---
tag: 2B8KYPCPAS
kind: concept
name: Topological transformation group
msc: [54H15, 22F05]
specializes: [0WCY42EV9V]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_group_action
same_as:
  wikidata: Q16951242
  wikipedia: Continuous group action
---

## Summary

A topological transformation group is a [[topological group|92700QAB5Y]] acting on a topological space so that the action map is continuous.
It is the topological version of a [[group action|0WCY42EV9V]], and it is the basic object of topological dynamics.
Each group element then acts by a homeomorphism of the space.
Homogeneous spaces, flows and actions of Lie groups on manifolds are all examples.

## Explanations

### L1 human

A group can move the points of a space around: rotations move the points of a sphere, for example.
In a topological transformation group, the motion depends continuously both on the group element and on the point.
Rotating by a slightly different angle, or starting at a slightly different point, changes the result only slightly.
The rotation group acting on the sphere is the standard picture.