I 'm trying to understand the classical blow up given by $$X=\{(x,[y])\in \mathbb{R}^n \times \mathbb{P}_N / \hspace{0.2cm} \exists \lambda \in \mathbb{R} \hspace{0.3cm} \text{such that} \hspace{0.3cm} x=\lambda y \} $$ and $$\Pi_X: X \longrightarrow \mathbb{R}^n \hspace{0.4cm} \text{the first projection restricted to X}$$
I have proved that $\Pi^{-1}(\{0\})= \{0\}\times \mathbb{P}_N \subset X$ and that $\Pi_{X\setminus (\{0\}\times \mathbb{P}_N)}$ is homeomorphism (in fact diffeomorphism). But I realized I should also proved that $X$ is a submanifold of the manifold $\mathbb{R}^n \times \mathbb{P}_N$.
My question is: how to do it? If $X$ were an open subset it would be easy, just cutting the charts of $\mathbb{R}^n \times \mathbb{P}_N $ with $X$, but $X$ is not open (in fact it is closed). I consider cutting the charts of $\mathbb{R}^n \times \mathbb{P}_N$ with $X$, but if I did it well, the result is not charts for $X$, because the cut sets of the charts are not homeomorphic to open sets in $\mathbb{R}^n$. Then I thought using $\Pi$ for building charts in $X$, but the $\Pi$ is not a homeomorphism if the set $\{0\}\times \mathbb{P}_N$ (which is closed in $X$) is considered in its domain.
Cover $X$ by the open subsets $U_i=\{y_i \neq 0\}\subset X$ and show that $X\cap U_i\subset U_i$ is a submanifold.
Explicitly:
For $i=0$, say, we have the identification $U_0=\mathbb R^{N+1}\times \mathbb R^N$ where in that identification $(y_1,\cdots,y_N)\in \mathbb R^N$ is identified with $[1:y_1\cdots:y_N]\in \mathbb P^N(\mathbb R)$.
In this set-up $ X\cap U_0$ consists of the pairs $((x_0,x_1,\cdots,x_N),(y_1,\cdots,y_N))\in \mathbb R^{N+1}\times \mathbb R^N$ such that $x_i=x_0y_i$ for $i=1,\cdots,N$, so that $X\cap U_0$ consists of all pairs $$((x_0,x_0y_1,\cdots,x_0y_N),(y_1,\cdots,y_N))\in \mathbb R^{N+1}\times \mathbb R^N$$ These pairs clearly form a submanifold of $\mathbb R^{N+1}\times \mathbb R^N$, parametrized by $x_0,y_1,\cdots,y_N$ and isomorphic to $\mathbb R^{N+1}$ .
We have thus proved that $X\cap U_0$ is a submanifold of $U_0=\mathbb R^{N+1}\times \mathbb R^N$ and a similar proof shows that $X\cap U_i$ is a submanifold of $U_i$ for $i=1,\cdots,N$.