Simple and compact direct topologies

✍ Scribed by Niel Shell

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 177 KB
• Volume: 103
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00164-3

✦ Synopsis

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.