---
abstract: 'Given an increasing sequence $(G_n)$ of topological groups, we study the topologies of the direct limits of the sequence $(G_n)$ in the categories of topological groups and of uniform spaces and find conditions under which these two direct limit topologies coincide.'
address:
- 'Instytut Matematyki, Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego, Świȩtokrzyska 15, Kielce, Poland'
- 'Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia 1000'
author:
- Taras Banakh
- Dušan Repovš
title: Direct limit topologies in the categories of topological groups and of uniform spaces
---

[^1]

Introduction
============

Given a tower $$G_0\subset G_1\subset G_2\subset\cdots$$of topological groups, we study in this paper the topological structure of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ of the tower $(G_n)$ in the category of topological groups. By definition, ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is the union $G=\bigcup_{n\in{\omega}} G_n$ endowed with the strongest (not necessarily Hausdorff) topology that turns $G$ into a topological group and makes the identity inclusions $G_n\to G$, $n\in{\omega}$, continuous.

Besides the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$, the union $G=\bigcup_{n\in{\omega}}G_n$ carries the topology of the direct limit ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ of the tower $(G_n)_{n\in{\omega}}$ in the category of topological spaces. The topology of ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is the strongest topology on $G$ making the identity inclusions $G_n\to G$, $n\in{\omega}$, continuous.

The definitions of the direct limits ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ and ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ imply that the identity map $${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$$is continuous. This map is a homeomorphism if and only if ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is a topological group. It was observed in [@Ba98] and [@TSH] that the group operation on $G={\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is not necessarily continuous with respect to the topology ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$. Moreover, if each group $G_n$, $n\in{\omega}$, is metrizable and closed in $G_{n+1}$, then the topological direct limit ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ is a topological group if and only if either all groups $G_n$ are locally compact or some group $G_n$ is open in all groups $G_m$, $m\ge n$ (see [@BaZ] or [@Yama]).

Thus in many interesting cases (in particular, those considered in [@Dierolf], [@Floret], [@Glo03], [@Glo06], [@Glo08]), the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ differs from the topology of the topological direct limit ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$. However, in contrast with the topology of ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ which has an explicit description (as the family of all subsets $U\subset \bigcup_{n\in{\omega}}G_n$ that have open traces $U\cap G_n$ on all spaces $G_n$) the topological structure of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is not so clear. The problem of explicit description of the topological structure of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ was discussed in [@Eda], [@HSTH], [@Glo03], [@Glo06], [@Glo08], [@TSH].

In this paper we shall show that under certain conditions on a tower of topological groups $(G_n)_{n\in{\omega}}$ the topology of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with one (or all) of four simply described topologies $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$ or $\RL[\tau]$ on the group $G=\bigcup_{n\in{\omega}}G_n$. These topologies are considered in Section \[s:top\]. In Sections \[s:PTA\] and \[s1:SIN\] we study two properties (${\mathrm{PTA}}$ and the balanced property) of a tower of topological groups $(G_n)_{n\in{\omega}}$ implying that the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with the topology $\LR[\tau]$, which is the strongest among the four topologies on $G$. In Section \[s2:SIN\] we define another (bi-balanced) property of the tower $(G_n)$ guaranteeing that the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with the topology $\RL[\tau]$, which is the weakest among the four topologies on $G$. In Section \[s:ulim-top\] we reveal the uniform nature of the topologies $\LA[\tau]$ and $\RA[\tau]$ and show that they coincide with the topologies of the uniform direct limits ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$ and ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^R$ of the groups $G_n$ endowed with the left and right uniformities. In Section \[s:OP\] we sum up the results obtained in this paper and pose some open problems.

The semitopological groups $\LA[G]$, $\RA[G]$, $\RL[G]$ and $\LR[G]$ {#s:top}
====================================================================

In this section, given a tower of topological groups $$G_0\subset G_1\subset G_2\subset\cdots$$we define four topologies $\LA[\tau]$, $\RA[\tau]$, $\RL[\tau]$ or $\LR[\tau]$ on the group $G=\bigcup_{n\in{\omega}}G_n$.

Given a sequence of subsets $(U_n)_{n\in{\omega}}$ of the group $G$, consider their directed products in $G$: $$\begin{aligned}
&{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n=\bigcup_{m\in{\omega}}{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n\le m}U_n\mbox{ \ where \ }{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k\le n\le m}U_n=U_k U_{k+1}\cdots U_m,\\
&{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n=\bigcup_{m\in{\omega}}{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{0\le n\le m}U_n\mbox{ \ where \ }{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k\le n\le m}U_n=U_m\cdots U_{k+1}U_k,\\
&{\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n=\bigcup_{m\in{\omega}}{\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{0\le n\le m}U_n\mbox{ where }
{\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{k\le n\le m}U_n=U_m\cdots U_kU_k\cdots U_m.
\end{aligned}$$ Observe that $$\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)^{-1}={\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n^{-1}\mbox{ and }{\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n=
\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)\cdot\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big).$$

In each topological group $G_n$ fix a base ${\mathcal B}_n$ of open symmetric neighborhoods $U=U^{-1}\subset G_n$ of the neutral element $e$.

The topologies $\LA[\tau]$, $\RA[\tau]$, $\RL[\tau]$ and $\LR[\tau]$ on the group $G=\bigcup_{n\in{\omega}}G_n$ are generated by the bases: $$\begin{aligned}
&\LA[{\mathcal B}]=\{({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)\cdot x {\, ;\;}x\in G,\;(U_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}\limits_{n\in{\omega}}{\mathcal B}_n\},\\
&\RA[{\mathcal B}]=\{x\cdot ({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big) {\, ;\;}x\in G,\;(U_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal B}_n\},\\
&\RL[{\mathcal B}]=\{x\cdot({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)\cdot y{\, ;\;}x,y\in G,\;(U_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal B}_n\},\\
&\LR[{\mathcal B}]=\{x\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}xU_n\big)\cap \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)y{\, ;\;}x,y\in G,\;(U_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal B}_n\}.
\end{aligned}$$ By $\LA[G]$, $\RA[G]$, $\RL[G]$, $\LR[G]$ we denote the groups $G$ endowed with the topologies $\LA[\tau]$, $\RA[\tau]$, $\RL[\tau]$, $\LR[\tau]$, respectively. It is easy to check that $\LA[G]$, $\RA[G]$, $\RL[G]$, $\LR[G]$ are semitopological groups having the families $$\begin{aligned}
&\LA[{\mathcal B}]_e=\{{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n{\, ;\;}(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal B}_n\},\\
&\RA[{\mathcal B}]_e=\{{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n{\, ;\;}(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal B}_n\},\\
&\RL[{\mathcal B}]_e=\{{\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n{\, ;\;}(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal B}_n\}=\{U^{-1} U{\, ;\;}U\in\LA[{\mathcal B}]_e\},\\
&\LR[{\mathcal B}]_e=\{\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)\cap \big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big){\, ;\;}(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal B}_n\}=\{U\cap  U^{-1}{\, ;\;}U\in\LA[{\mathcal B}]_e\}
\end{aligned}$$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 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.  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 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}
==============================

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 $\mathcal B_n$ at the identity $e$, consisting of open symmetric neighborhoods $U\subset G_n$ such that for every $m\ge n$ and every neighborhood $V\subset G_m$ of $e$ there is a neighborhood $W\subset G_m$ of $e$ such that $WU\subset UV$.

It was proved in [@HSTH] and [@TSH] that for a tower of topological groups $(G_n)_{n\in{\omega}}$ satisfying ${\mathrm{PTA}}$, the semitopological group $\RL[G]$ is a topological group, which can be identified with the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$.

The following theorem says a bit more:

\[t3.2\] If a tower of topological groups $(G_n)_{n\in{\omega}}$ satisfies ${\mathrm{PTA}}$, then the semitopological group $\LA[G]$ is a topological group and hence the conditions (1) through (7) of Theorem  hold. In particular the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with any of the topologies: $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$, $\RL[\tau]$.

Since the tower $(G_n)_{n\in{\omega}}$ satisfies ${\mathrm{PTA}}$, each topological group $G_n$ admits a neighborhood base ${\mathcal B}_n$ at $e$ that consists of open sets $U=U^{-1}$ such that for every $m\ge n$ and a neighborhood $V\subset G_m$ of $e$ there is a neighborhood $W\subset G_m$ of $e$ such that $WU\subset UV$.

In order to show that the semitopological group $\LA[G]$ is a topological group, it suffices to check the continuity of the multiplication and of the inversion at the neutral element $e$.

The continuity of the multiplication at $e$ will follow as soon as for every neighborhood ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\in\LA[{\mathcal B}]_e$ we find a neighborhood ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\in\LA[{\mathcal B}]_e$ such that $\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\big)^2\subset \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\big)$.

For every $n\in{\omega}$ find a neighborhood $U_n\in{\mathcal B}_n$ with $U_nU_n\subset W_n$. Put $V_n^{(0)}=U_n$ and using PTA, for every $0<i\le n$ find a neighborhood $V_n^{(i)}\in{\mathcal B}_n$ such that

-   $V_n^{(i)}\subset U_n$;

-   $V_n^{(i)}U_{n-i}\subset U_{n-i} V_n^{(i-1)}$.

