Polar functions—II: completion classes of archimedean f-algebras vs. covers of compact spaces

Abstract

There is an inclusion preserving bijection between the class of completion classes of uniformly complete real f-algebras with identity and the partially ordered class of covering classes of compact Hausdorff spaces. In this setting a completion class A is a hull class of uniformly complete f-algebras, with the additional feature that G∈ A if and only if G ∗∈ A . Using an idempotent invariant polar function X and the covering function K derived from it, the main theorem of this article states that the covering class associated with the uniformly complete f-algebras having no proper X -splitting extensions is the class of compact spaces X which equal their K -cover.