---
tag: 7GKCGQ7215
kind: concept
name: Hausdorff topological group
msc: [22A05, 54H11]
specializes: [92700QAB5Y]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Summary

A Hausdorff topological group is a [[topological group|92700QAB5Y]] whose topology is [[Hausdorff|0S1FJ8VEX9]].
For topological groups this is a mild condition: it holds as soon as the one-point set of the identity is closed.
Many authors include it in the definition of a topological group.
Every topological group has a largest Hausdorff quotient, obtained by dividing by the closure of the identity.

## Explanations

### L1 human

In a Hausdorff space two different points can be put in two disjoint open sets.
For a topological group, this follows already from a much weaker looking condition: the identity alone forms a closed set.
The real line under addition and every matrix group are Hausdorff.
A group with the indiscrete topology, where the only open sets are the empty set and the whole group, is not Hausdorff once it has two elements.