Topological groups, Lie groups Topological groups: basic notions Proposition

Open subgroups are closed

Builds on Open subgroup Closed set Left coset Index of a subgroup Topological group

Statement

Let $G$ be a topological group and let $H$ be an open subgroup of $G$ . Then $H$ is a closed set in $G$ .

Remarks

The left cosets $gH$ are open, because left translation by $g$ is a homeomorphism. The complement of $H$ is the union of the cosets $gH$ with $g \notin H$ , so it is open, and $H$ is closed. Conversely, a closed subgroup of finite index is open, since its complement is a finite union of closed cosets. Example: in the $p$ -adic integers $\Z_p$ , each subgroup $p^n \Z_p$ is open, hence also closed. The converse of the statement fails: $\Z$ is a closed subgroup of $\R$ that is not open.

Going deeper

For a student

A first consequence: a connected topological group has no open subgroup except itself, since an open subgroup is a nonempty set that is both open and closed. For instance, a connected group is generated by any neighbourhood of the identity, because the subgroup generated by an open set is open. In profinite groups the open subgroups are exactly the closed subgroups of finite index, which links the topology with finite quotients.

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Tag	`DKX3EAVQX6`
URI	`m2://DKX3EAVQX6`
Kind	statement, proposition
Writing	W0 stub
MSC	22A05
Status	This copy of the site has no build data. Run `m2 status DKX3EAVQX6`.
About	Open subgroup
Uses	Open subgroup, Closed set, Left coset, Index of a subgroup, Topological group
Changed	2 November 2026