---
tag: GNWE3XR0ZG
kind: concept
name: Prime end
msc: [30D40, 30C35]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_end
same_as:
  wikidata: Q7243367
  wikipedia: Prime end
---

## Summary

Prime ends are the ideal boundary points of a simply connected plane domain, introduced by Constantin Carathéodory.
A prime end is described by a shrinking sequence of cross cuts that close in on the boundary.
For a bounded simply connected domain, a conformal map from the unit disc matches the points of the unit circle with the prime ends.
They are the right notion of boundary when the topological boundary is too irregular for the conformal map to extend continuously.

## Explanations

### L1 human

Imagine the unit disc with a slit cut along a radius.
Seen from inside the domain, a point on the slit can be approached from above or from below, and these feel like two different places.
Prime ends make this precise: each way of closing in on the boundary through smaller and smaller cuts is one prime end.
So each inner point of the slit carries two prime ends, while each point of the circle carries one.