Infinite Hausdorff spaces may lack cellular families or discrete subsets of cardinality ℵ0

Abstract

In set theory without the Axiom of Choice (AC), we investigate the deductive strength of the statements “every infinite Hausdorff space has a countably infinite cellular family” and “every infinite Hausdorff space has a countably infinite relatively discrete subspace”, and of variants of the above statements for certain classes of Hausdorff spaces, their mutual relationship, as well as their relationship with various weak choice principles.Among several results, we construct a model N of ZF (i.e., of Zermelo–Fraenkel set theory without AC) in which there exists a dense-in-itself, zero-dimensional, Hausdorff topology on ω such that c(ω)=s(ω)=ω (where c(ω) and s(ω) are the cellularity and the spread of ω, respectively), but ω has no infinite discrete subsets in N, and thus neither has infinite cellular families in N.