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On sequentially compact subspaces of \(\mathbb R\) without the axiom of choice. (English) Zbl 1071.03035
Summary: We show that the property of sequential compactness for subspaces of \(\mathbb R\) is countably productive in ZF. Also, in the language of weak choice principles, we give a list of characterizations of the topological statement ‘sequentially compact subspaces of \(\mathbb R\) are compact’. Furthermore, we show that forms 152 (= every non-well-orderable set is the union of a pairwise disjoint well-orderable family of denumerable sets) and 214 (= for every family \(A\) of infinite sets there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a nonempty subset of \(y\) and \(|f(y)|=\aleph_0)\) of Howard and Rubin [P. Howard and J. E. Rubin, Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)] are equivalent.

MSC:
03E25 Axiom of choice and related propositions
54D30 Compactness
54D55 Sequential spaces
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