Remarks on the Stone Spaces of the Integers and the Reals without ${\bf AC}$
Volume 59 / 2011
Bulletin Polish Acad. Sci. Math. 59 (2011), 101-114
MSC: 03E25, 54B10, 54D30, 54D80.
DOI: 10.4064/ba59-2-1
Abstract
In $\mathbf{ZF}$, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice $\mathbf{AC}$, we investigate the relationship between the Tychonoff product $\mathbf{2}^{\mathcal{P}(X)}$, where $\mathbf{2}$ is $2=\{0,1\}$ with the discrete topology, and the Stone space $S(X)$ of the Boolean algebra of all subsets of $X$, where $X=\omega,\mathbb{R}$. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.
