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Equidistant sets in plane triodic continua. (English) Zbl 0754.54025
For each positive integer \(n\), a metric space \(X\) is said to have the \(n\)-point midset property (shortly \(n\)-MP) if for every two points \(x\) and \(y\) in \(X\) the set of all points of \(X\) equidistant from \(x\) and \(y\) consists of \(n\) points. Generalizing earlier results, the main theorem of the paper states that if a continuum in the Euclidean plane has the \(n\)- MP for \(n\geq 1\), then it must either be a simple closed curve or an arc. It is remarked at the end of the paper that in a forthcoming paper the authors have proved even a stronger result: if a planar continuum \(X\) has the \(n\)-MP for \(n\geq 1\), then either \(n=1\) and \(X\) is an arc, or \(n=2\) and \(X\) is a simple closed curve.

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