Show that the real projective space $\mathbb{R}P^n$ is an $n$-manifold.
We need to show that $\mathbb{R}P^n$ is second countable, locally Euclidean and Hausdorff.
Second countability simply follows from second countability of $\mathbb{R}^{n+1} \setminus \{0\}$.
To prove the locally Euclidean property, I follow a hint and consider the sets $U_i = \{ (x_0,...,x_n) \in \mathbb{R}^{n+1} : x_i \neq 0 \}$. Then we can construct the maps $$\varphi_i : U_i \to \mathbb{R}^n, (x_0,...,x_n) \mapsto \left(\frac{x_0}{x_i},...,\frac{x_{i-1}}{x_i}, \frac{x_{i+1}}{x_i},...,\frac{x_n}{x_i}\right).$$ I don't know how to proceed now. What can we use these maps for? Are they homeomorphisms?
I also haven't been able to prove the Hausdorff property.
Thank you for every hint.
For the Hausdorff property,
For your first question note that the images of the $U_i$ in the quotient are open sets, and that these form an open cover. Your maps $\varphi_i$ pass to the quotient and induce homeomorphisms onto their images (write down an inverse map!). If you need more details, please tell me so.