Direct limit topologies in the categories of topological groups and of uniform spaces
Taras Banakh, Dusan Repovs

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 of a tower of topological groups in the category of topological groups and show that under some conditions on the tower the topology of coincides with the topology of the direct limit of the groups endowed with the Roelcke uniformity in the category of uniform spaces.
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