Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 1 / a / ii / Solution

For a Sierpiński decomposition of the plane, $A,B$ partition ${\mathbb{R}}^2$, and one has countable vertical sections while the other has countable horizontal sections. With the orientation fixed as follows,for every real $x,y$. Interchanging the coordinate directions and the names $A,B$ gives the equivalent convention. Here “countable” includes finite sets.