discontinuous action

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

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

is finite. The action is called discontinuous if it is discontinuous atevery point.

Remark 1.

If $G$ acts discontinuously then the orbits of the action have noaccumulation 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 $X$ islocally 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 thecompact-open topology.If $\rho :G\to \mathrm{Aut}(X)$ defines a discontinuous actionthen the image$\rho (G)$ is a discrete subset of $\mathrm{Aut}(X)$.