---
tag: QYCFH68RD1
kind: concept
name: Open subgroup
msc: [22A05]
specializes: [0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Summary

An open subgroup of a [[topological group|92700QAB5Y]] is a [[subgroup|0X1JXT47GA]] that is an open set.
[[Open subgroups are also closed|DKX3EAVQX6]], so they are unions of connected components.
They are central in the study of totally disconnected groups.
In a profinite group the open normal subgroups form a base of neighbourhoods of the identity.
A connected topological group has no open subgroup other than itself.

## Explanations

### L1 human

An open subgroup is a subgroup that contains a whole neighbourhood of each of its points.
Because the group can be shifted around, it is enough that it contains a neighbourhood of the identity.
For example, in the $p$-adic integers the multiples of $p^n$ form an open subgroup for each $n$.
In the real line under addition, the only open subgroup is the whole line.