I'm dealing with a problem of identifying a quotient space and find out if this map $f$ is closed (or open) would be really helpfull.
Let $f:(z,w)\in S^1\times S^1 \rightarrow f(z,w)=(z^2,w)\in S^1\times S^1$
with
$S^1=\left \{ z \in \mathbb{C} : \left | z \right |=1\right \}$
I guess it's closed because it's a continuous map defined in compact spaces, but how can I show that more detailed?
Every closed subset of a compact metric space is compact. And a continuous map maps compact sets onto compact sets. And, finally, every compact subset of a metric space is closed. So, yes, your map is compact.
Indeed, $f$ is a closed map. Take a closed subset $F\subseteq \mathbb S^1\times\mathbb S^1$. Since $\mathbb S^1\times \mathbb S^1$ is compact, it follows that $F$ is compact. Then, by continuity, $f(F)$ is compact. Finally, since $f(F)$ is compact in a Hausdorff space (because $\mathbb S^1\times\mathbb S^1$ is a Hausdorff space), $f(F)$ is closed.
