No 𝑛-point set is 𝜎-compact

No $n$-point set is $\sigma$-compact
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by Khalid Bouhjar, Jan J. Dijkstra and R. Daniel Mauldin PDF

Proc. Amer. Math. Soc. 129 (2001), 621-622 Request permission

Let $n$ be an integer greater than 1. We prove that there exist no $F_\sigma$-subsets of the plane that intersect every line in precisely $n$ points.

References

V. J. Baston and F. A. Bostock, On a theorem of Larman, J. London Math. Soc. (2) 5 (1972), 715–718. MR 315589, DOI 10.1112/jlms/s2-5.4.715
K. Bouhjar, J. J. Dijkstra, and J. van Mill, Three-point sets, Topology Appl., to appear.
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387
D. G. Larman, A problem of incidence, J. London Math. Soc. 43 (1968), 407–409. MR 231724, DOI 10.1112/jlms/s1-43.1.407
R. Daniel Mauldin, On sets which meet each line in exactly two points, Bull. London Math. Soc. 30 (1998), no. 4, 397–403. MR 1620829, DOI 10.1112/S0024609397004268
Wacław Sierpiński, Cardinal and ordinal numbers, Polska Akademia Nauk. Monografie Matematyczne, Tom 34, Państwowe Wydawnictwo Naukowe, Warsaw, 1958. MR 0095787