Quasi P-spaces are defined to be those Tychonoff spaces X such that each prime z-ideal of C(X) is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of P-spaces. The compact quasi P-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi P-spaces is given. If X is a cozero-complemented space and every nowhere dense zeroset is a z-embedded P-space, then X is a quasi P-space. Conversely, if X is a quasi P-space and F is a nowhere dense z-embedded zeroset, then F is a P-space. On the other hand, there are examples of countable quasi P-spaces with no P-points at all. If a product X times Y is normal and quasi P, then one of the factors must be a P-space. Conversely, if one of the factors is a compact quasi P-space and the other a P-space then the product is quasi P. If X is normal and X and Y are cozero-complemented spaces and f: X → Y is a closed continuous surjection which has the property that f-1(Z) is nowhere dense for each nowhere dense zeroset Z, then if X is quasi P, so is Y. The converse fails even with more stringent assumptions on the map f. The paper then closes with a number of open questions, amongst which the most glaring is whether the free union of quasi P-spaces is always quasi P.
© 2003 Charles University in Prague
Henriksen, Melvin, Jorge Martínez, and R. Woods. "Spaces X in which all prime z-ideals of C(X) are minimal or maximal." Commentationes Mathematicae Universitatis Carolinae 44.2 (2003): 261-294.