### On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product ${2}^{\mathbb{R}}$

We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets ($A{C}^{WO}$) does not imply “the Tychonoff product ${2}^{\mathbb{R}}$, where 2 is the discrete space 0,1, is countably compact” in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets of reals imply...