Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ David L. Fearnley L. Fearnley J.W. Lamoreaux 13 February 2006 2006 2 13 477 480 723-5241-FeFeLa-2005 10.wxyz/C2005V72P3p477 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5241-FeFeLa/ \par We show that, for any positive integers $n$ and $m$, if a set $S \subset R^m$ intersects every $m-1$ dimensional affine hyperplane in $R^m$ in exactly $n$ points, then $S$ is not an $F_{\sigma }$ 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 $n=m$. 54B05, 54H05, 54F45 K. Bouhjar, J.J. Dijkstra and R.D. Mauldin Proc. Amer. Math. Soc. MR1800242 K. Bouhjar, J.J. Dijkstra and R.D. Mauldin; No $n$-point set is $\sigma $-compact, \textit{Proc. Amer. Math. Soc.} \textbf{129} (2001), pp.~621--622. J.J. Dijkstra and J. van Mill Bull. London Math. Soc. MR1887709 J.J. Dijkstra and J. van Mill; On sets that meet every hyperplane in $n$-space in at most $n$ points, \textit{Bull. London Math. Soc.} \textbf{34} (2002), pp.~361--368. D.L. Fearnley, L. Fearnley and J.W. Lamoreaux Proc. Amer. Math. Soc. MR1963773 D.L. Fearnley, L. Fearnley and J.W. Lamoreaux; Every three-point space is zero dimensional, \textit{Proc. Amer. Math. Soc.} \textbf{131} (2003), pp.~2241--2245. D.L. Fearnley, L. Fearnley and J.W. Lamoreaux Topology Appl. MR1955662 D.L. Fearnley, L. Fearnley and J.W. Lamoreaux; On the dimension of $n$-point sets, \textit{Topology Appl.} \textbf{129} (2003), pp.~15--18. J. Kulesza Proc. Amer. Math. Soc. MR1093599 J. Kulesza; A two point set must be zero dimensional, \textit{Proc. Amer. Math. Soc.} \textbf{116} (1992), pp.~551--553. R.D. Mauldin Open Problems in Topology North-Holland R.D. Mauldin; Is there a Borel set M in $R^2$ which meets each straight line in exactly two points?, in \textit{Open Problems in Topology} (North-Holland, Amsterdam, 1990), pp.~619--629. S. Mazurkiewicz (Polish) C. R. Varsovie 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 S. Mazurkiewicz; O pewnej mnogsci plaskiej, ktora ma z kazda prosta dwa i tylkp dwa punkty wspolne, (Polish), \textit{C. R. Varsovie} \textbf{7} (1914), pp.~382--384. 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