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

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 Gset membership, variantA if and only if G*set membership, variantA. Using an idempotent invariant polar function image and the covering function image derived from it, the main theorem of this article states that the covering class associated with the uniformly complete f-algebras having no proper image-splitting extensions is the class of compact spaces X which equal their image-cover.