Ordered Field Valued Continuous Functions with Countable Range

For a Hausdorff zero-dimensional topological space X and a totally ordered field F with interval topology, let Cc(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 ofR, then the structure space ofCc(X, F) is β0X, the Banaschewski Compactification of X. The ideals {Op,F c : p ∈ β0X} in Cc(X, F) are introduced as modified countable analogue of the ideals {Op : p ∈ βX} in C(X). It is realized that Cc(X, F) ∩ CK (X, F) = p∈β0X\X Op,F c , and this may be called a countable analogue of the well-known formula CK (X) = p∈βX\X Op in C(X). Furthermore, it is shown that the hypothesis Cc(X, F) is a Von-Neumann regular ring is equivalent to amongst others the condition that X is a P-space.