Observe that for $i=n-k$ the latter inclusion yields $$\label{PTA2a}
V_n^{(n-k)}U_k\subset U_kV_n^{(n-k-1)}.$$

We claim that $\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n^{(n)}\big)^2\subset \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\big)$. Since $V_n^{(n)}\subset U_n$, this inclusion will follow as soon as we check that $$\label{PTA2c}
{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}V_n^{(n)}\cdot {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}U_n\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}W_n$$ for every $m>0$.

For every non-negative integer $k\le m+1$ consider the subset $$\Pi_k={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}W_n\cdot{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k\le n\le m}V_n^{(n-k)}\cdot{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k\le n\le m}U_n$$of the group $G_m$. Observe that (\[PTA2c\]) is equivalent to the inclusion $\Pi_0\subset\Pi_{m+1}$. The last inclusion will follow as soon as we check that $\Pi_{k}\subset \Pi_{k+1}$ for every $k\le  m$.

By induction on $k$ we can deduce from (\[PTA2a\]) the following inclusion: $$\label{PTA2d}
({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k)})\cdot U_k\subset U_k\cdot{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k-1)}.$$

This inclusion combined with $V_k^{(0)}U_k=U_kU_k\subset W_k$ yields the desired inclusion: $$\begin{aligned}
\Pi_{k}&={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k} W_n\cdot {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k\le n\le m}V_n^{(n-k)}\cdot {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k\le n\le m}U_n=\\
&=\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k} W_n\Big)\cdot V_k^{(0)}\cdot \Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k)}\Big)\cdot U_k\cdot 
{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}U_n\\
&\subset\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}W_n\Big)\cdot V_k^{(0)}\cdot \Big(U_k\cdot{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k-1)}\Big)\cdot{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}U_n\\
&\subset\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}W_n\Big)\cdot W_k\cdot {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k-1)}\cdot{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n\le m}U_n=\Pi_{k+1}.
\end{aligned}$$

Next, we verify the continuity of the inversion at $e$. Given a set ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\in\LA[{\mathcal B}]_e$, we need to find a set ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\in\LA[{\mathcal B}]_e$ such that $\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\Big)^{-1}\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$.

For every $n\in{\omega}$ put $V_n^{(0)}=U_n$ and using ${\mathrm{PTA}}$, for every $0<i\le n$ choose a neighborhood $V_n^{(i)}\in{\mathcal B}_n$ such that $V_n^{(i)}U_{n-i}\subset U_{n-i}V_n^{(i-1)}$. The so-defined sets satisfy the inclusions $$\label{PTA3a}
V_n^{(n-k)}U_k\subset U_kV_n^{(n-k-1)},\quad 0\le k<n.$$

We claim that $$\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n^{(n)}\Big)^{-1}\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n.$$This inclusion will follow as soon as we check that $$\label{PTA3b}
 \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}V_n^{(n)}\Big)^{-1}={\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\le m}V_n^{(n)}\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}U_n$$ for all $m\in{\omega}$. The left-hand equality follows from the symmetry of the neighborhoods $V_n^{(n)}\in{\mathcal B}_n$.

For the proof of the right-hand inclusion, for every $k\le m+1$ consider the subset $$\Pi_k={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}U_n\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k\le n\le m}V_n^{(n-k)}$$ of the group $G_m$, and observe that (\[PTA3b\]) is equivalent to the inclusion $\Pi_0\subset\Pi_{m+1}$. So it suffices to check that $\Pi_k\subset\Pi_{k+1}$ for every $k\le m+1$.

By induction on $k\le m+1$ we can show that (\[PTA3a\]) implies $$\Big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k)}\Big)\cdot U_k\subset U_k\cdot {\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k-1)}.$$

Now the inclusion $\Pi_k\subset\Pi_{k+1}$ can be seen as follows: $$\begin{aligned}
\Pi_k&={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}U_n\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k\le n\le m}V_n^{(n-k)}=
{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}U_n\cdot \Big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k)}\Big)\cdot V_k^{(0)}=\\
&={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}U_n\cdot \Big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k)}\Big)\cdot U_k\subset\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0\le n<k}U_n\Big)\cdot U_k\cdot {\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n\le m}V_n^{(n-k-1)}=\Pi_{k+1}.
\end{aligned}$$

Balances triples and towers of groups {#s1:SIN}
=====================================

In this section we introduce another condition guaranteeing that the topology of the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ of a tower of topological groups $(G_n)_{n\in{\omega}}$ coincides with the topologies $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$ and $\RL[\tau]$.

Let us observe that a tower of topological groups $(G_n)_{n\in{\omega}}$ satisfies ${\mathrm{PTA}}$ if each group $G_n$, $n\in{\omega}$, is [*balanced*]{}. The latter means that $G_n$ has a neighborhood base at $e$ consisting of $G$-invariant neighborhoods, see [@AT p.69].

We define a subset $U\subset G$ of a group $G$ to be [*$H$-invariant*]{} for a subgroup $H\subset G$ if $xUx^{-1}=U$ for all $x\in H$. Observe that for any subset $U\subset G$ the set $$\sqrt[H]{U}=\{x\in G{\, ;\;}x^H\subset U\}$$ is the largest $H$-invariant subset of $U$. Here $x^H=\{hxh^{-1}{\, ;\;}h\in H\}$ stands for the conjugacy class of a point $x\in G$.

Observe that a topological group $G$ is balanced if and only if for every neighborhood $U\subset G$ of $e$ the set $\sqrt[G]{U}$ is a neighborhood of $e$.

A triple $(G,\Gamma,H)$ of topological groups $H\subset \Gamma\subset G$ is called [*balanced*]{} if for any neighborhoods $V\subset \Gamma$ and $U\subset G$ of the neutral element $e$ of $G$ the product $V\cdot\sqrt[H]{U}$ is a neighborhood of $e$ in $G$.

A tower of topological groups $(G_n)_{n\in{\omega}}$ is called [*balanced*]{} if each triple $(G_{n+2},G_{n+1},G_n)$, $n\in{\omega}$, is balanced.

\[t4.2\] If a tower of topological groups $(G_n)_{n\in{\omega}}$ is balanced, then the semitopological group $\LA[G]$ is a topological group and hence all the conditions (1) through (7) of Theorem  hold. In particular the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with any of the topologies: $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$, $\RL[\tau]$.

In order to show that the semitopological group $\LA[G]$ is a topological group, it suffices to check the continuity of the multiplication and of the inversion at the neutral element $e$.

In order to check the continuity of multiplication at $e$, fix a neighborhood ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\in\LA[{\mathcal B}]_e$. For every $n\in{\omega}$, find a symmetric neighborhood $W_n$ of $e$ in the group $G_n$ such that $W_n\cdot W_n\subset U_n$ and let $$Z_n=\sqrt[G_{n-2}]{W_n}=\{x\in G_n{\, ;\;}x^{G_{n-2}}\subset W_n\}$$ be the largest $G_{n-2}$-invariant subset of $W_n$ (here we assume that $G_{k}=\{e\}$ for $k<0$).

Let $V_0=U_0\cap W_1$ and $V_1\subset G_1$ be a symmetric neighborhood of $e$ such that $V_1^2\subset W_1$. Next, for each $n\ge 2$ by induction choose a neighborhood $V_n\subset G_n$ so that

-   $V_{n}^2\subset V_{n-1}\cdot Z_{n}$, and

-   $V_{n}\subset W_{n+1}$,

The condition (a) can be satisfied because the triple $(G_{n},G_{n-1},G_{n-2})$ is balanced according to our hypothesis.

We claim $\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}} V_n\Big)^2\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$. This inclusion will follow as soon as we check that $$({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}V_n)\cdot ({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}V_n)\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m+1}U_n$$ for every $m\ge 2$.

