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Locally functionally countable subalgebra of \(\mathcal{R}(L)\). (English) Zbl 07250674
Summary: 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.
MSC:
06D22 Frames, locales
54C05 Continuous maps
54C30 Real-valued functions in general topology
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