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.

对于Hausdorff零维拓扑空间X和具有区间拓扑的全序字段F，令\（C_c（X，F）\）是X上具有可数范围的所有F值连续函数的环。证明如果F是\（{\ mathbb {R}} \）的不可数字段或可计数子字段，则\（C_c（X，F）\）的结构空间为\（\ beta _0X \），X的Banaschewski压缩。引入\（C_c（X，F）\）中的理想\（\ {O ^ {p，F} _c：p \在\ beta _0X \} \中作为理想的修饰可数类似物C（X）中的\（\ {O ^ p：p \ in \ beta X \} \中。认识到\（C_c（X，F）\ cap C_K（X，F）= \ bigcap _ {p \ in \ beta _0X {\ setminus} X} O ^ {p，F} _c \），并且可称为众所周知的公式的一个可数类似物\（{在\测试X p \ {\ setminus} X} C_K（X）= \ bigcap _ö^ p \）在ç（X）。此外，还表明，假设\（C_c（X，F）\）是一个Von-Neumann正则环，除其他外，它还等于X是P-空间的条件。