RING EPIMORPHISMS AND C(X)

ABSTRACT. This paper studies the homomorphism of rings of continuous functions ρ: C(X) → C(Y), Y a subspace of a Tychonoff space X, induced by restriction. We ask when ρ is an epimorphism in the categorical sense. There are several appropriate categories: we look at CR, all commutative rings, and R/N, all reduced commutative rings. When X is first countable and perfectly normal (e.g., a metric space), ρ is a CRepimorphism if and only if it is a R/N-epimorphism if and only if Y is locally closed in X. It is also shown that the restriction of ρ to C ∗ (X) → C ∗ (Y), when X is normal, is a CR-epimorphism if and only if it is a surjection. In general spaces the picture is more complicated, as is shown by various examples. Information about Spec ρ and Spec ρ restricted to the proconstructible set of prime zideals is given. 1