Topological groups, Lie groups Definition

Locally compact abelian group

Builds on Topological group Abelian group Locally compact space

Definition

A locally compact abelian group is a topological group $G$ such that:

the group $G$ is abelian;
the topology of $G$ is Hausdorff;
the topology of $G$ is locally compact, that is, each point of $G$ has a compact neighbourhood.

Remarks

Example: $\R^n$ with addition and the usual topology is an LCA group. It is abelian, Hausdorff, and each point $x$ has the compact neighbourhood $\set{y : \norm{y - x} \le 1}$ . Non-example: the additive group $\Q$ of rational numbers with the topology from $\R$ is abelian and Hausdorff, but it is not locally compact. Each neighbourhood of $0$ contains a set $(-\varepsilon, \varepsilon) \cap \Q$ , hence a sequence of rationals that converges in $\R$ to an irrational number. This sequence has no subsequence that converges in $\Q$ , so no neighbourhood of $0$ is contained in a compact subset of $\Q$ . The usual abbreviation is "LCA group". Many authors include the Hausdorff condition in the definition of a topological group.

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Abelian group concept
Locally compact space concept
Topological group concept
Locally compact abelian group definition

Tag	`Z6TBXH0MHR`
URI	`m2://Z6TBXH0MHR`
Kind	definition
Writing	W0 stub
MSC	22B05, 43A40
Defines	Locally compact abelian group
Uses	Topological group, Abelian group, Locally compact space
Changed	2 November 2026

Locally compact abelian group

Definition

Remarks

Reading path

Further reading

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