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ON SETS THAT MEET EVERY HYPERPLANE IN n-SPACE IN AT MOST n POINTS

Published online by Cambridge University Press:  22 May 2002

JAN J. DIJKSTRA  and
JAN VAN MILL
JAN J. DIJKSTRA
Affiliation:
Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350, [email protected] Vrije Universiteit, Faculty of Sciences, Division of Mathematics and Computer Science, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
JAN VAN MILL
Affiliation:
Vrije Universiteit, Faculty of Sciences, Division of Mathematics and Computer Science, De Boelelaan 1081a, 1081 HV Amsterdam, The [email protected]
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Abstract

A simple proof that no subset of the plane that meets every line in precisely two points is an Fσ-set in the plane is presented. It was claimed that this result can be generalized for sets that meet every line in either one point or two points. No proof of this assertion is known, however. The main results in this paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set that meets every line in the plane in at least one but at most two points must be zero-dimensional, and that if it is σ-compact then it must be a nowhere dense Gδ-set in the plane. Generalizations for similar sets in higher-dimensional Euclidean spaces are also presented.

Type
PAPERS
Information
Bulletin of the London Mathematical Society , Volume 34 , Issue 3 , May 2002 , pp. 361 - 368
Copyright
© The London Mathematical Society 2002

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