Selective Survey on Spaces of Closed Subgroups of Topological Groups

Abstract

We survey different topologizations of the set $\mathcal{S}\left(G\right)$ of closed subgroups of a topological group G and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics.

1. Chabauty Spaces

1.1. From Minkowski to Chabauty

We recall that a lattice L in ${\mathbb{R}}^n$ is a discrete subgroup of rank n. We define $minL$ as the length of the shortest nonzero vector from L, and we define $vol({\mathbb{R}}^n/L)$ as the volume of a basic parallelepiped of L.

A sequence ${\left(L_m\right)}_{m\in \omega }$ of lattices in ${\mathbb{R}}^n$ converges to the lattice L if, for each $m\in \omega$, one can choose a basis $a_1\left(m\right),\dots ,a_n\left(m\right)$ of $L_m$ and a basis $a_1,\dots ,a_n$ of L such that the sequence ${\left(a_i\left(m\right)\right)}_{m\in \omega }$ converges to $a_i$ for each $i\in \{1,\dots ,n\}$. This convergence of lattices was introduced by H. Minkowski [1], and its usage in Geometry of Numbers (see [2]) is based on the following theorem from K. Mahler [3].

Theorem 1.

Let $\mathcal{M}$ be a set of lattices in ${\mathbb{R}}^n$. Every sequence in $\mathcal{M}$ has a convergent subsequence if and only if there exist two constants, $C>0$ and $c>0$, such that $minL>c$, $vol({\mathbb{R}}^n\backslash L)<C$ for each $L\in \mathcal{M}$.

What we know now is that Chabauty topology was invented by C. Chabauty in [4] in order to extend Theorem 1 to lattices in connected Lie groups.A discrete subgroup L of a connected Lie group G is called a lattice if the quotient space $G/L$ is compact.

Let X be a Hausdorff locally compact space, and let $expX$ denote the set of all closed subsets of X. The sets

$$\{F\in expX:F\cap K=\varnothing \},\{F\in expX:F\cap U\ne \varnothing \},$$

where K runs over all compact subsets of X and U runs over all open subsets of X, form the subbase of the Chabauty topology on $expX$. The space $expX$ is compact and Hausdorff. If X is discrete, then $expX$ is homeomorphic to the Cantor cube ${\{0,1\}}^{\left|X\right|}$.

We note also that a net ${\left(F_{\alpha }\right)}_{\alpha \in \mathcal{I}}$ converges in $expX$ to F if and only if

• for every compact K of X such that $K\cap F=\varnothing$, there exists $\beta \in \mathcal{I}$ such that $F_{\alpha }\cap K=\varnothing$ for each $\alpha >\beta$;
• for every $x\in F$ and every neighborhood U of x, there exists $\gamma \in \mathcal{I}$ such that $F_{\alpha }\cap U\ne \varnothing$ for each $\alpha >\gamma$.

If G is a locally compact group, then $\mathcal{S}\left(G\right)$ is a closed subspace of $expG$ (so, $\mathcal{S}\left(G\right)$ is compact); $\mathcal{S}\left(G\right)$ is called the Chabauty space of G.

Theorem 2.

[4]. Let G be a connected unimodular Lie group. A set $\mathcal{M}$ of lattices in G is relatively compact in $\mathcal{M}$ if and only if there exists a constant $C>0$ and a neighborhood U with the identity e of G such that $L\cap U=\left\{e\right\}$ and $vol(G/L)<C$ for each $L\in \mathcal{M}$.

There was some technical improvement made in [5] and the paper [4], which is included in [6], Chapter 8.

1.2. Pontryagin–Chabauty Duality

This duality was established in [7] and detailed in [8]. We use the standard abbreviation LCA for a locally compact Abelian group. Let G be an LCA-group $G^{\wedge }$, let denote its dual group $G^{\wedge }=Hom(G,\mathbb{R}/\mathbb{Z})$, and let $\phi$ denote the canonical bijection $\mathcal{S}\left(G\right)⟶\mathcal{S}\left(G^{\wedge }\right)$, $\phi \left(X\right)=\{f\in G^{\wedge }:X\subseteq kerf\}$.

Theorem 3.

For every LCA-group G, the bijection $\phi :\mathcal{S}\left(G\right)⟶\mathcal{S}\left(G^{\wedge }\right)$ is a homeomorphism.

Typically, Theorem 3 is applied to replace $\mathcal{S}\left(G\right)$ by $\mathcal{S}\left(G^{\wedge }\right)$ in the case of a compact Abelian group G.

