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 <math>x\in X</math> is freely discontinuous if there exists a neighborhood U of x such that <math>g(U)\cap U=\varnothing</math> for all <math>g\in G</math>, 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 <math>\Omega=\Omega(G)</math>. Note that <math>\Omega</math> 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 <math>\Omega /G</math> is a Hausdorff space.
[edit] Examples
The open set
- <math>\Omega(\Gamma)=\{\tau\in H: |\tau|>1 , |\tau +\overline\tau| <1\}</math>
is the free regular set of the modular group <math>\Gamma</math> on the upper half-plane H. This set is called the fundamental domain on which modular forms are studied.
