positive functions on the real line.

It is well-known that the Whitney topology turns the homeomorphism group $G={\mathcal H}_c({\mathbb R})$ into a topological group, see e.g., [@BMSY]. This group can be written as the countable union $G=\bigcup_{n\in{\omega}}G_n$ of the closed subgroups $$G_n=\big\{h\in G{\, ;\;}{\mathrm{supp}}(h)\subset[-n,n]\big\}.$$ Each subgroup $G_n$ can be identified with the group ${\mathcal H}_+({\mathbb I}_n)$ of orientation-preserving homeomorphisms of the closed interval ${\mathbb I}_n=[-n,n]$. The Whitney topology of the group $G$ induces on each subgroup $G_n$ the compact-open topology, generated by the sup-metric $\|f-h\|=\sup_{x\in{\mathbb R}}|f(x)-h(x)|$.

In the following theorem we shall show that the topology of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with the topology $\LA[\tau]$ on $G$ but the tower $(G_n)_{n\in{\omega}}$ does not satisfy ${\mathrm{PTA}}$. This answers Problem 17.3 [@Glo06] of H. Glöckner.

\[t6.1\]

1.  The tower of the homeomorphism groups $\big({\mathcal H}_+({\mathbb I}_n)\big)_{n\in{\omega}}$ is balanced

2.  the tower $\big({\mathcal H}_+({\mathbb I}_n)\big)_{n\in{\omega}}$ does not satisfy ${\mathrm{PTA}}$

3.  The Whitney topology on ${\mathcal H}_c({\mathbb R})$ coincides with the topologies $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$, $\RL[\tau]$ and those topologies coincide with the topology of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}{\mathcal H}_+({\mathbb I}_n)$.

Let $G={\mathcal H}_c({\mathbb I})$ and $G_n={\mathcal H}_+({\mathbb I}_n)\subset G$ for $n\in{\omega}$. For a constant ${\varepsilon}>0$ the ${\varepsilon}$-ball $B({\mathrm{id}}_{\mathbb R},{\varepsilon})=\{f\in{\mathcal H}_c({\mathbb R}){\, 