Further thoughts on the ring ${\mathcal {R}}_c (L)$ in frames

Abstract

Let C(X) denote the ring of all real-valued continuous functions on a topological space X and ${\mathcal {R}} (L)$ be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring ${\mathcal {R}}_c (L)$ is introduced as a sub-f-ring of ${\mathcal {R}} ( L)$ as a pointfree analogue to the subring $C_c(X)$ of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring ${\mathcal {R}}_c (L)$. In order to do so we introduce the set $R_{\alpha } := \{ r \in {\mathbb {R}} : {{\,\mathrm{coz}\,}}(\alpha - \mathbf{r}) \not = \top \}$ for every $\alpha \in {\mathcal {R}} (L)$. We prove that $R_{\alpha }$ is a countable subset of ${\mathbb {R}}$ for every $\alpha \in {\mathcal {R}}_c (L)$. Next, we show that if L is a compact frame, then $R_{\alpha }$ is a finite subset of ${\mathbb {R}}$ for every $\alpha \in {\mathcal {R}}_c (L)$. Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and $C_c(X) \cong C_c(Y )$ in pointfree topology. Finally, we prove that, for some frame L, the ring ${\mathcal {R}}_c (L)$ may not be isomorphic to ${\mathcal {R}} (M)$, for any given frame M.