On Densely Complete Metric Spaces and Extensions of Uniformly Continuous Functions in ZF

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)]\vskip-2mm Every densely complete (connected) metric space $\mathbf{X}$ is complete. \item[(2)]\vskip-2mm For every pair of metric spaces $\mathbf{X}$ and $\mathbf{Y}$, if $$ is complete and $\mathbf{S}$ is a dense subspace of $\mathbf{X}$, while $f\colon \mathbf{S}\rightarrow \mathbf{Y}$ is a uniformly continuous function, then there exists a uniformly continuous extension $F\colon \mathbf{X}\to$ of $f$. \item[(3)]\vskip-2mm Complete subspaces of metric spaces have complete closures. \item[(4)]\vskip-2mm Complete subspaces of metric spaces are closed. \end{itemize} \vskip-1mm It is also shown that the restriction of (i) to subsets of the real line is equivalent to the restriction $\mathbf{CAC}(\mathbb{R})$ of $\mathbf{CAC}$ to subsets of $\mathbb{R}$. However, the restriction of (ii) to subsets of $$ 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 $$, 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.