What is the set of points in the middle?
by
Gaston Brouwer
University of Alabama at Birmingham
Coauthors: J. Aarts and L. Oversteegen

Harold Bell has shown that the set of points which is equidistant to two disjoint continua in the plane is a 1-manifold. Our goal is to generalize this result as much as possible. We will first study the set of points which is equidistant to two disjoint closed sets in the plane. These results are similar to results obtained by Morton Brown for the boundary of an \epsilon-ball around a compact subset of the plane. In addition we introduce the notion of non-interlaced closed sets. We will show that each component of the set of points which is equidistant between two disjoint closed and non-interlaced sets in the plane is always a 1-manifold. Subsequently we introduce the notion of two disjoint (and not necessarily compact) non-interlaced sets in the plane and obtain the same results.

These results are sufficient to show that for each dense channel into a plane continuum X, there does indeed exists a ray which runs down this channel and is equidistant to the opposite sides of the channel (even though these two sides may compactify on each other).

Date received: March 2, 2003