---
tag: P65JDBW4FH
kind: definition
name: Topological group
msc: [22A05, 54H11]
defines: 92700QAB5Y
uses: [G5S1M9, 0CQHFPGJ45, ZZK2GM1V15, 0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
---

## Definition

A *topological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that:

1. the multiplication $G \times G \to G$, $(x, y) \mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]], where $G \times G$ carries the product topology;
2. the inversion $G \to G$, $x \mapsto x^{-1}$, is [[continuous|0VGFHTM75Y]].

## Remarks

Example: $(\R, +)$ with the usual topology.
Addition $(x, y) \mapsto x + y$ is continuous on $\R^2$, and $x \mapsto -x$ is continuous.
Non-example: $(\R, +)$ with the lower limit topology, generated by the intervals $[a, b)$.
Addition is still continuous, but inversion fails clause 2: the preimage of the open set $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open.
The two clauses together are equivalent to the single condition that $(x, y) \mapsto x y^{-1}$ is continuous from $G \times G$ to $G$.
Every group with the discrete topology, and every group with the indiscrete topology, is a topological group.
Many authors also require the topology to be Hausdorff.