Total Boundedness and the Axiom of Choice

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: 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);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.