Let $X, Y$ be topological spaces and $X \times Y$ their product set. Let $p_{Y}: X \times Y \rightarrow Y$ be the projection map.

(i) Define the product topology on $X \times Y$. Prove that if a subset $Z \subset X \times Y$ is open then $p_{Y}(Z)$ is open in $Y$.

(ii) Give an example of $X, Y$ and a closed set $Z \subset X \times Y$ such that $p_{Y}(Z)$ is not closed.

(iii) When $X$ is compact, show that if a subset $Z \subset X \times Y$ is closed then $p_{Y}(Z)$ is closed