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topological transformation group∗
CWoo†
2013-03-21 22:06:28
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 α : G × X → X, where G × X is given the
product topology
2. α(1, x) = x, and
3. α(g1 g2 , x) = α(g1 , α(g2 , x)).
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 (g1 g2 )x = g1 (g2 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 hg : X → X by
hg (x) = gx for each g ∈ G so that hg is the identity function precisely when
g = 1.
Some Examples.
1. Let X = Rn , and G be the group of n × n matrices over 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, α : G × G → G 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.
∗ hTopologicalTransformationGroupi
created: h2013-03-21i by: hCWooi version: h38955i
Privacy setting: h1i hDefinitioni h54H15i h22F05i
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