continuous posets
DOI: 10.1016/J.TOPOL.2011.05.024
description: scientific article; zbMATH DE number 5935899