the identity maps between the semitopological groups $\LR[G],\LA[G],\RA[G],\RL[G]$ are homeomorphisms. Consequently, $\LA[G]$ is a quasitopological group because so is $\LR[G]$ or $\RL[G]$. To see that $\LA[G]$ is a topological group, observe that the multiplication map $\LA[G]\times\LA[G]\to\LA[G]$, $(x,y)\mapsto xy$, is continuous, being continuous as a map $\RA[G]\times\LA[G]\to\RL[G]$.

$(5){\Rightarrow}(7)$ If the identity map $\LR[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism, then the identity map ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n\to\RL[G]$ is continuous being the composition of two continuous maps ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n\to\LR[G]\to\RL[G]$. The continuity of the inverse (identity) map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ was proved in Proposition \[p2.1\].

$(7){\Rightarrow}(6)$ If the identity map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism, then $\RL[G]$ is a topological group because so is ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$.

The final implication $(6){\Rightarrow}(7)$ can be proved by analogy with $(3){\Rightarrow}(5)$.

The topology $\RL[\tau]$ on the union $G=\bigcup_{n\in{\omega}}G_n$ of a tower of topological groups $(G_n)$ was introduced in [@TSH] and called the [*bamboo-shoot*]{} topology. This topology was later discussed in [@Eda], [@Glo03], [@Glo06], [@Glo08], [@HSTH].

The Passing Through Assumption {#s:PTA}
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In this section we shall discuss implications of ${\mathrm{PTA}}$, the Passing Through Assumption, introduced by Tatsuuma, Shimomura, and Hirai in [@TSH].

A tower of topological groups $(G_n)_{n\in{\omega}}$ is defined to satisfy ${\mathrm{PTA}}$ if each group $G_n$ has a neighborhood base $\mathcal 