Melvin Henriksen, Biswajit Mitra
$C(X)$ can sometimes determine $X$ without $X$ being realcompact

Comment.Math.Univ.Carolinae 46,4 (2005) 711-720.

Abstract: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 $C(X)$ {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.

Keywords: nearly realcompact space, fast set, SRM ideal, continuous functions with pseudocompact support, locally compact, locally pseudocompact
AMS Subject Classification: Primary 54C40; Secondary 46E25