Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2008 / A / 1 / b

Choose countable dense sets $D_X\subseteq X$ and $D_Y\subseteq Y$. Their Cartesian product is countable. Every nonempty basic open set of the product topology has the form $U\times V$ with $U,V$ nonempty open in their respective topological spaces. Density gives $d_X\in D_X\cap U$ and $d_Y\in D_Y\cap V$, hence $(d_X,d_Y)\in (D_X\times D_Y)\cap (U\times V)$.

Every nonempty open set in the product contains such a basic open set. Thus $D_X\times D_Y$ is dense in $X\times Y$, proving topological separability of the product. If one factor is empty the product is empty and the conclusion holds directly.