# Free regular set

In mathematics, a **free regular set** is a subset of a topological space that is acted upon disjointly under a given group action.[1]

To be more precise, let *X* be a topological space. Let *G* be a group of homeomorphisms from *X* to *X*. Then we say that the action of the group *G* at a point
is **freely discontinuous** if there exists a neighborhood *U* of *x* such that
for all
, excluding the identity. Such a *U* is sometimes called a *nice neighborhood* of *x*.

The set of points at which G is freely discontinuous is called the **free regular set** and is sometimes denoted by
. Note that
is an open set.

If *Y* is a subset of *X*, then *Y*/*G* is the space of equivalence classes, and it inherits the canonical topology from *Y*; that is, the projection from *Y* to *Y*/*G* is continuous and open.

Note that is a Hausdorff space.

## Examples

The open set

is the free regular set of the modular group
on the upper half-plane *H*. This set is called the fundamental domain on which modular forms are studied.

## References

- Maskit, Bernard (1987).
*Discontinuous Groups in the Plane, Grundlehren der mathematischen Wissenschaften Volume 287*. Springer Berlin Heidelberg. pp. 15–16. ISBN 978-3-642-64878-6.