Real Algebraic and Analytic Geometry |

On a class of pseudocompact spaces derived from ring epimorphisms.

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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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