  In the absence of the axiom of choice, the set-theoretic status of many
natural statements about metrizable compact spaces is investigated. Some of the
statements are provable in $\mathbf{ZF}$, some are shown to be independent of
$\mathbf{ZF}$. For independence results, distinct models of $\mathbf{ZF}$ and
permutation models of $\mathbf{ZFA}$ with transfer theorems of Pincus are
applied. New symmetric models are constructed in each of which the power set of
$\mathbb{R}$ is well-orderable, the Continuum Hypothesis is satisfied but a
denumerable family of non-empty finite sets can fail to have a choice function,
and a compact metrizable space need not be embeddable into the Tychonoff cube
$[0, 1]^{\mathbb{R}}$.
