Real Algebraic and Analytic Geometry |

When the Hewitt realcompactification and the P-coreflection commute.

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Submission: 2006, December 15.

*Abstract:
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. We show that if $A$ is a von
Neumann regular ring for which $C(X) \subseteq A \subseteq C(X_{\delta} )$
and for which there is a Tychonoff space $Y$ such that $A$ is
ring-isomorphic to $C(Y)$, then $A = C (X_{\delta})$ if and only if
$\upsilon (X_{\delta }) = (\upsilon X)_{\delta }$. Applications are given
to the epimorphic hull of $C(X)$.*

Mathematics Subject Classification (1991): 54D60, 54G10, 16S60.

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