Morita conjectures

2020 Mathematics Subject Classification: Primary: 54D [MSN][ZBL]

Three conjectures in general topology due to K. Morita:

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 conjecture (3).

References

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