Let $X_1$ and $X_2$ be infinite sets and $T_1$ and $T_2$ be the finite-closed topology on $X_1$ and $X_2$, respectively. Show that the product topology, $T$, on $X_2 \times X_2$ is not the finite closed topology.
I know that I have to show that there exist a set $X \backslash F$ such that it's not open in $T$. So: $$(X_1 \times X_2) \backslash (F_1 \times F_2) = (X_1 \backslash F_1) \times X_2 \bigcup X_1 \times (X_2\backslash F_2). $$ But here I'm stuck.