AbstractIn 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 $$\mathbf {ZF}$$
ZF
, some are shown to be independent of $$\mathbf {ZF}$$
ZF
. For independence results, distinct models of $$\mathbf {ZF}$$
ZF
and permutation models of $$\mathbf {ZFA}$$
ZFA
with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$
ZF
are constructed in each of which the power set of $$\mathbb {R}$$
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 $$[0, 1]^{\mathbb {R}}$$
[
0
,
1
]
R
.
Several results on compact metrizable spaces in $$\mathbf {ZF}$$
