Disasters in topology without the axiom of choice (Q1407509)

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).