There are no n-point Fσ sets in Rm

Abstract

We show that, for any positive integers n and m, if a set S ⊂ Rm intersects every m − 1 dimensional affine hyperplane in Rm in exactly n points, then S is not an Fσ 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.