The 𝑛-separated-arc property for homeomorphisms

Abstract

Let f f be a function defined in the open unit disk D D whose range is in the Riemann sphere W W , and let C C denote the unit circle. We show that if f f is a homeomorphism of D D onto a Jordan domain, then the set of points p ∈ C p \in C at which f f has the n n -separated-arc property ( n ≧ 2 ) (n \geqq 2) is a subset of the set of ambiguous points of f f and is thus countable.