On the locally countable subalgebra of C(X) whose local domain is cocountable

In this paper, we present a new subring of $C(X)$ that contains the subring $C_c(X)$, the set of all continuous functions with countable image. Let $L_{cc}(X)=\{ f\in C(X)\,:\, |X\backslash C_f|\leq \aleph_0 \}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq \aleph_0$. We observe that $L_{cc}(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$. It is shown that any hereditary lindel\"{o}f scattered space is functionally countable.Spaces $X$ such that $L_{cc}(X)$ is regular (von Neumann) are characterized and it is shown that $\aleph_0$-selfinjectivity and regularity of $L_{cc}(X)$ coincide.