The closed graph theorem is one of the cornerstones of linear functional
analysis in Frechet spaces, and the extension of this result to more general topological
vector spaces is a difficult problem comprising a great deal of technical difficulty.
However, the theory of convergence vector spaces provides a natural framework for
closed graph theorems. In this paper we use techniques from convergence vector space
theory to prove a version of the closed graph theorem for order bounded operators on
Archimedean vector lattices. This illustrates the usefulness of convergence spaces in
dealing with problems in vector lattice theory, problems that may fail to be amenable
to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.