On the dimension of \(n\)-point sets
type: Item
instance of: scholarly article
title: On the dimension of \(n\)-point sets
zbMATH Open document ID: 1023.54015
DOI: 10.1016/S0166-8641(02)00134-7
author: D. L. Fearnley
author: L. Fearnley
author: J. Lamoreaux
published in: Topology and its Applications
publication date: 2003-04-28T00:00:00Z
review text: The paper is related to the following old result (S. Mazurkiewicz, 1914): There exists a subset of the plane which intersects every straight line in exactly two points. A subset \(S\) of the plane is an \(n\)-point set (a partial \(n\)-point set) provided that \(S\) intersects every straight line in exactly \(n\) points (at most \(n\) points). The authors answer a question asked by Bouhjar and Dijkstra, whether or not every one-dimensional partial \(n\)-point set contains an arc. The answer is affirmative; moreover, they prove the following (Th. 2.3). Let \(S\) be a subset of the plane which intersects every straight line in only a finite number of points. Then \(S\) is one-dimensional if and only if \(S\) contains an arc.
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54H05
zbMATH DE Number: 1901919
zbMATH Keywords: : \(n\)-point sets in \({\text{ R}}^2\)
zbMATH Keywords: one-dimensional sets
zbMATH Keywords: sets containing arcs
reviewed by: Maria Moszyńska
MaRDI profile type: Publication
cites work: On the structure of \(n\)-point sets
cites work: Three-point sets
cites work: Every three-point set is zero dimensional
cites work: A Two-Point Set Must be Zero-Dimensional
full work available at URL: s0166-8641(02)00134-7
OpenAlex ID: W2017939902
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description: scientific article; zbMATH DE number 1901919