discontinuous action

Let X be a topological spaceMathworldPlanetmath and G a group that acts on X by homeomorphismsPlanetmathPlanetmath. The action of G is said to be discontinuousMathworldPlanetmath at xX if there is a neighborhoodMathworldPlanetmathPlanetmath U of x such that the set


is finite. The action is called discontinuous if it is discontinuous at every point.

Remark 1.

If G acts discontinuously then the orbits of the action have no accumulation pointsMathworldPlanetmathPlanetmath, i.e. if {gn} is a sequence of distinct elements of G and xX then the sequence {gnx} has no limit pointsMathworldPlanetmath. If X is locally compact then an action that satisfies this condition is discontinuous.

Remark 2.

Assume that X is a locally compact Hausdorff spacePlanetmathPlanetmath and let Aut(X) denote the group of self homeomorphisms of X endowed with the compact-open topologyMathworldPlanetmath. If ρ:GAut(X) defines a discontinuous action then the image ρ(G) is a discrete subset of Aut(X).

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