Frames of continuous functions (Q2310784)

In this paper, the authors propose a new approach to representing a topological space via a frame of continuous functions with values in what they call a topological frame. A topological frame is a frame \(\mathbb{F}\) equipped with a topology such that the operations \[\wedge\colon \mathbb{F}\times \mathbb{F} \to \mathbb{F}\colon (a,b)\mapsto a\wedge b\] and \[\sup_{i\in I}\colon \mathbb{F}^I\to \mathbb{F}\colon (a_i)_{i\in I}\mapsto \sup_{i\in I} a_i\] are continuous. The idea extends that of pointfree topology of investigating topological spaces via their open-set lattices (which are frames of continuous functions to the Sierpinski space). The authors investigate properties of a topological space \(X\) via the frame of continuous functions from \(X\) to a topological frame \(\mathbb{F}\), namely the associated notion of sobriety (\(\mathbb{F}\)-sobriety). One of the interesting results provides conditions on \(\mathbb{F}\) ensuring that a Hausdorff topological space is \(\mathbb{F}\)-sober. These conditions are fulfilled as soon as \(\mathbb{F}\) is a chain with \(0\neq 1\) equipped with the Scott topology. Further, \(\mathbb{F}\)-spectra of \(\mathbb{F}\)-function frames are computed for various spaces \(X\) and frames \(\mathbb{F}\) and a number of spaces that are not \(\mathbb{F}\)-sober are exhibited, showing in particular that the Hausdorff condition in the aforementioned result cannot be relaxed to classical sobriety. A final section discusses the relation between \(\mathbb{F}\)-sobriety and the notion of \(\mathbb{F}\)-fuzzy sobriety as considered in [\textit{D. Zhang} and \textit{Y. Liu}, Fuzzy Sets Syst. 76, No. 2, 259--270 (1995; Zbl 0852.54008)]. The paper ends with a brief outline of some open problems.