In what follows, we use the following notations: ${\mathbb{C}}_n$ is a cyclic group of order n, ${\mathbb{C}}_{p^{\infty }}$ is a quasi-cyclic (or Pr$\ddot{u}$fer) p-group, $\mathbb{Z}$ is a discrete group of integers, ${\mathbb{Z}}_p$ is the group of p-adic integers, and ${\mathbb{Q}}_p$ is an additive group of a field of p-adic numbers.

1.3. $S\left(G\right)$ for Compact G

The following two lemmas from [9] are the basic technical tools in this area.

Lemma 1.

If $G,H$ are compact groups and $\phi :G⟶H$ is a continuous surjective homomorphism, then the mapping $\mathcal{S}\left(G\right)⟶\mathcal{S}\left(H\right),$ $X⟼\phi \left(X\right)$ is continuous and open.

The continuity is easily deduced, but to prove the openness, we need

Lemma 2.

Let G be a compact group, $X\in \mathcal{S}\left(G\right)$. Then, the following subsets form a base of the neighborhoods of X is $\mathcal{S}\left(G\right)$:

$${\mathcal{N}}_X(U,N,x_1,\dots ,x_n)=\{u^{-1}Yu:u\in U,Y\in \mathcal{S}\left(G\right),Y\subseteq XN,Y\cap x_{1}U\ne \varnothing ,\dots ,Y\cap x_{n}U\ne \varnothing ,\}$$

where U is a neighborhood of the identity of G, N is closed normal subgroup such that $G/N$ is a Lie group, and $x_1,\dots ,x_n$ are arbitrary elements of X, $n\in \mathbb{N}$.

In particular, if G is a compact Lie group, then Lemma 2 states that there is a neighborhood $\mathcal{N}$ of X such that each subgroup $Y\in \mathcal{N}$ is conjugated to some subgroup of X. The Montgomery–Yang theorem on tubes [10] (see also ([11], Theorem 5.4, Chapter 2)) plays the key role in the proof of Lemma 2.

We recall that the cellularity (or Souslin number) $c\left(X\right)$ of a topological space X is the supremum of cardinalities of disjoint families of open subsets of X. A topological space X is called dyadic if X is a continuous image of some Cantor cube ${\{0,1\}}^{\kappa }$.

The weight $w\left(X\right)$ of a topological space X is the minimal cardinality of the open bases of X.

Theorem 4.

[9]. For every compact group G, we have $c\left(\mathcal{S}\left(G\right)\right)\le {\aleph }_0$. In addition, if $w\left(G\right)\le {\aleph }_1$, then $\mathcal{S}\left(G\right)$ is dyadic.

Theorem 5.

[12]. Let a group G be either profinite or compact and Abelian. If $w\left(G\right)>{\aleph }_2$, then the space $\mathcal{S}\left(G\right)$ is not dyadic.

Theorem 6.

[12]. Let G be an infinite compact Abelian group such that $w\left(G\right)\le {\aleph }_1$. Then, the space $\mathcal{S}\left(G\right)$ is homeomorphic to the Cantor cube ${\{0,1\}}^{w\left(G\right)}$ if and only if $\mathcal{S}\left(G\right)$ has no isolated points.

An Abelian group G is called Artinian if every increasing chain of subgroups of G is finite; every such group is isomorphic to the direct $sum{\oplus }_{p\in F}{\mathbb{C}}_{p^{\infty }}\oplus K$, where F is a finite set of primes, and K is a finite subgroup. An Abelian group G is called minimax if G has a finitely generated subgroup N such that $G/N$ is Artinian.

Theorem 7.

[12]. For a compact Abelian group G, the space $\mathcal{S}\left(G\right)$ has an isolated point if and only if the dual group $G^{\wedge }$ is minimax.

1.4. $S\left(G\right)$ for LCA G

The space $\mathcal{S}\left(\mathbb{R}\right)$ is homeomorphic to the segment $[0,1]$. By [13], $\mathcal{S}\left({\mathbb{R}}^2\right)$ is homeomorphic to the sphere ${\mathbf{S}}^4$. For $n\ge 3$, $\mathcal{S}\left({\mathbb{R}}^n\right)$ is not a topological manifold and its structure is far from understood (see [14]).

Theorem 8.

[15]. The space $\mathcal{S}\left(G\right)$ of an LCA-group G is connected if and only if G has a subgroup topologically isomorphic to $\mathbb{R}$.

