A Counterexample of Koebe's for Slit

1. We refer to a region £2 of the extended z-plane as a (parallel) slit domain if =o £Í2, and if the components of the boundary, dß, are either points, or segments ("slits") parallel to a common line, which without loss of generality will be assumed to be the y-axis 'z = x-\-iy). It was originally conjectured by Koebe that if two slit domains ßi and Í22 are conformally equivalent, that is, if there exists a function/, schlicht in fí2, such that/(co) = oo, /(ß2) =Qi, then, unless/ is linear, at most one of the two sets, Ei = dQ,i, E2=dQ2, has area zero. Later on Koebe [5] outlined the construction of a counterexample in which (using the present notation) (a) the components of 7¿i are not all points, (b) the projection of Ei onto the x-axis has linear Lebesgue measure zero, (c) E2 is a compact, totally disconnected subset of the x-axis. Although Koebe's example, and variants therefore, have been applied repeatedly in connection with various counterexamples in complex variable theory (see, for instance, [7]) it does not appear to have been previously noted in the literature that the reasoning in [5] contains a gap. The statement containing the word "offenbar" in the last paragraph of page 62 of [5] is incorrect. If P denotes the intersection of Koebe's Si with a line parallel to the y-axis, then P is denumerable, and supposedly closed. However, it is not difficult to show that the set of points in P that are two-sided limit points of P must be dense in itself. In the present note we fill this gap by obtaining the following slightly more general result.