(ON THE FUNCTIONALLY SEPRABLE SUBALGEBRA OF C(X

‎Let $C_c(X)=\{f\in C(X)‎ : ‎|f(X)|\leq \aleph_0\}$‎, ‎$C^F(X)=\{f\in C(X)‎: ‎|f(X)| \infty\}$‎. ‎If $C_c(X)=C(X)$ ($C^F(X)=C(X)$)‎, ‎then $X$ is called functionally countable (finite)‎. ‎We define functionally seprable subalgebra of $C(X)$ as $C_{cd}(X)=\{f\in C(X)‎: ‎|f(Y)|\leq\aleph_0\}$ where $Y$ is a dense subset of $X$‎. ‎We observe that $C_c(X)\subseteq C_{cd}(X)\subseteq C_c(Y)$ and $C_{cd}(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$‎. ‎Spaces $X$ such that $C_c(X)=C_{cd}(X)$ are investigated‎. ‎If $X$ is a functionally countable or seprable space then $C_{cd}(X)=C(X)$‎. ‎Let $X$ be pseudocompact (i.e.‎, ‎$C(X)=C^*(X)$) and $\beta X$ be seprable‎, ‎then $C_{cd}(X)=C(X)$‎. ‎Conversely if $C_{cd}(X)=C(X)$ and any $G_{\delta}$-set has nonempty interior‎, ‎then $X$ is functionally countable‎. ‎We also observe that for a pseudocompact space $X$‎, ‎$C_{cd}(X)=C(X)$ if and only if $C_{cd}(\beta X)=C(\beta X)$‎.