If F is a non-solvable finite group, then $\mathcal{S}(\mathbb{R}\times F)$ is not connected ([8], Proposition 8.6).

Theorem 9.

[8]. The space $\mathcal{S}\left(G\right)$ of an LCA-group G is totally disconnected if and only if G is either totally disconnected or each element of G belongs to a compact subgroup.

Some more information on $\mathcal{S}\left(G\right)$ for LCA G can be found in [8] and the references therein, particularly on the topological dimension of $\mathcal{S}\left(G\right)$.

By Theorems 3 and 4, $c\left(\mathcal{S}\left(G\right)\right)\le {\aleph }_0$ for every discrete Abelian group. We say that a topological space X has the Shanin number $\omega$ if any uncountable family $\mathcal{F}$ of the non-empty open subsets of X has an uncountable subfamily ${\mathcal{F}}^{\prime }$ such that $\cap {\mathcal{F}}^{\prime }\ne \varnothing$. Evidently, if a space X has the Shanin number $\omega$, then $c\left(X\right)\le {\aleph }_0$. By [16], Theorem 1, for every discrete Abelian group G, the space $\mathcal{S}\left(G\right)$ has the Shanin number $\omega$. By [16], Theorem 3, for every infinite cardinal $\tau$, there exists a solvable discrete group G such that $c\left(\mathcal{S}\right(G\left)\right)=\left|G\right|=\tau$.

1.5. $S\left(G\right)$ as a Lattice

The set $S\left(G\right)$ has the natural structure of a lattice with the operations ∨ and ∧, where $A\wedge B=A\cap B$, and $A\vee B$ is the smallest closed subgroup of G containing A and B. In this subsection, we formulate some results from [17] about the interrelations between the topological and lattice structures on $S\left(G\right)$.

For $g\in G$, $\overline{<g>}$ denotes the subgroup of G topologically generated by g. A totally disconnected locally compact group G is called periodic if $\overline{<g>}$ is compact for each $g\in G$. In this case, $\pi \left(G\right)$ denotes the set of all prime numbers such that $p\in \pi \left(G\right)$ if and only if $g\in G$ such that $\overline{<g>}$ is topologically isomorphic either to ${\mathbb{C}}_{p^n}$ or to ${\mathbb{Z}}_p$; this g is called a topological p-element.

Theorem 10.

For a compact group G, the following statements are equivalent:

(i) 

∧ is continuous;

(ii) 

∧ and ∨ are continuous;

(iii) 

G is the semidirect product $K⋋P$, where K is profinite with finite Sylow p-subgroups, P is Abelian profinite and each Sylow p-subgroup of G is ${\mathbb{Z}}_p$, $\pi \left(K\right)\cap \pi \left(P\right)=\varnothing$, and the centralizer of each Sylow p-subgroup of G has a finite index in G.

Theorem 11.

For a locally compact group G, the operation ∧ is continuous if and only if the following conditions are satisfied:

(i) 

G is either discrete or periodic;

(ii) 

∧ is continuous in $\mathcal{S}\left(H\right)$ for each compact subgroup H of G;

(iii) 

the centralizer of each topological p-element of G is open.

We recall that a torsion group G is layer-finite if the set $\{g\in G:g^n=e\}$ is finite for each $n\in \mathbb{N}$. A layer-finite group G is called thin if each Sylow p-subgroup of G is finite (equivalently, G has no subgroup isomorphic to ${\mathbb{C}}_{p^{\infty }}$).

Theorem 12.

Let G be a locally compact group. The operations ∧ and ∨ are continuous if and only if G is periodic and topologically isomorphic to $A\times B\times (C⋋D)$, where C has a dense thin layer-finite subgroup; $A,B,D$ are Abelian with Sylow p-subgroups ${\mathbb{C}}_{p^{\infty }}$, ${\mathbb{Q}}_p$, or ${\mathbb{Z}}_p$; the sets $\pi \left(A\right)$, $\pi \left(B\right)$, $\pi \left(G\right)$, $\pi \left(D\right)$ are pairwise disjoint; and the centralizer of each Sylow p-subgroup of G is open.

1.6. From Chabauty to Local Method

A topological group G is called topologically simple if each closed normal subgroup of G is either G or $\left\{e\right\}$. Every topologically simple LCA-group is discrete, and either $G=\left\{e\right\}$ or G is isomorphic to ${\mathbb{C}}_p$.

