Morita conjectures

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

The Morita conjectures in topology ask

  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].

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]