Bull. Austral. Math. Soc. 72(3) pp.477--480, 2005.

There are no n-point Fσ sets in Rm

David L. Fearnley

L. Fearnley

J.W. Lamoreaux

Received: 29th August, 2005



We show that, for any positive integers n and m, if a set S $ \subset $ Rm intersects every m - 1 dimensional affine hyperplane in Rm in exactly n points, then S is not an F$\scriptstyle \sigma $ set. This gives a natural extension to results of Khalid Bouhjar, Jan J. Dijkstra, and R. Daniel Mauldin, who have proven this result for the case when m = 2, and also Jan J. Dijkstra and Jan van Mill, who have shown this result for the case when n = m.

Click to download PDF of this article (free access until July 2006)

or get the no-frills version

[an error occurred while processing this directive]
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199649 Z'blatt-MATH: 1091.54003


  1. K. Bouhjar, J.J. Dijkstra and R.D. Mauldin;
    No n-point set is σ-compact,
    Proc. Amer. Math. Soc. 129 (2001), pp. 621--622. MR1800242
  2. J.J. Dijkstra and J. van Mill;
    On sets that meet every hyperplane in n-space in at most n points,
    Bull. London Math. Soc. 34 (2002), pp. 361--368. MR1887709
  3. D.L. Fearnley, L. Fearnley and J.W. Lamoreaux;
    Every three-point space is zero dimensional,
    Proc. Amer. Math. Soc. 131 (2003), pp. 2241--2245. MR1963773
  4. D.L. Fearnley, L. Fearnley and J.W. Lamoreaux;
    On the dimension of n-point sets,
    Topology Appl. 129 (2003), pp. 15--18. MR1955662
  5. J. Kulesza;
    A two point set must be zero dimensional,
    Proc. Amer. Math. Soc. 116 (1992), pp. 551--553. MR1093599
  6. R.D. Mauldin;
    Is there a Borel set M in R2 which meets each straight line in exactly two points?,
    in Open Problems in Topology (North-Holland, Amsterdam, 1990), pp. 619--629. MR1078636
  7. S. Mazurkiewicz;
    O pewnej mnogsci plaskiej, ktora ma z kazda prosta dwa i tylkp dwa punkty wspolne,
    C. R. Varsovie 7 (1914), pp. 382--384. MR0250248
    French translation: {Sur un ensemble plan qui a avec chaque droit deut at seulement points cummuns}, Stefan Mazurkiewicz, in Traveaux de Topologie et ses Applications (PWN, Warsaw, 1969), pp. 46--47

ISSN 0004-9727