Montel's concept of normal families, which became the standard method of proof in textbooks. Carathéodory continued in 1913 by resolving the additional question of whether the Riemann mapping between the domains can be extended to a homeomorphism of the boundaries (see Carathéodory's theorem). Carathéodory's proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way that did not require them. Another proof, due to Lipót Fejér and to Frigyes Riesz, was published in 1922 and it was rather shorter than the previous ones. In this proof, like in Riemann's proof, the desired mapping was obtained as the solution of an extremal problem. The Fejér–Riesz proof was further simplified by Alexander Ostrowski and by Carathéodory. Importance The following points detail the uniqueness and power of the Riemann mapping theorem: Even relatively simple Riemann mappings (for example a map from the interior of a circle to the interior of a square) have no explicit formula using only elementary functions. Simply connected open sets in the plane can be highly complicated, for instance, the boundary can be a nowhere-differentiable fractal curve of infinite length, even if the set itself is bounded. One such example is the Koch curve. The fact that such a set can be mapped in an angle-preserving manner to the nice and regular unit disc seems counter-intuitive. The analog of the Riemann mapping theorem for more complicated domains is not true. The next simplest case is of doubly connected domains (domains with a single hole). Any doubly connected domain except for the punctured disk and the punctured plane is conformally equivalent to some annulus with , however there are no conformal maps between annuli except inversion and multiplication by constants so the annulus is not conformally equivalent to the annulus (as can be proven using extremal length). The analogue of the Riemann mapping theorem in three or more real dimensions is not true. The family of conformal maps in three dimensions is very poor, and essentially contains only Möbius transformations (see Liouville's theorem). Even if arbitrary homeomorphisms in higher dimensions are permitted, contractible manifolds can be found that are not homeomorphic to the ball (e.g., the Whitehead continuum). The analogue of the Riemann mapping theorem in several complex variables is also not true. In (), the ball and polydisk are both simply connected, but there is no biholomorphic map between them. Proof via normal