---
tag: A34VR4FMRE
kind: concept
name: Topological homogeneity
msc: [54H15, 22F30]
sources:
  - url: https://en.wikipedia.org/wiki/Homogeneous_space
---

## Summary

A topological space is homogeneous when for any two of its points some [[homeomorphism|0QT00PB5G2]] of the space carries the first point to the second.
So the space looks the same near every point.
Every [[topological group|92700QAB5Y]] is homogeneous, because translations are homeomorphisms.
Homogeneity is therefore a first test of whether a space can carry the structure of a topological group.

## Explanations

### L1 human

A homogeneous space has no special points: whatever you can say about the space near one point, you can say near any other.
The circle is homogeneous, since rotating it moves any point to any other.
A closed interval $[0, 1]$ is not homogeneous: an endpoint is different from an inner point.
Removing an endpoint leaves the interval connected, while removing an inner point cuts it in two.
So a closed interval cannot be made into a topological group.