$C(X)$ 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 $C^F(X)$.