Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Abstract

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ℓ-ideals contained in the maximal ℓ-ideal M x = {f ∈ C(X):f(x) = 0}. The space X has finite rank if there is an n ∈ N such that every point x ∈ X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ∈ X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.