No continuum in \(E^ 2\) has the TMP. II: Triodic continua
[For Part I see Proc. Am. Math. Soc. 110, No. 4, 1119-1128 (1990; Zbl 0731.54024).] In a metric space \((X,\rho)\), the midset of a pair of points \(x, y \in X\) is the set of points \(m \in X\) such that \(\rho(x,m) = \rho(y,m)\). For a subset \(X\) of the Euclidean plane \(E^ 2\) the midset consists of the intersection of \(X\) and the perpendicular bisector of the line segment \(\overline{xy}\) joining \(x\) and \(y\). It is conjectured that a continuum (= compact, connected metric space) in which the midset of each pair of points consists of exactly two points is a simple closed curve or a point and that a continuum in which the midset of each pair of points consists of exactly three points is a point. In the previous paper, the author verified the first conjecture for subcontinua of the plane. The paper being reviewed provides the final steps needed to establish the second conjecture for planar continua as well.
scientific article; zbMATH DE number 168040