Ordered field valued continuous functions with countable range

TL;DR

This paper characterizes the structure space of continuous functions with countable range on zero-dimensional spaces over ordered fields, introducing new ideals and linking algebraic properties to topological conditions.

Contribution

It establishes the structure space of $C_c(X,F)$ as the Banaschewski Compactification and introduces modified ideals, extending classical results to countable range functions.

Findings

Structure space of $C_c(X,F)$ is $eta_0X$ for certain fields.

Introduces ideals $O^{p,F}_c$ as countable analogues of classical ideals.

Shows $C_c(X,F)$ is Von-Neumann regular iff $X$ is a $P$-space.

Abstract

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 $C_c(X,F)\cap C_K(X,F)=\bigcap_{p\in\beta_0X\texttt{\textbackslash}X} O^{p,F}_c$, this may be called a countable analogue of the well-known formula $C_K(X)=\bigcap_{p\in\beta X\texttt{\textbackslash}X}O^p$ 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.