section we introduce another condition guaranteeing that the topology of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ of a tower of topological groups $(G_n)_{n\in{\omega}}$ coincides with the topologies $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$ and $\RL[\tau]$.

Let us observe that a tower of topological groups $(G_n)_{n\in{\omega}}$ satisfies ${\mathrm{PTA}}$ if each group $G_n$, $n\in{\omega}$, is [*balanced*]{}. The latter means that $G_n$ has a neighborhood base at $e$ consisting of $G$-invariant neighborhoods, see [@AT p.69].

We define a subset $U\subset G$ of a group $G$ to be [*$H$-invariant*]{} for a subgroup $H\subset G$ if $xUx^{-1}=U$ for all $x\in H$. Observe that for any subset $U\subset G$ the set $$\sqrt[H]{U}=\{x\in G{\, ;\;}x^H\subset U\}$$ is the largest $H$-invariant subset of $U$. Here $x^H=\{hxh^{-1}{\, ;\;}h\in H\}$ stands for the conjugacy class of a point $x\in G$.

Observe that a topological group $G$ is balanced if and only if for every neighborhood $U\subset G$ of $e$ the set $\sqrt[G]{U}$ is a neighborhood of $e$.

A triple $(G,\Gamma,H)$ of topological groups $H\subset \Gamma\subset G$ is called [*balanced*]{} if for any neighborhoods $V\subset \Gamma$ and $U\subset G$ of the neutral element $e$ of $G$ the product $V\cdot\sqrt[H]{U}$ is a neighborhood of $e$ in $G$.

A tower of topological groups $(G_n)_{n\in{\omega}}$ is called [*balanced*]{} 