## You are here

Homehomogeneous topological space

## Primary tabs

# homogeneous topological space

# Definitions

A topological space $X$ is said to be *homogeneous*
if for all $a,b\in X$ there is a homeomorphism $\phi\colon X\to X$
such that $\phi(a)=b$.

A topological space $X$ is said to be *bihomogeneous*
if for all $a,b\in X$ there is a homeomorphism $\phi\colon X\to X$
such that $\phi(a)=b$ and $\phi(b)=a$.

# Examples

The long line (without initial point) is homogeneous, but it is not bihomogeneous as its self-homeomorphisms are all order-preserving. This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous.

Every topological group is bihomogeneous. To see this, note that if $G$ is a topological group and $a,b\in G$, then $x\mapsto ax^{{-1}}b$ defines a homeomorphism interchanging $a$ and $b$.

Every connected topological manifold without boundary is homogeneous. This is true even if we do not require our manifolds to be paracompact, as any two points share a Euclidean neighbourhood, and a suitable homeomorphism for this neighbourhood can be extended to the whole manifold. In fact, except for the long line (as mentioned above), every connected topological manifold without boundary is bihomogeneous. This is for essentially the same reason, except that the argument breaks down for $1$-manifolds.

## Mathematics Subject Classification

54D99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections