In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action.
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 Ω = Ω(G). Note that Ω 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 Ω / G is a Hausdorff space.
Examples
The open set
is the free regular set of the modular group Γ on the upper half plane H. This set is called the fundamental domain on which modular forms are studied.
See also