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.
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 formula_1 is freely discontinuous if there exists a neighborhood "U" of "x" such that formula_2 for all formula_3, 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 formula_4. Note that formula_5 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 formula_6 is a Hausdorff space.
Examples.
The open set 
is the free regular set of the modular group formula_8 on the upper half-plane "H". This set is called the fundamental domain on which modular forms are studied.