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Throughout C(X) will denote the ring of all continuous real-valued functions on a Tychonoff space X, and C*(X) will denote the subring of bounded elements of C(X). The real line is denoted by R, and N denotes the (discrete) subspace of positive integers. A subset S of X such that the map f → f|s is an epimorphism of C(X) (resp. C*(X)) is said to be C-embedded (resp. C*-embedded) in X. As is well-known, every f Є C*(X) has a unique continuous extension βf over its Stone-Čech compactification βX [GJ, Chapter 6]. That is, X is C*-embedded in βX.

In [NR] , L. Nel and D. Riordan introduced the subset C#(X) of C(X) consisting of all f such that for every maximal ideal M of C(X), there is an r Є R such that (f-r) Є M, and they noted that C#(X) is a subalgebra and sublattice of C(X) containing the constant functions. They show how C#(X) determines a compactification of X in a number of cases and leave the impression that it always does. In [Cl], E. Choo notes that this is true if X is locally compact and seems to conjecture that it need not be the case otherwise. In [SZ 1], o. Stefani and A. Zanardo show that every f Є C#(Rω) is a constant function, where Rω denotes a countably infinite product of copies of R. In [SZ 2] they show that C#(X) determines a compactification of X in case X is locally compact, pseudo compact, or zero-dimensional, and they describe the compactifications so determined when X is realcompact [GJ, Chapter 8].

In this paper, I show that under certain restrictions on X, the ring C#(X) determines the Freudenthal compactification of X [Il, pp. 109-120], I observe that, at least in disguised form, C#(X) has been considered by a number of authors other than those named above, and some conditions are given that are either necessary or sufficient for X to determine a compactification of X. In particular, it is shown that if X is realcompact, and C#(X) determines a compactification of X, then X is rimcompact and it determines the Freudenthal compactification ΦX of X. There are realcompact rimcompact spaces X for which C#(X) does not determine a compactification of X, but C#(X) does determine φX if every point of x has either a compact neighborhood, or a base of open and closed neighborhoods. Other sufficient conditions are given for C#{X) to determine ΦX. I close with some remarks and open problems.


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© 1977 Topology Proceedings, Auburn University Department of Mathematics

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