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

2021/09/01 by Keremedis, Kyriakos, Tachtsis, Eleftherios, Wajch, Eliza
#03E25 #03E35 #54A35 #54B10 #54D30 #54E35 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2109.00640

Abstract

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 new ones) are shown in which a countable 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 2^ℵ₀ 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 2^ℵ₀ but a denumerable family of denumerable sets need not have a multiple choice function.

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