Several amazing discoveries about compact metrizable spaces in ZF
Kyriakos Keremedis, Eleftherios Tachtsis, Eliza Wajch

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 , some are shown to be independent of . For independence results, distinct models of and permutation models of with transfer theorems of Pincus are applied. New symmetric models are constructed in each of which the power set of 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 .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Fuzzy and Soft Set Theory · Advanced Banach Space Theory
