Direct limit topologies in the categories of topological groups and of uniform spaces

Taras Banakh, Dusan Repovs

Published 2009-11-20, updated 2010-05-28Version 2

We study the topological structure of the direct limit $\glim G_n$ of a tower of topological groups $(G_n)$ in the category of topological groups and show that under some conditions on the tower $(G_n)$ the topology of $\glim G_n$ coincides with the topology of the direct limit $\ulim G_n$ of the groups $G_n$ endowed with the Roelcke uniformity in the category of uniform spaces.