---
tag: A4E24P47KP
kind: definition
name: Hausdorff topological group
msc: [22A05, 54H11]
defines: 7GKCGQ7215
uses: [92700QAB5Y, 0S1FJ8VEX9, 0ZCK8DQYN1]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Definition

A *Hausdorff topological group* is a [[topological group|92700QAB5Y]] $G$ whose topology is [[Hausdorff|0S1FJ8VEX9]].
That is, for any two distinct $x, y \in G$ there are disjoint open sets $U \ni x$ and $V \ni y$.

## Remarks

For a topological group $G$ with identity $e$, the following are equivalent: $G$ is Hausdorff; $G$ is $T_1$; $G$ is $T_0$; the set $\set{e}$ is [[closed|0ZCK8DQYN1]].
They are also equivalent to: the intersection of all neighbourhoods of $e$ is $\set{e}$.
Example: $(\R, +)$ with the usual topology, which comes from a metric and so is Hausdorff.
Non-example: $\Z/2\Z$ with the indiscrete topology is a topological group, but its two points have no disjoint open neighbourhoods.
In fact a topological group is Hausdorff if and only if it is a Tychonoff space.