On Sequential Compactness and Related Notions of Compactness of Metric Spaces in $\mathbf {ZF}$

Kyriakos Keremedis¹

¹ Department of Mathematics University of the Aegean Karlovassi, Samos 83200, Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 64 (2016) no. 1, pp. 29-46.

Résumé

We show that: (i) If every sequentially compact metric space is countably compact then for every infinite set $X,$ $[X]^{ \lt \omega }$ is Dedekind-infinite. In particular, every infinite subset of $\mathbb {R}$ is Dedekind-infinite. (ii) Every sequentially compact metric space is compact iff every sequentially compact metric space is separable. In addition, if every sequentially compact metric space is compact then: every infinite set is Dedekind-infinite, the product of a countable family of compact metric spaces is compact, and every compact metric space is separable. (iii) The axiom of countable choice implies that every sequentially bounded metric space is totally bounded and separable, every sequentially compact metric space is compact, and every uncountable sequentially compact, metric space has size $|\mathbb {R}|$. (iv) If every sequentially bounded metric space is totally bounded then every infinite set is Dedekind-infinite. (v) The statement: “Every sequentially bounded metric space is bounded” implies the axiom of countable choice restricted to the real line. (vi) The statement: “For every compact metric space $\mathbf {X}$ either $|X|\leq |\mathbb {R}|$, or ${|\mathbb {R}|\leq |X|}$” implies the axiom of countable choice restricted to families of finite sets. (vii) It is consistent with $\mathbf {ZF}$ that there exists a sequentially bounded metric space whose completion is not sequentially bounded. (viii) The notion of sequential boundedness of metric spaces is countably productive.

Détail

DOI : 10.4064/ba8054-5-2016

Keywords: every sequentially compact metric space countably compact every infinite set omega dedekind infinite particular every infinite subset mathbb dedekind infinite every sequentially compact metric space compact every sequentially compact metric space separable addition every sequentially compact metric space compact every infinite set dedekind infinite product countable family compact metric spaces compact every compact metric space separable iii axiom countable choice implies every sequentially bounded metric space totally bounded separable every sequentially compact metric space compact every uncountable sequentially compact metric space has size nbsp mathbb every sequentially bounded metric space totally bounded every infinite set dedekind infinite statement every sequentially bounded metric space bounded implies axiom countable choice restricted real line statement every compact metric space mathbf either leq mathbb mathbb leq implies axiom countable choice restricted families finite sets vii consistent mathbf there exists sequentially bounded metric space whose completion sequentially bounded viii notion sequential boundedness metric spaces countably productive