homogeneous topological space

initial point

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