Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

W is the category of archimedean l-groups with distinguished weak order unit, with l-group homomorphisms which preserve unit. This category includes all rings of continuous functions C(X) and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in W. There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each W– objectB, we define a topology τ B and a convergence B −→. These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject Aof B is dense iff A is epically embedded. The topology is T1, but only sometimes Hausdorff or an l-group topology. The convergence is a Hausdorff l-group convergence, but only sometimes topological. The associations of B to τ B, and to B −→, are functorial.