For every $1\le k\le m$ consider the subset $$\Pi_k=({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m-k}V_n)\cdot V_{m-k+1}^2\cdot({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m-k}V_n)\cdot ({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{m-k< n<m} Z_{n+1}V_n)\cdot V_m.$$

We claim that $\Pi_k\subset\Pi_{k+1}$. Indeed, $$\begin{aligned}
\Pi_k&=\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m-k}V_n\Big)\cdot V_{m-k+1}^2\cdot\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m-k}V_n\Big)\cdot \Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{m-k< n<m} Z_{n+1}V_n\Big)\cdot V_m   \subset\\
&\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m-k}V_n\Big)\cdot V_{m-k}\cdot Z_{m-k+1}\cdot\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m-k}V_n\Big)\cdot V_{m-k}\cdot \Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{m-k< n<m} Z_{n+1}V_n\Big)\cdot V_m=\\
&\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m-k}V_n\Big)\cdot V^2_{m-k}\cdot\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m-k}V_n\Big)\cdot Z_{m-k+1}\cdot V_{m-k}\cdot\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{m-k<n<m}Z_{n+1}V_n\Big)\cdot V_m=\\
&\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m-k}V_n\Big)\cdot V^2_{m-k}\cdot\Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m-k}V_n\Big)\cdot \Big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{m-k\le n<m}Z_{n+1}V_n\Big)\cdot V_m=
\Pi_{k+1}.
\end{aligned}$$ Now we see that $$\begin{aligned}
\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{i\le m}V_i\big)^2&\subset({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{i\le m-1}V_i\big)\cdot V^2_m\cdot  \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{i\le m}V_i\big)=\Pi_1\subset\Pi_m=V_0V_1^2V_0\cdot\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0<n<m}Z_{n+1}V_n\big)V_m\subset\\
&U_0W_1W_1\cdot\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0<n<m}W_{n+1}W_{n+1}\big)W_{m+1}\subset U_0U_1\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{0<n<m}U_{n+1}\big)\cdot U_{m+1}={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m+1}U_n.
\end{aligned}$$

Now we check that the inversion is continuous at $e$ with respect to the topology $\LA[\tau]$. Given any basic set ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\in\LA[{\mathcal B}]_e$, we need to find a basic set ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\in\LA[{\mathcal B}]_e$ such that $\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)^{-1}\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n$.

For every $n\in{\omega}$ let $Z_{n+2}=\sqrt[G_{n}]{W_{n+2}}$ be the largest $G_n$-invariant subset of $W_{n+2}$. For each non-negative number $n<2$ pick a symmetric neighborhood $V_n\subset G_n$ such that $V_n^2\subset W_n$. For $n\ge 2$ by induction choose a symmetric neighborhood $V_n\subset G_n$ of $e$ such that $V_n^2\subset W_n\cap(V_{n-1}\cdot Z_n)$. Such a neighborhood $V_n$ exists by the balanced property of the triple $(G_n,G_{n-1},G_{n-2})$. Finally, for every $n\in{\omega}$ put $U_n=V_n\cap V_{n+1}$.

We claim that $\big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\big)^{-1}\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n$. This inclusion will follow as soon as we check that ${\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}W_n$ for every $m\in{\omega}$. By induction we shall prove a bit more: $$\label{eq:inv}
V_m\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}W_n$$ for every $m\in{\mathbb N}$.

For $m=1$ the inclusion (\[eq:inv\]) is true: $V_1U_0\subset V_1^2\subset W_1\subset W_0W_1$. Assume that the inclusion (\[eq:inv\]) has been proved for some $m=k\ge 1$. Then $$\begin{aligned}
V_{m+1}\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\le m}U_n&\subset V_{m+1}\cdot U_m\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\subset V_{m+1}^2\cdot{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\subset V_mZ_{m+1}{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n=\\&=V_m\cdot \big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_n\big)\cdot Z_{m+1}\subset \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}W_n\big)\cdot Z_{m+1}\subset {\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m+1}W_n,
\end{aligned}$$which means that the inclusion (\[eq:inv\]) holds for $m=k+1$.

Bi-balanced triples and towers of groups {#s2:SIN}
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In this section we introduce the bi-balanced property of a tower $(G_n)$, which is weaker than the balanced property and implies that the semitopological group $\RL[G]$ is a topological group.

A triple $(G,\Gamma,H)$ of topological groups $H\subset \Gamma\subset G$ is called [*bi-balanced*]{} if for any neighborhoods $V\subset \Gamma$ and $U\subset G$ of the neutral element $e$ of $G$ the product $\sqrt[H]{U}\cdot V\cdot\sqrt[H]{U}$ is a neighborhood of $e$ in $G$.

A tower of topological groups $(G_n)_{n\in{\omega}}$ is called [*bi-balanced*]{} if each triple $(G_{n+2},G_{n+1},G_n)$, $n\in{\omega}$, is bi-balanced.

\[t5.2\] If a tower of topological groups $(G_n)_{n\in{\omega}}$ is bi-balanced, then the identity map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism and hence the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with the topology $\RL[\tau]$.

By Theorem \[t2.2\], it suffices to show that $\RL[G]$ is a topological group. Since $\RL[G]$ is a quasitopological group, it suffices to check the continuity of multiplication at the neutral element. Given a basic neighborhood ${\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n\in\RL[{\mathcal B}]_e$, we should find a neighborhood ${\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\in\RL[{\mathcal B}]_e$ such that $\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\big)^2\subset {\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n$.

For every $n\in{\omega}$, find a symmetric neighborhood $U_n\subset G_n$ of $e$ such that $U_n^2\subset W_n$ and let $Z_{n}=\sqrt[G_{n-2}]{U_{n}}$ be the maximal $G_{n-2}$-invariant subset of $U_{n}$ (here we assume that $G_k=\{e\}$ for $k<0$). Let $\widetilde U_0=W_0$ and by induction for every $n\in{\mathbb N}$ choose a symmetric neighborhood $\widetilde U_n\subset U_n$ of $e$ such that $\widetilde U_n^3\subset Z_n\widetilde U_{n-1}Z_n$. The choice of the neighborhood $\widetilde U_n$ is possible because the set $Z_n\widetilde U_{n-1}Z_n$ is a neighborhood of $e$ in the group $G_n$ by the bi-balanced property of the triple $(G_n,G_{n-1},G_{n-2})$. Finally, for every $n\in{\omega}$ let $V_n=G_n\cap \widetilde U_{n+1}$.

We claim that ${\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n$ is the required neighborhood with $\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}V_n\big)^2\subset {\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}W_n$. This inclusion will follow as soon as we check that $$\label{SIN2b}
\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<m}V_n\big)^2\subset {\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\le m}W_n$$ for all $m\in{\omega}$.

For $m=0$ this inclusion is trivial. Assume that the inclusion (\[SIN2b\]) has been proved for some $m=p\in{\omega}$. We shall prove it for $m=p+1$. For every non-negative $k<m$ consider the subset $$\Pi_{k}=\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k\le n<m}V_nZ_{n+1}\big)\cdot\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot \widetilde U_{k}\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n< k}V_n\big)\cdot \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k\le n<m}Z_{n+1}V_n\big)$$ of the group $G_m$. The following chain of inclusions guarantees that $\Pi_{k+1}\subset\Pi_{k}$: $$\begin{aligned}
\Pi_{k+1}&=\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n<m}V_nZ_{n+1}\big)\cdot\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\le k}V_n\big)\cdot \widetilde U_{k+1}\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\le k}V_n\big)\cdot \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n<m}Z_{n+1}V_n\big)\subset\\
&\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n<m}V_nZ_{n+1}\big)\cdot V_k\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot V_k\cdot \widetilde U_{k+1}\cdot V_k\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot V_k\cdot  \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n<m}Z_{n+1}V_n\big)\subset\\
&\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n<m}V_nZ_{n+1}\big)\cdot V_k\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot \widetilde U_{k+1}^3\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot V_k\cdot  \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n<m}Z_{n+1}V_n\big)\subset\\
&\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n<m}V_nZ_{n+1}\big)\cdot V_k\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot Z_{k+1}\cdot\widetilde U_{k}\cdot Z_{k+1}\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot V_k\cdot  \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n<m}Z_{n+1}V_n\big)=\\
&\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k<n<m}V_nZ_{n+1}\big)\cdot V_k\cdot Z_{k+1}\cdot\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot\widetilde U_{k}\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot  Z_{k+1}\cdot V_k\cdot  \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k<n<m}Z_{n+1}V_n\big)=\\
&\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{k\le n<m}V_nZ_{n+1}\big)\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot\widetilde U_{k}\cdot \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<k}V_n\big)\cdot  \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{k\le n<m}Z_{n+1}V_n\big)=\Pi_{k}.
\end{aligned}$$ Now we see that $$\begin{aligned}
\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<m}V_n\big)^2&\subset \big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<m}V_n\big)\cdot\widetilde U_m\cdot\big({\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n<m}V_n\big)=\Pi_{m}\subset\Pi_0=
\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}V_nZ_{n+1}\big)\cdot\widetilde U_0\cdot \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m}Z_{n+1}V_n\big)\\
&\subset
\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}U_{n+1}^2\big)\cdot \widetilde U_0\cdot \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m}U_{n+1}^2\big)\subset
\big({\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n<m}W_{n+1}\big)\cdot W_0^2\cdot \big({\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n<m}W_{n+1}\big)={\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\le m}W_m.
\end{aligned}$$

The independence of ${\mathrm{PTA}}$ and the balanced property {#PTA-SIN}
==============================================================

Looking at Theorems \[t3.2\] and \[t4.2\] (which have the same conclusion) the reader can ask about the interplay between ${\mathrm{PTA}}$ and the balanced property. These two properties are independent.

First we present an example of a tower of topological groups $(G_n)_{n\in{\omega}}$ that is balanced but does not satisfy ${\mathrm{PTA}}$.

