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 of \(\mathbf {ZF}\) 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 \([0, 1]^{\mathbb {R}}\).

在没有选择公理的情况下，研究了许多关于可度量紧空间的自然陈述的集合论状态。有些陈述在\(\mathbf {ZF}\) 中是可证明的，有些陈述是独立于\(\mathbf {ZF}\)。对于独立性结果，应用了不同的\(\mathbf {ZF}\)模型和具有 Pincus 传递定理的\(\mathbf {ZFA}\)置换模型。\(\mathbf {ZF}\) 的新对称模型被构造在每个模型中，其中\(\mathbb {R}\)的幂集是良序的，连续统假设得到满足，但非空有限集的可数族可能没有选择函数，并且紧凑的可度量空间不需要嵌入 Tychonoff 立方体\([0, 1]^{ \mathbb {R}}\)。