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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
##### Keywords:
bisector; triod; midset property; simple closed curve; arc
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##### References:
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