$\def\R{{\mathbb R}}$
May I please receive help with the following problem? It is from Munkrees Topology Textbook.
We know the projection to $X$ or $Y$ of an open subset of $X\times Y$ with the product topology is necessarily an open set.
(ii) Prove $X$ and $Y$ are compact topological spaces, then the projection of a closed set of $X\times Y$ to $Y$ is a closed set.
$\textbf{Solution:}$ If $\mu \subseteq X \times Y$ is closed, it must also be compact because closed sets in compact spaces are compact. Being the image of a compact set, $p(\mu)$ is compact. Because $p(\mu) \subseteq Y$ is Hausdorff, and compact sets of Hausdorff spaces are closed, $p(\mu)$ must be closed.
Best Answer
As others in the comments have said, your example for i. is good. I'd just be careful when you say
This could be construed to mean that because $p_2(A)$ is open, it is not closed. It's possible for sets to be both open and closed, so you may want to re-write it (this is very nitpicky).
Your proof of ii. is good, if a bit unclear. Because you say you're not sure about it, I'll spell it out in more detail with links to relevant theorems.
If $\mu \subseteq X \times Y$ is closed, it must also be compact, because closed sets in compact spaces are compact. Being the image of a compact set, $p(\mu)$ is compact. Because $p(\mu) \subseteq Y$ is hausdorff, and compact sets of hausdorff spaces are closed, $p(\mu)$ must be closed.