标题： 点无形式的$C_c (X)$的Cozero部分 显示英文标题

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

摘要： 设$\mathcal C_{c}(L):= \{\alpha\in \mathcal{R}(L) \mid R_{\alpha} \, \text{ is a countable subset of } \, \mathbb R \}$，其中$R_\alpha:=\{r\in\mathbb R \mid {\mathrm{coz}}(\alpha-r)\neq\top\}$对于每个$\alpha\in\mathcal R (L).$ 通过使用幂等元素，将证明对于每个完全正则框架$L,$，${{\mathrm{Coz}}}_c[L]:= \{{\mathrm{coz}}(\alpha) \mid \alpha\in\mathcal{C}_c (L) \}$是一个$\sigma$-框架，并由此得出它是正则的，仿紧的，完美正则的，并且是一个Alexandroff代数框架，使得它的每个覆盖都是可收缩的。 此外，我们证明当且仅当 $ L$ 是一个$c$-完全正则框架时，$L$是一个零维框架。 显示英文摘要

摘要： 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.