Ordered Field Valued Continuous Functions with Countable Range

Abstract

For a Hausdorff zero-dimensional topological space X and a totally ordered field F with interval topology, let $C_c(X,F)$ be the ring of all F-valued continuous functions on X with countable range. It is proved that if F is either an uncountable field or countable subfield of ${\mathbb {R}}$, then the structure space of $C_c(X,F)$ is $\beta _0X$, the Banaschewski Compactification of X. The ideals $\{O^{p,F}_c:p\in \beta _0X\}$ in $C_c(X,F)$ are introduced as modified countable analogue of the ideals $\{O^p:p\in \beta X\}$ in C(X). It is realized that $C_c(X,F)\cap C_K(X,F)=\bigcap _{p\in \beta _0X{\setminus } X} O^{p,F}_c$, and this may be called a countable analogue of the well-known formula $C_K(X)=\bigcap _{p\in \beta X{\setminus } X}O^p$ in C(X). Furthermore, it is shown that the hypothesis $C_c(X,F)$ is a Von-Neumann regular ring is equivalent to amongst others the condition that X is a P-space.