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

Abstract

Let Lc(X) = {f ∈ C(X) : C f = X}, where C f is the union of all open subsets U ⊆ X such that |f (U)| ℵ 0. In this paper, we present a pointfree topology version of Lc(X), named R c (L). We observe that R c (L) enjoys most of the important properties shared by R(L) and Rc(L), where Rc(L) is the pointfree version of all continuous functions of C(X) with countable image. The interrelation between R(L), R c (L), and Rc(L) is examined. We show that Lc(X) ∼ = R c O(X) for any space X. Frames L for which R c (L) = R(L) are characterized.