Characterization of equivalent uniformities in topological groups

AbstractWe study the concept of equal uniformities on locally compact topological groups. In Section 1 we collect some classical characterizations for T0 groups and metric groups, along with some new ones. In Section 2 and 3 we find characterizations for the locally compact metric case and the σ-compact case. The Gδ condition is introduced in Section 3 and plays a crucial part in the characterizations obtained for the locally compact case. Section 4 contains a brief survey of inductive limit systems and includes a reinterpretation of a famous theorem of Kakutani and Kodaira, together with some slight generalizations and consequences. Section 5 contains structure theorems for locally compact groups with equal uniformities using Lie group approximation methods. These results generalize a theorem of Grosser and Moskowitz and answer a question posed by G. Itzkowitz. It is shown that if every open σ-compact subgroup of a locally compact group G has equal uniformities then so does G. The structure theorems yield a number of interesting characterizations of equal uniformities and include the characterization that every real valued left uniformly continuous function on G is right uniformly continuous. Section 6 contains an independent characterization for totally disconnected locally compact groups which motivated the method of proof used in the general case