On metrizability and compactness of certain products without the Axiom of Choice

Abstract

We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) in which there is a compact, metrizable, non-second countable, Cantor cube. This answers in the affirmative an open question by E. Wajch (2018) [15].Furthermore, we strengthen a result in the above paper, namely “Countable products of metrizable spaces are quasi-metrizable implies van Douwen's Choice Principle”, by first observing that the above topological statement implies the weak choice principle MC(ℵ0,ℵ0), i.e. “Every countably infinite set of countably infinite sets has a multiple choice function”, and then by establishing that van Douwen's choice principle is strictly weaker than MC(ℵ0,ℵ0) in ZF. The above independence result also resolves an open problem in P. Howard and J.E. Rubin (1998) [7], and strengthens a result by P. Howard, K. Keremedis, H. Rubin, and J.E. Rubin (1998) [6].