topological transformation group
Let $G$ be a topological group^{} and $X$ any topological space^{}. We say that $G$ is a topological transformation group of $X$ if $G$ acts on $X$ continuously, in the following sense:

1.
there is a continuous function^{} $\alpha :G\times X\to X$, where $G\times X$ is given the product topology

2.
$\alpha (1,x)=x$, and

3.
$\alpha ({g}_{1}{g}_{2},x)=\alpha ({g}_{1},\alpha ({g}_{2},x))$.
The function $\alpha $ is called the (left) action of $G$ on $X$. When there is no confusion, $\alpha (g,x)$ is simply written $gx$, so that the two conditions above read $1x=x$ and $({g}_{1}{g}_{2})x={g}_{1}({g}_{2}x)$.
If a topological transformation group $G$ on $X$ is effective, then $G$ can be viewed as a group of homeomorphisms on $X$: simply define ${h}_{g}:X\to X$ by ${h}_{g}(x)=gx$ for each $g\in G$ so that ${h}_{g}$ is the identity function precisely when $g=1$.
Some Examples.

1.
Let $X={\mathbb{R}}^{n}$, and $G$ be the group of $n\times n$ matrices over $\mathbb{R}$. Clearly $X$ and $G$ are both topological spaces with the usual topology. Furthermore, $G$ is a topological group. $G$ acts on $X$ continuous if we view elements of $X$ as column vectors^{} and take the action to be the matrix multiplication^{} on the left.

2.
If $G$ is a topological group, $G$ can be considered a topological transformation group on itself. There are many continuous actions that can be defined on $G$. For example, $\alpha :G\times G\to G$ given by $\alpha (g,x)=gx$ is one such action. It is continuous, and satisfies the two action axioms. $G$ is also effective with respect to $\alpha $.
Title  topological transformation group 

Canonical name  TopologicalTransformationGroup 
Date of creation  20130322 16:43:58 
Last modified on  20130322 16:43:58 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54H15 
Classification  msc 22F05 
Defines  effective topological transformation group 