More on spaces X for which all prime z-ideals of C(X) are minimal or maximal
by
Melvin Henriksen
Harvey Mudd College

A (Tychonoff) space X is called a quasi P-space if all prime z-ideals of C(X) are minimal or maximal. (Most of the terminology used may be found in the Giilman-Jerison text "Rings of Continuous Functions"). Some of the results were reported on earlier, and a sample of new results follow. If a locally compact space is quasi P, then it is scattered of Cantor-Bendixson index no larger than 2, and the converse holds for normal spaces. A compact space is quasi P iff it is a finite free union of one-point compactifications of discrete spaces. A space with only finitely many nonP-points is quasi P, but no space containing a copy of the Stone-Cech compactifcation of omega is quasi P. A perfectly normal or a normal ccc-space is quasi P iff each of its nowhere dense zerosets is a P-space, but there are countable quasi P-spaces that have no P-points. If the product of two spaces is normal and quasi P, then one of the is a P-space. If X is a compact quasi P-space and Y is a P-space, then XxY is quasi P. Pertinent examples and unsolved problems will be mentioned. This is part of joint research with J. Martinez, R. Wilson, and R.G. Woods.

Date received: January 31, 2002