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The quotient of an LCA group by a closed subgroup is LCA

Statement

Let $G$ be a locally compact Hausdorff abelian topological group (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, 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 closed subgroup (Subgroup). Then the quotient group $G / H$ with the quotient topology (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ , The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection) is a locally compact Hausdorff abelian topological group. No choice principle is used.

Facts & Assumptions

Given: A locally compact Hausdorff abelian group $G$ , a closed subgroup $H \leq G$ , and the quotient map $q : G \to G / H$ .

[F1]

$G / H$ is the abelian group of cosets with $(x + H) + (y + H) = x + y + H$ and the quotient topology, the finest topology making $q$ continuous; a set $V \subseteq G / H$ is open exactly when $q^{- 1} (V)$ is open in $G$ . (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ , The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, Subgroup, Every quotient group of an abelian group is abelian)

[F2]

Translations and inversion in $G$ are homeomorphisms and the group operations are continuous; for open $U$ and $h \in H$ the translate $U + h$ is open. (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 continuous image of a compact set is compact; a compact subset of a Hausdorff space is closed; every point of a locally compact space has a compact neighbourhood. (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, 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, 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, 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)

[F4]

If $q : X \to Y$ is a continuous open surjection then $q$ and $q \times q$ are quotient maps, and a map out of $Y$ is continuous exactly when its composite with $q$ is; composites and coordinatewise maps into products are handled by the universal properties. (A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps, For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map, A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice)

Proof

1.1F1F2

The quotient map $q$ is open: for open $U \subseteq G$ one has $q^{- 1} (q (U)) = U + H = ⋃_{h \in H} (U + h)$ , a union of open translates, hence open in $G$ ; by the definition of the quotient topology $q (U)$ is open in $G / H$ .

2.1F1F2step 1.1

$G / H$ is Hausdorff: let $q (x) \neq q (y)$ , so $x - y \notin H$ . Since $H$ is closed, choose an open neighbourhood $W$ of $x - y$ with $W \cap H = \emptyset$ ; by continuity of subtraction there are open neighbourhoods $U, V$ of $0$ with $(x - y) + U - V \subseteq W$ . Then $q (x + U)$ and $q (y + V)$ are open by step 1.1 and are disjoint: if $x + u + H = y + v + H$ with $u \in U$ , $v \in V$ , then $x - y + u - v \in H \cap W = \emptyset$ , a contradiction.

3.1F3step 1.1step 2.1

$G / H$ is locally compact: given $x \in G$ , choose a compact neighbourhood $N$ of $x$ and an open $U$ with $x \in U \subseteq N$ . Then $q (N)$ is compact as a continuous image of $N$ , and it is closed because $G / H$ is Hausdorff by step 2.1; $q (U)$ is an open neighbourhood of $q (x)$ contained in $q (N)$ . Hence $q (N)$ is a compact neighbourhood of $q (x)$ .

4.1F1F2F4step 1.1step 2.1step 3.1∎

The quotient operations are continuous: the product $q \times q : G \times G \to G / H \times G / H$ is a continuous open surjection (images of basic open rectangles are open rectangles), hence a quotient map, and $m_{G / H} \circ (q \times q) = q \circ m_{G}$ where $m_{G}$ and $m_{G / H}$ are the respective addition maps. The quotient universal property therefore makes addition on $G / H$ continuous, and inversion descends in the same way from inversion in $G$ . Thus $G / H$ is an abelian topological group which is Hausdorff and locally compact, as claimed.

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Sources

Lynn H. Loomis, Introduction to Abstract Harmonic Analysis, D. Van Nostrand, 1953 (Harvard-hosted full scan) (standard reference, not scraped)
T. W. Koerner, Topological Groups (author lecture notes) (standard reference, not scraped)