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 a circle in an appropriate way.
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Formal definition
The set of prime ends of the domain B is the set of equivalence classes of chains of arcs converging to a point on the boundary of B.
In this way, a point in the boundary may correspond to many points in the prime ends of B, and conversely, many points in the boundary may correspond to a point in the prime ends of B.
Applications
Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows:
If ƒ maps the unit disk conformally and one-to-one onto the domain B, it induces a one-to-one mapping between the points on the unit circle and the prime ends of B.