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Disasters in topology without the axiom of choice. (English) Zbl 1027.03040
The author shows that some well-known theorems of topology which are proved in ZFC (e.g., (1) Countable products of metrizable spaces are metrizable. (2) Countable products of second countable spaces are second countable. (3) Countable products of first countable spaces are first countable. (4) Countable products of separable \(T_2\) spaces are separable.) are not provable in ZF\(^-\) without AC. However, the countable axiom of choice CAC implies each of (2) and (4), and the countable multiple choice axiom CMC implies (1) and (3).

MSC:
03E25 Axiom of choice and related propositions
54A35 Consistency and independence results in general topology
54D30 Compactness
54E52 Baire category, Baire spaces
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