Discontinuous group

A discontinuous group is a mathematical concept relating to mappings in topological space.

Definition

Let $T$ be a topological space of points $\tau$, and let $\tau \to f(\tau ,x)$, $x\in G$, be an open continuous representation of the topological group $G$ as a transitive group of homeomorphic mappings of $T$ on itself. The representation $\tau \to f(\tau ,a)$ $a\in H$ of the discrete subgroup $H\subset G$ in $T$ is called discontinuous, if no sequence $f(\tau ,a_{n})$ ($n=1,2,\ldots$) converges to a point in $T$, as $a_{n}$ runs over distinct elements of $H$.