Topological groups, Lie groups Definition

Hausdorff topological group

Builds on Topological group Hausdorff space Closed set

Definition

A Hausdorff topological group is a topological group $G$ whose topology is Hausdorff. 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. 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.

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Tag	`A4E24P47KP`
URI	`m2://A4E24P47KP`
Kind	definition
Writing	W0 stub
MSC	22A05, 54H11
Defines	Hausdorff topological group
Uses	Topological group, Hausdorff space, Closed set
Changed	2 November 2026

Hausdorff topological group

Definition

Remarks

Reading path

Further reading

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