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No continuum in \(E^ 2\) has the TMP. II: Triodic continua. (English) Zbl 0792.54033
[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.

54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
54F15 Continua and generalizations
51K05 General theory of distance geometry
54F65 Topological characterizations of particular spaces
51M05 Euclidean geometries (general) and generalizations
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