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.