If $X$ is a Tychonoff space then its
$P$-coreflection $X_{\delta } $ is a Tychonoff space that is a dense
subspace of the realcompact space $( \upsilon X)_{\delta }$, where $
\upsilon X$ denotes the Hewitt realcompactification of $X$.  We
investigate under what conditions $X_{\delta }$ is $C$-embedded in $(
\upsilon X)_{\delta }$, i.e. under what conditions $ \upsilon (X_{\delta
}) = (\upsilon X )_{\delta}$.  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 $C(X_{\delta})$ without being isomorphic to $C(X_{\delta})$. 
This cannot occur if $X$ is realcompact or more generally if $ \upsilon
(X_{\delta }) = (\upsilon X )_{\delta}$. Applications are given to the
epimorphic hull of $C(X)$.