Abstract
If X is a Tychonoff space then its P -coreflection Xδ is a Tychonoff space that is a dense subspace of the realcompact space (υX)δ, where υX denotes the Hewitt realcompactification of X. We investigate under what conditions Xδ is C-embedded in (υX)δ, i.e. under what conditions υ(Xδ )=( υX)δ. An example shows that this can fail for the product of a compact space and a P -space. It is possible for a von Neumann regular ring A to be isomorphic to a C(Y ) and lie between C(X) and C(Xδ) without being isomorphic to C(Xδ). This cannot occur if X is realcompact or more generally if υ(Xδ )=( υX)δ. Applications are given to the epimorphic hull of C(X).
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