The intersection of the free maximal ideals in a complete space

Abstract

Introduction. In [1, p. 123] Gillman aiid Jerison prove that if X is realcompact, the intersection of the free maximal ideals in C(X) is CK(X), the ring of functions with compact support. Other authors [2], [3] have proved this result for discrete spaces and p-spaces, respectively. This note provides a proof of the fact that the result holds for any space admitting a complete uniform structure. We point out also that unlike that of the Gillman-Jerison theorem, our proof requires no prior construction of ,BX, the Stone-Cech compactification of X. Since a realcompact space is complete in the structure generated by C(X), our result extends the Gillman-Jerison theorem. (Whether every complete space is realcompact is an unsettled question: its resolution would require a proof of the existence or nonexistence of measurable cardinals.) We will employ the same terminology as in [1] and assume all spaces are completely regular and all uniform structures Hausdorff. Also, we take our uniform structures to be defined by pseudometrics. For a topological space X, C(X) denotes its ring of real valued continuous functions, and for f E C(X), Zf = { x: f(x) = 0 1. Thus, the support of f is clx [X-Zf ] and CK(X) is the ring of functions with compact support. An ideal IC C(X) is free provided nf {Zf:f I} = 0 and a maximal ideal which is free is called a free maximal ideal. A subset SCX is C-embedded in X provided each fE C(S) admits a continuous extension to all of X.