The following open rectangles form a <basis of a topology> for the <product topology>:
$$
\boxed{\{U\times V:U\in\mathcal T_X,\ V\in\mathcal T_Y\}.}
$$
These <open rectangles in a finite product> cover $X\times Y$, and intersections remain rectangles since $(U_1\times V_1)\cap(U_2\times V_2)=(U_1\cap U_2)\times(V_1\cap V_2)$. One may equivalently take products of elements from a chosen <basis of a topology> in each factor.

Solution

Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / E / 1 / b / i / Solution

 Solution (source code)