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No  -point set is  -compact
Author(s):
Khalid
Bouhjar;
Jan
J.
Dijkstra;
R.
Daniel
Mauldin
Journal:
Proc. Amer. Math. Soc.
129
(2001),
621-622.
MSC (2000):
Primary 54H05, 57N05, 54G99
Posted:
September 27, 2000
MathSciNet review:
1800242
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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.
References:
-
- 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:
10.1090/S0002-9939-00-05869-X
PII:
S 0002-9939(00)05869-X
Received by editor(s):
November 30, 1999
Received by editor(s) in revised form:
January 7, 2000
Posted:
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
Copyright of article:
Copyright
2000,
American Mathematical Society
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