Second category E with each \(\Pr oj({\mathbb{R}}^ 2\backslash E^ 2)\) dense (Q1092189)

The following theorem is proved: Theorem. The continuum hypothesis (CH) implies that there exists a set \(E\subseteq {\mathbb{R}}\) such that (i) E has non-empty intersection with every uncountable closed set and (ii) the projection of \(E\times E\) on any line contains no interval. Remark. Property (i) implies that E is both of full outer measure and of second category in every nondegenerate interval. Property (ii) is equivalent to the existence of a dense set of lines in every direction, disjoint from \(E\times E\).