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Keywords:
nearly realcompact space; fast set; SRM ideal; continuous functions with pseudocompact support; locally compact; locally pseudocompact
nearly realcompact space; fast set; SRM ideal; continuous functions with pseudocompact support; locally compact; locally pseudocompact
Summary:
As usual $C(X)$ will denote the ring of real-valued continuous functions on a Tychonoff space $X$. It is well-known that if $X$ and $Y$ are realcompact spaces such that $C(X)$ and $C(Y)$ are isomorphic, then $X$ and $Y$ are homeomorphic; that is $C(X)$ {\it determines\/} $X$. The restriction to realcompact spaces stems from the fact that $C(X)$ and $C(\upsilon X)$ are isomorphic, where $\upsilon X$ is the (Hewitt) realcompactification of $X$. In this note, a class of locally compact spaces $X$ that includes properly the class of locally compact realcompact spaces is exhibited such that $C(X)$ determines $X$. The problem of getting similar results for other restricted classes of generalized realcompact spaces is posed.
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