Following the algebraic tradition, we say that a group G is locally nilpotent (solvable) if every finitely generated subgroup is nilpotent (solvable).

In [18], Problem 1.76, V. Platonov asked whether there exists a non-Abelian, topologically simple, locally compact, locally nilpotent group. Here, we present the negative answer to this question for the locally solvable group obtained in [19].

Let G be a locally compact, locally solvable group. We take $g\in G\backslash \left\{e\right\}$, choose a compact neighborhood U of G, and denote by $\mathcal{F}$ the family of all topologically finitely generated subgroups of G containing g. We may assume that G is not topologically finitely generated, so $\mathcal{F}$ is directed by the inclusion ⊂. For each $F\in \mathcal{F}$, we choose $A_F$, $B_F\in \mathcal{S}\left(F\right)$ such that $B_F\subset A_F$; $A_F$ and $B_F$ are normal in F, $A_F\cap U\ne \varnothing$, $B_F\cap U=\varnothing$, and $A_F/B_F$ is Abelian. Since $\mathcal{S}\left(G\right)$ is compact, we can choose two subnets ${\left(A_{\alpha }\right)}_{\alpha \in \mathcal{I}}$, ${\left(B_{\alpha }\right)}_{\alpha \in \mathcal{I}}$ of the nets ${\left(A_F\right)}_{F\in \mathcal{F}}$, ${\left(B_F\right)}_{F\in \mathcal{I}}$ which converge to $A,B\in \mathcal{S}\left(G\right)$. Then $A,B$ are normal in G, and $A/B$ is Abelian. Moreover, $x\notin B$ and $A\cap U\ne \varnothing$. If $A\ne \left\{G\right\}$, then A is a proper normal subgroup of G; otherwise, $G/B$ is Abelian.

In [20], the Chabauty topology was defined on some systems of closed subgroups of a locally compact group G. A system $\mathfrak{A}$ of closed subgroups of G is called subnormal if

• $\mathfrak{A}$ contains $\left\{e\right\}$ and G;
• $\mathfrak{A}$ is linearly ordered by the inclusion ⊂;
• for any subset $\mathfrak{M}$ of $\mathfrak{A}$, the closure of ${\bigcup }_{F\in \mathfrak{M}}F\in \mathfrak{A}$ and ${\bigcap }_{F\in \mathfrak{M}}F\in \mathfrak{A}$;
• whenever A and B comprise a jump in $\mathfrak{A}$ (i.e., $B\subset A$, and no members of $\mathfrak{A}$ lie between B and A), B is a normal subgroup of A.

If the subgroups $A,B$ form a jump, then $A/B$ is called a factor of G. The system is called normal if each $A\in \mathfrak{A}$ is normal in G.

A group G is called an RN-group if G has a normal system with Abelian factors. Among the local theorems from [20], one can find the following: if every topologically finitely generated subgroup of a locally compact group G is an RN-group, then G is an RN-group. In particular, every locally compact, locally solvable group is an RN-group.

In 1941 (see ([21], pp. 78–83), A.I. Mal’tsev obtained local theorems for discrete groups as applications of the following general local theorem: if every finitely generated subsystem of an algebraic system A satisfies some property $\mathcal{P}$, which can be defined by some quasi-universal second-order formula, then A satisfies $\mathcal{P}$.

In [22], Mal’tsev’s local theorem was generalized on a topological algebraic system. The part of the model-theoretical Compactness Theorem in Mal’tsev’s arguments employs some convergents of closed subsets. A net ${\left(F_{\alpha }\right)}_{\alpha \in \mathcal{I}}$ of closed subsets of a topological space X S-converges to a closed subset F if

• for every $x\in F$ and every neighborhood U of x, there exists $\beta \in \mathcal{I}$ such that $F_{\alpha }\cap U\ne \varnothing$ for each $\alpha >\beta$;
• for every $y\in X\backslash F$, there exist a neighborhood $\mathcal{V}$ of y and a $\gamma \in \mathcal{I}$ such that $F_{\alpha }\cap \mathcal{V}=\varnothing$ for each $\alpha >\gamma$.

Every net of closed subsets of an arbitrary (!) topological space has a convergent subnet. If X is a Hausdorff locally compact space, then the S-convergence coincides with the convergence in the Chabauty topology.

