# Homogeneous space

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

## Contents

## Definition

### Symbol-free definition

A topological space is said to be **homogeneous** if given any two points in it, there is a homeomorphism from the topological space to itself that maps the first point to the second. In other words, the self-homeomorphism group is transitive on the points of the topological space.

### Definition with symbols

A topological space is said to be **homogeneous** if it satisfies the following equivalent conditions:

- For any points , there is a homeomorphism such that .
- The self-homeomorphism group of is transitive on .

## Examples

### Extreme examples

- The empty space is homogeneous for trivial reasons.
- The one-point space is homogeneous for trivial reasons.
- The discrete topology and trivial topology both give homogeneous spaces.

### Mainstream examples

- The real line, Euclidean space, connected manifolds, and the underlying spaces of topological groups are all homogeneous.

### Non-examples

- The closed interval is not homogeneous, because there is no homeomorphism sending to any point in the open interval .
- A pair of intersecting lines is not homogeneous because every homeomorphism fixes the point of intersection.

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Underlying space of topological group | underlying space for a topological group | underlying space of topological group implies homogeneous | ||

Underlying space of T0 topological group | underlying space for a T0 topological group | |||

Connected manifold | connected manifold implies homogeneous | |FULL LIST, MORE INFO | ||

Compactly homogeneous space |