Generalized metric spaces are paracompact

A generalized metric space is a triple (X, d, G), where X is a set, G an ordered Abelian group and d: X x X → G is a function which satisfies the metric axioms. The generalized metric d defines a topology on X in the same way as an ordinary metric does for a metric space. These spaces are studied by Stevenson & Ton(5) and Sikorski (4) amongst others. A survey of these spaces is the subject of an M.Phil. thesis by the author (6). Generalized metric spaces satisfy most of the separation properties of ordinary metric spaces (T2, regular, etc.). We prove here that these spaces are also paracompact. The proof is a generalization of the proof given by Ruffin (2) for metric spaces. The result given here follows, in fact, from a theorem proved by Hayes (1). He proved that uniformities with totally ordered bases have paracompact topologies. It is known (Stevenson & Ton (5); Shock (3); Souppouris (6), chapter 5) that these spaces are identical to generalized metric spaces. The proof given here uses metric space language.