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?^[1]
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 ^[2] 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.^[3]

Notes

1. 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
2. K. Chiba, T.C. Przymusiński, M.E. Rudin, "Normality of products and Morita's conjectures" Topol. Appl. 22 (1986) 19–32
3. Z. Balogh, Non-shrinking open covers and K. Morita's duality conjectures, Topology Appl., 115 (2001) 333-341

References

• A.V. Arhangelskii, K.R. Goodearl, B. Huisgen-Zimmerman, Kiiti Morita 1915-1995, Notices of the AMS, June 1997