Lower and upper regularizations of frame semicontinuous real functions
type: Item
instance of: scholarly article
title: Lower and upper regularizations of frame semicontinuous real functions
zbMATH Open document ID: 1181.06003
author: Javier Gutiérrez García
author: Tomasz Kubiak
author: Jorge Picado
published in: Algebra Universalis
publication date: 2010-01-04T00:00:00Z
review text: The main gist of this paper is a characterization of extremally disconnected frames (also known as De Morgan frames) in terms of extendibility of bounded real-valued continuous functions on open sublocales. The main theorem states that a frame \(L\) is extremally disconnected if and only if, for every open sublocale \(S\), every bounded real-valued continuous function on \(S\) extends to a bounded real-valued continuous function on \(L\). The authors establish this result by first defining the pointfree concepts of lower and upper regularizations of frame semicontinuous real-valued functions. The stated theorem is a much strengthened version of an earlier effort in this regard by \textit{Y.-M. Li} and \textit{Z.-H. Li} [``Constructive insertion theorems and extension theorems on extremally disconnected frames'', Algebra Univers. 44, No. 3--4, 271--281 (2000; Zbl 1013.06010)], which was demonstrated by \textit{J. Picado} [``A new look at localic interpolation theorems'', Topology Appl. 153, No. 16, 3203--3218 (2006; Zbl 1104.06007)] to have shortcomings emanating from a treatment of pointfree semicontinuity which was not fully reflective of its point-sensitive counterpart. The preparatory results in the present paper are a continuation of the authors' study of semicontinuity in pointfree topology.
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54G05
zbMATH DE Number: 5652276
zbMATH Keywords: frame
zbMATH Keywords: locale
zbMATH Keywords: sublocale
zbMATH Keywords: frame of reals
zbMATH Keywords: frame semicontinuous real functions
zbMATH Keywords: lower and upper regularizations
zbMATH Keywords: insertion theorem
zbMATH Keywords: extremally disconnected
MaRDI profile type: Publication
full work available at URL: s00012-009-2102-8
OpenAlex ID: W2047017534
DOI: 10.1007/S00012-009-2102-8
description: scientific article; zbMATH DE number 5652276