International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 2, Pages 289-304

The order topology for function lattices and realcompactness

W. A. Feldman and J. F. Porter

Department of Mathematics, University of Arkansas, Fayetteville 72701, AR, USA

Received 12 June 1980

Copyright © 1981 W. A. Feldman and J. F. Porter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.