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''boundary elements, prime ends, of a domain''

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 [[#References|[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 [[#References|[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|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 $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,   "Ueber die Begrenzung einfach zusammenhängender Gebiete"  ''Math. Ann.'' , '''73'''  (1913)  pp. 323–370</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Koebe,   "Abhandlungen zur Theorie der konformen Abbildung. I"  ''J. Reine Angew. Math.'' , '''145'''  (1915)  pp. 177–223</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Suvorov,   "Families of plane topological mappings" , Novosibirsk  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>

====Comments====
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|Attainable boundary point]]). See also [[Conformal mapping, boundary properties of a|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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Ohtsuka,   "Dirichlet problem, extremal length and prime ends" , v. Nostrand  (1967)</TD></TR></table>