topological transformation group

Let G be a topological groupMathworldPlanetmath and X any topological spaceMathworldPlanetmath. We say that G is a topological transformation group of X if G acts on X continuously, in the following sense:

  1. 1.

    there is a continuous functionMathworldPlanetmathPlanetmath α:G×XX, where G×X is given the product topology

  2. 2.

    α(1,x)=x, and

  3. 3.


The function α is called the (left) action of G on X. When there is no confusion, α(g,x) is simply written gx, so that the two conditions above read 1x=x and (g1g2)x=g1(g2x).

If a topological transformation group G on X is effective, then G can be viewed as a group of homeomorphisms on X: simply define hg:XX by hg(x)=gx for each gG so that hg is the identity function precisely when g=1.

Some Examples.

  1. 1.

    Let X=n, and G be the group of n×n matrices over . 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 vectorsMathworldPlanetmath and take the action to be the matrix multiplicationMathworldPlanetmath on the left.

  2. 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, α:G×GG given by α(g,x)=gx is one such action. It is continuous, and satisfies the two action axioms. G is also effective with respect to α.

Title topological transformation group
Canonical name TopologicalTransformationGroup
Date of creation 2013-03-22 16:43:58
Last modified on 2013-03-22 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