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Version 1 |
| \PMlinkescapeword{argument} |
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| \PMlinkescapeword{euclidean} |
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| \PMlinkescapeword{even} |
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| \PMlinkescapephrase{initial point} |
\PMlinkescapephrase{initial point} |
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\PMlinkescapeword{self-homeomorphisms} % until we can link to something correct |
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| \section*{Definitions} |
\section*{Definitions} |
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| A topological space $X$ is said to be \emph{homogeneous} |
A topological space $X$ is said to be \emph{homogeneous} |
| if for all $a,b\in X$ there is a homeomorphism $\phi\colon X\to X$ |
if for all $a,b\in X$ there is a homeomorphism $\phi\colon X\to X$ |
| such that $\phi(a)=b$. |
such that $\phi(a)=b$. |
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| A topological space $X$ is said to be \emph{bihomogeneous} |
A topological space $X$ is said to be \emph{bihomogeneous} |
| if for all $a,b\in X$ there is a homeomorphism $\phi\colon X\to X$ |
if for all $a,b\in X$ there is a homeomorphism $\phi\colon X\to X$ |
| such that $\phi(a)=b$ and $\phi(b)=a$. |
such that $\phi(a)=b$ and $\phi(b)=a$. |
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| \section*{Examples} |
\section*{Examples} |
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| The long line (without initial point) is homogeneous, |
The long line (without initial point) is homogeneous, |
| but it is not bihomogeneous |
but it is not bihomogeneous |
| as its self-homeomorphisms are all order-preserving. |
as its self-homeomorphisms are all order-preserving. |
| This can be considered a pathological example, |
This can be considered a pathological example, |
| as most homogeneous topological spaces encountered in practice |
as most homogeneous topological spaces encountered in practice |
| are also bihomogeneous. |
are also bihomogeneous. |
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| Every topological group is bihomogeneous. |
Every topological group is bihomogeneous. |
| To see this, note that if $G$ is a topological group and $a,b\in G$, |
To see this, note that if $G$ is a topological group and $a,b\in G$, |
| then $x\mapsto ax^{-1}b$ defines a homeomorphism interchanging $a$ and $b$. |
then $x\mapsto ax^{-1}b$ defines a homeomorphism interchanging $a$ and $b$. |
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| Every connected topological manifold without boundary is homogeneous. |
Every connected topological manifold without boundary is homogeneous. |
| This is true even if we do not require our manifolds to be paracompact, |
This is true even if we do not require our manifolds to be paracompact, |
| as any two points share a Euclidean neighbourhood, |
as any two points share a Euclidean neighbourhood, |
| and a suitable homeomorphism for this neighbourhood |
and a suitable homeomorphism for this neighbourhood |
| can be extended to the whole manifold. |
can be extended to the whole manifold. |
| In fact, except for the long line (as mentioned above), |
In fact, except for the long line (as mentioned above), |
| every connected topological manifold without boundary is bihomogeneous. |
every connected topological manifold without boundary is bihomogeneous. |
| This is for essentially the same reason, |
This is for essentially the same reason, |
| except that the argument breaks down for $1$-manifolds. |
except that the argument breaks down for $1$-manifolds. |
