Simple and compact direct topologies

Abstract

We characterize the compact and locally compact Hausdorff topological groups and rings that are completions of groups with a topology induced by a direct system, and we characterize the topological groups and rings whose topologies are induced by simple direct topologies. The fact that the p-adic topology on the additive group of the rational field is not a direct topology (for which Zobel gave a long intricate proof) is a special case of an immediate corollary of our characterization of locally compact completions of direct topologies. We show that a direct topology has a neighborhood base at zero consisting of open subgroups if and only if it is induced by a direct system consisting of subgroups.