No continuum in $E^ 2$ has the TMP. I. Arcs and spheres
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- by L. D. Loveland PDF
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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.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 1119-1128
- MSC: Primary 54F15; Secondary 54F50
- DOI: https://doi.org/10.1090/S0002-9939-1990-1027099-X
- MathSciNet review: 1027099