Total Boundedness and the Axiom of Choice

A metric space is Totally Bounded (also called preCompact) if it has a finite epsilon-net for every epsilon > 0 and it is preLindelof if it has a countable epsilon-net for every epsilon > 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 preLindelof space if and only if it is a Lindelof space. In the absence of CC, it is not clear anymore what should the definition of preLindelofness be. There are two distinguished options. One says that a metric space X is: preLindelof if, for every epsilon > 0, there is a countable cover of X by open balls of radius ?? (Keremedis, Math. Log. Quart. 49, 179-186 2003); Quasi Totally Bounded if, for every epsilon > 0, there is a countable subset A of X such that the open balls with centers in A and radius epsilon 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 preLindelof spaces and Quasi Totally Bounded spaces, and other related classes, namely the Lindelof spaces. Although it follows directly from the definitions that every pseudometric Lindelof space is preLindelof, 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 Lindelof space is Quasi Totally Bounded iff Countable Choice holds in general or fails even for families of subsets of (Theorem 3.5).