The topology most often used on a totally ordered group (G, <) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.
© 1987 Cambridge Philosophical Society
Henriksen, M., R. Kopperman, and F. A. Smith. "Ordered products of topological groups." Mathematical Proceedings of the Cambridge Philosophical Society 102.2 (September 1987): 281-295. DOI: 10.1017/S030500410006730X