# Article

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Keywords:
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.
References:
[BD92] Blair R., van Douwen E.: Nearly realcompact spaces. Topology Appl. 47 (1992), 209-221. MR 1192310 | Zbl 0772.54021
[C68] Comfort W.W.: On the Hewitt realcompactification of a product space. Trans. Amer. Math. Soc. (1968), 107-118. MR 0222846 | Zbl 0157.53402
[GJ76] Gillman L., Jerison M.: Rings of Continuous Functions. Springer, New York, 1976. MR 0407579 | Zbl 0327.46040
[JM73] Johnson D., Mandelker M.: Functions with pseudocompact support. General Topology and Appl. 3 (1973), 331-338. MR 0331310 | Zbl 0277.54009
[Ma71] Mandelker M.: Supports of continuous functions. Trans. Amer. Math. Soc 156 (1971), 73-83. MR 0275367 | Zbl 0197.48703
[Mi82] Misra P.R.: On isomorphism theorems for $C(X)$. Acta Math. Acad. Sci. Hungar. 39 (1982), 379-380. MR 0653849 | Zbl 0479.54011
[S94] Schommer J.: Fast sets and nearly realcompact spaces. Houston J. Math. 20 (1994), 161-174. MR 1272569 | Zbl 0802.54016
[SS01] Schommer J., Swardson M.A.: Almost* realcompactness. Comment. Math. Univ. Carolinae 42 (2001), 385-394. MR 1832157

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