Morita conjectures

The Morita conjectures in topology ask


 * 1) If X &times; Y is normal for every normal space Y, is X discrete?
 * 2) If X &times; Y is normal for every normal P-space Y, is X metrizable ?
 * 3) If X &times; Y is normal for every normal countably paracompact space Y, is X metrizable and sigma-locally compact?

Here a normal P-space Y is characterised by the property that the product with every metrizable X is normal; it is thus conjectured that the converse holds.

K. Chiba, T.C. Przymusiński and M.E. Rudin proved conjecture (1) and showed that conjecture (2) is true if the axiom of constructibility V=L, holds.

Z. Balogh proved conjectures (2) and (3).