B_n$ at the identity $e$, consisting of open symmetric neighborhoods $U\subset G_n$ such that for every $m\ge n$ and every neighborhood $V\subset G_m$ of $e$ there is a neighborhood $W\subset G_m$ of $e$ such that $WU\subset UV$.

It was proved in [@HSTH] and [@TSH] that for a tower of topological groups $(G_n)_{n\in{\omega}}$ satisfying ${\mathrm{PTA}}$, the semitopological group $\RL[G]$ is a topological group, which can be identified with the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$.

The following theorem says a bit more:

\[t3.2\] If a tower of topological groups $(G_n)_{n\in{\omega}}$ satisfies ${\mathrm{PTA}}$, then the semitopological group $\LA[G]$ is a topological group and hence the conditions (1) through (7) of Theorem  hold. In particular the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with any of the topologies: $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$, $\RL[\tau]$.

Since the tower $(G_n)_{n\in{\omega}}$ satisfies ${\mathrm{PTA}}$, each topological group $G_n$ admits a neighborhood base ${\mathcal B}_n$ at $e$ that consists of open sets $U=U^{-1}$ such that for every $m\ge n$ and a neighborhood $V\subset G_m$ of $e$ there is a neighborhood $W\subset G_m$ of $e$ such that $WU\subset UV$.

In order to show that the semitopological group $\LA[G]$ is a topological group, it suffices to check the continuity of the multiplication and of the inversion 