Metric spaces in choiceless set theory

Abstract

We answer open questions from Keremedis (2016) [12] and Keremedis and Tachtsis (2022) [16], and properly strengthen some results from the above papers as well as from Keremedis et al. (2023) [19]. In particular, and among other results, we establish the following:

• 1.
  The Boolean Prime Ideal Theorem does not imply “For every sequentially compact metric space $〈X,d〉$, $\left|X\right|\le 2^{{\aleph }_0}$” in ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)).
• 2.
  “Every linearly ordered set can be well ordered” ∧ “The union of a well-orderable family of well-orderable sets is well orderable” ∧ “For every uncountable sequentially compact metric space $〈X,d〉$, $\left|X\right|=2^{{\aleph }_0}$” does not imply the Axiom of Countable Choice in ZFA (ZF with atoms).
• 3.
  “For every uncountable compact metric space $〈X,d〉$, $\left|X\right|\ge 2^{{\aleph }_0}$” does not imply “For every uncountable sequentially compact metric space $〈X,d〉$, $\left|X\right|\ge 2^{{\aleph }_0}$” in ZFA”.
• 4.
  “For every uncountable sequentially compact metric space $〈X,d〉$ $\left|X\right|\ge 2^{{\aleph }_0}$” does not imply “For every uncountable compact metric space $〈X,d〉$, $\left|X\right|=2^{{\aleph }_0}$” in ZFA.
• 5.
  “For every uncountable sequentially compact metric space $〈X,d〉$ $\left|X\right|\ge 2^{{\aleph }_0}$” does not imply the Axiom of Countable Multiple Choice for countable sets (${\mathrm{CMC}}_{\omega }$) in ZFA.
• 6.
  “Every linearly ordered set can be well ordered” does not imply “Every infinite compact metric space has an infinite scattered subspace” ∨ “Axiom of Countable Choice for finite sets” in ZFA.

We also address the open problem of the deductive strength of Stone's Theorem “Every metric space is paracompact” (ST) and provide a non-trivial partial answer to the open question whether ST implies ${\mathrm{CMC}}_{\omega }$. In particular, we show that the formally weaker statement “For every metric space $〈X,d〉$, $\left|X\right|\ge 2^{{\aleph }_0}$ or $〈X,d〉$ is paracompact” does not imply ${\mathrm{CMC}}_{\omega }$ in ZFA. We also show that, for every uncountable regular cardinal κ, the above formally weaker form of ST is not implied by “For all infinite well-ordered cardinals $\lambda <\kappa$, the Principle of Dependent Choices for λ holds” in ZF.