Question: I have $(X,d) \text{ and } (Y,d')$ two metric spaces. I define the metric space $(X \times Y, d\_{\infty})$ and the projections $\pi\_1:X \times Y \rightarrow X$, $\pi\_2:X \times Y \rightarrow Y$. I've already shown that these functions are continous and open, but can't find an example for where it's not closed

Answer: 
A classical example is $X=Y=\mathbb R$. The sup metric on $\mathbb R^2$ induces the Euclidean topology.

Let now $C=\{xy=1\}$. It is closed in $\mathbb R^2$ but it's project on the X-axis is $\{x\neq0\}$, which is not closed.
Let $X=Y=\mathbb{R}$ and let $C=\{(x,y)\in (0,\infty)\times (0,\infty)\;|y\geq 1/x \}$

Then, $C$ is closed. Indeed, if $(x\_n,y\_n)\in C$ with limit $(x,y)$, then either $x\neq 0$, in which case $(x,y)\in C$ by continuity of $x\mapsto \frac{1}{x}$, or $x=0$, in which case the $y\_n$ sequence is unbounded and thus, not convergent. We conclude that $C$ is closed.

However, $\pi\_1(C)=(0,\infty),$ which clearly isn't closed.