identity inclusions $G_n\to G$, $n\in{\omega}$, continuous.

The definitions of the direct limits ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ and ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ imply that the identity map $${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$$is continuous. This map is a homeomorphism if and only if ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is a topological group. It was observed in [@Ba98] and [@TSH] that the group operation on $G={\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is not necessarily continuous with respect to the topology ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$. Moreover, if each group $G_n$, $n\in{\omega}$, is metrizable and closed in $G_{n+1}$, then the topological direct limit ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is a topological group if and only if either all groups $G_n$ are locally compact or some group $G_n$ is open in all groups $G_m$, $m\ge n$ (see [@BaZ] or [@Yama]).

Thus in many interesting cases (in particular, those considered in [@Dierolf], [@Floret], [@Glo03], [@Glo06], [@Glo08]), the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ differs from the topology of the topological direct limit ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$. However, in contrast with the topology of ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ which has an explicit description (as the family of all subsets $U\subset \bigcup_{n\in{\omega}}G_n$ that have open traces $U\cap G_n$ on all spaces $G_n$) the topological structure of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is not so clear. The problem of explicit description of the topological 