In mathematics, the '''prime end''' compactification is a method to 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 maps 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 {{mvar|B}} is the set of equivalence classes of chains of arcs converging to a point on the boundary of {{mvar|B}}.
In this way, a point in the boundary may correspond to many points in the prime ends of {{mvar|B}}, and conversely, many points in the boundary may correspond to a point in the prime ends of {{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 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 {{mvar|B}}, it induces a one-to-one mapping between the points on the unit circle and the prime ends of {{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)
{{topology-stub}}