> [!NOTE] > Write $\mathbb{R}^{n+\ell}$ as the [[Direct Product of Groups|direct product]] $\mathbb{R}^n \oplus \mathbb{R}^{\ell}$, that is $\mathbb{R}^{n+\ell}=\left\{(x, y): x \in \mathbb{R}^n, y \in \mathbb{R}^{\ell}\right\}.$ > Denote by $\pi_1$ and $\pi_2$ the two projections of $\mathbb{R}^{n+\ell}$ onto $\mathbb{R}^n$ and $\mathbb{R}^{\ell}$ respectively:$\pi_1(x, y):=x, \quad \pi_2(x, y):=y, \quad x \in \mathbb{R}^n, y \in \mathbb{R}^{\ell} .$ > > Then $\pi_1$ and $\pi_2$ are [[Continuous maps|continuous]]. ###### Proof Proof. Fix $\left(x_0, y_0\right) \in \mathbb{R}^{n+\ell}$ and, given $\varepsilon>0$, choose $\delta=\varepsilon$. Then $ \left|(x, y)-\left(x_0, y_0\right)\right|<\delta \Rightarrow\left|\pi_1(x, y)-\pi_1\left(x_0, y_0\right)\right|=\left|x-x_0\right| \leqslant\left|(x, y)-\left(x_0, y_0\right)\right|<\varepsilon $ that is, $\pi_1$ is continuous. The continuity of $\pi_2$ is proved similarly.