I would like to show that the real projective space $P^n$ is locally homeomorphic to an open subset of $\mathbb{R}^{n-1}$.
I would just like to sketch the proof before diving into details. Let $[x]$ be a linear subspace in $P^n$, then there exists at least one element $x \in \mathbb{R}^n$ and $x \in [x]$ such that the $i$th component is equals to one.
Take an open ball $B \subset \mathbb{R}^n$ around the point $x$. And if we restrict the quotient map $q: \mathbb{R}^n \times\{0) \rightarrow P^n$ that defines $P^n$ and its topology to $B \cap (\mathbb{R}^{n} \cap (x \in \mathbb{R}^n |x_i = 1))$.
Then we can show that this restriction is a homeomorphism between $B \cap (\mathbb{R}^{n} \cap (x \in \mathbb{R}^n |x_i = 1))$ and its image.
Is this feasible?
I checked and I don't think anyone else used a similar proof for showing $P^n$ is a manifold, so I want to know if my strategy is valid.
thank you