Completely normal frames and real-valued functions
type: Item
instance of: scholarly article
title: Completely normal frames and real-valued functions
zbMATH Open document ID: 1190.06008
author: Maria João Ferreira
author: Javier Gutiérrez García
author: Jorge Picado
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/11233
review text: In an earlier publication [J. Pure Appl. Algebra 213, No.~6, 1064--1074 (2009; Zbl 1187.06005)] \textit{J. Gutiérrez García, T. Kubiak} and \textit{J. Picado} define the pointfree version of real-valued (not necessarily continuous or semicontinuous) functions. In the paper under review, the authors use this notion to characterize completely normal locales. Given a locale \(L\), take its sublocale lattice and put it on its head, so that you have a frame \(\mathcal{S}(L)\) instead of a co-frame. Dually to the point-sensitive case, two sublocales of \(L\) are said to be separated if each is co-disjoint (in the frame \(\mathcal{S}(L)\)) from the closure of the other. Further, two sublocales \(S\) and \(T\) of \(L\) are said to be separated by open sublocales if there are co-disjoint open sublocales \(U\) and \(V\) such that \(S\geq U\) and \(T\geq V\). The authors then define a locale to be completely normal in case every pair of separated sublocales is separated by open sublocales. As expected, a locale is called hereditarily normal if every sublocale is normal. The authors give numerous characterizations of complete normality in terms of elements and sublocales, including the following: A locale is completely normal iff it is hereditarily normal iff every open sublocale is normal. After recalling the notions of lower and upper regularization of a real-valued function on a locale, which were introduced in [\textit{J. Gutiérrez García, T. Kubiak} and \textit{J. Picado}, Algebra Univers. 60, No.~2, 169--184 (2009; Zbl 1181.06003 )], the authors proceed to characterize completely normal locales as those with the property that if two real-valued functions on them are comparable (and satisfy some other feature related to upper and lower regularizations), then a lower semicontinuous function and its upper regularization can be inserted between the two functions. The paper contains many other interesting results, and is written in the authors' inimitable style that makes it e…
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54G05
zbMATH DE Number: 5625489
zbMATH Keywords: localic real function
zbMATH Keywords: completely normal frame
zbMATH Keywords: completely normal space
zbMATH Keywords: extremally disconnected frame
zbMATH Keywords: insertion
zbMATH Keywords: upper and lower regularization
zbMATH Keywords: completely normal locales
zbMATH Keywords: sublocale lattice
MaRDI profile type: Publication
OpenAlex ID: W2150850582
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DOI: 10.1016/J.TOPOL.2008.12.042
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description: scientific article; zbMATH DE number 5625489