The order topology for function lattices and realcompactness

Abstract

A lattice K ( X , Y ) of continuous functions on space X is associated to each compactification Y of X . It is shown for K ( X , Y ) that the order topology is the topology of compact convergence on X if and only if X is realcompact in Y . This result is used to provide a representation of a class of vector lattices with the order topology as lattices of continuous functions with the topology of compact convergence. This class includes every C ( X ) and all countably universally complete function lattices with 1. It is shown that a choice of K ( X , Y ) endowed with a natural convergence structure serves as the convergence space completion of V with the relative uniform convergence.