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 $$H\sub G$$ 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 $$H$$.