Compact families of noncompact subgroups and clopen partial orders

Problem

Does every scattered compact space embed in the space of closed noncompact subgroups of some locally compact Hausdorff group, equipped with the full Vietoris topology?

X\text{ scattered compact} \quad\stackrel{?}{\Longrightarrow}\quad \exists G:\ X\hookrightarrow\mathcal N(G).

The topology includes both upper conditions $H\subseteq U$ and lower conditions $H\cap U\ne\varnothing$ for every open $U\subseteq G$ .

Setup and result

Let $G$ be a locally compact Hausdorff group, and write $\mathcal S(G)$ for the set of its closed subgroups. The full Vietoris topology on $\mathcal S(G)$ has subbase

\tag{1} U^+=\{H:H\subseteq U\},\qquad U^-=\{H:H\cap U\ne\varnothing\},

where $U$ ranges over all open subsets of $G$ . Write $\mathcal N(G)$ for the subspace consisting of noncompact subgroups. This is the $E$ -topology of Kourovka Problem 9.47 (3). The problem, posed by Protasov, asks whether every scattered compact space embeds in $\mathcal N(G)$ for a suitable locally compact group $G$ .

The use of arbitrary open sets in the upper subbase is essential. It gives a countable separation property for noncompact subgroups. Compactness then turns this property into an order-theoretic obstruction: inclusion ideals in a compact family are clopen. Conversely, a partial order with clopen principal ideals determines a continuous family of subgroups of a discrete abelian group.

Theorem 1. For a compact Hausdorff space $X$ , the following conditions are equivalent.

$X$ embeds in $\mathcal N(G)$ for some locally compact Hausdorff group $G$ .
There is a partial order $\le$ on $X$ such that $\mathord{\downarrow}x=\{y:y\le x\}$ is clopen for every $x\in X$ .
$X$ embeds in $\mathcal N(D_X)$ , where
$D_X=\mathbb Z\oplus\bigoplus_{t\in X}\mathbb Z/2\mathbb Z$
has the discrete topology.

The compact-family theory underlying the implication (i) $\Rightarrow$ (ii) goes back to Protasov (5); see also the compactness criterion in his survey (6, §2). The neighbourhood property for inclusion ideals is stated explicitly in Muldagaliev and Ermekqali (4, Lemma 2). We give a proof through a general lemma on compact partially ordered spaces. The reverse implication uses an explicit family of subgroups with one common infinite cyclic summand.

The obstruction in (ii) can be tested on an existing scattered compact space. The collapsed ordinal square below appears in the work of Abraham, Bonnet and Rubin (1); Banakh, Bonnet and Kubiś (2, Proposition 5.4 of the preprint) give its topological formulation. We use this space to obtain the following consequence.

Theorem 2. Let $X$ be the quotient of $[0,\omega_1]^2$ obtained by identifying all points of

\Delta=\{(\alpha,\beta):\beta\le\alpha\}

to one point. Then $X$ is compact, Hausdorff and scattered, but does not embed in $\mathcal N(G)$ for any locally compact Hausdorff group $G$ .

Here $\omega_1$ is the first uncountable ordinal, and ordinal intervals carry their order topology. A space is scattered if each of its nonempty subspaces has an isolated point.

Separation in the subgroup space

Inclusion is a closed partial order on $\mathcal S(G)$ . Indeed, if $A\nsubseteq B$ , choose $a\in A\setminus B$ and an open neighbourhood $V$ of $a$ with $\overline V\cap B=\varnothing$ . Then

V^-\times(G\setminus\overline V)^+

is a neighbourhood of $(A,B)$ containing no pair $(A',B')$ with $A'\subseteq B'$ . In particular, $\mathcal S(G)$ is Hausdorff: its diagonal is the intersection of the inclusion relation with its inverse.

Lemma 3. Let $H,K$ be closed subgroups of $G$ , with $K$ noncompact and $K\nsubseteq H$ . For every compact set $D\subseteq G$ , there is a point in $K\setminus(H\cup D)$ .

Proof. Suppose $K\setminus H\subseteq D$ , and choose $a\in K\setminus H$ . If $x\in K\cap H$ , then $ax\in K\setminus H\subseteq D$ , so $x\in a^{-1}D$ . Hence

K\subseteq D\cup a^{-1}D.

Since $K$ is closed, it is compact, a contradiction. ◻

Lemma 4. Let $H\in\mathcal S(G)$ and $K_n\in\mathcal N(G)$ , with $K_n\nsubseteq H$ for every $n\ge0$ . There is an open set $U\supseteq H$ such that $K_n\nsubseteq U$ for every $n$ . Consequently,

H\notin\overline{\{K_n:n\ge0\}}.

Proof. Choose an open identity neighbourhood $W$ with compact closure, and set

Q=(\overline W)^{-1}\overline W.

Using Lemma 3 recursively, choose

x_n\in K_n\setminus H,\qquad x_n\notin\bigcup_{i<n}x_iQ.

