# 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 |