discontinuous action

Let $X$ be a topological space and $G$ a group that acts on $X$ by homeomorphisms. The action of $G$ is said to be discontinuous at $x\in X$ if there is a neighborhood $U$ of $x$ such that the set

$$\{g\in G|gU\cap U\ne \mathrm{\varnothing }\}$$

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 points, i.e. if $\{g_n\}$ is a sequence of distinct elements of $G$ and $x\in X$ then the sequence $\{g_{n}x\}$ has no limit points. If $X$ is locally compact then an action that satisfies this condition is discontinuous.

Remark 2.

Assume that $X$ is a locally compact Hausdorff space and let $\mathrm{Aut}(X)$ denote the group of self homeomorphisms of $X$ endowed with the compact-open topology. If $\rho :G\to \mathrm{Aut}(X)$ defines a discontinuous action then the image $\rho (G)$ is a discrete subset of $\mathrm{Aut}(X)$.

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