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).
