C(X) Versus Cc(X)

Let C c (X) (resp. C F (X)) denote the subring of C(X) consisting of functions with countable (resp. finite) image and C (X) be the socle of C(X). If X is any topological space there is a zero-dimensional space Y such that C F c (X) C c (Y). We characterize spaces X with C (X) = C (X), which generalizes a celebrated result due to Rudin, Pelczynnski and Semadeni. Two zero-dimensional compact spaces X, Y are homeomorphic if and only if C c (X) C c (Y) (resp. C F (X) C c F (Y)). The well-known algebraic property of C(X), where X is realcompact, is extended to C (X). In contrast to the fact that C F (X) is never prime in C(X), we characterize spaces X for which C c F (X) is a prime ideal in C (X). It is observed for these spaces, C (X) coincides with its own socle (a fact, which is never true for C(X)). Finally, we show that a space X is the one-point compactification of a discrete space if and only if C F c (X) is a unique proper essential ideal in C F (X), see [9], [10]. A similar characterization, as the Gelfand-Kolmogoro Theorem for the maximal ideals in C(X), is given for the maximal ideals of C c (X), see [4]. The subalgebra L c (X) = f f 2 C(X) : C f = Xg of C(X), where C is the union of all open subsets U X such that j f (U)j @ 0 , which is C c (X) L c (X) C(X), see [13]. f c