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.


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.

See also


  1. 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.
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