Let X be a compact topological space, and let D be a subset of X. Let Y be a Hausdorff topological space. Let f be a continuous map of the closure of D to Y such that f(D) is open. Let E be any connected subset of the complement (to Y) of the image f(∂D) of the boundary ∂D of D. Then f(D) either contains E or is contained in the complement of E.
Applications of this dichotomy principle are given, in particular for holomorphic maps, including maximum and minimum modulus principles, an inverse boundary correspondence, and a proof of Haagerup's inequality for the absolute power moments of linear combinations of independent Rademacher random variables. (A three-line proof of the main theorem of algebra is also given.) More generally, the dichotomy principle is naturally applicable to conformal and quasiconformal mappings.