Striking differences between ZF and ZF+weak choice in view of metric spaces

Abstract

Well-known properties held by metric spaces with the countable chain condition (ccc) such as separability, second countability, Lindelöf, paracompactness may fail in the absence of the axiom of choice AC. In the long catalogue of weak choice principles, see [10], we find those which are sufficient (and in some cases necessary) to establish well known equivalences in ZFC between the statements CCMX, where CCMX stands for: ccc metric spaces have the property X, X ∈ {separable, second countable, Lindelöf, paracompact, size ≤ 2 ^ω }. In addition, we show: (1) The weak form of AC, DC(ℵ₁), implies the weaker version of Stone's theorem ccc metric spaces are paracompact and that the latter statement does not imply back DC(ℵ₁) in ZF⁰ (Zermelo-Fraenkel set theory minus the axiom of regularity). (2) In [11] it was shown that Stone's theorem, i.e. metric spaces are paracompact, is deducible from the statement every metric space has a σ-locally finite base and the validity of the reverse implication was in question. We close this gap by showing that the answer to the above question is affirmative.