NationMaster - Encyclopedia: 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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, groups are often used to describe symmetries of objects. ...

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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This word should not be confused with homomorphism. ... Neighbourhood is also a term in topology. ...

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. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of 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 $Ω / G$ is a Hausdorff space. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

Examples

The open set

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. In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... Modular form - Wikipedia /**/ @import /skins-1. ...

Categories: Topological groups | Group theory In mathematics, specifically topology, a covering map is a continuous surjective map p : C â†’ X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... In mathematics, a Clifford-Klein form is a double coset space Γ/G H, where G is a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space G H. A suitable discrete subgroup Γ may...

Results from FactBites:

regular expression: Definition and Much More from Answers.com (4204 words)
	A regular expression (abbreviated as regexp or regex, with plural forms regexps, regexes, or regexen) is a string that describes or matches a set of strings, according to certain syntax rules.
	The set of utilities (including the editor sed and the filter grep) provided by Unix distributions were the first to popularize the concept of regular expressions.
	Regular expressions correspond to the type 3 grammars of the Chomsky hierarchy and may be used to describe a regular language.

Fundamental domain - Wikipedia, the free encyclopedia (889 words)
	The set of orbits of the symmetry group define a partitioning of space.
	One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in G with a set of measure zero, for the Haar measure on G.
	Each triangular region is a free regular set of H/Γ; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.

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