- If X × Y is normal for every normal space Y, is X discrete?
- If X × Y is normal for every normal P-space Y, is X metrizable?
- 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 Mary Ellen 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).
- K. Morita, "Some problems on normality of products of spaces" J. Novák (ed.) , Proc. Fourth Prague Topological Symp. (Prague, August 1976) , Soc. Czech. Math. and Physicists , Prague (1977) pp. 296–297
- K. Chiba, T.C. Przymusiński, M.E. Rudin, "Normality of products and Morita's conjectures" Topol. Appl. 22 (1986) 19–32
- Z. Balogh, Non-shrinking open covers and K. Morita's duality conjectures, Topology Appl., 115 (2001) 333-341
- A.V. Arhangelskii, K.R. Goodearl, B. Huisgen-Zimmerman, Kiiti Morita 1915-1995, Notices of the AMS, June 1997 
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