Remarks on the Stone Spaces of the Integers and the Reals without AC

摘要

Summary. In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2~(P(X)) where 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 = ω, R. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.