Locally functionally countable subalgebra of $\mathcal{R}(L)$

Elyasi, M. ; Estaji, A. A. ; Robat Sarpoushi, M.

Archivum mathematicum, Tome 56 (2020) no. 3, pp. 127-140.

Résumé

Let $L_c(X)= \lbrace f \in C(X) \colon \overline{C_f}= X\rbrace $, where $C_f$ is the union of all open subsets $U \subseteq X$ such that $\vert f(U) \vert \le \aleph _0$. In this paper, we present a pointfree topology version of $L_c(X)$, named $\mathcal{R}_{\ell c}(L)$. We observe that $\mathcal{R}_{\ell c}(L)$ enjoys most of the important properties shared by $\mathcal{R}(L)$ and $\mathcal{R}_c(L)$, where $\mathcal{R}_c(L)$ is the pointfree version of all continuous functions of $C(X)$ with countable image. The interrelation between $\mathcal{R}(L)$, $\mathcal{R}_{\ell c}(L)$, and $\mathcal{R}_c(L)$ is examined. We show that $L_c(X) \cong \mathcal{R}_{\ell c}\big (\mathfrak{O}(X)\big )$ for any space $X$. Frames $L$ for which $\mathcal{R}_{\ell c}(L)=\mathcal{R}(L)$ are characterized.

Détail

DOI : 10.5817/AM2020-3-127

Classification : 06D22, 54C05, 54C30
Keywords: functionally countable subalgebra; locally functionally countable subalgebra; sublocale; frame