  A cuf space (set, resp.) is a space (set, resp.) which is a countable union
of finite subspaces (subsets, resp.). It is proved in $\mathbf{ZF}$ (with the
absence of the axiom of choice) that all countable unions of cuf (denumerable,
resp.) sets are cuf sets iff all countable products of cofinite cuf
(denumerable, resp.) spaces are quasi-metrizable iff all countable products of
one-point Hausdorff compactifications of infinite cuf (denumerable, resp.)
spaces are quasi-metrizable. A countable product of one-point Hausdorff
compactifications of denumerable discrete spaces is first-countable iff it is
quasi-metrizable. A model of $\mathbf{ZF}$ is shown in which a countable
product two-point Hausdorff compactifications of denumerable discrete spaces is
first-countable without being quasi-metrizable. Other relevant independence
results are also proved.
