  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 boundary 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.