Let $G={\mathcal H}_c({\mathbb R})$ be the group of all homeomorphisms $h:{\mathbb R}\to{\mathbb R}$ having compact support ${\mathrm{supp}}(h)={\mathrm{cl}}_{\mathbb R}\{x\in{\mathbb R}{\, ;\;}h(x)\ne x\}.$

The homeomorphism group $G={\mathcal H}_c({\mathbb R})$ is endowed with the Whitney topology whose base at a homeomorphism $h\in {\mathcal H}_c({\mathbb R})$ consists of the sets $$B(h,{\varepsilon})=\{f\in {\mathcal H}_c({\mathbb R}){\, ;\;}|f-h|<{\varepsilon}\}$$where ${\varepsilon}:{\mathbb R}\to(0,1)$ runs over continuous 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 $B({\mathrm{id}}_{\mathbb R},{\varepsilon})=\{f\in{\mathcal H}_c({\mathbb R}){\, ;\;}\|f-{\mathrm{id}}\|<{\varepsilon}\}\subset G$ centered at the identity homeomorphism ${\mathrm{id}}_{\mathbb R}$ will be denoted by $B({\varepsilon})$.

1\. We need to show that for every $n\in{\omega}$ the triple $(G_{n+2},G_{n+1},G_n)$ is balanced. This will follow as soon as we check that for every neighborhood $U\subset G_{n+1}$ of the identity homeomorphism ${\mathrm{id}}_{\mathbb R}$ and any neighborhood $W\subset G_{n+2}$ of ${\mathrm{id}}_{\mathbb R}$ the set $U\cdot\sqrt[G_n]{W}$ is a neighborhood of ${\mathrm{id}}_{\mathbb R}$ in $G_{n+2}$. Since the Whitney topology on the subgroup $G_{n+2}={\mathcal H}_+({\mathbb I}_{n+2})$ is generated by the sup-metric, the neighborhood $W\subset G_{n+2}$ contains the ${\varepsilon}$-ball $G_{n+2}\cap B({\varepsilon})$ for some positive constant ${\varepsilon}<1$. The constant ${\varepsilon}$ can be chosen so small that $G_{n+1}\cap B({\varepsilon})\subset U$.

Consider the closed subgroup $$H=\{h\in G_{n+2}{\, ;\;}{\mathrm{supp}}(h)\subset{\mathbb I}_{n+2}\setminus{\mathbb I}_{n}\}$$ of $G_{n+2}$ and observe that $W\cap H\subset \sqrt[G_n]{W}$. Now it suffices to check that $U\cdot (W\cap H)$ contains the ball $G_{n+2}\cap B({\varepsilon}/2)$. Take any homeomorphism $h\in G_{n+2}\cap B({\varepsilon}/2)$ and observe that $h$ maps the interval ${\mathbb I}_n=[-n,n]$ into the interval $[-n-{\varepsilon}/2,n+{\varepsilon}/2]$. So, we can consider the homeomorphism $g\in G_{n+1}$, which is equal to $h$ on the interval ${\mathbb I}_n$ and is linear on the intervals $[n,n+1]$ and $[-n-1,-n]$. It is clear that $\|g-{\mathrm{id}}\|\le\|h-{\mathrm{id}}\|<{\varepsilon}/2$ and $\|g^{-1}-{\mathrm{id}}\|=\|g-{\mathrm{id}}\|$. Let $f=h\circ g^{-1}\in G_{n+2}$. The equality $g|{\mathbb I}_n=h|{\mathbb I}_n$ implies $f|{\mathbb I}_n={\mathrm{id}}|{\mathbb I}_n$ and thus $f\in H$. It follows that $\|f-{\mathrm{id}}\|=\|h\circ g^{-1}-{\mathrm{id}}\|\le\|h\circ g^{-1}-g^{-1}\|+\|g^{-1}-{\mathrm{id}}\|<{\varepsilon}/2+{\varepsilon}/2={\varepsilon}$. Now we see that the elements $g\in G_{n+1}\cap B({\varepsilon}/2)\subset U$ and $f=g^{-1}\circ h\in H\cap B({\varepsilon})\subset \sqrt[G_n]{W}$ yield $h=g\circ f\subset U\cdot\sqrt[G_n]{W}$, which establishes the required inclusion $G_{n+2}\cap B({\varepsilon}/2)\subset U\cdot\sqrt[G_n]{W}$.

2\. Assuming that the tower $(G_n)_{n\in{\omega}}$ satisfies ${\mathrm{PTA}}$, we can find a neighborhood $U\subset G_1$ of ${\mathrm{id}}_{\mathbb R}$ such that for every neighborhood $V\subset G_2$ there is a neighborhood $W\subset G_2$ such that $WU\subset UV$.

Find ${\varepsilon}\in(0,1)$ such that $U\supset G_1\cap B({\varepsilon})$. Then for the neighborhood $V=G_2\cap B({\varepsilon})$ there is a neighborhood $W\subset G_2$ of ${\mathrm{id}}_{\mathbb R}$ with $WU\subset UV$. Find a positive constant $\delta<{\varepsilon}$ with $G_2\cap B(\delta)\subset W$. It follows that $(G_2\cap B(\delta))\cdot (G_1\cap B({\varepsilon}))\subset WU\subset UV\subset G_1\cdot (G_2\cap B({\varepsilon}))$ and after inversion, $(G_1\cap B({\varepsilon}))\cdot(G_2\cap B(\delta))\subset (G_2\cap B({\varepsilon}))\cdot G_1$. Take a homeomorphisms $f\in G_2\cap B(\delta)$ such that $f(1)=1-\delta/2$ and a homeomorphism $g\in G_1\cap B({\varepsilon})$ such that $g(1-\delta/2)=1-{\varepsilon}$. Then $g\circ f(1)=1-{\varepsilon}$ which is not possible as $g\circ f\in (G_2\cap B({\varepsilon}))\cdot G_1\subset\{h\in G_2:|h(1)-1|<{\varepsilon}\}$.

3\. Since the tower $(G_n)_{n\in{\omega}}$ is balanced, the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ coincides with the topologies $\LA[\tau]$, $\RA[\tau]$, $\LR[\tau]$, and $\RL[\tau]$ according to Theorem \[t4.2\]. Since ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ carries the strongest group topology inducing the original topology on each group $G_n$, we conclude that the Whitney topology is weaker that the topology $\LA[\tau]$. In order to show that these two topologies coincide, it suffices to check that each basic neighborhood ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$ of $e$ in the topology $\LA[\tau]$ is a neighborhood of ${\mathrm{id}}_{\mathbb R}$ in the Whitney topology. Here for every $n\in{\omega}$, $U_n$ is an open symmetric neighborhood of ${\mathrm{id}}_{\mathbb R}$ in the group $G_n={\mathcal H}_+({\mathbb I}_n)$. Since the Whitney topology on the subgroup $G_n$ is generated by the sup-metric, we can find a positive constant ${\varepsilon}_n<1/2$ such that $U_n\supset G_n\cap B({\varepsilon}_n)$.

Choose a continuous function ${\varepsilon}:{\mathbb R}\to(0,\tfrac12)$ such that $$\label{E1}
\sup\{{\varepsilon}(x){\, ;\;}x\in{\mathbb I}_{n}\setminus {\mathbb I}_{n-4}\}<{\varepsilon}_n/2\mbox{ \ for all $n\in{\omega}$}.$$ Here we assume that ${\mathbb I}_k=\emptyset$ for all negative $k$.

The function ${\varepsilon}$ determines the neighborhood $B({\varepsilon})=\{h\in G{\, ;\;}|h-{\mathrm{id}}_{\mathbb R}|<{\varepsilon}\}$ of the identity map ${\mathrm{id}}_{\mathbb R}$ in the Whitney topology.

We claim that $B({\varepsilon})\subset{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$. Fix any homeomorphism $h\in B({\varepsilon})$ and for every $n\in{\mathbb N}$ consider the homeomorphism $h_n\in G_{n}$ such that $h_n|{\mathbb I}_{n-1}=h|{\mathbb I}_{n-1}$ and $h_n$ is linear on the intervals $[n,n+1]$ and $[-n-1,-n]$. For $n\le 0$ we put $h_n={\mathrm{id}}_{\mathbb R}$. It is clear that $h_{m}=h$ for some $m\in{\mathbb N}$.

For every $n\in{\omega}$ consider the homeomorphism $g_{n}=h_{n-1}^{-1}\circ h_{n}\in G_n$. Then $h=h_{m}={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}g_n$. It remains to prove that each homeomorphism $g_n$ belongs to the neighborhood $U_n$. This will follow as soon as we check that $|g_n(x)-x|<{\varepsilon}_n$ for any $x\ne g_n(x)$.

Since $h_{n-1}|{\mathbb I}_{n-2}=h_{n}|{\mathbb I}_{n-2}=h|{\mathbb I}_{n-2}$, we conclude that $x\in {\mathbb I}_{n}\setminus{\mathbb I}_{n-2}$. It follows from $h_{n}\in G_n\cap B(1/2)$ that the point $y=h_{n}(x)$ belongs to the set ${\mathbb I}_{n}\setminus{\mathbb I}_{n-3}$. Since $h_{n-1}\in G_{n-1}\cap B(1/2)$, the point $z=h_{n-1}^{-1}(y)$ belongs to ${\mathbb I}_{n}\setminus{\mathbb I}_{n-4}$.

