# Morita conjectures

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 54D [MSN][ZBL]

Three conjectures in general topology due to K. Morita:

- If $X \times Y$ is normal for every normal space $Y$, is $X$ discrete?
- If $X \times Y$ is normal for every normal P-space $Y$, is $X$ metrizable?
- 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

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Morita conjectures.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Morita_conjectures&oldid=35012