On the Functionally Countable Subalgebra of $C(X)$

Abstract

Let $C_c\left(X\right)=\{f\in C(X):f(X)is\; countable\}$. Similar to $C(X)$ it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring $C_c\left(X\right)$ is either $C_c\left(X\right)$ or a semiprime (resp. prime) ideal in $C_c\left(X\right)$ . For an ideal $I$ in $C_c\left(X\right)$ , it is observed that $I$ and $\sqrt{I}$ have the same largest $z_c$-ideal. If $X$ is any topological space, we show that there is a zero-dimensional space $Y$ such that $C_c\left(X\right)\cong C_c\left(Y\right)$ . Consequently, if $X$ has only countable number of components, then $C_c\left(X\right)\cong C(Y)$ for some zero-dimensional space $Y$. Spaces X for which $C_c\left(X\right)$ is regular (called $CP$-spaces) are characterized both algebraically and topo log ically and it is shown that $P$-spaces and $CP$-spaces coincide when $X$ is zero-dimensional. In contrast to $C^{\ast }(X)$, we observe that $C_c\left(X\right)$ enjoys the algebraic properties of regularity, ${\aleph }_{{}_0}$-selfinjectivity and some others, whenever $C(X)$ has these properties. Finally an example of a space $X$ such that $C_c\left(X\right)$ is not isomorphic to any $C(Y)$ is given.