We claim that $$\label{H1}
|h_{n-1}(z)-z|<{\varepsilon}_{n}/2.$$ If $z\in {\mathbb I}_{n-2}\setminus{\mathbb I}_{n-4}$, then $|h_{n-1}(z)-z|=|h(z)-z|<{\varepsilon}(z)\le {\varepsilon}_{n}/2$ by the condition (\[E1\]) from the definition of the function ${\varepsilon}$. If $z\in {\mathbb I}_{n-1}\setminus{\mathbb I}_{n-2}$, then the linearity of $h_{n-1}$ on the two intervals composing the set ${\mathbb I}_{n-1}\setminus{\mathbb I}_{n-2}$ implies that $$|h_{n-1}(z)-z|\le\max_{t\in\partial{\mathbb I}_{n-2}}|h(t)-t|<\max_{t\in\partial{\mathbb I}_{n-2}}{\varepsilon}(t)\le
{\varepsilon}_{n}/2$$by the definition of the function ${\varepsilon}$. Here $\partial{\mathbb I}_{k}=\{k,-k\}$ stands for the boundary of the interval ${\mathbb I}_{k}=[-k,k]$ in ${\mathbb R}$.

By a similar argument we can prove the inequality $$\label{H2}
|h_{n}(x)-x|<{\varepsilon}_{n}/2.$$

Unifying (\[H1\]) and (\[H2\]) we obtain the desired inequality: $$\begin{aligned}
|g_{n}(x)-x|&=|h_{n-1}^{-1}\circ h_{n}(x)-x|\le |h_{n-1}^{-1}\circ h_{n}(x)-h_{n}(x)|+|h_{n}(x)-x|=\\
&=|z-h_{n-1}(z)|+|h_{n}(x)-x|<\frac12{\varepsilon}_{n}+\frac12{\varepsilon}_n={\varepsilon}_n.
\end{aligned}$$

Next, we present an example of a tower $(G_n)_{n\in{\omega}}$ that satisfies ${\mathrm{PTA}}$ but is not (bi-) balanced.

Let $(e_n)_{n\in{\omega}}$ be an orthonormal basis of the separable Hilbert space $l_2$ and $\mathcal B(l_2)$ be the Banach algebra of bounded linear operators on $l_2$. For every $n\in{\mathbb N}$ let $G_n$ be the subgroup of $\mathcal B(l_2)$ consisting of invertible linear operators $T:l_2\to l_2$ such that

-   $Te_0\in (0,+\infty)\cdot e_0$;

-   $Te_i\in e_i+{\mathbb R}\cdot e_0$ for all $1\le i\le n$;

-   $Te_i=e_i$ for all $i>n$.

The tower $(G_n)_{n\in{\mathbb N}}$ satisfies ${\mathrm{PTA}}$ because each group $G_n$ is locally compact, see [@TSH], [@HSTH]. On the other hand, for every $n\in{\mathbb N}$ the triple $(G_n,G_{n+1},G_{n+2})$ is not bi-balanced. The reason is that for the neighborhood $W=\{T\in G_{n+2}:Te_{n+2}\in e_{n+2}+(-1,1)e_0\}$ of the identity ${\mathrm{id}}$ in $G_{n+2}$ the set $Z=\sqrt[G_n]{W}$ lies in the subgroup $G_{n+1}$. Then for each neighborhood $V\subset G_{n+1}$, the product $ZVZ\subset G_{n+1}$ fails to be a neighborhood of ${\mathrm{id}}$ in the group $G_{n+2}$.

It is clear that each balanced triple of groups is bi-balanced. The converse implication is not true.

In the group $G=GL(3,{\mathbb R})$ of non-degenerated $3\times 3$-matrices consider the subgroups $$\Gamma=\left\{\left(\begin{array}{ccc}
a_{11}&a_{12}&0\\
a_{21}&a_{22}&0\\
0&0&a_{33}\end{array}\right)\in G\right\}\mbox{ \ \ and \ \ }
H=\left\{\left(\begin{array}{ccc}
a_{11}&0&0\\
0&1&0\\
0&0&1\end{array}\right)\in G\right\}.$$ It is easy to check that the triple $(G,\Gamma,H)$ is bi-balanced but not balanced.

Direct limits in the category of uniform spaces {#s:ulim}
===============================================

In this section we shall discuss the notion of the direct limit in the category of uniform spaces and their uniformly continuous maps. In Section \[s:ulim-top\] we shall apply those results to show that for a tower $(G_n)_{n\in{\omega}}$ of topological groups the topologies $\LA[\tau]$ and $\RA[\tau]$ on the union $G=\bigcup_{n\in{\omega}}G_n$ are generated by uniformities of direct limits of the groups $G_n$ endowed with the left and right uniformities.

Fundamenta of the theory of uniform spaces can be found in [@En Ch.8]. Uniformities on groups are thoroughly discussed in [@RD] and [@AT §1.8]. In the sequel, for a uniform space $X$ by ${\mathcal U}_X$ we shall denote the uniformity of $X$.

Let $$X_0\to X_1\to X_2\to\cdots$$be a sequence of uniform spaces and their injective uniformly continuous maps. We shall identify each space $X_n$ with a subset of the uniform space $X_{n+1}$, carrying its own uniformity, which is stronger than that inherited from $X_{n+1}$. By the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}X_n$ of the sequence of uniform spaces $(X_n)_{n\in{\omega}}$ we understand the union $X=\bigcup_{n\in{\omega}}X_n$ endowed with the strongest (not necessarily separated) uniformity turning the identity inclusions $X_n\to X$, $n\in{\omega}$, into uniformly continuous maps.

A sequence $$X_0\to X_1\to X_2\to\cdots$$ of uniform spaces is called a [*tower*]{} of uniform spaces if each uniform space $X_n$ is a subspace of the uniform space $X_{n+1}$, so the identity inclusion $X_n\to X_{n+1}$ is a uniform embedding.

The uniformity of the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}X_n$ of a tower $(X_n)_{n\in{\omega}}$ of uniform spaces was described in [@BaR] with help of uniform pseudometrics.

Let us recall that a pseudometric on a uniform space $Y$ is [*uniform*]{} if for every ${\varepsilon}>0$ the set $$\{d<{\varepsilon}\}:=\{(x,y)\in Y{\, ;\;}d(x,y)<{\varepsilon}\}$$ belongs to the uniformity ${\mathcal U}_Y$ of $Y$. By [@En 8.1.10], the uniformity ${\mathcal U}_Y$ of a uniform space $Y$ is generated by the family ${\mathcal{P\!M}}_Y$ of all uniform pseudometrics on $Y$ in the sense that the sets $\{d<1\}$, $d\in{\mathcal{P\!M}}_Y$, form a base of the uniformity ${\mathcal U}_Y$.

Let $(X_n)_{n\in{\omega}}$ be a tower of uniform spaces. A sequence of pseudometric $(d_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}$ is called [*monotone*]{} if $d_n\le d_{n+1}|X_n^2$ for every $n\in{\omega}$. Let $${\operatornamewithlimits{\bigwedge}}_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}=\{(d_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}{\, ;\;}(d_n)_{n\in{\omega}}\mbox{ is monotone}\}$$be the subspace of Cartesian product, consisting of monotone sequences of uniform pseudometrics on the uniform spaces $X_n$.

A family ${\mathcal A}\subset{\operatornamewithlimits{\bigwedge}}\limits_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}$ is defined to be [*adequate*]{} if for each sequence of entourages $(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal U}_{X_n}$ there is a monotone sequence of uniform pseudometrics $(d_n)_{n\in{\omega}}\in{\mathcal A}$ such that $\{d_n<1\}\subset U_n$ for all $n\in{\omega}$.

The following proposition proved in [@BaR] shows that adequate families exist.

For any tower $(X_n)_{n\in{\omega}}$ of uniform spaces the family ${\mathcal A}={\operatornamewithlimits{\bigwedge}}\limits_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}$ is adequate.

For a point $x\in X=\bigcup_{n\in{\omega}}X_n$ let $|x|=\min\{n\in{\omega}{\, ;\;}x\in X_n\}$ be the [*height*]{} of $x$ in $X$. For two points $x,y\in X$ put $|x,y|=\max\{|x|,|y|\}$. Now we define a limit operator ${\varinjlim}$ assigning to each sequence of pseudometrics $(d_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}$ the pseudometric $d_\infty={\varinjlim}d_n$ on $X$ defined by the formula $$d_\infty(x,y)=\inf\Big\{\sum_{i=1}^md_{|x_{i-1},x_i|}(x_{i-1},x_i){\, ;\;}x=x_0,x_1,\dots,x_n=y\Big\}.$$ In fact, the pseudometric ${\varinjlim}d_n$ is well-defined for any functions $d_n:X_n\times X_n\to [0,\infty)$, $n\in{\omega}$, such that $d_n(x,x)=0$ and $d_n(x,y)=d_n(y,x)$ for all $x,y\in X_n$.

