No -point set is -compact

Authors:
Khalid Bouhjar, Jan J. Dijkstra and R. Daniel Mauldin

Journal:
Proc. Amer. Math. Soc. **129** (2001), 621-622

MSC (2000):
Primary 54H05, 57N05, 54G99

DOI:
https://doi.org/10.1090/S0002-9939-00-05869-X

Published electronically:
September 27, 2000

MathSciNet review:
1800242

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an integer greater than 1. We prove that there exist no -subsets of the plane that intersect every line in precisely points.

**1.**V. J. Baston and F. A. Bostock,*On a theorem of Larman*, J. London Math. Soc. (2)**5**(1972), 715-718. MR**47:4138****2.**K. Bouhjar, J. J. Dijkstra, and J. van Mill,*Three-point sets*, Topology Appl., to appear.**3.**H. Federer,*Geometric Measure Theory*, Springer Verlag, New York, 1969. MR**41:1976****4.**E. Hewitt and K. Stromberg,*Real and Abstract Analysis*, Springer Verlag, New York, 1965. MR**32:5826****5.**D. G. Larman,*A problem of incidence*, J. London Math. Soc.**43**(1968), 407-409. MR**38:52****6.**R. D. Mauldin,*On sets which meet each line in exactly two points*, Bull. London Math. Soc.**30**(1998), 397-403. MR**99f:28010****7.**W. Sierpinski,*Cardinal and Ordinal Numbers*, PWN, Warsaw, 1958. MR**20:2288**

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Additional Information

**Khalid Bouhjar**

Affiliation:
Faculty of Sciences, Division of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081 A, 1081 HV Amsterdam, The Netherlands

Email:
kbouhjar@cs.vu.nl

**Jan J. Dijkstra**

Affiliation:
Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350

Email:
jdijkstr@obelix.math.ua.edu

**R. Daniel Mauldin**

Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203

Email:
mauldin@dynamics.math.unt.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05869-X

Received by editor(s):
November 30, 1999

Received by editor(s) in revised form:
January 7, 2000

Published electronically:
September 27, 2000

Additional Notes:
The second author was supported in part by a grant from the Research Advisory Committee of the University of Alabama.

The third author was supported in part by NSF grant DMS-9801583.

Communicated by:
Alan Dow

Article copyright:
© Copyright 2000
American Mathematical Society