Countable products and countable 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 $\mathbf{ZF}$, 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 $2^{\aleph_0}$ 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 $2^{\aleph_0}$ but a denumerable family of denumerable sets need not have a multiple choice function.