Geodesic
$C(X)$ can sometimes determine $X$ without $X$ being realcompact
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 711-720

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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.
Classification : 46E25, 54C35, 54C40
Keywords: nearly realcompact space; fast set; SRM ideal; continuous functions with pseudocompact support; locally compact; locally pseudocompact 
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     title = {$C(X)$ can sometimes determine $X$ without $X$ being realcompact},
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Henriksen, Melvin; Mitra, Biswajit. $C(X)$ can sometimes determine $X$ without $X$ being realcompact. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 711-720. http://geodesic.mathdoc.fr/item/CMUC_2005__46_4_a9/