Countable products and countable direct sums of compact metrizable spaces in the absence of the Axiom of Choice

• Published in: Annals of pure and applied logic 2023-07, Vol.174 (7), p.103283, Article 103283
• Main Authors: Keremedis, Kyriakos, Tachtsis, Eleftherios, Wajch, Eliza
• Format: Article
• Language: English
• Subjects: Compact metrizable space; Countable direct sum; Countable product; Fraenkel-Mostowski models; Weak forms of the Axiom of Choice; ZF-models
• ISSN: 0168-0072
• DOI: 10.1016/j.apal.2023.103283

The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of ZF, statements: “Every countable product of compact metrizable spaces is separable (respectively, compact)” and “Every countable product of compact metrizable spaces is metrizable”. Statements related to the above-mentioned ones are also studied. Permutation models (among them two new ones) are shown in which a countable direct sum (also a countable product) of metrizable spaces need not be metrizable, countable unions of countable sets are countable and there is a countable family of non-empty sets of size at most 2ℵ0 which does not have a choice function. A new permutation model is constructed in which every uncountable compact metrizable space is of size at least 2ℵ0 but a denumerable family of denumerable sets need not have a multiple choice function.