Is a direct limit of topological groups always a topological group?

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).

Solution 1:

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.

Solution 2:

Examples of direct systems $(G_i,f_{i,j})$ of topological groups for which the direct limit topology DOES NOT make $G:=\lim_{i \to \infty} G_i$ a topological group are described in the following paper. N. Tatsuma, H. Shimomura and T. Hirai, On group topologies and unitary representations of inductive limits of topological groups and the case of the groups of diffeomorphisms, J. Math. Kyoto Univ., 38-3 (1998) 551-578. However, it is established in the same paper (Theorem 2.7) that, if the index set is countable and the groups $G_i$ locally compact, then $G$ IS a topological group.