On a class of pseudocompact spaces derived from ring epimorphisms

A Tychonoff space X is RG if the embedding of C ( X ) → C ( X δ ) is an epimorphism of rings. Compact RG-spaces are known and easily described. We study the pseudocompact RG-spaces. These must be scattered of finite Cantor Bendixon degree but need not be locally compact. However, under strong hypotheses, (countable compactness, or small cardinality) these spaces must, indeed, be compact. The main theorems shows, how to construct a suitable maximal almost disjoint family, and apply it to obtain examples of RG-spaces that are almost compact, locally compact, non-compact, almost-P, and of Cantor Bendixon degree 2. More complicated examples of pseudocompact non-compact RG-spaces ensue.