1.7. Spaces of Marked Groups

Let $F_k$ be the free group of rank k, with the free generators $x_1,\dots ,x_k$, and let ${\mathcal{G}}_k$ denote the set of all normal subgroups of $F_k$. In the metric form, the Chabauty topology on ${\mathcal{G}}_k$ was introduced in [23] as a reply to Gromov’s idea of the topologizations of some sets of groups [24].

Let G be a group generated by $g_1,\dots ,g_k$. The bijection $x_i⟼g_i$ $g_1,\dots ,g_n$ can be extended to the homomorphism $f:F_k⟶G$. With the correspondence $G⟼kerf$, ${\mathcal{G}}_k$ is called the space marked k-generated groups.

A couple of papers in development by [23] are aimed at understanding how large, in the topological sense, are the well-known classes of finitely generated groups, or how a given k-generated group is placed in ${\mathcal{G}}_k$ (see [25]). Among the applications of ${\mathcal{G}}_k$, we mention the construction of topologizable Tarski Monsters in [26].

1.8. Dynamical Development

Every locally compact group G acts on the Chabauty space $\mathcal{S}\left(G\right)$ by the rule: $(g,H)⟼g^{-1}Hg$. Under this action, every minimal closed invariant subset of $\mathcal{S}\left(G\right)$ is called a uniformly recurrent subgroup (URS). The study of URSs was initiated by Glasner and Weiss [27] with the following observation.

Let G be a locally compact group G acting on a compact X so that is G minimal, i.e., the orbit of each point $x\in X$ is dense. We consider the mapping $Stab:X⟶\mathcal{S}\left(G\right)$, defined by $Stab\left(x\right)=\{g\in G:gx=x\}$. Then, there is the unique URS contained in the closure of $Stab\left(X\right)$. This URS is called the stabilizer URS. Glasner and Weiss asked whether every URS of a locally compact group G arises as the stabilizer URS of a minimal action of G on a compact space. This question was answered in the affirmative in [28].

2. Vietoris Spaces

For a topological space X, the Vietoris topology on the set $expX$ of all closed subsets of X is defined by the subbase of the open sets

$$\{F\in expX:F\subseteq U\},\{F\in expX:F\cap V\ne \varnothing \},$$

where $U,\mathcal{V}$ run over all open subsets of X.

A net ${\left(F_{\alpha }\right)}_{\alpha \in \mathcal{I}}$ converges to F in $expX$ if and only if

• for each open subset U of X such that $F\subseteq U$, there exists $\beta \in \mathcal{I}$ such that $F_{\alpha }\subseteq U$ for each $\alpha >\beta$;
• for each $x\in F$ and each neighborhood $\mathcal{V}$ of x, there exists $\gamma \in \mathcal{I}$ such that $F_{\alpha }\cap \mathcal{V}\ne \varnothing$ for each $\alpha >\gamma$.

If X is regular, then $\mathcal{S}\left(G\right)$ is closed in $expG$. To my knowledge, the spaces $\mathcal{S}\left(G\right)$, where G needs not be compact, endowed with Vietoris topologies appeared in [29] with the characterization of LCA-groups G such that the canonical mapping $\phi :\mathcal{S}\left(G\right)⟶\mathcal{S}\left(G^{\wedge }\right)$ is a homeomorphism.

2.1. Compactness

We cannot ask for a constructive description of arbitrary topological groups G with compact space $\mathcal{S}\left(G\right)$ because we know nothing about G with $S\left(G\right)=2$.

Theorem 13.

[30]. Let G be a locally compact group. The space $\mathcal{S}\left(G\right)$ is compact if and only if G is one of the following groups:

(i) 

G is compact;

(ii) 

${\mathbb{C}}_{p_1^{\infty }}\times \dots \times {\mathbb{C}}_{p_n^{\infty }}\times K$, where $p_1,\dots ,p_n$ are distinct prime numbers, K is finite, and each $p_i$ is not a divisor of $\left|K\right|$;

(iii) 

$Q_p\times K$, where K is finite and p does not divide $\left|K\right|$.

A similar characterization of groups with compact $\mathcal{S}\left(G\right)$ is given in [31], provided that G has a base of neighborhoods at the identity consisting of subgroups.

Theorem 14.

[32]. Let G be a locally compact group. A closed subset $\mathcal{F}$ of $\mathcal{S}\left(G\right)$ is compact if and only if the following conditions are satisfied:

(i) every descending chain of non-compact subgroups from F is finite;

(ii) every closed subset ${\mathcal{F}}^{\prime }$ of $\mathcal{F}$ has only a finite number of non-compact subgroups maximal in $\mathcal{F}$;

(iii) if a closed subset ${\mathcal{F}}^{\prime }$ of $\mathcal{F}$ has no non-compact subgroups, then $\cup {\mathcal{F}}^{\prime }$ is compact.

