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No continuum in $ E\sp 2$ has the TMP. I. Arcs and spheres

Author: L. D. Loveland
Journal: Proc. Amer. Math. Soc. 110 (1990), 1119-1128
MSC: Primary 54F15; Secondary 54F50
MathSciNet review: 1027099
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Abstract: A subset $ X$ of the Euclidean plane $ {E^2}$ is said to have the triple midset property (TMP) if, for each pair of points $ x$ and $ y$ of $ X$, the perpendicular bisector of the segment joining $ x$ and $ y$ intersects $ X$ at exactly three points. In this paper it is proved that no arc or simple closed curve in $ {E^2}$ can have the TMP. In a subsequent paper these results are used to prove that no planar continuum can have the TMP.

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Keywords: Arc, bisector, continuum, equidistant set, midset, simple closed curve, triple midset property
Article copyright: © Copyright 1990 American Mathematical Society

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