---
tag: DKX3EAVQX6
kind: statement
role: proposition
name: Open subgroups are closed
msc: [22A05]
uses: [QYCFH68RD1, 0ZCK8DQYN1, 03DEH43F4H, 0DTXBQPA2S, 92700QAB5Y]
about: [QYCFH68RD1]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Statement

Let $G$ be a [[topological group|92700QAB5Y]] and let $H$ be an [[open subgroup|QYCFH68RD1]] of $G$.
Then $H$ is a [[closed set|0ZCK8DQYN1]] in $G$.

## Remarks

The [[left cosets|03DEH43F4H]] $gH$ 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|0DTXBQPA2S]] 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.

## Explanations

### L1 human

If a subgroup of a topological group is open, it is automatically closed as well.
The reason is that the group splits into copies of the subgroup, all shifted, and each copy is open.
The part outside the subgroup is a union of these open copies, so it is open too.

### L2 human

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.