Carathéodory domain

A bounded simply-connected domain $G$ in the complex plane such that its boundary is the same as the boundary of the domain $G_\infty$ which is the component of the complement of $G$ containing the point $\infty$. A domain bounded by a Jordan curve is an example of a Carathéodory domain. Every Carathéodory domain is representable as the kernel of a decreasing convergent sequence of simply-connected domains $\{ G_n\}$:

$$\overline{G}\subset G_{n+1}\subset\overline{G}_{n+1}\subset G_n,\quad n=1,2,\ldots,$$

and every domain $G$ for which there exists such a sequence is a Carathéodory domain (Carathéodory's theorem, see [Ca]).

Let $G_n$ be a sequence of simply-connected domains in the complex plane. Suppose that each contains a fixed disc $D$ with centre $z_0$. Let

$$ E=\{ z:\mathrm{there is a neighbourhood }N\mathrm{ such that }N\subset G_n\mathrm{ for all large enough }n\}.$$

Then $E$ is open. Let $G_{z_0}$ be the component of $E$ containing $z_0$. This domain is called the kernel of the sequence $\{ G_n\}$ (relative to the point $z_0$). The sequence $\{ G_n\}$ is said to converge to $G_{z_0}$ if every subsequence of $\{ G_n\}$ has the same kernel relative to $z_0$ as $\{ G_n\}$ itself. See [Ma].

References

[Ca]	C. Carathéodory, "Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten" Math. Ann. , 72 (1912) pp. 107–144
[Ma]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)