On uniformly continuous functions between pseudometric spaces and the axiom of countable choice (Q1734263)

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 \(M,N\) be pseudometric spaces and assume \(M\) to be sequentially compact (i.e., every sequence in \(M\) has a convergent subsequence converging to a point in \(M\)). Then for every function \(f:M\rightarrow N\), \(f\) is continuous, if and only if, it is uniformly continuous. \item[4.] Sequentially compact, pseudometric spaces are UC. \end{itemize}