Free regular set
Encyclopedia
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
. 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
.
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 form
s are studied.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a free regular set is a subset of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
that is acted upon disjointly under a given group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
.
To be more precise, let X be a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. 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 setOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
.
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 spaceHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
Examples
The open setis the free regular set of the modular group
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
on the upper half-plane H. This set is called the fundamental domainFundamental domain
In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...
on which modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s are studied.
See also
- Covering mapCovering mapIn mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...
- Klein geometryKlein geometryIn mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.For background and motivation...
- Homogenous space
- Clifford–Klein form
- G-torsor