Can someone explain why a open set lets say of $\mathbb{R}^{n+l}$ is still open after being with its first n coordinates projected on the $\mathbb{R}^n$? Thank you for your help.
My informal argument would be, that we might find for every point in $\mathbb{R}^n$ several corresponding points in $\mathbb{R}^{n+l}$ but that is not very formal tough.