Stranger things about the cardinality of compact metric spaces without AC

Abstract

In set theory without the Axiom of Choice ($\mathbf{AC}$), we investigate the open problem of determining the possible cardinalities of compact metric spaces and bounds on the cardinality of such spaces. We establish the following surprising and unexpected results:

• 1. There exists a permutation model $\mathcal{N}$ in which there is a crowded compact metric space $\langle X,d \rangle$ with $|X|> |\mathbb{R}|$ in $\mathcal{N}$ and, for every compact metric space $\langle Y,\rho\rangle$ in $\mathcal{N}$, $|Y|$ is comparable to $|\mathbb{R}|$ in $\mathcal{N}$. This resolves an open problem from Keremedis and Tachtsis [9], Keremedis [8] and Keremedis, Tachtsis and Wajch [11]. Using the proof-technique of the above result, we also provide a general criterion for permutation models to satisfy the proposition ''The cardinality of a compact metric space is comparable to $|\mathbb{R}|$''.

• 2. It is relatively consistent with $\mathbf{ZF}$ that there exist at least $|\mathbb{R}|$ compact metric spaces with pairwise incomparable cardinalities, each of which is incomparable with $|\mathbb{R}|$.

• 3. It is relatively consistent with $\mathbf{ZF}$ that there exists a crowded compact metric space $\langle X,d\rangle$ such that the set $[X]^{<\omega}$ of finite subsets of X has no denumerable (i.e. countably infinite) subsets and $|X|$ is incomparable with $|\mathbb{R}|$; this will yield that ''There exists a crowded compact metric space having no infinite scattered subspaces'' is relatively consistent with $\mathbf{ZF}$. The latter result resolves an open problem from Keremedis, Tachtsis and Wajch [10]. For the proof, we construct a new permutation model having the required properties and then we transfer the result into $\mathbf{ZF}$ via the Jech–Sochor First Embedding Theorem.