Ordered Field Valued Continuous Functions with Countable Range

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