Products of perfectly meagre sets

Abstract

We show that there exists a perfect set D ⊆ 2 ω × 2 ω D \subseteq {2^\omega } \times {2^\omega } such that for every Luzin set in D D both projections of it are perfectly meagre. It follows (under CH) that the product of two perfectly meagre sets need not be perfectly meagre (or even have the Baire property in the restricted sense). This provides an answer to a 55-year-old question of Marczewski.