An additivity theorem for uniformly continuous functions
type: Item
instance of: scholarly article
title: An additivity theorem for uniformly continuous functions
zbMATH Open document ID: 1078.54014
author: Alessandro Berarducci
author: Dikran Dikranjan
author: Jan Pelant
published in: Topology and its Applications
publication date: 2005-02-22T00:00:00Z
review text: A metric space \(X\) is said to be straight (2-straight) if for any finite (resp. two set) closed cover of \(X\) any continuous real-valued function on \(X\) is uniformly continuous if and only if its restriction to each of the closed sets of the cover is uniformly continuous. It is shown that 2-straight is equivalent to straight. For a locally connected metric space, \(X\), it is also shown that \(X\) is straight if and only if \(X\) is uniformly locally connected i.e. for all \(\varepsilon> 0\), there is a \(\delta> 0\) such that any two points less than \(\delta\) apart lie in a connected set of diameter less than \(\varepsilon\). Moreover, for a totally disconnected space, \(X\), (i.e. a space for which the intersection of all open sets containing a point is that point) it is proved that \(X\) is straight if and only if \(X\) is a UC space i.e. if and only if all real-valued functions on \(X\) are uniformly continuous.
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54E50
zbMATH DE Number: 2136475
zbMATH Keywords: Uniform continuity
zbMATH Keywords: Metric spaces
zbMATH Keywords: Local connectedness
zbMATH Keywords: Total disconnectedness
zbMATH Keywords: Atsuji space
MaRDI profile type: Publication
full work available at URL: j.topol.2003.05.007
OpenAlex ID: W2022745863
cites work: Uniform continuity of continuous functions of metric spaces
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cites work: Q3994935
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cites work: On Uniform Continuity and Compactness in Metric Spaces
DOI: 10.1016/J.TOPOL.2003.05.007
description: scientific article; zbMATH DE number 2136475