Total Boundedness and the Axiom of Choice

Abstract

A metric space is Totally Bounded (also called preCompact) if it has a finite ε-net for every ε > 0 and it is preLindelöf if it has a countable ε-net for every ε > 0. Using the Axiom of Countable Choice (CC), one can prove that a metric space is topologically equivalent to a Totally Bounded metric space if and only if it is a preLindelöf space if and only if it is a Lindelöf space. In the absence of CC, it is not clear anymore what should the definition of preLindelöfness be. There are two distinguished options. One says that a metric space X is:

• (a) preLindelöf if, for every ε > 0, there is a countable cover of X by open balls of radius ?? (Keremedis, Math. Log. Quart. 49, 179–186 2003);

• (b) Quasi Totally Bounded if, for every ε > 0, there is a countable subset A of X such that the open balls with centers in A and radius ε cover X.

As we will see these two notions are distinct and both can be seen as a good generalization of Total Boundedness. In this paper we investigate the choice-free relations between the classes of preLindelöf spaces and Quasi Totally Bounded spaces, and other related classes, namely the Lindelöf spaces. Although it follows directly from the definitions that every pseudometric Lindelöf space is preLindelöf, the same is not true for Quasi Totally Bounded spaces. Generalizing results and techniques used by Horst Herrlich in [8], it is proven that every pseudometric Lindelöf space is Quasi Totally Bounded iff Countable Choice holds in general or fails even for families of subsets of $\mathbb {R}$ (Theorem 3.5).