$$C(X)$$ C ( X ) Versus its Functionally Countable Subalgebra

Abstract

Let $$C_c(X)$$ (resp. $$C^F(X)$$ ) denote the subring of $$C(X)$$ consisting of functions with countable (resp. finite) image and $$C_F(X)$$ be the socle of $$C(X)$$ . We characterize spaces X with $$C^*(X)=C_c(X)$$ , which generalizes a celebrated result due to Rudin, Pelczynnski, and Semadeni. Two zero-dimensional compact spaces X, Y are homeomorphic if and only if $$C_c(X)\cong C_c(Y)$$ (resp. $$C^F(X)\cong \ C^F(Y)$$ ). The spaces X for which $$C_c(X)=C^F(X)$$ are characterized. The socles of $$C_c(X)$$ , $$C^F(X)$$ , which are observed to be the same, are topologically characterized and spaces X for which this socle coincides with $$C_F(X)$$ are determined, too. A certain well-known algebraic property of $$C(X)$$ , where X is real compact, is extended to $$C_c(X)$$ . In contrast to the fact that $$C_F(X)$$ is never prime in $$C(X)$$ , we characterize spaces X for which $$C_F(X)$$ is a prime ideal in $$C_c(X)$$ . It is observed for these spaces, $$C_c(X)$$ coincides with its own socle (a fact, which is never true for $$C(X)$$ , unless X is finite). Finally, we show that a space X is the one-point compactification of a discrete space if and only if $$C_F(X)$$ is a unique proper essential ideal in $$C^F(X)$$ .