In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. They are important to the theory of topological spaces, generally exemplary of automorphism groups and topologically invariant in the group isomorphism sense.
Properties and examples
There is a natural group action of the homeomorphism group of a space on that space. Let <math>X</math> be a topological space and denote the homeomorphism group of <math>X</math> by <math>G</math>. The action is defined as follows:
<math>\begin{align}
G\times X &\longrightarrow X\\
(\varphi, x) &\longmapsto \varphi(x)
\end{align}</math>
This is a group action since for all <math>\varphi,\psi\in G</math>,
<math>\varphi\cdot(\psi\cdot x)=\varphi(\psi(x))=(\varphi\circ\psi)(x)</math>,
where <math>\cdot</math> denotes the group action, and the identity element of <math>G</math> (which is the identity function on <math>X</math>) sends points to themselves. If this action is transitive, then the space is said to be homogeneous.
Topology
As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the compact-open topology.
In the case of regular, locally compact spaces the group multiplication is then continuous.
If the space is compact and Hausdorff, the inversion is continuous as well and <math>\operatorname{Homeo}(X)</math> becomes a topological group. If <math>X</math> is Hausdorff, locally compact, and locally connected this holds as well.
Some locally compact separable metric spaces exhibit an inversion map that is not continuous, resulting in <math>\text{Homeo}(X)</math> not forming a topological group.