if each triple $(G_{n+2},G_{n+1},G_n)$, $n\in{\omega}$, is balanced.

\[t4.2\] If a tower of topological groups $(G_n)_{n\in{\omega}}$ is balanced, then the semitopological group $\LA[G]$ is a topological group and hence all 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: $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$, $\RL[\tau]$.

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 at the neutral element $e$.

In order to check the continuity of multiplication at $e$, fix a neighborhood ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\in\LA[{\mathcal B}]_e$. For every $n\in{\omega}$, find a symmetric neighborhood $W_n$ of $e$ in the group $G_n$ such that $W_n\cdot W_n\subset U_n$ and let $$Z_n=\sqrt[G_{n-2}]{W_n}=\{x\in G_n{\, ;\;}x^{G_{n-2}}\subset W_n\}$$ be the largest $G_{n-2}$-invariant subset of $W_n$ (here we assume that $G_{k}=\{e\}$ for $k<0$).

Let $V_0=U_0\cap W_1$ and $V_1\subset G_1$ be a symmetric neighborhood of $e$ such that $V_1^2\subset W_1$. Next, for each $n\ge 2$ by induction choose a neighborhood $V_n\subset G_n$ so that

-   $V_{n}^2\subset V_{n-1}\cdot Z_{n}$, and

-   $V_{n}\subset W_{n+1}$,

The condition (a) can be satisfied because the triple $(G_{n},G_{n-1},G_{n-2})$ is balanced according to our hypothesis.

We claim $\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}} V_n\Big)^2\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$. This inclusion will 