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

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.