Long colimits of topological groups I: Continuous maps and homeomorphisms

Given a directed family of topological groups, the finest topology on their union making each injection continuous need not be a group topology, because the multiplication may fail to be jointly continuous. This begs the question of when the union is a topological group with respect to this topology. If the family is countable, the answer is well known in most 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.