The following theorem proved in [@BaR] describes the uniformity of uniform direct limits.

\[t7.2\] For a tower of uniform spaces $(X_n)_{n\in{\omega}}$ and an adequate family ${\mathcal A}\subset{\operatornamewithlimits{\bigwedge}}\limits_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}$ the uniformity of the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}X_n$ is generated by the family of pseudometrics $\{{\varinjlim}d_n{\, ;\;}(d_n)_{n\in{\omega}}\in{\mathcal A}\}.$

Theorem \[t7.2\] implies a simple description of the topology of the uniform limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}X_n$ also given in [@BaR]. Given two subsets $U,V\subset X^2$ of the square of $X$, consider their composition (as relations): $$A\circ B=\{(x,z)\in X^2{\, ;\;}\mbox{ there is $y\in X$ such that $(x,y)\in A$ and $(y,z)\in B$ }\}.$$ This operation can be extended to finite and infinite sequences of subsets $(A_n)_{n\in{\omega}}$ of $X^2$ by the formula $$\sum_{n\ge k}A_n=\bigcup_{n\ge k}A_k\circ A_{k+1}\cdots \circ A_n.$$

For a point $x$ of a set $X$ and a subset $U\subset X^2$ let $B(x,U)=\{y\in X{\, ;\;}(x,y)\in U\}$ be the $U$-ball centered at $x$. We recall that for a point $x$ of the union $X=\bigcup_{n\in{\omega}}X_n$ of a tower $(X_n)$ by $|x|=\min\{n\in{\omega}{\, ;\;}x\in X_n\}$ we denote the height of $x$ in $X$.

\[t7.3\] The topology of the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}X_n$ of a tower of uniform spaces $(X_n)$ is generated by the base $$\mathcal B=\Big\{B\big(x; \sum_{n\ge|x|}U_n\big){\, ;\;}x\in X,\;\;(U_n)_{n\ge|x|}\in\prod\limits_{n\ge|x|}{\mathcal U}_{X_n}\Big\}.$$

Uniformities on groups {#s:ug}
======================

In this section we discuss some natural uniformities on topological groups. For more information on this subject, see [@RD] and [@AT §1.8].

Let us recall that each topological group $G$ carries four natural uniformities:

-   the [*left uniformity*]{} ${\mathcal U}^{{\mathrm{L}}}$, generated by the entourages $U^{{\mathrm{L}}}=\{(x,y)\in G{\, ;\;}x\in yU\}$ where $U\in{\mathcal B}_e$;

-   the [*right uniformity*]{} ${\mathcal U}^{{\mathrm{R}}}$, generated by the entourages $U^{{\mathrm{R}}}=\{(x,y)\in G{\, ;\;}x\in Uy\}$ where $U\in{\mathcal B}_e$;

-   the [*two-sided uniformity*]{} ${\mathcal U}^{{\mathrm{LR}}}$, generated by the entourages $U^{{\mathrm{LR}}}=\{(x,y)\in G{{\, ;\;}}\allowbreak x\in yU\cap Uy\}$ with $U\in{\mathcal B}_e$;

-   the [*Roelcke uniformity*]{} ${\mathcal U}^{{\mathrm{RL}}}$, generated by the entourages $U^{{\mathrm{RL}}}=\{(x,y)\in G{\, ;\;}\allowbreak x\in UyU\}$ with $U\in{\mathcal B}_e$.

Here ${\mathcal B}_e$ stands for the family of open symmetric neighborhoods $U=U^{-1}\subset G$ of the neutral element $e$ in the topological group $G$.

The group $G$ endowed with the uniformity ${\mathcal U}^{{\mathrm{L}}}$, ${\mathcal U}^{{\mathrm{R}}}$, ${\mathcal U}^{{\mathrm{LR}}}$ or ${\mathcal U}^{{\mathrm{RL}}}$ will be denoted by $G^{{\mathrm{L}}}$, $G^{{\mathrm{R}}}$, $G^{{\mathrm{LR}}}$ or $G^{{\mathrm{RL}}}$, respectively. It follows from the definition of those uniformities that the identity maps in the following diagram are uniformly continuous:

(200,90)(-130,-10) (5,30)[$G^{{\mathrm{LR}}}$]{} (30,28)[(3,-2)[30]{}]{} (30,38)[(3,2)[30]{}]{} (65,0)[$G^{{\mathrm{L}}}$]{} (85,6)[(3,2)[30]{}]{} (65,60)[$G^{{\mathrm{R}}}$]{} (85,58)[(3,-2)[30]{}]{} (120,30)[$G^{{\mathrm{RL}}}$]{}

Any isomorphic topological embedding $H\hookrightarrow G$ of topological groups induces uniform embeddings $$H^{{\mathrm{L}}}\hookrightarrow G^{{\mathrm{L}}},\;\;H^{{\mathrm{R}}}\hookrightarrow G^{{\mathrm{R}}}, \;\; H^{{\mathrm{LR}}}\hookrightarrow G^{{\mathrm{LR}}}$$ of the corresponding uniform spaces, see Proposition 1.8.4 of [@AT]. For the Roelcke uniformity the induced map $H^{{\mathrm{RL}}}\to G^{{\mathrm{RL}}}$ is merely uniformly continuous, but is not necessarily a uniform embedding, see [@Us].

Let us observe that $G^{{\mathrm{LR}}}$, $G^{{\mathrm{L}}}$, $G^{{\mathrm{R}}}$, $G^{{\mathrm{RL}}}$ are groups endowed with uniformities which are tightly connected with their algebraic structure.

By analogy with semitopological and quasitopological groups, let us define a group $G$ endowed with a uniformity to be a

-   [*semiuniform group*]{} if left and right shifts on $G$ are uniformly continuous;

-   [*quasiuniform group*]{} if $G$ is a semiuniform group with uniformly continuous inversion;

-   [*uniform group*]{} if $G$ is a quasiuniform group with uniformly continuous multiplication $G\times G\to G$, $(x,y)\mapsto xy$.

The groups $G^{{\mathrm{L}}}$, $G^{{\mathrm{R}}}$, $G^{{\mathrm{LR}}}$, $G^{{\mathrm{RL}}}$ are basic examples of groups endowed with a uniformity. Some elementary properties of those groups are presented in the following two propositions whose proof is left to the interested reader (cf. Corollary 1.8.16 [@AT]).

For any topological group $G$

1.  $G^{{\mathrm{L}}}$ and $G^{{\mathrm{R}}}$ are semiuniform topological groups

2.  $G^{{\mathrm{LR}}}$ and $G^{{\mathrm{RL}}}$ are quasiuniform topological groups.

For a topological group $G$ the following conditions are equivalent:

1.  $G^{{\mathrm{L}}}$ is a quasiuniform group

2.  $G^{{\mathrm{L}}}$ is a uniform group

3.  $G^{{\mathrm{R}}}$ is a quasiuniform group

4.  $G^{{\mathrm{R}}}$ is a uniform group

5.  $G^{{\mathrm{LR}}}$ is a uniform group

6.  $G^{{\mathrm{RL}}}$ is a uniform group

7.  the left and right uniformities on $G$ coincide

8.  the group $G$ is balanced.

The uniform structure of the semitopological groups $\LA[G]$, $\RA[G]$, and $\LR[G]$ {#s:ulim-top}
====================================================================================

Each tower of topological groups $(G_n)_{n\in{\omega}}$ induces four ascending sequences of uniform spaces $(G^{{\mathrm{L}}}_n)_{n\in{\omega}}$, $(G_n^{{\mathrm{R}}})_{n\in{\omega}}$, $(G_n^{{\mathrm{LR}}})_{n\in{\omega}}$, $(G_n^{{\mathrm{RL}}})_{n\in{\omega}}$. The direct limits of these sequences in the category of uniform spaces are denoted by $${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}},\quad{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{R}}},\quad{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}},\mbox{ \ and \ }{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}},$$respectively.

These uniform spaces endowed with the group operation inherited from $G=\bigcup_{n\in{\omega}}G_n$ are semiuniform groups. The uniform continuity of the left and right shifts follows from the uniform continuity of the left and right shifts on the semiuniform groups $G_n^{{\mathrm{L}}}$, $G_n^{{\mathrm{R}}}$, $G_n^{{\mathrm{LR}}}$, $G_n^{{\mathrm{RL}}}$, $n\in{\omega}$. Moreover, the semiuniform groups ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}$ and ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ are quasiuniform because so are the groups $G_n^{{\mathrm{LR}}}$ and $G_n^{{\mathrm{RL}}}$, $n\in{\omega}$.

