Fundamental theorems of welfare economics: Difference between revisions

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fixed summary of local non-satiation assumption to refer to bundles of goods, not individual goods
 
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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]].<ref name=Maskit1987>{{cite book|last=Maskit|first=Bernard|title=Discontinuous Groups in the Plane, Grundlehren der mathematischen Wissenschaften Volume 287|year=1987|publisher=Springer Berlin Heidelberg|isbn=978-3-642-64878-6|pages=15–16}}</ref>
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To be more precise, let ''X'' be a [[topological space]].  Let ''G'' be a group of [[homeomorphism]]s 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 [[Neighbourhood (mathematics)|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]].
 
==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 form]]s are studied.
 
==See also==
* [[Covering map]]
* [[Klein geometry]]
* [[Homogenous space]]
* [[Clifford–Klein form]]
* [[G-torsor]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Free Regular Set}}
[[Category:Topological groups]]
[[Category:Group actions]]

Latest revision as of 02:15, 2 December 2014

Hello. Let me introduce the author. Her title is Refugia Shryock. California is our beginning location. It's not a typical factor but what she likes doing is base leaping and now she is trying to earn cash with it. Hiring is his profession.

my site: std testing at home