On uniformly continuous functions between pseudometric spaces and the axiom of countable choice
type: Item
instance of: scholarly article
title: On uniformly continuous functions between pseudometric spaces and the axiom of countable choice
zbMATH Open document ID: 1475.03095
author: Samuel Gomes da Silva
published in: Archive for Mathematical Logic
publication date: 2019-03-27T00:00:00Z
review text: A (pseudo)metric space is said to be \textit{Atsuji} or UC if every continuous real-valued function defined on the space is uniformly continuous. \textit{M. Atsuji} [Pac. J. Math. 8, 11--16 (1958; Zbl 0082.16207)] was the first to provide equivalent conditions under which continuous functions of a metric space are uniformly continuous. Recently, in the context of ZF set theory (i.e., Zermelo-Fraenkel set theory without the axiom of choice), the property of being UC in the class of metric spaces has been studied by \textit{K. Keremedis} [Topology Appl. 210, 366--375 (2016; Zbl 1355.54035); Bull. Pol. Acad. Sci., Math. 65, No. 2, 113--124 (2017; Zbl 1436.03256); Topol. Proc. 52, 73--93 (2018; Zbl 1398.54046)]. In the paper under review, the author investigates (within ZF) the property UC in the class of pseudometric spaces and obtains some interesting equivalents of the axiom of countable choice \(\mathrm{AC}_{\omega}\) (i.e., every countably infinite family of non-empty sets has a choice function); one of those equivalents being a well-known characterization -- via pairs of sequences -- of uniform continuity of real-valued continuous functions defined on pseudometric spaces (see (2) below). The main result of this paper is that \(\mathrm{AC}_{\omega}\) is equivalent to each of the following statements: \begin{itemize} \item[1.] If \(M,N\) are pseudometric spaces and \(f:M\rightarrow N\) is a continuous function, \(f\) is uniformly continuous, if and only if, for every pair of (not necessarily convergent) sequences \((x_n)\), \((y_n)\) in \(M\) such that \(d_{M}(x_n,y_n)\rightarrow 0\) one has \(d_{N}(f(x_n),f(y_n))\rightarrow 0\). \item[2.] If \(M\) is a pseudometric space and \(f:M\rightarrow\mathbb{R}\) is a continuous function, \(f\) is uniformly continuous, if and only if, for every pair of (not necessarily convergent) sequences \((x_n)\), \((y_n)\) in \(M\) such that \(d_{M}(x_n,y_n)\rightarrow 0\) one has \(|f(x_n)-f(y_n)|\rightarrow 0\). \item[3.] Let \(M,N\) …
reviewed by: Eleftherios Tachtsis
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 7043051
zbMATH Keywords: axiom of countable choice
zbMATH Keywords: pseudometric spaces
zbMATH Keywords: uniform continuity
MaRDI profile type: Publication
full work available at URL: s00153-018-0643-2
OpenAlex ID: W2884768194
Wikidata QID: Q114018306
cites work: Countable choice and pseudometric spaces
cites work: Axiom of choice
cites work: On the metric reflection of a pseudometric space in ZF
cites work: Q4210469
cites work: On Weierstrass compact pseudometric spaces and a weak form of the axiom of choice
cites work: On metric spaces where continuous real valued functions are uniformly continuous in \(\mathbf{ZF}\)
cites work: Uniform continuity and normality of metric spaces in $\mathbf{ZF}$
DOI: 10.1007/S00153-018-0643-2
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description: scientific article; zbMATH DE number 7043051