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.
The "standard" example: 

Take $(X,d)=(\Bbb R, d\_e)=(Y,d')$ where $d\_e(x,y)=|x-y|$ the usual Euclidean metric. $d\_\infty$ is equivalent to the usual metric on $\Bbb R^2$.

Show that $A=\{(x,y) \in \Bbb R^2: xy=1\}$ is a closed set (the inverse image of $\{1\}$ under a continuous map) and $\pi\_1[A]=\pi\_2[A]= \Bbb R\setminus \{0\}$ which are not closed in the factor spaces. So $\pi\_1$ nor $\pi\_2$ are closed.

If $X$ is compact, $\pi\_2$ is a closed map, and if $Y$ is, so is $\pi\_1$ (projections along a compact space are closed). So for such an example we need non-compact spaces.