Morita conjectures

The Morita conjectures in general topology are certain problems about normal spaces, now solved in the affirmative. They asked

1. If X × Y is normal for every normal space Y, is X discrete?
2. If X × Y is normal for every normal P-space Y, is X metrizable?
3. If X × Y is normal for every normal countably paracompact space Y, is X metrizable and sigma-locally compact?

The answers were believed to be affirmative. Here a normal P-space Y is characterised by the property that the product with every metrizable X is normal; thus the conjecture was that the converse holds.

K. Chiba, T.C. Przymusiński and M.E. Rudin proved conjecture (1) and showed that conjectures (2) and (3) cannot be proven false under the standard ZFC axioms for mathematics (specifically that the conjectures hold under the axiom of constructibility V=L).

Fifteen years later, Z. Balogh succeeded in proving conjectures (2) and (3) true.