Two corollaries: Every compact subset of $\mathcal{S}\left(G\right)$ consisting of non-compact subgroups is scattered; a subset $\mathcal{F}$ is compact if and only if $\mathcal{F}$ is countably compact.

For locally compact groups with the $\sigma$-compact space $\mathcal{S}\left(G\right)$ (see [33]), a description of the LCA-groups with locally compact space $\mathcal{S}\left(G\right)$ can be obtained in [34].

A topological group G is called inductively compact if every finite subset of G is contained in a compact subgroup. For a group G, $K\left(G\right)$ and $IK\left(G\right)$ denote the sets of all compact and closed inductively compact subgroups.

Theorem 15.

[35]. For every locally compact group G, $IK\left(G\right)$ is the closure of $K\left(G\right)$.

Two corollaries: If G is a connected Lie group, then $K\left(G\right)$ is closed; $\mathcal{S}\left(G\right)$ is a k-space for each locally compact group G of countable weight, i.e., the topology of $\mathcal{S}\left(G\right)$ is uniquely determined by the family of all compact subsets of $\mathcal{S}\left(G\right)$.

2.2. Metrizability and Normality

LCA-groups G with metrizable and normal space $\mathcal{S}\left(G\right)$ were characterized by S. Panasyuk in the candidate thesis Normality and metrizability of the space of closed subgroups, Kyiv University, 1989.

Theorem 16.

For a discrete Abelian group G, the following statements are equivalent:

(i) $\mathcal{S}\left(G\right)$ is metrizable;

(ii) $\mathcal{S}\left(G\right)$ is normal;

(iii) G has a finitely generated subgroup H such that $G/H={\mathbb{C}}_{p_1^{\infty }}\times \dots \times {\mathbb{C}}_{p_n^{\infty }}$, where $p_1,\dots ,p_n$ are distinct primes.

In the general case, metrizability and normality of $\mathcal{S}\left(G\right)$ are not equivalent, but if G is a connected semisimple Lie group, then $\mathcal{S}\left(G\right)$ is metrizable if and only if $\mathcal{S}\left(G\right)$ is normal if and only if G is compact (see [36,37]). The space $\mathcal{S}\left(G\right)$ for every connected solvable Lie group is metrizable [36].

2.3. Some Cardinal Invariants

We remind the reader that $c\left(X\right)$ denotes the cellularity of X.

Theorem 17.

[9]. For every infinite locally compact group G, we have $c\left(\mathcal{S}\right(G\left)\right)\le c\left(G\right)$.

Theorem 18.

[38]. For every locally compact group G, the following conditions are equivalent:

(i) $\mathcal{S}\left(G\right)$ is of countable pseudocharacter;

(ii) $\mathcal{S}\left(G\right)$ is of countable tightness;

(iii) $\mathcal{S}\left(G\right)$ is sequential;

(iv) $w\left(G\right)\le {\aleph }_0$.

3. Other Topologizations

3.1. Bourbaki Uniformities

Let $(X,\mathcal{U})$ be a uniform space. The uniformity $\mathcal{U}$ induces the uniformity $\tilde{\mathcal{U}}$ on the set $\mathcal{F}\left(X\right)$ of all non-empty closed subsets of X which have as a base the family of sets of the form

$$\left\{\right(A,B)\in \mathcal{F}(X)\times \mathcal{F}(X):B\subseteq U(A),A\subseteq U(B\left)\right\},$$

whenever $U\in \mathcal{U}$. The uniformity $\tilde{\mathcal{U}}$ was introduced in [39] (Chapter 2, § 1), and $\tilde{\mathcal{U}}$ is called the Bourbaki (sometimes, Hausdorff–Bourbaki) uniformity.

Let G be a topological group. We endow G with the left uniformity L and $F\left(G\right)$ with the Bourbaki uniformity $\tilde{L}$. We denote by $\mathcal{L}\left(G\right)$ and $\mathcal{B}\left(G\right)$ the subspaces of $\mathcal{F}\left(G\right)$ consisting of all subgroups and all totally bounded subsets of G.

Theorem 19.

