Abstract
A metric space is Totally Bounded (also called preCompact) if it has a finite e-net for every e > 0 and it is preLindelof if it has a countable e-net for every e > 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: (a) preLindelof if, for every e > 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 e > 0, there is a countable subset A of X such that the open balls with centers in A and radius e cover X.
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