Equidistant sets in plane triodic continua

Abstract:

Let $x$ and $y$ be two points in a metric space $(X,\rho )$. The equidistant set or midset $M(x,y)$ of $x$ and $y$ is the set $\{ p \in X|\rho (x,p) = \rho (y,p)\}$. If the midset of each pair of points of $X$ consists of a finite number of points then the metric space $X$ is said to have the finite midset property, and if the midsets of pairs of points in $X$ are pairwise homeomorphic then $X$ 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.