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.

Keywords

• order topology
• relative uniform convergence
• realcompactness
• universally complete function lattice
• convergence space completion.