Editing Homeomorphism group (section)

==Properties and examples==
There is a natural [[Group action (mathematics)|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 group action|transitive]], then the space is said to be [[Homogeneous space|homogeneous]].

===Topology===
{{Expand section|date=March 2009}}
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 space|regular]], [[locally compact space]]s the group multiplication is then continuous.

If the space is [[Compact space|compact]] and [[Hausdorff space|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 space|locally connected]] this holds as well.<ref name="vu.nl">{{citation
 | last = Dijkstra | first = Jan J.
 | doi = 10.2307/30037630
 | issue = 10
 | journal = [[The American Mathematical Monthly]]
 | mr = 2186833
 | pages = 910–912
 | title = On homeomorphism groups and the compact-open topology
 | url = https://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf
 | volume = 112
 | year = 2005| jstor = 30037630
 }}</ref> 
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.<ref name="vu.nl"/>