A Galois connection related to restrictions of continuous real functions

TL;DR

This paper explores a Galois connection between families of continuous functions and hereditary families of closed sets, analyzing how properties of these lattices depend on the function family.

Contribution

It introduces a Galois connection linking continuous functions and closed sets, characterizes the resulting lattices, and provides exact descriptions in special cases.

Findings

Properties of the lattices depend on the function family al G

Complete lattice structures are characterized and described

Exact lattice descriptions are obtained in special cases

Abstract

Given a family of continuous real functions $\mathcal{G}$, let $R_\mathcal{G}$ be a binary relation defined as follows: a continuous function $f\colon\mathbb{R}\to\mathbb{R}$ is in the relation with a closed set $E\subseteq\mathbb{R}$ if and only if there exists $g\in\mathcal{G}$ such that $f\upharpoonright E = g\upharpoonright E$. We consider a Galois connection between families of continuous functions and hereditary families of closed sets of reals naturally associated to $R_\mathcal{G}$. We study complete lattices determined by this connection and prove several results showing the dependence of the properties of these lattices on the properties of $\mathcal{G}$. In some special cases we obtain exact description of these lattices.