Descrip.
The theory rings of continuous functions has been studied extensively. Certain important connections between compact, real compact spaces and various structures of C(X) have been discovered. For compact space X, Gelfand and Kolmogroff , showed that C(X) as a ring characterizes X; M.H. Stone showed that as a lattice alone, C(X) characterizes X. Shirota proved that for real compact space X, C(X) as lattice also characterizes X.
