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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\C$ be a bounded [[simply connected domain|075C629XTH]] with [[boundary|0JJ9J1XZ8G]] $\\partial\\Omega$.\nA *crosscut* of $\\Omega$ is a set $C \\subset \\Omega$ that is homeomorphic to the open interval $(0, 1)$ and whose closure is $C \\cup \\set{a, b}$ with $a, b \\in \\partial\\Omega$.\nThen $\\Omega \\setminus C$ has exactly two [[connected components|0R37DFJKFW]].\n\nA *null chain* is a sequence $(C_n)_{n \\ge 0}$ of crosscuts of $\\Omega$ such that:\n\n1. the closures $\\overline{C_n}$ and $\\overline{C_{n+1}}$ are disjoint for all $n$;\n2. for each $n \\ge 1$, the crosscut $C_n$ separates $C_0$ from $C_{n+1}$ in $\\Omega$;\n3. the diameter of $C_n$ tends to $0$.\n\nFor a null chain, let $V_n$ be the component of $\\Omega \\setminus C_n$ that does not contain $C_0$, so $V_1 \\supset V_2 \\supset \\cdots$.\nTwo null chains $(C_n)$ and $(C'_n)$, with domains $V_n$ and $V'_n$, are *equivalent* if for each $m$ there is an $n$ with $V_n \\subseteq V'_m$ and $V'_n \\subseteq V_m$.\nA *prime end* of $\\Omega$ is an equivalence class of null chains.\n\n## Remarks\n\nExample: 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.\nTheir closures are disjoint, each arc separates the previous one from the next, and their diameters tend to $0$.\nEvery 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.\nNon-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.\nThe *impression* of a prime end is the intersection of the closures $\\overline{V_n}$; it is a nonempty compact connected subset of $\\partial\\Omega$.\nCarathéodory'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 $\\Omega$.\n",
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