论文标题
有序场有价值的连续功能，可计数范围
Ordered field valued continuous functions with countable range
论文作者
论文摘要
对于Hausdorff零维拓扑空间$ x $和带有间隔拓扑的完全有序的字段$ f $，让$ c_c（x，f）$是所有$ f- $估值的连续功能的环，$ x $具有可数范围。事实证明，如果$ f $是$ \ mathbb {r} $的无数字段或可计数子字段，则$ c_c（x，f）$的结构空间为$β_0x$，banaschewski compactification $ x $。理想$ \ {o^{o^{p，f} _c：p \ in $ c_c（x，f）$ inβ_0x\} $ in $ c_c（x，f）$作为理想的可计数类似物$ \ {o^p：p \ inp：p \ inp：p \inβx\} $ in $ c（x）$。人们意识到$ c_c（x，f）\ cap c_k（x，f）= \ bigcap_ {p \inβ_0x\ texttt {\ textbackslash} x} x} x} o^{p，f} $ c_k（x）= \ bigcap_ {p \inβx\ texttt {\ textbackslash} x} x} o^p $ in $ c（x）$。此外，可以证明假设$ c_c（x，f）$是von-neumann常规戒指，等同于其他条件，条件是$ x $是$ p-$ space。
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 $β_0X$, the Banaschewski Compactification of $X$. The ideals $\{O^{p,F}_c:p\in β_0X\}$ in $C_c(X,F)$ are introduced as modified countable analogue of the ideals $\{O^p:p\inβX\}$ in $C(X)$. It is realized that $C_c(X,F)\cap C_K(X,F)=\bigcap_{p\inβ_0X\texttt{\textbackslash}X} O^{p,F}_c$, this may be called a countable analogue of the well-known formula $C_K(X)=\bigcap_{p\inβX\texttt{\textbackslash}X}O^p$ 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.
