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In [[mathematics]], the '''prime end''' compactification is a method to [[Compactification (mathematics)|compactify]] a [[topological disc]] (i.e. a simply connected open set in the plane) by adding the boundary circle in an appropriate way.

==Historical notes==
The concept of prime ends was introduced by [[Constantin Carathéodory]] to describe the boundary behavior of [[conformal map]]s in the [[complex plane]] in geometric terms.<ref>{{harv|Epstein|1981|p=385}}.</ref>  The theory has been generalized to more general open sets.<ref name="Epst.1981.par2">{{harv|Epstein|1981|loc=§2}}.</ref> The expository paper of {{harvtxt|Epstein|1981}} provides a good account of this theory with complete proofs: it also introduces a definition which make sense in any open set and dimension.<ref name="Epst.1981.par2" /> {{harvtxt|Milnor|2006}} gives an accessible introduction to prime ends in the context of complex dynamical systems.

==Formal definition==
The set of prime ends of the domain&nbsp;{{mvar|B}} is the set of equivalence classes of chains of arcs converging to a point on the boundary of&nbsp;{{mvar|B}}.

In this way, a point in the boundary may correspond to many points in the prime ends of&nbsp;{{mvar|B}}, and conversely, many points in the boundary may correspond to a point in the prime ends of&nbsp;{{mvar|B}}.<ref>A more precise and formal definition of the concepts of "chains of arcs" and of their equivalence classes is given in the [[Prime end#References|references]] cited.</ref>

==Applications==

Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows:

If {{mvar|ƒ}} maps the [[unit disk]] conformally and one-to-one onto the domain&nbsp;{{mvar|B}}, it induces a [[one-to-one mapping]] between the points on the [[unit circle]] and the prime ends of&nbsp;{{mvar|B}}.

==Notes==
{{reflist|50em}}

==References==
{{More footnotes|date=May 2010}}
{{Citizendium|title=Prime ends}}
*{{Citation
|last =Epstein
|first =D. B. A.
|author-link =David B. A. Epstein
|title =Prime Ends
|journal =[[Proceedings of the London Mathematical Society]]
|volume =s3–42
|issue =3
|place=Oxford
|publisher =[[Oxford University Press]]
|pages =385–414
|date =3 May 1981
|doi =10.1112/plms/s3-42.3.385
|mr =0614728
|zbl =0491.30027
}}.
*{{Citation
|last =Milnor
|first =John
|author-link =John Milnor
|title =Dynamics in one complex variable
|place =Princeton, NJ
|publisher =[[Princeton University Press]]
|series =Annals of Mathematics Studies
|volume =160
|orig-year =1999
|year =2006
|edition =3rd
|pages =viii+304
|doi =10.1515/9781400835539
|isbn =0-691-12488-4
|mr =2193309
|zbl =1281.37001
}}, {{isbn|978-0-691-12488-9}},
*{{Springer|id=l/l058860|title=Limit elements}}

[[Category:Compactification (mathematics)]]


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