homogeneous topological space


initial point


A topological spaceMathworldPlanetmath X is said to be homogeneousPlanetmathPlanetmath if for all a,bX there is a homeomorphismPlanetmathPlanetmath ϕ:XX such that ϕ(a)=b.

A topological space X is said to be bihomogeneous if for all a,bX there is a homeomorphism ϕ:XX such that ϕ(a)=b and ϕ(b)=a.


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 groupMathworldPlanetmath is bihomogeneous. To see this, note that if G is a topological group and a,bG, then xax-1b defines a homeomorphism interchanging a and b.

Every connectedPlanetmathPlanetmath topological manifoldMathworldPlanetmathPlanetmath 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