Limit elements

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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 [1] 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 [2]. 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 .


[1] C. Carathéodory, "Ueber die Begrenzung einfach zusammenhängender Gebiete" Math. Ann. , 73 (1913) pp. 323–370
[2] P. Koebe, "Abhandlungen zur Theorie der konformen Abbildung. I" J. Reine Angew. Math. , 145 (1915) pp. 177–223
[3] G.D. Suvorov, "Families of plane topological mappings" , Novosibirsk (1965) (In Russian)
[4] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)
[5] 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.

The (only) point of a prime end of the first kind is an accessible boundary point (cf. Attainable boundary point). See also Conformal mapping, boundary properties of a.

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)
How to Cite This Entry:
Limit elements. E.G. Goluzina (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098