Equidistant sets in plane triodic continua

Abstract

Let x x and y y be two points in a metric space ( X , ρ ) (X,\rho ) . The equidistant set or midset M ( x , y ) M(x,y) of x x and y y is the set { p ∈ X | ρ ( x , p ) = ρ ( y , p ) } \{ p \in X|\rho (x,p) = \rho (y,p)\} . If the midset of each pair of points of X X consists of a finite number of points then the metric space X X is said to have the finite midset property, and if the midsets of pairs of points in X X are pairwise homeomorphic then X X is said to have uniform midsets. Generalizing earlier results, the main theorem states that no continuum in the Euclidean plane can have both finite and uniform midsets if it contains a triod. It follows that a plane continuum with finite, uniform midsets must be either an arc or a simple closed curve.