2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58599We prove that there is a one-to-one, bounded, holomorphic function on a region $Ω$ iff $S^{2} - Ω$ is not totally disconnected. This paper has been withdrawn by the author since Theorem 3 is incorrect.This paper has been withdrawn by the author since Theorem 3 is incorrectComplex Variables30A99text