My ultimate goal is to show that the real projective space $\mathbb{P}^n_{\mathbb{R}}$ is an $n$-manifold. But first I'd like to understand the topological structure of $\mathbb{P}^n_{\mathbb{R}}$.
The quotient topology on $\mathbb{P}^n_{\mathbb{R}}$ is defined via the quotient map $q \colon \mathbb{R}^{n+1} \backslash \{0\} \to \mathbb{P}^n_{\mathbb{R}}$ sending $x \mapsto [x]$, its linear span. Thus open subsets of $\mathbb{P}^n_{\mathbb{R}}$ are collections of lines through the origin in $\mathbb{R}^{n+1}$ (correct)? Then $\beta := \{U_i\}_{0 \le i \le n}$ is a basis for the topology on $\mathbb{P}^n_{\mathbb{R}}$ (correct?), where $U_i = \{[(x_0, \ldots, x_n)]: x_i = 1 \}$ since each $U_i$ contains every line through the origin for a fixed $i$; so $\mathbb{P}^n_{\mathbb{R}}$ is second countable.
Proving $\mathbb{P}^n_{\mathbb{R}}$ is Hausdorff is where I am having a bit of trouble. I understand that this should be true since distinct points in $\mathbb{P}^n_{\mathbb{R}}$ are just distinct lines in $\mathbb{R}^{n+1}$, but I cannot think of a way to say this precisely. What is an open neighborhood of a point in $\mathbb{P}^n_{\mathbb{R}}$? I think it is the line itself with a "bundle" of the lines contained in a circle around it. But again, I am unsure of how to make this precise. This is my attempt: let $[v]$ and $[w]$ be distinct points in $\mathbb{P}^n_{\mathbb{R}}$. Let $d(v,w) = \delta$ (the Euclidean metric on $\mathbb{R}^{n+1}$), then $U = \{[u]: d(u,v) < \frac{\delta}{2} \}$ and $Z = \{[z] : d(z,w) < \frac{\delta}{2}\}$ are disjoint open neighborhoods of $[v]$ and $[w]$, respectively.
I think I am okay with the locally Euclidean property.
Suggestions/corrections would be appreciated.