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.