---
abstract: 'Given an increasing sequence $(G_n)$ of topological groups, we study the topologies of the direct limits of the sequence $(G_n)$ in the categories of topological groups and of uniform spaces and find conditions under which these two direct limit topologies coincide.'
address:
- 'Instytut Matematyki, Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego, Świȩtokrzyska 15, Kielce, Poland'
- 'Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia 1000'
author:
- Taras Banakh
- Dušan Repovš
title: Direct limit topologies in the categories of topological groups and of uniform spaces
---

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Introduction
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Given a tower $$G_0\subset G_1\subset G_2\subset\cdots$$of topological groups, we study in this paper the topological structure of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ of the tower $(G_n)$ in the category of topological groups. By definition, ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is the union $G=\bigcup_{n\in{\omega}} G_n$ endowed with the strongest (not necessarily Hausdorff) topology that turns $G$ into a topological group and makes the identity inclusions $G_n\to G$, $n\in{\omega}$, continuous.

Besides the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$, the union $G=\bigcup_{n\in{\omega}}G_n$ carries the topology of the direct limit ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ of the tower $(G_n)_{n\in{\omega}}$ in the category of topological spaces. The topology of ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is the strongest topology on $G$ making the 