# Morita conjectures

From Citizendium, the Citizens' Compendium

The **Morita conjectures** in topology ask

- 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^{[1]}? - 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]}.

## 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
- ↑ 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-Zimmermann,
*Kiiti Morita 1915-1995*, Notices of the AMS, June 1997 [1]