Purity of the ideal of continuous functions with pseudocompact support

Abstract

Let CΨ(X) be the ideal of functions with pseudocompact support and let kX be the set of all points in υX having compact neighborhoods. We show that CΨ(X) is pure if and only if βX−kX is a round subset of βX, CΨ(X) is a projective C(X)-module if and only if CΨ(X) is pure and kX is paracompact. We also show that if CΨ(X) is pure, then for each f∈CΨ(X) the ideal (f) is a projective (flat) C(X)-module if and only if kX is basically disconnected (F′-space).