Real Algebraic and Analytic Geometry
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198. R. Raphael, R. Grant Woods:
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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