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When is $$\mathbb N$$ Lindelöf? (English) Zbl 0938.54008
From the authors’ abstract:
Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $$\mathbb N$$ is a Lindelöf space, (2) $$\mathbb Q$$ is a Lindelöf space, (3) $$\mathbb R$$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $$\mathbb R$$ is separable, (6) in $$\mathbb R$$, a point $$x$$ is in the closure of a set $$A$$ iff there exists a sequence in $$A$$ that converges to $$x$$, (7) a function $$f:\mathbb R \rightarrow \mathbb R$$ is continuous at a point $$x$$ iff $$f$$ is sequentially continuous at $$x$$, (8) in $$\mathbb R$$, every unbounded set contains a countable, unbounded set, (9) the axiom of countable choice holds for subsets of $$\mathbb R$$.
Reviewer: L.Skula (Brno)

##### MSC:
 54A35 Consistency and independence results in general topology 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54-02 Research exposition (monographs, survey articles) pertaining to general topology 03E25 Axiom of choice and related propositions
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