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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)

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
Full Text: EuDML