Is product of two closed sets closed?
Solution 1:
$A$ is closed in $X$, so $A^c$ is open, likewise for $B$ in $Y$.
Moreover, $A^c \times Y$ and $X \times B^c$ are both open in $X \times Y$.
Thus $$(A \times B)^c = (A^c \times Y) \cup (X \times B^c) $$ is open. Hence $A \times B$ is closed.
Solution 2:
Let $\pi_i$ denote the projection on $i$-th coordinate
Product topology: $X×B^c = π_2^{-1}(B^c)$ is open in $X×Y$, and $A^c×Y = π_1^{-1}(A^c)$ is open in $X×Y$. And $(A×B)^c = ?$