Rings and subrings of continuous functions with countable range

TL;DR

This paper studies intermediate rings of continuous functions with countable range on zero-dimensional spaces, characterizing their structure, ideals, and topological properties, including conditions for $P$-spaces and pseudocompactness.

Contribution

It introduces and analyzes the structure of intermediate rings of countable-range continuous functions, linking their properties to space classifications like $P$-spaces and pseudocompactness.

Findings

Structure space of each intermediate ring is the Banaschewski compactification.

Characterization of $P$-spaces via closed ideals in $C_c(X)$.

Equivalent conditions for pseudocompactness using various topologies.

Abstract

Intermediate rings of real valued continuous functions with countable range on a Hausdorff zero-dimensional space $X$ are introduced in this article. Let $\Sigma_c(X)$ be the family of all such intermediate rings $A_c(X)$'s which lie between $C_c^*(X)$ and $C_c(X)$. It is shown that the structure space of each $A_c(X)$ is $\beta_0X$, the Banaschewski compactification of $X$. $X$ is shown to be a $P$-space if and only if each ideal in $C_c(X)$ is closed in the $m_c$-topology on it. Furthermore $X$ is realized to be an almost $P$-space when and only when each maximal ideal/ $z$-ideal in $C_c(X)$ becomes a $z^0$-ideal. Incidentally within the family of almost $P$-spaces, $C_c(X)$ is characterized among all the members of $\Sigma_c(X)$ by virtue of either of these two properties. Equivalent descriptions of pseudocompact condition on $X$ are given via $U_c$-topology, $m_c$-topology and norm on $C_c(X)$. The article ends with a result which essentially says that $z^0$-ideals in a typical $A_c(X)$ $\in$ $\Sigma_c(X)$ are precisely the contraction of $z^0$-ideals in $C_c(X)$.