The uniform continuity of the identity maps

(200,80)(-100,-2) (5,30)[$G_n^{{\mathrm{LR}}}$]{} (30,28)[(3,-2)[30]{}]{} (30,38)[(3,2)[30]{}]{} (65,0)[$G_n^{{\mathrm{L}}}$]{} (85,6)[(3,2)[30]{}]{} (65,60)[$G_n^{{\mathrm{R}}}$]{} (85,58)[(3,-2)[30]{}]{} (120,30)[$G_n^{{\mathrm{RL}}}$]{} (145,32)[(1,0)[30]{}]{} (185,30)[$({\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n)^{{\mathrm{RL}}}$]{}

for all $n\in{\omega}$ implies the uniform continuity of the identity maps:

(200,90)(-100,-10) (-10,30)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}$]{} (25,23)[(2,-1)[25]{}]{} (25,42)[(2,1)[25]{}]{} (50,0)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$]{} (80,12)[(2,1)[25]{}]{} (50,60)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{R}}}$]{} (80,53)[(2,-1)[25]{}]{} (100,30)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$]{} (150,32)[(1,0)[30]{}]{} (190,30)[$({\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n)^{{\mathrm{RL}}}$]{}

Theorems \[t7.2\] and \[t7.3\] imply the following description of the uniform and topological structure of the uniform limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$.

For a tower of topological groups $(G_n)_{n\in{\omega}}$

1.  the topology of the semiuniform group ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$ coincides with the topology $\LA[\tau]$ on the group $G=\bigcup_{n\in{\omega}}G_n$

2.  the uniformity of ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$ is generated by the family of pseudometrics $\{{\varinjlim}d_n{\, ;\;}(d_n)_{n\in{\omega}}\in{\mathcal A}\}$ for any adequate family ${\mathcal A}\subset{\operatornamewithlimits{\bigwedge}}\limits_{n\in{\omega}}{\mathcal{P\!M}}_{G_n}$.

This theorem allows us to identify the semitopological group $\LA[G]$ with the semiuniform group ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$. In the same way we shall identify the semitopological group $\RA[G]$ with the semiuniform group ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{R}}}$.

The semitopological group $\LR[G]$ is a quasiuniform group with respect to the uniformity inherited from the product $\RA[G]\times\LA[G]$ by the diagonal embedding $$\LR[G]\hookrightarrow \RA[G]\times\LA[G],\quad x\mapsto (x,x).$$

The uniform continuity of the identity maps $${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}=\LA[G]\mbox{ \  and \ }
{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{R}}}=\RA[G]$$ yields the uniform continuity of the identity map ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}\to\LR[G]$.

Now we discuss the interplay between the semitopological group $\RL[G]$ and the semiuniform group ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$. Since the topological embedding $G_n^{{\mathrm{RL}}}\to G_{n+1}^{{\mathrm{RL}}}$ in general is not a uniform embedding, Theorems \[t7.2\] and \[t7.3\] cannot be applied to describing the uniform and topological structures of the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$. So, this case requires a special treatment.

Given a pseudometric $d$ on a group $H$, let $d^{-1}$ be the [*mirror pseudometric*]{} defined by $$d^{-1}(x,y)=d(x^{-1},y^{-1})\mbox{ \ for $x,y\in H$}.$$

For a tower of topological groups $(G_n)_{n\in{\omega}}$,

1.  the uniformity of the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is generated by the family of pseudometrics $\big\{{\varinjlim}\min\{d_n,d^{-1}_n\}{\, ;\;}(d_n)_{n\in{\omega}}\in{\operatornamewithlimits{\bigwedge}}_{n\in{\omega}}{\mathcal{P\!M}}_{G_n^{{\mathrm{L}}}}\big\};$

2.  the uniformity of ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ coincides with the strongest uniformity on the group $G=\bigcup_{n\in{\omega}}G_n$ such that the identity maps $\LA[G]\to G$ and $\RA[G]\to G$ are uniformly continuous

3.  the identity map $\RL[G]\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is continuous

4.  the identity map $\RL[G]\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is a homeomorphism if $\RL[G]$ is a topological group or if each identity inclusion $G_n^{{\mathrm{RL}}}\to G_{n+1}^{{\mathrm{RL}}}$, $n\in{\omega}$, is a uniform embedding

5.  ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is a topological group if and only if the identity map ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism.

1\. First we show that for any monotone sequence of pseudometrics $(d_n)_{n\in{\omega}}\in{\operatornamewithlimits{\bigwedge}}\limits_{n\in{\omega}}{\mathcal{P\!M}}_{G_n^{{\mathrm{L}}}}$ the pseudometric $d_\infty={\varinjlim}\min\{d_n,d_n^{-1}\}$ is uniform on $G={\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$. For this it suffices to check that $d_\infty$ is uniform on each quasiuniform group $G_n^{{\mathrm{RL}}}$, $n\in{\omega}$.

Fix any ${\varepsilon}>0$. Since each pseudometric $d_n\in{\mathcal{P\!M}}_{G_n^{{\mathrm{L}}}}$ is uniform with respect to the left uniformity on the topological group $G_n$, there is an open symmetric neighborhood $U_n\subset G_n$ of $e$ such that $U_n^{{\mathrm{L}}}\subset\{d_n<{\varepsilon}/2\}$. After inversion, we get the inclusion $U_n^{{\mathrm{R}}}\subset\{d_n^{-1}<{\varepsilon}/2\}$. We claim that $U_n^{{\mathrm{RL}}}\subset\{d_\infty|G_n^2<{\varepsilon}\}$. Take any points $(x,y)\in U_n^{{\mathrm{RL}}}$. Then $y=uxv$ for some $u,v\in U_n$. Consider the chain of points $x_0=x$, $x_1=ux$, $x_2=uxv=y$ and observe that $$\begin{aligned}
d_\infty(x,y)&\le \min\{d_{|x_0,x_1|}(x_0,x_1),d^{-1}_{|x_0,x_1|}(x_0,x_1)\}+
\min\{d_{|x_1,x_2|}(x_1,x_2),d^{-1}_{|x_1,x_2|}(x_1,x_2)\}\\
&\le \min\{d_n(x,ux),d^{-1}_n(x,ux)\}+\min\{d_n(ux,uxv),d^{-1}_n(ux,uxv)\}\\   &\le d_n(x^{-1},x^{-1}u^{-1})+d_n(ux,uxv)<\frac{\varepsilon}2+\frac{\varepsilon}2={\varepsilon},
\end{aligned}$$ witnessing that the pseudometric $d_\infty$ is uniform.

Now given any entourage $U\subset G^2$ that belongs to the uniformity of the space ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$, we shall find a monotone sequence $(d_n)_{n\in{\omega}}\in{\operatornamewithlimits{\bigwedge}}_{n\in{\omega}}{\mathcal{P\!M}}_{G_n^{{\mathrm{L}}}}$ such that $\{d_\infty<1\}\subset U$ for the limit pseudometric $d_\infty={\varinjlim}\min\{d_n,d_n^{-1}\}$.

By [@En 8.1.10], there is a uniform pseudometric $d$ on ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ such that $\{(d<1\}\subset U$. Since the inversion is uniformly continuous on ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$, the pseudometric $\rho=\max\{d,d^{-1}\}$ on ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is uniform. Now observe that for every $n\in{\omega}$ the pseudometric $d_n=\rho|(G_n^{{\mathrm{RL}}})^2$ on $G_n^{{\mathrm{RL}}}$ is uniform and the sequence $(d_n)_{n\in{\omega}}$ is monotone. The triangle inequality and the definition of the pseudometric $d_\infty={\varinjlim}\min\{d_n,d_n^{-1}\}={\varinjlim}d_n$ implies that $d_\infty=\rho$. In this case $\{d_\infty<1\}=\{\rho<1\}\subset\{d<1\}\subset U.$

2\. Let ${\mathcal U}$ be the largest uniformity on $G$ such that the identity maps $\LA[G]\to(G,{\mathcal U})$ and $\RA[G]\to(G,{\mathcal U})$ are uniformly continuous. The definition of ${\mathcal U}$ implies that ${\mathcal U}$ coincides with its mirror uniformity ${\mathcal U}^-$ consisting of the sets $U^-=\{(x^{-1},y^{-1}){\, ;\;}(x,y)\in U\}$, $U\in{\mathcal U}$.

We need to show that ${\mathcal U}$ coincides with the uniformity of ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$. The uniform continuity of the identity maps from $\LA[G]$ and $\RA[G]$ into ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ implies that ${\mathcal U}$ is larger then the uniformity of ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$. It remains to prove that each entourage $U\in{\mathcal U}$ belongs to the uniformity of ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$. By [@En 8.1.10], there is a uniform pseudometric $d$ on $(G,{\mathcal U})$ such that $\{d<1\}\subset U$. Since the inversion on $G$ is uniformly continuous with respect to the uniformity ${\mathcal U}$, the mirror pseudometric $d^{-1}$ is uniform on $(X,{\mathcal U})$ and so is the pseudometric $\rho=\max\{d,d^{-1}\}$. Now we see that for each $n\in{\omega}$ the restriction $d_n=\rho|G^2_n$ belongs to the family ${\mathcal{P\!M}}_{G_n^{{\mathrm{L}}}}$ and is equal to its mirror pseudometric $d_n^{-1}$. The sequence $(d_n)_{n\in{\omega}}$ belongs to ${\operatornamewithlimits{\bigwedge}}\limits_{n\in{\omega}}{\mathcal{P\!M}}_{G_n^{{\mathrm{L}}}}$, and the definition of the pseudometric $d_\infty$ implies that $d_\infty=\rho$. By the first item, the pseudometric $d_\infty$ is uniform on ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}$ and consequently, the entourage $$U\supset\{d<1\}\supset\{\rho<1\}=\{d_\infty<1\}$$belongs to the uniformity of the space ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$.