[40]. Let G be a group with a base at the identity consisting of subgroups. The space $\mathcal{L}\left(G\right)$ is compact if and only if G is totally bounded and $K\bigcap G$ is dense in K for each closed subgroup K from the completion of G.

In particular, if $\mathcal{L}\left(G\right)$ is compact, then G is totally minimal.

Theorem 20.

[40]. If a group G is complete in the left uniformity, then $\mathcal{B}\left(G\right)$ is complete.

We recall that a topological group G is almost metrizable if each neighborhood of e contains a compact subgroup K such that the set of all open subsets containing K have a countable base. Every metrizable and every locally compact topological group is almost metrizable.

Theorem 21.

[40]. If an almost metrizable group G is complete in the left uniformity, then $\mathcal{F}\left(G\right)$ is complete.

In [41], Theorem 21 is proved with the bilateral uniformity on G (and so on $\mathcal{F}\left(G\right)$) in place of the left uniformity.

3.2. Functionally Balanced Groups

For a topological group G, the set $\mathcal{F}\left(G\right)$ has the natural structure of a semigroup with the operation $(A,B)⟼clAB$.

Theorem 22.

[42]. For a topological group G, the following statements are equivalent:

(i) $\mathcal{F}\left(G\right)$ is a topological semigroup;

(ii) for every subset X of G and every neighborhood U of e, there exists a neighborhood V of e such that $VX\subseteq XU$;

(iii) every bounded left uniformly continuous function on G is right uniformly continuous.

A topological group G is called balanced (or a SIN-group) if the left and right uniformities of G coincide. A group G is called functionally balanced if G satisfies $\left(iii\right)$ of Theorem 22. The study of functionally balanced groups was initiated by G. Itzkowitz [43].

The equivalence of $\left(ii\right)$ and $\left(iii\right)$ in Theorem 22 is a criterion for a topological group to be functionally balanced. In [44], this criterion was used to show that each almost-metrizable functionally balanced group is balanced.

3.3. Lattice Topologies

These topologies on a complete lattice $\mathcal{L}\left(G\right)$ of closed subgroups are algebraically defined by the lattice structure of $\mathcal{L}\left(G\right)$.

For example, a net ${\left(A_{\alpha }\right)}_{\alpha \in \mathcal{I}}$ in $\mathcal{L}\left(G\right)$ order-converges to $A\in \mathcal{L}\left(G\right)$ if there exist two nets ${\left(B_{\alpha }\right)}_{\alpha \in \mathcal{I}}$, ${\left(C_{\alpha }\right)}_{\alpha \in \mathcal{I}}$ in $\mathcal{L}\left(G\right)$ such that, for each $\alpha \in \mathcal{I},$$B_{\alpha }\subseteq A_{\alpha }\subseteq C_{\alpha }$ and ${\vee }_{\alpha \in \mathcal{I}}{B}_{\alpha }={\wedge }_{\alpha \in \mathcal{I}}{C}_{\alpha }=A$. By [45], for a compact group G, every net in $\mathcal{L}\left(G\right)$ has an order-convergent subset if and only if $\mathcal{L}\left(G\right)$ endowed with the Shabauty topology is a topological lattice (see Theorem 10).

More on the lattices’ topologies on $\mathcal{L}\left(G\right)$ in the case of a compact G can be found in [46].

3.4. Segment Topologies

Let G be a topological group; ${\mathcal{P}}_G$ is the family of all subsets of G, and ${\left[G\right]}^{<\omega }$ is the family of all finite subsets of G. Each pair $\mathcal{A},\mathcal{B}$ of subsets of ${\mathcal{P}}_G$ closed under finite unions defines the segment topology on $\mathcal{L}\left(G\right)$ with a base consisting of the segments

$$[A,G\backslash B]=\{X\in \mathcal{L}(G):A\subseteq X\subseteq G\backslash B\},A\in \mathcal{A},B\in \mathcal{B}.$$

These topologies are described in [47] in the following three cases: $\mathcal{A}=\mathcal{B}={\left[G\right]}^{<\omega }$; $\mathcal{A}={\mathcal{P}}_G$ and $\mathcal{B}={\left[G\right]}^{<\omega }$; $\mathcal{A}={\left[G\right]}^{<\omega },$ $\mathcal{B}={\mathcal{P}}_G$

3.5. $(\Sigma ,\Theta )$-Topologies

This general construction for topologizations of the set $\mathcal{L}\left(G\right)$ of closed subgroups of a topological group G from [48] produces Chabauty, Vietoris, and Bourbaki topologies, along with plenty of other topologies.

