from Γ to the unit circle S1.
Such a region is called a Jordan domain. Equivalently, this theorem states that for such sets U there is a homeomorphism
F : cl(U) → cl(D)
Another standard formulation of Carathéodory's theorem states that for any pair of simply connected open sets U and V bounded by Jordan curves Γ1 and Γ2, a conformal map
extends to a homeomorphism
This version can be derived from the one stated above by composing the inverse of one Riemann map with the other.
Intuitively, Carathéodory's theorem says that compared to general simply connected open sets in the complex plane C, those bounded by Jordan curves are particularly well-behaved.
Carathéodory's theorem is a basic result in the study of boundary behavior of conformal maps, a classical part of complex analysis. In general it is very difficult to decide whether or not the Riemann map from an open set U to the unit disk D extends continuously to the boundary, and how and why it may fail to do so at certain points.
While having a Jordan curve boundary is sufficient for such an extension to exist, it is by no means necessary. For example, the map
from the upper half-plane H to the open set G that is the complement of the positive real axis is holomorphic and conformal, and it extends to a continuous map from the real line R to the positive real axis R+; however, the set G is not bounded by a Jordan curve.