
Journal of Convex Analysis 27 (2020), No. 4, 10991122 Copyright Heldermann Verlag 2020 On Densely Complete Metric Spaces and Extensions of Uniformly Continuous Functions in ZF Kyriakos Keremedis Dept. of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece kker@aegean.gr Eliza Wajch Institute of Mathematics, Faculty of Exact and Natural Sciences, Siedlce University of Natural Sciences and Humanities, 08110 Siedlce, Poland eliza.wajch@wp.pl [Abstractpdf] A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D$ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the countable axiom of choice, $\mathbf{CAC}$ for abbreviation, is equivalent to the following statements: \begin{itemize} \item[(1)]\vskip2mm Every densely complete (connected) metric space $\mathbf{X}$ is complete. \item[(2)]\vskip2mm For every pair of metric spaces $\mathbf{X}$ and $\mathbf{Y}$, if $% \mathbf{Y}$ is complete and $\mathbf{S}$ is a dense subspace of $\mathbf{X}$, while $f\colon \mathbf{S}\rightarrow \mathbf{Y}$ is a uniformly continuous function, then there exists a uniformly continuous extension $F\colon \mathbf{X}\to% \mathbf{Y}$ of $f$. \item[(3)]\vskip2mm Complete subspaces of metric spaces have complete closures. \item[(4)]\vskip2mm Complete subspaces of metric spaces are closed. \end{itemize} \vskip1mm It is also shown that the restriction of (i) to subsets of the real line is equivalent to the restriction $\mathbf{CAC}(\mathbb{R})$ of $\mathbf{CAC}$ to subsets of $\mathbb{R}$. However, the restriction of (ii) to subsets of $% \mathbb{R}$ is strictly weaker than $\mathbf{CAC}(\mathbb{R})$ because it is equivalent to the statement that $\mathbb{R}$ is sequential. Moreover, among other relevant results, it is proved that, for every positive integer $% n$, the space $\mathbb{R}^n$ is sequential if and only if $\mathbb{R}$ is sequential. It is also shown that $\mathbb{R}\times\mathbb{Q}$ is not densely complete if and only if $\mathbf{CAC}(\mathbb{R})$ holds. Keywords: Countable axiom of choice, complete metric spaces, connected metric spaces, sequential spaces. MSC: 03E25, 54E35, 54E50, 54C20, 54D55. [ Fulltextpdf (186 KB)] for subscribers only. 