Definitions

A topological space 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 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 is a topological group and $a,b\in G$ , then $x\mapsto ax^{-1}b$ defines a homeomorphism interchanging and .

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 -manifolds.

"homogeneous topological space" is owned by yark.

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Other names:

homogeneous space

Also defines:

homogeneous, bihomogeneous, bihomogeneous space, bihomogeneous topological space

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Cross-references: neighbourhood, points, paracompact, manifolds, boundary, topological manifold, connected, topological group, pathological, self-homeomorphisms, long line, homeomorphism, topological space
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This is version 2 of homogeneous topological space, born on 2006-10-07, modified 2006-10-14.
Object id is 8428, canonical name is HomogeneousTopologicalSpace.
Accessed 3220 times total.

Classification:

AMS MSC:

54D99 (General topology :: Fairly general properties :: Miscellaneous)

Pending Errata and Addenda

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