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Transcript
```homogeneous topological space∗
yark†
2013-03-21 21:17:42
initial point
Definitions
A topological space X is said to be homogeneous if for all a, b ∈ X there is a
homeomorphism φ: X → X such that φ(a) = b.
A topological space X is said to be bihomogeneous if for all a, b ∈ X there
is a homeomorphism φ: X → X such that φ(a) = b and φ(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 ∈ G, then x 7→ ax−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.
∗ hHomogeneousTopologicalSpacei
created: h2013-03-21i by: hyarki version: h38428i
Privacy setting: h1i hDefinitioni h54D99i
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