Prime end

Remarks

Example: for the unit disc $\mathbb{D}$ and a point $\zeta$ with $\abs{\zeta} = 1$, the arcs $C_n = \set{z \in \mathbb{D} : \abs{z - \zeta} = 2^{-n}}$ for $n \ge 1$, relabelled from $0$, form a null chain. Their closures are disjoint, each arc separates the previous one from the next, and their diameters tend to $0$. Every null chain of $\mathbb{D}$ shrinks to a single boundary point, and the prime ends of $\mathbb{D}$ correspond to the points of the unit circle. Non-example: in $\mathbb{D}$, the vertical chords $C_n = \set{z \in \mathbb{D} : \operatorname{Re} z = 1/(n+2)}$ satisfy clauses 1 and 2, but their diameters $2\sqrt{1 - (n+2)^{-2}}$ tend to $2$, so clause 3 fails. The impression of a prime end is the intersection of the closures $\overline{V_n}$; it is a nonempty compact connected subset of $\partial\Omega$. Carathéodory's prime end theorem says that a conformal map of $\mathbb{D}$ onto $\Omega$ induces a bijection from the unit circle onto the set of prime ends of $\Omega$.