The topology of close approximation
type: Item
instance of: scholarly article
title: The topology of close approximation
zbMATH Open document ID: 0715.54007
DOI: 10.1016/0166-8641(90)90111-E
author: Eric K. van Douwen
author: A. H. Stone
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: Let \({\mathcal L}\subset R^ R\) denote the set of all Lebesgue measurable real-valued functions defined on R, and let \({\mathcal L}^+\) denote \(\{f\in {\mathcal L}:\) \(f(x)>0\) for all \(x\in R\}\). The topology of close approximation, \({\mathcal T}_ c\), is defined on \(R^ R\) by taking as a neighbourhood base of \(f\in R^ R\) the sets \(\hat C(f,\epsilon)=\{g\in R^ R|\) \(g(x)-f(x)| <\epsilon (x)\) for all \(x\in R\}\), where \(\epsilon\in {\mathcal L}^+\). The uniform topology, \({\mathcal T}_ u\), is defined on \(R^ R\) by taking as neighbourhood base of \(f\in R^ R\) the sets \(\hat U(f,\epsilon)=(g\in R^ R:| g(x)-f(x)| <\epsilon\) for all \(x\in R\}\), where \(\epsilon \in R^+\), the set of positive real numbers. The Lusin topology, \({\mathcal T}_ L\), of convergence in measure is defined by the neighborhood base of \(f\in R^ R\) \(\hat L(f,\epsilon)=\{g\in R^ R:\lambda^*\{x\in R| g(x)-f(x)| \geq \epsilon \}<\epsilon\}\), where \(\epsilon \in R^+\) and \(\lambda^*\) denotes the Lebesgue outer measure. Clearly \({\mathcal T}_ L\subset {\mathcal T}_ u\subset {\mathcal T}_ c.\) The authors discuss the properties of the spaces (\({\mathcal L},{\mathcal T}_ c)\), (\({\mathcal L},{\mathcal T}_ u)\) and (\({\mathcal L},{\mathcal T}_ L)\) interacting on each other, and the situation of their subspaces, which are summarized as below: 1. \({\mathcal L}\) is a closed \(G_{\delta}\)-set in \(R^ R\), in each of \({\mathcal T}_ L\), \({\mathcal T}_ u\) and \({\mathcal T}_ c.\) Let \({\mathcal L}'\) be the set \(\{\) \(f\in {\mathcal L}:\int_{R}| f(x)| d\lambda(x)<\infty\}\) and \({\mathcal C}\) the set of all continuous functions from R to R. 2. All four spaces \(R^ R\), \({\mathcal L}\), \({\mathcal L}'\) and \({\mathcal C}\) are Baire spaces in all three topologies \({\mathcal T}_ L\), \({\mathcal T}_ u\) and \({\mathcal T}_ c\), except for (\({\mathcal L}',{\mathcal T}_ L)\), (\({\mathcal L}',{\mathcal T}_ u)\) and (\({\mathcal C},{\mathcal T}_ L).\) 3. (\({\mathca…
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 26A99
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 28A20
zbMATH DE Number: 4178081
zbMATH Keywords: Lebesgue measurable real-valued functions
zbMATH Keywords: topology of close approximation
zbMATH Keywords: uniform topology
zbMATH Keywords: Lusin topology
zbMATH Keywords: Baire spaces
Wikidata QID: Q106043543
reviewed by: Akihiro Okuyama
MaRDI profile type: Publication
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full work available at URL: 0166-8641(90)90111-e
OpenAlex ID: W2167715569
description: scientific article; zbMATH DE number 4178081