I have to show that the real projective space $\mathbb{RP}^n$ is an n-manifold. I will use the set label $\mathbb{P}^n$ instead of $\mathbb{RP}^n$. $\mathbb{P}^n$ the set of all the lines passing through the origin in $\mathbb{R}^{n+1}$ and is equipped with the quotient topology induced by the quotient map $q: \mathbb{R^{n+1}}-{0} \to \mathbb{P}^n$. Here q takes each point in $\mathbb{R^{n+1}}-{0} $ to its span.
There are a few solutions available on the net but I have thought of a different way to do this. Please let me know if it is correct and how I can make it mathematically rigorous.
Now, the main part arrives. Consider the $\mathbb{R^2}$ plane and any line passing through the origin. Consider the lines $x=1$ and $y=1$. Any line through origin will cut intersect $x=1$ at one point or $y=1$ at one point. There is just one line through the origin that intersects both. We observe that if I define f: $\mathbb{P^1} \to \mathbb{R^2}$ such that f sends any element of $\mathbb{P^1}$ to the point on $x=1$ or $y=1$ where this element intersects it then f is an injection.
Now restrict the co-domain to just the lines $x=1$, $y=1$ (the wedge sum of these 2 lines and they meet at (1,1)). Then, the restricted f is surjective as well. So, it is a bijection now. Moreover, it seems continuous (However, I am not able to prove this). Even its inverse also seems continuous.
So, this f with its codomain restricted to this wedge sum should be a homeomorphism.
Now, consider again $x=1$ and $y=1$ in $\mathbb{R^2}$ and equip them with subspace topology. While considering the wedge sum of these, equip them with the disjoint union topology. Now, $x=1$ and $y=1$ are Hausdorff so their wedge sum is also Hausdorff. They are second countable as well. Now, for each point in this wedge sum, I can find a neighborhood which is homeomorphic to an open interval in R. So, this is locally euclidean of dimension 1. Hence, this wedge sum will be a 1-manifold. So, $\mathbb{P^1}$ will be a 1-manifold (since it is homeomorphic to this space). Then, I can do this thing in higher dimenision to conclude that $\mathbb{P^n}$ is a n-manifold.
Is this proof correct? If it is, then how do I make it mathematically rigorous?