m}W_n$ for every $m\in{\omega}$. By induction we shall prove a bit more: $$\label{eq:inv}
V_m\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}W_n$$ for every $m\in{\mathbb N}$.

For $m=1$ the inclusion (\[eq:inv\]) is true: $V_1U_0\subset V_1^2\subset W_1\subset W_0W_1$. Assume that the inclusion (\[eq:inv\]) has been proved for some $m=k\ge 1$. Then $$\begin{aligned}
V_{m+1}\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\le m}U_n&\subset V_{m+1}\cdot U_m\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\subset V_{m+1}^2\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\subset V_mZ_{m+1}{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n=\\&=V_m\cdot \big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\big)\cdot Z_{m+1}\subset \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}W_n\big)\cdot Z_{m+1}\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m+1}W_n,
\end{aligned}$$which means that the inclusion (\[eq:inv\]) holds for $m=k+1$.

Bi-balanced triples and towers of groups {#s2:SIN}
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In this section we introduce the bi-balanced property of a tower $(G_n)$, which is weaker than the balanced property and implies that the semitopological group $\RL[G]$ is a topological group.

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

A tower of topological groups $(G_n)_{n\in{\omega}}$ is called [*bi-balanced*]{} if each triple $(G_{n+2},G_{n+1},G_n)$, $n\in{\omega}$, is bi-balanced.

\[t5.2\] If a tower of topological groups $(G_n)_{n\in{\omega}}$ is bi-balanced, then the identity map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism and hence the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with the topology $\RL[\tau]$.

By Theorem \[t2.2\], it suffices to show that $\RL[G]$ 