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

TL;DR

This paper investigates the topological structure of direct limits of topological groups, showing conditions under which their topology aligns with the direct limit topology induced by the Roelcke uniformity in uniform spaces.

Contribution

It establishes conditions where the direct limit topology of a tower of topological groups matches the topology from the Roelcke uniformity, linking group and uniform space structures.

Findings

Topology of direct limits coincides under certain conditions

Conditions relate to the structure of the tower of groups

Results unify topological and uniform space perspectives

Abstract

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.