Selections and sandwich-like properties via semi-continuous Banach-valued functions.
type: Item
instance of: scholarly article
title: Selections and sandwich-like properties via semi-continuous Banach-valued functions.
zbMATH Open document ID: 1039.54012
author: Haruto Ohta
author: Kaori Yamazaki
author: Valentin Gutev
published in: Journal of the Mathematical Society of Japan
publication date: 2003-06-30T00:00:00Z
review text: The classical Katetov-Tong insertion theorem says that a \(T_1\) space \(X\) is normal if and only if for every two functions \(g,h: X\to\mathbb R\) such that \(g\) is upper semicontinuous, \(h\) is lower semicontinuous and \(g\leq h\), there exists a continuous function \(f: X\to\mathbb R\) such that \(g\leq f\leq h\). In this paper the authors obtain the natural generalization that arises when the real line is replaced by the Banach space \(c_0(\lambda)\), where \(\lambda\) is an infinite cardinal and \(c_0(\lambda)\) is defined as the set of all functions from \(\mathbb R^\lambda\) to \(\mathbb R\) such that, for each \(\varepsilon> 0\), the set \(\{\alpha< \lambda:| s(\alpha)|\geq \varepsilon\}\) is finite. They prove that the corresponding statement is equivalent to the fact that, for every Banach space \(Y\), with \(\omega(Y)\leq \lambda\), every lower semicontinuous set-valued mapping \(\varphi: X\to{\mathcal C}_C(Y)\) admits a continuous selection, where \({\mathcal C}_C(Y)\) is the set of all nonempty compact convex sets in \(Y\). Results characterizing paracompact-like properties in terms of sandwich-like statements and selections are shown and some interesting questions are included.
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 1943322
zbMATH Keywords: selection
zbMATH Keywords: insertion
zbMATH Keywords: Banach space
zbMATH Keywords: semicontinuous collectionwise normal
zbMATH Keywords: perfectly normal
zbMATH Keywords: countably paracompact
zbMATH Keywords: paracompact
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191419128
OpenAlex ID: W2052538190
DOI: 10.2969/JMSJ/1191419128
description: scientific article; zbMATH DE number 1943322