The Cozero part of the pointfree version of $C_c (X)$

TL;DR

This paper investigates the properties of a specific subset of a pointfree version of $C_c(X)$, showing it forms a $\sigma$-frame with various topological properties and characterizing zero-dimensional frames.

Contribution

It introduces and analyzes the cozero part of the pointfree $C_c(X)$, proving it forms a $\sigma$-frame and characterizing zero-dimensional frames.

Findings

${ m Coz}_c[L]$ is a $\sigma$-frame for every completely regular frame $L$

${ m Coz}_c[L]$ is regular, paracompact, and perfectly normal

A frame $L$ is zero-dimensional iff it is $c$-completely regular

Abstract

Let $\mathcal C_{c}(L):= \{\alpha\in \mathcal{R}(L) \mid R_{\alpha} \, \text{ is a countable subset of } \, \mathbb R \}$, where $R_\alpha:=\{r\in\mathbb R \mid {\mathrm{coz}}(\alpha-r)\neq\top\}$ for every $\alpha\in\mathcal R (L).$ By using idempotent elements, it is going to prove that ${{\mathrm{Coz}}}_c[L]:= \{{\mathrm{coz}}(\alpha) \mid \alpha\in\mathcal{C}_c (L) \}$ is a $\sigma$-frame for every completely regular frame $L,$ and from this, we conclude that it is regular, paracompact, perfectly normal and an Alexandroff algebra frame such that each cover of it is shrinkable. Also, we show that $L$ is a zero-dimensional frame if and only if $ L$ is a $c$-completely regular frame.