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(uX) are isomorphic, where uX is the (Hewitt) realcompactifcation 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.
© 2005 Charles University in Prague
Henriksen, Melvin, and Biswajit Mitra. "C(X) can sometimes determine X without X being realcompact." Commentationes Mathematicae Universitatis Carolinae 46.4 (2005): 711-720.