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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Equidistant sets in plane triodic continua


Authors: L. D. Loveland and S. M. Loveland
Journal: Proc. Amer. Math. Soc. 115 (1992), 553-562
MSC: Primary 54F15; Secondary 54F50, 54F65
MathSciNet review: 1120508
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Abstract: Let $ x$ and $ y$ be two points in a metric space $ (X,\rho )$. The equidistant set or midset $ M(x,y)$ of $ x$ and $ y$ is the set $ \{ p \in X\vert\rho (x,p) = \rho (y,p)\} $. If the midset of each pair of points of $ X$ consists of a finite number of points then the metric space $ X$ is said to have the finite midset property, and if the midsets of pairs of points in $ X$ are pairwise homeomorphic then $ X$ is said to have uniform midsets. Generalizing earlier results, the main theorem states that no continuum in the Euclidean plane can have both finite and uniform midsets if it contains a triod. It follows that a plane continuum with finite, uniform midsets must be either an arc or a simple closed curve.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1120508-9
PII: S 0002-9939(1992)1120508-9
Keywords: Arc, bisector, continuum, equidistant set, midset, simple closed curve, triod, uniform midset, midset properties
Article copyright: © Copyright 1992 American Mathematical Society