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

$(1){\Rightarrow}(2)$ Assume that $\LA[G]$ is a topological group. Then the identity map $\LA[G]\to\RA[G]$ is continuous because each basic neighborhood ${\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\in\RA[{\mathcal B}]_e$ of $e$ in $\RA[G]$ is open in $\LA[G]$, being the inversion $({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)^{-1}$ of the basic neighborhood ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\in\LA[{\mathcal B}]_e$ of $e$ in the topological group $\LA[G]$. By the same reason, the identity map $\LA[G]\to\RA[G]$ is open. Consequently, the topologies $\LA[\tau]$ and $\RA[\tau]$ on $G$ coincide, and hence $\LR[G]$ is a topological group.

The implication $(2){\Rightarrow}(1)$ can be proved by analogy.

$(1){\Rightarrow}(3)$ If $\LA[G]$ is a topological group, then $\LA[\tau]=\RA[\tau]$ and then $\LR[\tau]=\LA[\tau]=\RA[\tau]$, by the definition of the topology $\LR[\tau]$. Consequently, $\LR[G]=\LA[G]$ is a topological group.

$(3){\Rightarrow}(5)$ If $\LR[G]$ is a topological group, then the identity map ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n\to\LR[G]$ is continuous by the definition of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ because all the identity homomorphisms $G_n\to \LR[G]$, $n\in{\omega}$, are continuous. The inverse (identity) map $\LR[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is always continuous by Proposition \[p2.1\]. So, it is a homeomorphism.

$(5){\Rightarrow}(4)$ If the identity map $\LR[G]\to {\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism, then the identity map $\RL[G]\to\LR[G]$ is continuous being the composition of two continuous maps $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n\to\LR[G]$.

$(4){\Rightarrow}(1)$ Assume that the identity map $\RL[G]\to\LR[G]$ is a homeomorphism. Then 