homogeneous topological space

initial point

Definitions

A topological space $X$ is said to be homogeneousif 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 bihomogeneousif 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 bihomogeneousas its self-homeomorphisms are all order-preserving.This can be considered a pathological example,as most homogeneous topological spaces encountered in practiceare 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 neighbourhoodcan 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.