On the set-theoretic strength of the $n$-compactness of generalized Cantor cubes
Volume 234 / 2016
Abstract
We investigate, in set theory without the Axiom of Choice $\mathbf{\mathsf{AC}}$, the set-theoretic strength of the statement
$Q(n)$: For every infinite set $X$, the Tychonoff product $2^X$, where $2=\{0,1\}$ has the discrete topology, is $n$-compact,
where $n=2,3,4,5$ (definitions are given in Section 1).
We establish the following results:
(1) For $n=3,4,5$, $Q(n)$ is, in $\mathbf{\mathsf{ZF}}$ (Zermelo–Fraenkel set theory minus $\mathbf{\mathsf{AC}}$), equivalent to the Boolean Prime Ideal Theorem $\mathbf{\mathsf{BPI}}$, whereas
(2) $Q(2)$ is strictly weaker than $\mathbf{\mathsf{BPI}}$ in $\mathbf{\mathsf{ZFA}}$ set theory (Zermelo–Fraenkel set theory with the Axiom of Extensionality weakened in order to allow atoms).
This settles the open problem in Tachtsis (2012) on the relation of $Q(n)$, $n=2,3,4,5$, to $\mathbf{\mathsf{BPI}}$.
