No continuum in \(E^ 2\) has the TMP. II: Triodic continua
type: Item
instance of: scholarly article
title: No continuum in \(E^ 2\) has the TMP. II: Triodic continua
zbMATH Open document ID: 0792.54033
author: L. D. Loveland
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-05-16T00:00:00Z
review text: [For Part I see Proc. Am. Math. Soc. 110, No. 4, 1119-1128 (1990; Zbl 0731.54024).] In a metric space \((X,\rho)\), the midset of a pair of points \(x, y \in X\) is the set of points \(m \in X\) such that \(\rho(x,m) = \rho(y,m)\). For a subset \(X\) of the Euclidean plane \(E^ 2\) the midset consists of the intersection of \(X\) and the perpendicular bisector of the line segment \(\overline{xy}\) joining \(x\) and \(y\). It is conjectured that a continuum (= compact, connected metric space) in which the midset of each pair of points consists of exactly two points is a simple closed curve or a point and that a continuum in which the midset of each pair of points consists of exactly three points is a point. In the previous paper, the author verified the first conjecture for subcontinua of the plane. The paper being reviewed provides the final steps needed to establish the second conjecture for planar continua as well.
reviewed by: John Walsh
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 51K05
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 51M05
zbMATH DE Number: 168040
zbMATH Keywords: TMP
zbMATH Keywords: bisector
zbMATH Keywords: triod
zbMATH Keywords: triple midset property
zbMATH Keywords: midset
zbMATH Keywords: simple closed curve
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072708
OpenAlex ID: W2032545247
cites work: Q5828113
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cites work: Q5586965
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DOI: 10.1216/RMJM/1181072708
description: scientific article; zbMATH DE number 168040