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

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 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.