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The main aim of the article is to show, in the absence of the Axiom ofChoice, 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 metrizablespaces is metrizable". Statements related to the above-mentioned ones are alsostudied. Permutation models (among them new ones) are shown in which acountable sum (also a countable product) of metrizable spaces need not bemetrizable, countable unions of countable sets are countable and there is acountable family of non-empty sets of size at most $2^{\aleph_0}$ which doesnot have a choice function. A new permutation model is constructed in whichevery uncountable compact metrizable space is of size at least $2^{\aleph_0}$but a denumerable family of denumerable sets need not have a multiple choicefunction.

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