A prime end describes an approach to the boundary of a simply connected domain through a nested sequence of crosscuts. A conformal map induces a correspondence between prime ends, even when the Euclidean boundary does not have a continuous one-to-one parametrization. This is the natural boundary interpretation of the endpoints and growing tip of a Loewner chain.
Choose a conformal map from a proper simply connected domain to the unit disc and pull back the topology of the closed disc. The resulting compactification is independent of the chosen map, since disc automorphisms extend to its boundary. Its intrinsic boundary distinguishes different approaches to a slit. Every conformal map between such domains extends to their intrinsic compactifications. This boundary agrees with the prime-end description in the simply connected setting.
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In mathematics, particularly in the field of complex analysis, the term "prime end" refers to a concept used in the study of conformal mappings and, more generally, in the theory of Riemann surfaces and potential theory. The notion was introduced by the mathematician Henri Poincaré. **Prime Ends**: Prime ends can be thought of as a way to extend the notion of boundary points in a domain in the complex plane.