boundary elements, prime ends, of a domain
Elements of a domain in the complex plane that are defined as follows. Let be a simply-connected domain of the extended complex plane, and let be the boundary of . A section of is defined as any simple Jordan arc , closed in the spherical metric, with ends and (including the cases and ), such that belong to , and such that the arc subdivides into two subdomains such that the boundary of each of them contains a point belonging to and different from and . A sequence of sections of a domain is called a chain if: 1) the diameter of tends to zero as ; 2) for each the intersection is empty; and 3) any path connecting a fixed point in with the section , , intersects . Two chains and in are equivalent if any section separates in the point from all sections , except for a finite number of them. An equivalence class of chains in is called a limit element, or prime end, of .
Let be a prime end of defined by a chain , and let be that one of the two subdomains into which is subdivided by which does not contain . The set 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 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 onto the unit disc there is a one-to-one correspondence between the points of the circle and the prime ends of , and each sequence of points of which converges to a prime end becomes a sequence of points in the unit disc which converge to a point , , this point being the image of .
|||C. Carathéodory, "Ueber die Begrenzung einfach zusammenhängender Gebiete" Math. Ann. , 73 (1913) pp. 323–370|
|||P. Koebe, "Abhandlungen zur Theorie der konformen Abbildung. I" J. Reine Angew. Math. , 145 (1915) pp. 177–223|
|||G.D. Suvorov, "Families of plane topological mappings" , Novosibirsk (1965) (In Russian)|
|||A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)|
|||E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9|
Instead of "section" the phrase cross cut or cut is also used.
Instead of prime end one also finds Carathéodory end in the literature.
There is a second, not entirely dissimilar notion in the literature which goes by the name "end of a topological spaceend" . This refers to the ends of a topological space.
|[a1]||M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand (1967)|
Limit elements. E.G. Goluzina (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Limit_elements&oldid=18944