;\;}(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal B}_n\},\\
&\RL[{\mathcal B}]_e=\{{\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n{\, ;\;}(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal B}_n\}=\{U^{-1} U{\, ;\;}U\in\LA[{\mathcal B}]_e\},\\
&\LR[{\mathcal B}]_e=\{\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)\cap \big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big){\, ;\;}(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal B}_n\}=\{U\cap  U^{-1}{\, ;\;}U\in\LA[{\mathcal B}]_e\}
\end{aligned}$$as neighborhood bases at the identity $e$. Since the inversion $(\cdot)^{-1}:G\to G$ is continuous with respect to the topologies $\RL[\tau]$ or $\LR[\tau]$, the semitopological groups $\RL[G]$ and $\LR[G]$ are quasitopological groups.

We recall that a group $H$ endowed with a topology is

-   a [*semitopological group*]{} if the binary operation $H\times H\to H$, $(x,y)\mapsto xy,$ is separately continuous;

-   a [*quasitopological group*]{} if $H$ is a semitopological group with continuous inversion $(\cdot)^{-1}:H\to H$, $(\cdot)^{-1}:x\mapsto x^{-1}$.

Now we see that for any tower $(G_n)_{n\in{\omega}}$ of topological groups we get the following five semitopological groups linked by continuous identity homomorphisms:

(200,90)(-100,-10) (19,29)[$\LR[G]$]{} (33,28)[(3,-2)[30]{}]{} (33,38)[(3,2)[30]{}]{} (69,2)[$\LA[G]$]{} (85,8)[(3,2)[30]{}]{} (69,58)[$\RA[G]$]{} (85,56)[(3,-2)[30]{}]{} (120,29)[$\RL[G]$]{} (135,32)[(1,0)[27]{}]{} (170,30)[${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$]{}

The continuity of the final map in the diagram is not trivial:

\[p2.1\] The identity map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is continuous.

Since $\RL[G]$ and ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ are semitopological groups, it suffices to prove the continuity of the identity map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ at the neutral element $e$.

Given a neighborhood $U\subset{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ of $e$, find an open neighborhood $V\subset{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ of $e$ such that $V^{-1}V\subset U$. Such a neighborhood exists because ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a 