Let $X$ and $Y$ be topological spaces.

(a) Define what is meant by the product topology on $X \times Y$. Define the projection maps $p: X \times Y \rightarrow X$ and $q: X \times Y \rightarrow Y$ and show they are continuous.

(b) Consider $\Delta=\{(x, x) \mid x \in X\}$ in $X \times X$. Show that $X$ is Hausdorff if and only if $\Delta$ is a closed subset of $X \times X$ in the product topology.