# Homogeneous space

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

## 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 $X$ is said to be homogeneous if it satisfies the following equivalent conditions:

1. For any points $x,y \in X$, there is a homeomorphism $\varphi:X \to X$ such that $\varphi(x) = y$.
2. The self-homeomorphism group of $X$ is transitive on $X$.

## 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.

### Non-examples

• The closed interval $[0,1]$ is not homogeneous, because there is no homeomorphism sending $0$ to any point in the open interval $(0,1)$.
• 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