A Galois connection related to restrictions of continuous real functions

Abstract

Given a family of continuous real functions G, let RG be a binary relation defined as follows: a continuous function f:R→R is in the relation with a closed set E⊆R if and only if there exists g∈G such that f↾E=g↾E. We consider a Galois connection between families of continuous functions and hereditary families of closed sets of reals naturally associated to RG. 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 G. In some special cases we obtain exact description of these lattices.