Cozero complemented spaces; when the space of minimal prime ideals of a C( X) is compact

Abstract

If X is a Tychonoff space, C( X) its ring of real-valued continuous functions, and f∈ C( X), then the cozeroset of f is coz( f)= {x∈X: f(x)≠0} . If, for every cozeroset V of X, there is a disjoint cozeroset V′ such that V∪ V′ is dense in X, then X is said to be cozero complemented. It has long been known that X is cozero complemented iff the space Min C( X) of minimal prime ideals of C( X) (in the hull-kernel or Zariski topology) is compact iff the classical ring of fractions of C( X) is von Neumann regular. While many characterizations of cozero complemented spaces are known, they seem not to be adequate to answer some natural questions about them raised by R. Levy and J. Shapiro in an unpublished preprint. These questions concern the relationship between a space being cozero complemented and certain kinds of subspaces having this property, and between a product of two spaces being cozero complemented and the factor spaces being cozero complemented. Also, some conditions are given that guarantee that a space that is locally cozero complemented has this property globally. In this paper partial answers are given to these questions. Sample results: If X is weakly Lindelöf and dense in T, then X is cozero complemented iff T is cozero complemented; if X× Y is weakly Lindelöf and cozero complemented, then X and Y are cozero complemented, but if D is an uncountable discrete space, then βD× βD is not cozero complemented even though βD is cozero complemented. If X is locally cozero complemented and either weakly Lindelöf or paracompact, then X is cozero complemented.