On Sets that Meet Every Hyperplane in n-Space in at Most n Points

A simple proof that no subset of the plane that meets everyline in precisely two points is an F-set in the plane is presented.It was claimed that this result can be generalized for setsthat meet every line in either one point or two points. No proofof this assertion is known, however. The main results in thispaper form a partial answer to the question of whether the claimis valid. In fact, it is shown that a set that meets every linein the plane in at least one but at most two points must bezero-dimensional, and that if it is -compact then it must bea nowhere dense G-set in the plane. Generalizations for similarsets in higher-dimensional Euclidean spaces are also presented.