Several amazing discoveries about compact metrizable spaces in ZF

TL;DR

This paper explores the set-theoretic properties of compact metrizable spaces without assuming the axiom of choice, establishing which statements are provable or independent within ZF and constructing models demonstrating these results.

Contribution

It provides new independence results and models showing the set-theoretic behavior of compact metrizable spaces in ZF without the axiom of choice.

Findings

Some statements are provable in ZF.

Some statements are independent of ZF.

Existence of models where certain compact spaces are not embeddable into Tychonoff cubes.

Abstract

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 $[0, 1]^{\mathbb{R}}$.