discontinuous action
Let be a topological space and a group that acts on by
homeomorphisms
. The action of is said to be
discontinuous
at if there is a neighborhood
of such
that the set
is finite. The action is called discontinuous if it is discontinuous at every point.
Remark 1.
If acts discontinuously then the orbits of the action have no
accumulation points, i.e. if is a sequence of distinct elements of
and
then the sequence has no limit points
. If is
locally compact then an action that satisfies this condition is discontinuous.
Remark 2.
Assume that is a locally compact Hausdorff space and let
denote the group of self homeomorphisms of endowed with the
compact-open topology
.
If defines a discontinuous action
then the image
is a discrete subset of .
Title | discontinuous action |
---|---|
Canonical name | DiscontinuousAction |
Date of creation | 2013-03-22 13:28:49 |
Last modified on | 2013-03-22 13:28:49 |
Owner | Dr_Absentius (537) |
Last modified by | Dr_Absentius (537) |
Numerical id | 7 |
Author | Dr_Absentius (537) |
Entry type | Definition |
Classification | msc 37B05 |
Related topic | PoperlyDiscontinuousAction |
Defines | discontinuous action |