3\. Since $\RL[G]$ and ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ are semitopological groups, the continuity of the identity map $\RL[G]\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is equivalent to its continuity at the neutral element $e$.

Given a neighborhood $O^{{\mathrm{RL}}}(e)\subset{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ of $e$, find a uniform pseudometric $d$ on ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ such that $\{x\in G{\, ;\;}d(x,e)<1\}\subset O^{{\mathrm{RL}}}(e)$. For every $n\in{\omega}$ the identity map $G_n^{{\mathrm{RL}}}\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is uniformly continuous, so we can find a symmetric neighborhood $U_n\subset G_n$ of $e$ such that $U_n^{{\mathrm{RL}}}\subset \big\{d|G_n^2<1/{2^{n+1}}\big\}$. By the definition of the topology $\RL[\tau]$ of the quasitopological group $\RL[G]$, the set ${\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$ is a neighborhood of $e$ in $\RL[G]$. We claim that ${\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n\subset O^{{\mathrm{RL}}}(e)$. Given any point $z\in {\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$, find two points $x\in{\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$ and $y\in{\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$ with $z=xy$.

By the definition of the directed products ${\operatornamewithlimits{\overleftarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$ and ${\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$, there are chains of points $e=x_0,x_1,\dots, x_m=x$ and $e=y_0,y_1,\dots,y_m$ in $G$ such that $x_{i+1}\in U_ix_i$ and $y_{i+1}\in y_iU_i$ for all $i<m$. Now consider the chain $e=x_0y_0,x_1y_1,\dots,x_my_m=xy$ linking the points $e$ and $z=xy$. Observe that for every $i<m$,   $x_{i+1}y_{i+1}\in U_ix_iy_{i}U_i$ implies $(x_{i+1}y_{i+1},x_iy_{i})\in U_i^{{\mathrm{RL}}}$ and hence $d(x_{i+1}y_{i+1},x_iy_{i+1})<1/{2^{i+1}}$ by the choice of the neighborhood $U_i$. Consequently, $$d(e,xy)\le\sum_{i<m}d(x_iy_i,x_{i+1}y_{i+1})\le\sum_{i<m}\frac1{2^{i+1}}<1$$ and $xy\in O^{{\mathrm{RL}}}(e)$ by the choice of the pseudometric $d$.

4a. If $\RL[G]$ is a topological group, then the identity map $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism by Theorem \[t2.2\]. By the preceding item, the identity map $\RL[G]\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is continuous, and has continuous inverse, which is the composition of two continuous maps ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n\to \RL[G]$.

4b. Assume that the identity maps $G_n^{{\mathrm{RL}}}\to G_{n+1}^{{\mathrm{RL}}}$, $n\in{\omega}$, are uniform embeddings. In this case Theorem \[t7.3\] implies that the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ has the family $$\Big\{B\big(e;\sum_{n\in{\omega}}U_n^{{\mathrm{RL}}}\big){\, ;\;}(U_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal B}_n\Big\}$$as a neighborhood base at $e$. Since each set $B(e;\sum_{n\in{\omega}}U_n^{{\mathrm{RL}}})$ coincides with ${\operatornamewithlimits{\overleftrightarrow{\textstyle\prod}}}_{n\in{\omega}}U_n$, we see that the topologies of the semitopological groups ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ and $\RL[G]$ coincide at $e$ and thus coincide everywhere.

5\. If ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is a topological group, then the identity map ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is continuous because its restrictions to the groups $G_n$ are continuous. The inverse identity map ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is continuous because it is uniformly continuous as the identity map into the topological group $({\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n)^{{\mathrm{RL}}}$ endowed with the Roelcke uniformity.

If the identity map ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism, then ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ is a topological group because ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a topological group.

Open Problems {#s:OP}
=============

Summing up, we conclude that for any tower of topological groups $(G_n)_{n\in{\omega}}$

-   the direct limit ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a topological group,

-   ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$ and $\RL[G]$ are quasitopological groups,

-   $\LA[G]={\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$ and $\RA[G]={\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{R}}}$ are semiuniform groups, and

-   $\LR[G]$, ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}$ and ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ are quasiuniform groups,

having the union $G=\bigcup_{n\in{\omega}}G_n$ as their underlying group.

The interplay between these semitopological and semiuniform groups are described in the following diagram. A simple (resp. double) arrow indicates that the corresponding identity map is continuous (resp. uniformly continuous).

(300,200)(-220,-100) (-35,-2)[$\LR[G]$]{} (-65,3)[(1,0)[23]{}]{} (-65,3)[(1,0)[20]{}]{} (-20,11)[(1,1)[12]{}]{} (-20,11)[(1,1)[10]{}]{} (-20,-7)[(1,-1)[12]{}]{} (-20,-7)[(1,-1)[10]{}]{} (8,-18)[(1,1)[13]{}]{} (8,22)[(1,-1)[13]{}]{}

(25,-2)[$\RL[G]$]{} (40,3)[(1,0)[25]{}]{} (-5,-28)[$\RA[G]$]{} (-5,25)[$\LA[G]$]{} (0,45)[(0,1)[20]{}]{} (0,45)[(0,1)[17]{}]{} (0,65)[(0,-1)[20]{}]{} (0,65)[(0,-1)[23]{}]{}

(-120,0)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}$]{} (-147,3)[(1,0)[20]{}]{}

(-90,15)[(4,3)[74]{}]{} (-90,15)[(4,3)[71]{}]{} (-90,-11)[(4,-3)[73]{}]{} (-90,-11)[(4,-3)[70]{}]{} (16,70)[(4,-3)[73]{}]{} (16,70)[(4,-3)[70]{}]{} (16,-65)[(4,3)[73]{}]{} (16,-65)[(4,3)[70]{}]{}

(70,0)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$]{} (125,3)[(1,0)[20]{}]{}

(-195,0)[${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n$]{} (152,0)[${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$]{} (-20,80)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{L}}}$]{} (-20,-80)[${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{R}}}$]{}

(0,-62)[(0,1)[22]{}]{} (0,-62)[(0,1)[25]{}]{} (0,-40)[(0,-1)[20]{}]{} (0,-40)[(0,-1)[23]{}]{}

Under certain conditions on the tower $(G_n)_{n\in{\omega}}$ some of the identity maps in this diagram are homeomorphisms. In particular,

-   ${\mathrm t\mbox{-}\kern-2pt\varinjlim}G_n\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism if all topological groups $G_n$, $n\in{\omega}$, are locally compact [@TSH];

-   ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism if all topological groups $G_n$, $n\in{\omega}$ are balanced [@BaR];

-   $\LR[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism if the tower $(G_n)_{n\in{\omega}}$ is balanced or satisfies ${\mathrm{PTA}}$ (Theorems \[t4.2\] and \[t3.2\]);

-   $\RL[G]\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ is a homeomorphism if the tower $(G_n)_{n\in{\omega}}$ is bi-balanced (Theorem \[t5.2\]).

Nonetheless many open questions related to this diagram remain unsolved.

Is the identity map $\LR[G]\to {\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}$ (uniformly) continuous?

What can be said about separation properties of the quasitopological group $\RL[G]$? Is it always Tychonoff? Is the identity map $\RL[G]\to{\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}$ a homeomorphism?

We define a topological space $X$ to be [*Tychonoff*]{} if for each closed subset $F\subset X$ and each point $x\in X\setminus F$ there is a continuous function $f:X\to{\mathbb R}$ with $f(x)=1$ and $f(F)\subset\{0\}$. It is known that each uniform (not necessarily separated) space is Tychonoff. In particular, each semiuniform group is Tychonoff.

Surprisingly, but we know no (natural) example of a tower of topological groups $(G_n)_{n\in{\omega}}$ for which the topology of ${\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ would be different from $\RL[\tau]$ or even $\LR[\tau]$. However, we expect counterexamples to the following problem.

Is the identity map ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{RL}}}\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$ a homeomorphism? What about the identity map ${\mathrm u\mbox{-}\kern-2pt\varinjlim}G_n^{{\mathrm{LR}}}\to{\mathrm g\mbox{-}\kern-2pt\varinjlim}G_n$?

Acknowledgment
==============

The authors express their thanks to the referee for many valuable remarks and suggestions improving the presentation.

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[^1]: This research was supported by Slovenian Research Agency grant P1-0292-0101, J1-9643-0101 and J1-2057-0101.
