Metric spaces on which continuous functions are ``almost'' uniformly continuous
type: Item
instance of: scholarly article
title: Metric spaces on which continuous functions are ``almost'' uniformly continuous
zbMATH Open document ID: 1379.54018
author: Kyriakos Keremedis
published in: Topology and its Applications
publication date: 2017-11-09T00:00:00Z
review text: In this paper, the author discusses the relationship among the notions of continuous functions, almost-bounded functions, almost-uniformly continuous functions and Cauchy-continuous functions, and proves that the image of a Cauchy sequence in a metric space under a continuous and almost-uniformly continuous function has a Cauchy subsequence, and the image of a totally bounded metric space under a continuous and almost-uniformly continuous function is totally bounded. Some characterizations of Lebesgue metric spaces are given. The author also considers some properties of \(AUC\) metric spaces. The following results are proved: (1) Every \(AUC\) metric space is complete; (2) A metric space is compact if and only if it is totally bounded and \(AUC\); (3) Let \(\mathbf{X}=(X,d)\) be a metric space. The following are equivalent: (i) \(\mathbf{X}\) is \(AUC\); (ii) \(\mathbf{X}\) is weakly Lebesgue; (iii) \(\mathbf{X}\) is weakly countably Lebesgue; (iv) For every metric space \(\mathbf{Y}=(Y,\rho)\), every continuous function \(f:\mathbf{X}\rightarrow\mathbf{Y}\) is almost-uniformly continuous.
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54C30
zbMATH DE Number: 6804710
zbMATH Keywords: compact
zbMATH Keywords: complete
zbMATH Keywords: totally bounded
zbMATH Keywords: uniformly continuous metric space
zbMATH Keywords: Lebesgue metric space
reviewed by: Shou Lin
MaRDI profile type: Publication
full work available at URL: j.topol.2017.10.009
OpenAlex ID: W2767193103
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DOI: 10.1016/J.TOPOL.2017.10.009
description: scientific article; zbMATH DE number 6804710