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Fundamental domain
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== Hints at a general definition == [[Image:Lattice torsion points.svg|right|thumb|300px| A lattice in the complex plane and its fundamental domain, with quotient a torus. ]] Given an [[Group action (mathematics)|action]] of a [[group (mathematics)|group]] ''G'' on a [[topological space]] ''X'' by [[homeomorphism]]s, a fundamental domain for this action is a set ''D'' of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. 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 ''X'' with a set of [[measure zero]], for a certain (quasi)invariant [[measure (mathematics)|measure]] on ''X''. A fundamental domain always contains a [[free regular set]] ''U'', an [[open set]] moved around by ''G'' into [[Disjoint sets|disjoint]] copies, and nearly as good as ''D'' in representing the orbits. Frequently ''D'' is required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero. This is a typical situation in [[ergodic theory]]. If a fundamental domain is used to calculate an [[integral]] on ''X''/''G'', sets of measure zero do not matter. For example, when ''X'' is [[Euclidean space]] '''R'''<sup>''n''</sup> of dimension ''n'', and ''G'' is the [[lattice (group theory)|lattice]] '''Z'''<sup>''n''</sup> acting on it by translations, the quotient ''X''/''G'' is the ''n''-dimensional [[torus]]. A fundamental domain ''D'' here can be taken to be <nowiki>[0,1)</nowiki><sup>''n''</sup>, which differs from the open set (0,1)<sup>''n''</sup> by a set of measure zero, or the [[closed set|closed]] unit cube <nowiki>[0,1]</nowiki><sup>''n''</sup>, whose [[boundary (topology)|boundary]] consists of the points whose orbit has more than one representative in ''D''.
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