If $(G_i,f_{ij})$ is a direct system of topological groups, is it always the case that the topological\group-theoretical direct limit $G:=\varinjlim_iG_i$ is a topological group? (The topology on $G$ is the final topology with respect to the canonical homomorphisms $\psi_i:G_i\rightarrow G$). It is immediate from the group structure on $G$ and the definition of its topology that the inversion map is continuous. Moreover, I'm pretty certain that if the transition maps $f_{ij}$ are open, then the $\psi_i$ are open, and this gives continuity of multiplication. It's not at all clear to me that this condition is necessary though.
I have always sort of assumed that this was true, but it's never really been an issue because I never start with a system of topological groups and take the direct limit; I always have a topological group and express it as a direct limit (e.g. the idele group of a global field, or the Cartier dual of a finite free $\mathbb{Z}_p$-module).
Best Answer
You can construct colimits by a rather general procedure. See my answer here.
The topologies mentioned by Agusti are not correct. The continuity of the group laws does not hold in general. The problem is that quotient topologies do not commute with product topologies.