# Limit elements

Elements of a domain $B$ in the complex plane that are defined as follows. Let $B$ be a simply-connected domain of the extended complex plane, and let $\partial B$ be the boundary of $B$. A section $c$ of $B$ is defined as any simple Jordan arc $c = \overline{ {ab }}\;$, closed in the spherical metric, with ends $a$ and $b$( including the cases $a = b$ and $b = \infty$), such that $a, b$ belong to $\partial B$, and such that the arc $c$ subdivides $B$ into two subdomains such that the boundary of each of them contains a point belonging to $\partial B$ and different from $a$ and $b$. A sequence $K$ of sections $c _ {n}$ of a domain $B$ is called a chain if: 1) the diameter of $c _ {n}$ tends to zero as $n \rightarrow \infty$; 2) for each $n$ the intersection $\overline{c}\; _ {n} \cap \overline{c}\; _ {n + 1 }$ is empty; and 3) any path connecting a fixed point $0 \in B$ in $B$ with the section $c _ {n}$, $n > 1$, intersects $c _ {n - 1 }$. Two chains $K = \{ c _ {n} \}$ and $K ^ \prime = \{ c _ {n} ^ \prime \}$ in $B$ are equivalent if any section $c _ {n}$ separates in $B$ the point $0$ from all sections $c _ {n} ^ \prime$, except for a finite number of them. An equivalence class of chains in $B$ is called a limit element, or prime end, of $B$.
Let $P$ be a prime end of $B$ defined by a chain $K = \{ c _ {n} \}$, and let $B _ {n}$ be that one of the two subdomains into which $B$ is subdivided by $c _ {n}$ which does not contain $0$. The set $I ( P) = \cap _ {n = 1 } ^ \infty \overline{B}\; _ {n}$ is called the impression or the support of the prime end. The impression of a prime end consists of boundary points and does not depend on the selection of the chain $K$ in the equivalence class. A principal point of a prime end is a point of it to which sections of at least one of the chains defining the prime end converge. A neighbouring, or subsidiary, point of a prime end is any point of it which is not a principal point of it. Any prime end contains at least one principal point. The principal points of a prime end form a closed set. The following is the Carathéodory classification  of prime ends: Elements of the first kind contain a single principal point and no subsidiary points; elements of the second kind contain one principal point and infinitely many subsidiary points; elements of the third kind contain a continuum of principal points and no subsidiary points; elements of the fourth kind contain a continuum of principal points and infinitely many subsidiary points.
Another, equivalent, definition was given by P. Koebe . It is based on equivalence classes of paths. The principal theorem in the theory of prime ends is the theorem of Carathéodory: Under a univalent conformal mapping of a domain $B$ onto the unit disc $| \zeta | \leq 1$ there is a one-to-one correspondence between the points of the circle and the prime ends of $B$, and each sequence of points of $B$ which converges to a prime end $P$ becomes a sequence of points in the unit disc which converge to a point $\zeta _ {0}$, $| \zeta _ {0} | = 1$, this point being the image of $P$.