Pointfree forms of Dowker's and Michael's insertion theorems
type: Item
instance of: scholarly article
title: Pointfree forms of Dowker's and Michael's insertion theorems
zbMATH Open document ID: 1154.06006
author: Jorge Picado
author: Javier Gutiérrez García
author: Tomasz Kubiak
published in: Journal of Pure and Applied Algebra
publication date: 2008-11-11T00:00:00Z
review text: The authors prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space, they are equivalent to the following insertion theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. (Dowker) A topological space \(X\) is normal and countably paracompact if and only if, given \(h,g:X\rightarrow {\mathbb R}\) such that \(h<g, h\) is upper semicontinuous and \(g\) is lower semicontinuous, there is a continuous \(f:X\rightarrow {\mathbb R}\) such that \(h<f<g.\) (Michael) A topological space \(X\) is perfectly normal if and only if, given \(h,g:X\rightarrow {\mathbb R}\) such that \(h\leq g, h\) is upper semicontinuous and \(g\) is lower semicontinuous, there is a continuous \(f:X\rightarrow {\mathbb R}\) such that \(h\leq f\leq g\) and \(h(x)<f(x)<g(x)\) whenever \(h(x)<g(x).\) The authors state that, however, in the pointfree context they have been unable to provide pointfree assertions corresponding exactly to these two classical insertion results. Furthermore, they present a study of the concept of perfect normality for frames.
reviewed by: Hans Peter Künzi
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 5364859
zbMATH Keywords: perfectly normal
zbMATH Keywords: insertion theorem
zbMATH Keywords: pointfree topology
zbMATH Keywords: frame
zbMATH Keywords: semicontinuous
zbMATH Keywords: normal
zbMATH Keywords: countably paracompact
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.05.010
OpenAlex ID: W2079808228
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DOI: 10.1016/J.JPAA.2008.05.010
description: scientific article; zbMATH DE number 5364859