Simple and compact direct topologies (Q1567308)

The article is devoted to the characterization of definite classes of Hausdorff topological groups and rings in terms of a direct system. The main results are for locally compact groups and rings. Theorem 2: If \(\{M_{in}\}\) is a direct system on the group (ring) \(G\), then the associated neighborhood base is a neighborhood base at zero for a Hausdorff group (ring) topology on \(G\). The sets \(U_n\), \(n\geq 2\), are clopen in this topology. Several statements are devoted to direct topological groups (rings), which are characterized by specific pavings of sequences of subsets. Few results are about relations of such topologies with non-Archimedean direct systems. Conditions on a simple direct group (ring) topology are given leading to locally compact completions.