Each translate $gW$ contains at most one of the points $x_n$ : otherwise $x_i^{-1}x_j\in W^{-1}W\subseteq Q$ for some $i<j$ . Thus $E=\{x_n:n\ge0\}$ is closed. More explicitly, if $g\notin E$ , deleting the at most one point of $E\cap gW$ from $gW$ gives an open neighbourhood of $g$ disjoint from $E$ .

Now $U=G\setminus E$ is open, contains $H$ , and contains none of the subgroups $K_n$ . The upper Vietoris neighbourhood $U^+$ of $H$ is disjoint from $\{K_n:n\ge0\}$ . ◻

A compact-order lemma

Let $P$ be a compact Hausdorff space with a closed partial order. Write $\mathord{\downarrow}x$ and $\mathord{\uparrow}x$ for its principal lower and upper sets. Assume that

\tag{9} x\notin\overline C \quad\text{whenever $C\subseteq P\setminus\mathord{\downarrow}x$ is countable.}

Lemma 5. Under condition (9), every principal ideal $\mathord{\downarrow}x$ is clopen.

Proof. We first establish three order properties.

The strict order is well founded. Otherwise, choose a strictly decreasing sequence $x_0>x_1>\cdots$ and a cluster point $x$ of this sequence. The closedness of principal ideals gives $x\le x_{n+1}<x_n$ for every $n$ . Thus all $x_n$ lie outside $\mathord{\downarrow}x$ , contrary to (9).

Every point of a nonempty closed set $F\subseteq P$ lies below a maximal element of $F$ . To apply Zorn’s lemma above a given point, note that the closed sets

F\cap\mathord{\uparrow}c\qquad(c\in C)

have the finite intersection property for every nonempty chain $C\subseteq F$ . Compactness supplies an upper bound in $F$ .

The set $M(F)$ of maximal elements of a closed set $F$ is finite. If not, choose a countably infinite subset $C\subseteq M(F)$ and an accumulation point $x\in F$ . For $y\in C\setminus\{x\}$ , maximality implies $y\nleq x$ . But $x\in\overline{C\setminus\{x\}}$ , contradicting (9).

These facts imply that whenever $x\in\overline B$ , some $m\in M(\overline B)$ satisfies

\tag{11} x\in\overline{B\cap\mathord{\downarrow}m}.

Indeed, the finitely many sets $B\cap\mathord{\downarrow}m$ , for $m\in M(\overline B)$ , cover $B$ .

Suppose now that $\mathord{\downarrow}x$ is not a neighbourhood of $x$ . Then $x\in\overline{P\setminus\mathord{\downarrow}x}$ . By (11) and well-foundedness, choose a minimal $m$ for which there is a set $B\subseteq P\setminus\mathord{\downarrow}x$ satisfying (11). Since $\mathord{\downarrow}m$ is closed, $x\le m$ ; equality would make $B\cap\mathord{\downarrow}m$ empty. Hence $x<m$ .

Choose a closed neighbourhood $N$ of $x$ with $m\notin N$ , and put

C=B\cap\mathord{\downarrow}m\cap N.

Then $x\in\overline C$ . Applying (11) again gives $m'\in M(\overline C)$ with $x\in\overline{C\cap\mathord{\downarrow}m'}$ . Since $\overline C\subseteq\mathord{\downarrow}m\cap N$ , we have $m'<m$ , contradicting the choice of $m$ .

Thus $\mathord{\downarrow}x$ is a neighbourhood of $x$ for every $x$ . If $y\le x$ , then $\mathord{\downarrow}y\subseteq\mathord{\downarrow}x$ , so $\mathord{\downarrow}x$ is a neighbourhood of each of its points. It is therefore open, and it is closed because the order is closed. ◻

Corollary 6. If $\mathcal K\subseteq\mathcal N(G)$ is compact, then

\{K\in\mathcal K:K\subseteq H\}

is clopen in $\mathcal K$ for every $H\in\mathcal K$ .

Proof. Equip $\mathcal K$ with inclusion. The order is closed, and Lemma 4 gives (9). Apply Lemma 5. ◻

Discrete abelian representations

Proof of Theorem 1. An embedding $e:X\to\mathcal N(G)$ pulls subgroup inclusion back to a partial order on $X$ . Its principal ideals are clopen by Corollary 6. This proves (i) $\Rightarrow$ (ii).

Assume (ii), and identify $D_X$ with the pairs $(n,f)$ , where $n\in\mathbb Z$ and $f:X\to\mathbb F_2$ has finite support. Set

\tag{14} H_x=\{(n,f)\in D_X:f(t)=0\text{ whenever }x\le t\}.

This is a subgroup, and it is closed because $D_X$ is discrete. It contains $\mathbb Z\oplus0$ , so it is infinite and hence noncompact.

Let $e_t$ denote the element with integer coordinate zero and second coordinate the basis vector at $t$ . Then

\tag{15} e_t\in H_x\ \Longleftrightarrow\ x\nleq t, \qquad H_x\subseteq H_y\ \Longleftrightarrow\ x\le y.

For the nontrivial direction of the second equivalence, if $x\nleq y$ , then $e_y\in H_x\setminus H_y$ . The other direction follows immediately from (14). In particular, $x\mapsto H_x$ is injective.

For a fixed $g=(n,f)\in D_X$ , the membership set is

\tag{16} \{x:g\in H_x\} =\bigcap_{t\in\mathrm{supp}(f)}(X\setminus\mathord{\downarrow}t),

and is clopen. Thus the inverse image of $U^-$ is open, being the union of these membership sets over $g\in U$ .

If $H_x\subseteq U$ , then $H_y\subseteq H_x\subseteq U$ for every $y\le x$ . The open neighbourhood $\mathord{\downarrow}x$ is therefore contained in the inverse image of $U^+$ . This proves continuity for the upper subbase as well. A continuous injection from compact $X$ into the Hausdorff space $\mathcal N(D_X)$ is an embedding, proving (ii) $\Rightarrow$ (iii). Finally, a discrete group is locally compact, so (iii) $\Rightarrow$ (i). ◻

The collapsed ordinal square

Put $Y=[0,\omega_1]^2$ and

U=\{(\alpha,\beta)\in Y:\alpha<\beta\}.

The space $U$ is open and locally compact. Realize $X$ as its one-point compactification $U\cup\{\infty\}$ . Define $q:Y\to X$ to be the identity on $U$ and to send $Y\setminus U=\Delta$ to $\infty$ . This map is continuous: away from $\infty$ use that $U$ is open, and at $\infty$ use that compact subsets of $U$ are closed in $Y$ . It is surjective, hence a quotient map from compact $Y$ to Hausdorff $X$ .

The space $Y$ is scattered. Given a nonempty subset, choose the least first coordinate $\alpha$ occurring in it and then the least second coordinate $\beta$ occurring above $\alpha$ . The open rectangle $[0,\alpha]\times[0,\beta]$ isolates that point in the subset. The same argument applies to subsets of $U$ . Every nonempty subset of $X$ either meets $U$ , where an isolated point has an open isolating neighbourhood in $X$ , or is $\{\infty\}$ . Thus $X$ is scattered.

Proposition 7. The space $X$ admits no partial order with clopen principal ideals.

Proof. Suppose that such an order $\preccurlyeq$ exists, and write $V_x=\{y:y\preccurlyeq x\}$ . Since $q^{-1}(V_\infty)$ is an open neighbourhood of $(\omega_1,\omega_1)$ , there is $c<\omega_1$ such that

\tag{18} q(\alpha,\beta)\preccurlyeq\infty \qquad(c<\alpha,\beta\le\omega_1).

For $\alpha<\omega_1$ , put $p_\alpha=q(\alpha,\omega_1)$ . Openness of $q^{-1}(V_{p_\alpha})$ at $(\alpha,\omega_1)$ gives a countable ordinal $g(\alpha)>\alpha$ with

\tag{19} q(\alpha,g(\alpha))\preccurlyeq p_\alpha.

Choose $\alpha_0=c+1$ and $\alpha_{n+1}=g(\alpha_n)$ . Then

\lambda=\sup_{n<\omega}\alpha_n<\omega_1, \qquad \alpha_n\longrightarrow\lambda, \qquad \alpha_{n+1}\longrightarrow\lambda.

By continuity of $q$ , $p_{\alpha_n}\to p_\lambda$ . Since $V_{p_\lambda}$ is open and contains $p_\lambda$ , eventually $p_{\alpha_n}\preccurlyeq p_\lambda$ . Equation (19) and transitivity yield

q(\alpha_n,\alpha_{n+1})\preccurlyeq p_\lambda \quad\text{eventually}.

The left-hand points converge to $q(\lambda,\lambda)=\infty$ . Closedness of $V_{p_\lambda}$ therefore gives $\infty\preccurlyeq p_\lambda$ . On the other hand, $\lambda>c$ , so (18) gives $p_\lambda\preccurlyeq\infty$ . Antisymmetry forces $p_\lambda=\infty$ , contrary to $\lambda<\omega_1$ . ◻

Proof of Theorem 2. The space $X$ is compact, Hausdorff and scattered by the preceding construction. Proposition 7 excludes condition (ii) of Theorem 1, and hence excludes every embedding into $\mathcal N(G)$ . ◻

References

Preprint · Lean (GitHub)

U. Abraham, R. Bonnet and M. Rubin, On a superatomic Boolean algebra which is not generated by a well-founded sublattice, Israel J. Math. 123 (2001), 221–239.

T. Banakh, R. Bonnet and W. Kubiś, Vietoris hyperspaces over scattered Priestley spaces, Israel J. Math. 249 (2022), 37–81. Preprint: arXiv:2007.12890.

E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st ed., September 2026 update, Problem 9.47, arXiv:1401.0300.

V. S. Muldagaliev and K. R. Ermekqali, Compacta in the space of subgroups of a topological group (Russian), in Taymanov Readings–2022, M. Utemisov West Kazakhstan University, Oral, 2022, 31–34.

I. V. Protasov, Compact subspaces in the space of subgroups of a topological group, Ukrainian Math. J. 38 (1986), 512–516.

I. V. Protasov, Selective survey on spaces of closed subgroups of topological groups, Axioms 7 (2018), no. 4, article 75.