# Long colimits of topological groups I: Continuous maps and homeomorphisms

## TL;DR

This paper investigates the topological properties of unions of long families of topological groups, focusing on when different natural topologies coincide, with specific results for groups of continuous maps and homeomorphisms.

## Contribution

It extends understanding of topological group unions beyond countable cases, providing new results for long families and examples involving continuous maps and homeomorphism groups.

## Key findings

- Identifies conditions under which the two topologies coincide for long families.
- Provides examples of long families of topological groups.
- Answers to the main question for families of continuous maps and homeomorphism groups.

## Abstract

The union of a directed family of topological groups can be equipped with two noteworthy topologies: the finest topology making each injection continuous, and the finest group topology making each injection continuous. This begs the question of whether the two topologies coincide. If the family is countable, the answer is well known in many cases. We study this question in the context of so-called long families, which are as far as possible from countable ones. As a first step, we present answers to the question for families of group-valued continuous maps and homeomorphism groups, and provide additional examples.