Let $X$ and $Y$ be topological spaces. Define the product topology on $X \times Y$ and show that if $X$ and $Y$ are Hausdorff then so is $X \times Y$.

Show that the following statements are equivalent.

(i) $X$ is a Hausdorff space.

(ii) The diagonal $\Delta=\{(x, x): x \in X\}$ is a closed subset of $X \times X$, in the product topology.

(iii) For any topological space $Y$ and any continuous maps $f, g: Y \rightarrow X$, the set $\{y \in Y: f(y)=g(y)\}$ is closed in $Y$.