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 modi...
