Exercise 4, § 5

Let $X$ and $Y$ be two topological spaces and let $A$ (resp. $B$) be a subset of $X$ (resp. $Y$).

a) $A \times B$ is nowhere dense (resp. perfect) in $X \times Y$ if and only if one of the two sets $A, B$ is nowhere dense (resp. $A$ and $B$ are closed and one of them is perfect).

b) $A \times B$ is thin in $X \times Y$ if and only if $A$ and $B$ are both thin sets.