Morita conjectures

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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?

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 ^[2] 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)^[3].

[edit] References

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

• A.V. Arhangelskii, K.R. Goodearl, B. Huisgen-Zimmermann, Kiiti Morita 1915-1995, Notices of the AMS, June 1997 [1]