Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2004, January 16.

Abstract:
A Tychonoff space $X$ is called $RG$ if the embedding of $C(X) \rightarrow C(X_{\delta} )$ 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 theorem 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, and of Cantor Bendixon degree $2$. More complicated examples ensue.

Mathematics Subject Classification (2000): 54C30.

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