The C-topology on lattice-ordered groups

Let A be a lattice-ordered group. Gusić showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusić’s theorem, and reveal the very nature of a “C-group” of Gusić in this paper. Moreover, we show that the C-topological groups are topological lattice-ordered groups, and prove that every archimedean lattice-ordered vector space is a T ₂ topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T ₂. A further example demonstrates that a T ₂ topological archimedean lattice-ordered group need not be C-archimedean, either.