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 set $\{(x,y)\in\mathbb R^2\mid xy\geq1\}$ is a closed subset of $\mathbb R^2$.

However it is projected to $\mathbb R-\{0\}$ which is not closed in $\mathbb R$.