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

TL;DR

This paper explores the relationships between countable products and sums of compact metrizable spaces without relying on the Axiom of Choice, revealing new models where certain properties fail.

Contribution

It introduces new permutation models demonstrating the independence of these properties from ZF and constructs a model with unique size and choice characteristics.

Findings

Countable product of compact metrizable spaces may not be metrizable

Countable sums of metrizable spaces may not be metrizable

Existence of models with specific size and choice properties

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.