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Conditions under which the ring C(X, R) of all real-valued continuous functions on a completely regular space X determines X have been investigated in a number of papers [l-5]. Gelfand and Kolmogoroff [2] have shown that if A is a compact completely regular space, then the space of maximal ideals in C(X, R) with the Stone topology is homeomorphic to X and hence C(X, R) determines X. The most general result for noncompact spaces has been obtained by F. W. Anderson [l ] who has shown that the lattice of all real-valued continuous functions on a topological space determines this space if it is a completely regular space all of whose points are G-delta sets. Since/(x) ^g(x) for all x if and only iff—g has square root in C(X, R), a ring isomorphism of C(X, R) implies a lattice isomorphism and Anderson's result is equivalent to the statement that the ring C(X, R) determines A if A is a completely regular space all of whose points are G-delta sets. In this paper we show that if X is completely regular, then X is topologically determined by the pair of rings consisting of the ring Rx of all real-valued functions on X and its subring C(X, R).

The ring 𝐶(𝑋,𝑅) considered as a subring of the ring of all real-valued functions