topological group. By induction, construct a sequence of open symmetric neighborhoods $V_n\subset{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ of $e$ such that $V_0^2\subset V$ and $V_{n+1}^2\subset V_n$ for all $n\in{\omega}$. By induction on $m\in{\omega}$ we shall prove the inclusion $$\label{ind1}
\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<m}V_n\big)\cdot V_m^2\subset V.$$ For $m=0$ this inclusion holds according to the choice of $V_0$. Assuming that for some $m$ the inclusion is true observe that $$\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n\le m}V_n\big)\cdot V_{m+1}^2=\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<m}V_n\big)\cdot V_mV_{m+1}^2\subset 
\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<m}V_n\big)\cdot V_mV_m\subset V$$by the inductive hypothesis. Then $${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n=\bigcup_{m\in{\omega}}{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n\le m}V_n\subset V.$$

For every $n\in{\omega}$ find a basic neighborhood $W_n\in{\mathcal B}_n$ in the group $G_n$ such that $W_n\subset V_n$ and observe that ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\subset V$ and hence $$\RL[{\mathcal B}]_e\ni{\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in w}W_n=\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\big)^{-1}\cdot {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\subset V^{-1}V\subset U$$witnessing the continuity of the identity map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ at $e$.

One may ask about conditions guaranteeing that the semitopological groups $\LA[G]$, $\RA[G]$, $\LR[G]$ or $\RL[G]$ are topological groups.

\[t2.2\] The following conditions (1) through (5) are equivalent

1.  $\LA[G]$ is a topological group

2.  $\RA[G]$ is a topological group

3.  $\LR[G]$ is a topological group

4.  the identity map $\RL[G]\to\LR[G]$ is continuous

5.  the identity map $\LR[G]\to {\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism.

The equivalent conditions (1) through (5) imply the following two equivalent conditions

1.  $\RL[G]$ is a topological group

2.  