---
tag: 92700QAB5Y
kind: concept
name: Topological group
msc: [22A05, 54H11]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_group
same_as:
  wikidata: Q1046291
  wikipedia: Topological group
---

## Summary

A topological group is a [[group|G5S1M9]] with a [[topology|0CQHFPGJ45]] in which multiplication and inversion are continuous.
It is the basic object where algebra and topology work together.
It is the setting of abstract harmonic analysis and of the theory of Lie groups.
Translations by group elements are homeomorphisms, so the topology near any point looks like the topology near the identity.
The real line under addition, the circle group, matrix groups and the $p$-adic integers are all topological groups.

## Explanations

### L1 human

A topological group is a group where it also makes sense to say that two elements are close.
The group law must respect closeness: if $x$ is near $x'$ and $y$ is near $y'$, then $xy$ is near $x'y'$, and $x^{-1}$ is near $x'^{-1}$.
Picture the real line under addition: shifting the whole line by a number does not tear it.
Another example is the circle of complex numbers of absolute value $1$ under multiplication, where multiplying means rotating.