= Solution
The <product topology> on $X\times Y$ has as a basis all open rectangles
$$
\boxed{U\times V,\qquad U\in\mathcal T_X,\quad V\in\mathcal T_Y.}
$$
Thus a subset $W\subseteq X\times Y$ is open precisely when, around every $(x,y)\in W$, there are open $U\ni x$ and $V\ni y$ with $U\times V\subseteq W$. These rectangles cover the product, and their pairwise intersections are rectangles because $(U\times V)\cap(U'\times V')=(U\cap U')\times(V\cap V')$. Hence they define a topology by part (b).
This is equivalently the coarsest <topology> making the two coordinate <projection maps> continuous: the preimages of factor-open sets are the open strips $U\times Y$ and $X\times V$, whose finite intersections are the basis rectangles.
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