# 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

 $\{g\in G\,|\,gU\cap U\neq\emptyset\}$

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 $\operatorname{Aut}(X)$ denote the group of self homeomorphisms of $X$ endowed with the compact-open topology. If $\rho\colon\thinspace G\to\operatorname{Aut}(X)$ defines a discontinuous action then the image $\rho(G)$ is a discrete subset of $\operatorname{Aut}(X)$.

Title discontinuous action DiscontinuousAction 2013-03-22 13:28:49 2013-03-22 13:28:49 Dr_Absentius (537) Dr_Absentius (537) 7 Dr_Absentius (537) Definition msc 37B05 PoperlyDiscontinuousAction discontinuous action