Monatshefte für Mathematik (2021) 196:67–102 https://doi.org/10.1007/s00605-021-01582-0 Several results on compact metrizable spaces in ZF Kyriakos Keremedis1 · Eleftherios Tachtsis2 · Eliza Wajch3 Received: 1 October 2020 / Accepted: 10 June 2021 / Published online: 29 June 2021 © The Author(s) 2021 Abstract In the absence of the axiom of choice, the set-theoretic status of many natural state- ments about metrizable compact spaces is investigated. Some of the statements are provable in ZF, some are shown to be independent of ZF. For independence results, distinct models of ZF and permutation models of ZFA with transfer theorems of Pin- cus are applied. New symmetric models of ZF are constructed in each of which the power set of R is well-orderable, the Continuum Hypothesis is satisfied but a denumer- able 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]R. Keywords Weak forms of the Axiom of Choice · Metrizable space · Totally bounded metric · Compact space · Permutation model · Symmetric model Mathematics Subject Classification 03E25 · 03E35 · 54A35 · 54E35 · 54D30 Communicated by S.-D. Friedman. B Eliza Wajch
[email protected] Kyriakos Keremedis
[email protected] Eleftherios Tachtsis
[email protected] 1 Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece 2 Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece 3 Institute of Mathematics, Faculty of Exact and Natural Sciences, Siedlce University of Natural Sciences and Humanities, ul. 3 Maja 54, 08-110 Siedlce, Poland 123 68 K.