Spaces X for which C(X) has a compact space of minimal prime ideals and every prime z-ideal is minimal or maximal.
by
Melvin Henriksen
Harvey Mudd College, Claremont CA
Coauthors: J. Martinez, R. G. Wilson, R. G. Woods

All spaces considered are Tychonoff. If every prime z-ideal of C(X) is minimal or maximal, X is called a qausi P-space. If: (*) C(X) has a compact space of minimal prime ideals (Equivalently, for each f in C(X) there is a g in C(X) such that cl(coz(f))=cl(int Z(g)).) Then X is quasi P if and only if every prime z-filter with an element with empty interior is maximal. Perfectly normal and ccc-spaces (e.g., spaces in which every family of pairwise disjoint open sets is countable) satisfy (*).

Theorem 1. A perfectly normal space is quasi P if and only if every nowhere dense closed subset is discrete.

Theorem 2. A normal ccc-space is qusi P if and only if it is scattered of Cantor-Bendixson index 1 or 2. This latter property also characterizes locally compact quasi P-spaces even if (*) fails to hold.

Pertinent examples will be mentioned.

If time permits, some remarks about other papers given at this conference may be made.

Date received: February 16, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caiu-06.