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On Lindelöf metric spaces and weak forms of the axiom of choice. (English) Zbl 0952.03060
The authors show that the countable axiom of choice (CAC) strictly implies the statements “Lindelöf metric spaces are second countable” and “Lindelöf metric space are separable”. It is also shown that CAC is equivalent to: “If \((X,T)\) is a Lindelöf topological space with respect to the base \({\mathcal B}\), then \((X,T)\) is Lindelöf”.

MSC:
03E25 Axiom of choice and related propositions
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A35 Consistency and independence results in general topology
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