# homogeneous topological space

## Definitions

A topological space^{} $X$ is said to be *homogeneous ^{}*
if for all $a,b\in X$ there is a homeomorphism

^{}$\varphi :X\to X$ such that $\varphi (a)=b$.

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

## 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 $x\mapsto a{x}^{-1}b$ 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.

Title | homogeneous topological space |
---|---|

Canonical name | HomogeneousTopologicalSpace |

Date of creation | 2013-03-22 16:18:21 |

Last modified on | 2013-03-22 16:18:21 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 5 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54D99 |

Synonym | homogeneous space |

Defines | homogeneous |

Defines | bihomogeneous |

Defines | bihomogeneous space |

Defines | bihomogeneous topological space |