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A locally compact subgroup of a Hausdorff topological group is closed

Statement

Let $G$ be a Hausdorff topological group (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, Topological group: multiplication and inversion are continuous) and let $H \leq G$ be a subgroup (Subgroup) which is locally compact in the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space). Then $H$ is closed in $G$ . Conversely, if $G$ is locally compact and $H$ is closed in $G$ , then $H$ is locally compact in the subspace topology. In particular, closed subgroups of locally compact Hausdorff abelian groups are again locally compact Hausdorff abelian.

No choice principle is used.

Facts & Assumptions

Given: A Hausdorff topological group $G$ , a subgroup $H \leq G$ locally compact in the subspace topology, and a point $x \in G$ .

[F1]

Local compactness of $H$ gives an $H$ -open neighbourhood $V$ of the identity $e$ whose $H$ -closure $C : = {c l}_{H} (V)$ is compact in $H$ . For a subspace $H$ of $G$ the closure traces exactly: ${c l}_{H} (V) = {c l}_{G} (V) \cap H$ , and $V = H \cap O$ for some $O$ open in $G$ . (In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, For $A \subseteq S \subseteq X$ the closure of $A$ in $S$ is ${\overline{A}}^{X} \cap S$ , while the interior only contains ${int}^{X} (A) \cap S$ , with equality when $S$ is open; and a dense subset of $X$ traces to a dense subset of every open $S$ , Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace)

[F2]

$G$ is a topological group: for fixed $a \in G$ the translations $x \mapsto a + x$ and $x \mapsto x + a$ and the inversion $x \mapsto - x$ are homeomorphisms, hence map open sets to open sets and preserve closures. (Topological group: multiplication and inversion are continuous, Left and right translations and inversion in a topological group are homeomorphisms, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological)

[F3]

A closed subset of a compact space is compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact). A subset compact in the subspace $H$ is compact in $G$ , and a compact subset of the Hausdorff space $G$ is closed. (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not)

[F4]

A point $x$ lies in ${c l}_{G} (H)$ if and only if every open neighbourhood of $x$ meets $H$ . If $O \subseteq G$ is open and $y \in {c l}_{G} (H) \cap O$ , then $y \in {c l}_{G} (H \cap O)$ : every open neighbourhood $N$ of $y$ has $N \cap O$ an open neighbourhood of $y$ , which meets $H$ , hence meets $H \cap O$ . (Interior, closure, boundary, exterior, derived set and isolated point in a topological space, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, For $A \subseteq S \subseteq X$ the closure of $A$ in $S$ is ${\overline{A}}^{X} \cap S$ , while the interior only contains ${int}^{X} (A) \cap S$ , with equality when $S$ is open; and a dense subset of $X$ traces to a dense subset of every open $S$ )

Proof

1.1F1F3

Choose an $H$ -open neighbourhood $V$ of $e$ with $C : = {c l}_{H} (V)$ compact in $H$ , and write $V = H \cap O$ with $O$ open in $G$ . Then $C$ is compact in $G$ and closed in $G$ , and $V \subseteq C \subseteq H$ with ${c l}_{G} (V) \subseteq C$ because $C$ is closed in $G$ and contains $V$ .

2.1F2F4step 1.1

Let $x \in {c l}_{G} (H)$ . The set $x - O = {x - o : o \in O}$ is an open neighbourhood of $x$ by [F2], so by the closure characterisation it meets $H$ : there are $h \in H$ and $o \in O$ with $h = x - o$ , that is $x = h + o$ .

3.1F4step 1.1step 2.1

With $h, o$ as in step 2.1, the point $o = (- h) + x$ lies in ${c l}_{G} (H)$ , because $- h + H = H$ and translations preserve closures; and $o \in O$ , so $o \in {c l}_{G} (H) \cap O \subseteq {c l}_{G} (H \cap O) = {c l}_{G} (V) \subseteq C \subseteq H$ , using $V = H \cap O$ and the inclusion of [F4].

4.1F1F2F3step 3.1step 2.1∎

Since $o \in H$ and $x = h + o$ with $h \in H$ , the subgroup $H$ contains $x$ . Hence every $x \in {c l}_{G} (H)$ lies in $H$ , that is ${c l}_{G} (H) = H$ and $H$ is closed in $G$ . Conversely, suppose $G$ is locally compact and $H$ is closed. For $h \in H$ , a compact neighbourhood $N$ of $h$ in $G$ gives a compact neighbourhood $N \cap H$ in $H$ : it is closed in the compact space $N$ and contains the trace on $H$ of an open neighbourhood of $h$ . The Hausdorff property and continuous group operations restrict to $H$ , as does abelianness. Thus a closed subgroup of an LCA group is LCA.

Remarks

The proof uses no compactness of $G$ , no abelianness, and no choice principle: the single compact set is ${c l}_{H} (V)$ , supplied by local compactness of $H$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Linus Kramer, Locally Compact Groups and Lie Groups, Chapter 1 (author lecture notes, 2020) (standard reference, not scraped)