On prime ends and local connectivity

Abstract

Let U ⊂ ℂ̂ be a simply connected domain whose complement K = ℂ̂\ U contains more than one point. We show that the impression of a prime end of U contains at most two points at which K is locally connected. This is achieved by establishing a characterization of local connectivity of K at a point z0 ∈ ∂ U in terms of the prime ends of U whose impressions contain z0, and then invoking a result of Ursell and Young. © 2008 London Mathematical Society.