Free regular set

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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 x\in X is freely discontinuous if there exists a neighborhood U of x such that g(U)\cap U=\varnothing for all g\in G, 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.

[edit] Examples

The open set

\Omega(\Gamma)=\{\tau\in H: |\tau|>1 , |\tau +\overline\tau| <1\}

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.

[edit] See also