We assume that, for each $H\in \mathcal{L}\left(G\right)$, $\Sigma \left(H\right)$ is some family of open subsets of G, $\Sigma ={\cup }_{H\in \mathcal{L}\left(G\right)}\Sigma \left(H\right)$, and the following conditions are satisfied:

• if $U,\mathcal{V}\in \Sigma \left(H\right)$, then $U\cap \mathcal{V}$ contains some $W\in \Sigma \left(H\right)$;
• for every $U\in \Sigma \left(H\right)$, there exists $\mathcal{V}\in \Sigma \left(H\right)$ such that $U\in \Sigma \left(K\right)$ for each $K\in \mathcal{L}\left(G\right)$, $K\subseteq \mathcal{V}$;
• ${\bigcap }_{U\in \Sigma \left(H\right)}\overline{U}=H$ for each $H\in \mathcal{L}\left(G\right).$

Then, the family $\{X\in \mathcal{L}(G):X\subseteq U\}$, $U\in \Sigma$, is a base for the $\Sigma$-topology on $\mathcal{L}\left(G\right)$.

Let $\tau$ denote the topology of G, and let ${\mathcal{P}}_{\tau }$ denote the family of all subsets of $\tau$. We assume that, for each $H\in \mathcal{L}\left(G\right)$, $\Theta \left(H\right)$ is some subset of ${\mathcal{P}}_{\tau }$ such that the following conditions are satisfied:

• for every $\alpha ,\beta \in \Theta \left(H\right)$, there is a $\gamma \in \Theta \left(H\right)$ such that $\alpha <\gamma$, $\beta <\gamma$ ($\alpha <\beta$ means that, for every $U\in \alpha$, there exists $V\in \beta$ such that $V\subseteq U$);
• for every $\alpha \in \Theta \left(H\right)$, there exists $\beta \in \Theta \left(H\right)$ such that if $K\in \mathcal{L}\left(G\right)$ and $K\cap V\ne \varnothing$ for each $V\in \beta$, then $\alpha <\gamma$ for some $\gamma \in \Theta \left(K\right)$;
• for each $H\in \mathcal{L}\left(G\right)$ and every neighborhood V of x, there exists $\alpha \in \Theta \left(H\right)$ such that $x\in U$, $U\subseteq V$ for some $U\in \alpha$.

Then, the family $\{X\in \mathcal{L}(G):X\cap U\ne \varnothing$ for each $U\in \alpha \}$, where $\alpha \in \Theta \left(H\right)$, $H\in \mathcal{L}\left(G\right)$, is a base for the $\Theta$-topology on $\mathcal{L}\left(G\right)$.

The upper bound of $\Sigma$- and $\Theta$-topologies is called the $(\Sigma ,\Theta )$-topology.

A net ${\left(H_{\alpha }\right)}_{\alpha \in \mathcal{I}}$ converges in $(\Sigma ,\Theta )$-topology to $H\in \mathcal{L}\left(G\right)$ if and only if

• for any $U\in \Sigma \left(H\right)$, there exists $\beta \in \mathcal{I}$ such that $H_{\alpha }\subseteq U$ for each $\alpha >\beta$;
• for any $\alpha \in \Theta \left(H\right)$, there exists $\gamma \in \mathcal{I}$ such that $H_{\alpha }\cap \mathcal{V}\ne \varnothing$ for each $\alpha >\gamma$.

In [48], one can find characterizations of G with a compact and discrete $\mathcal{L}\left(G\right)$ for some concrete $(\Sigma ,\Theta )$-topologies.

3.6. Hyperballeans of Groups

Let G be a discrete group. The set $\{Fg:g\in G,F\in {\left[G\right]}^{<\omega }\}$ is a family of balls in the finitary coarse structure on G. For definitions of coarse structures and balleans, see [49,50]. The finitary coarse structure on G induces the coarse structure on $\mathcal{L}\left(G\right)$ in which $\{X\in \mathcal{L}(G):X\subseteq FA,A\in FX\}$, $F\in {\left[G\right]}^{<\omega }$ is the family of balls centered at $A\in \mathcal{L}\left(G\right)$. The set $\mathcal{L}\left(G\right)$ endowed with the finitary coarse structure is called a hyperballean of G. Hyperballeans of groups, carefully studied in [51], can be considered as asymptotic counterparts of Bourbaki uniformities.