A Galois connection related to restrictions of continuous real functions (Q2324525)

Let \({\mathcal G}\) be a family of continuous real functions and \( R_{\mathcal G}\) be a binary relation defined in the following way: A continuous function \(f:{\mathbb R}\rightarrow {\mathbb R}\) is in the relation with a closed set \(E\subseteq {\mathbb R}\) if and only if there exists \(g\in {\mathcal G}\) such that \(f \upharpoonright E = g \upharpoonright E.\) The author considers a Galois connection between families of continuous functions and hereditary families of closed sets of reals naturally associated with \(R_{\mathcal G}\). Complete lattices determined by the connection are investigated. Results study how properties of these lattices depend on those of \({\mathcal G}.\) For some simple cases the lattices are described explicitly.