We have the following setting:
We let $$ X = Gr_k(\mathbb{R}^n) = \{k\text{-dimensional linear subgroups of $\mathbb{R}^n$}\}$$ To topologize this space, we pick a fixed $V \in Gr_k(\mathbb{R}^n)$. Then any $k$-dimensional subspace can be obtained by applying some linear map to $V$. So we obtain a surjection \begin{align*} GL_n(\mathbb{R}) &\to Gr_k(\mathbb{R}^n)\\ M &\mapsto M(V). \end{align*} So we can given $Gr_k(\mathbb{R}^n)$ the quotient (final) topology. For example, $$ Gr_1(\mathbb{R}^{n + 1}) = \mathbb{RP}^n.$$ Now to construct a vector bundle, we need to assign a vector space to each point in $X$. But a point in $Gr_k(\mathbb{R}^n)$ is a vector space, so we have an obvious definition $$ E = \{(V, v) \in Gr_k(\mathbb{R}^n) \times \mathbb{R}^n: v \in V\}. $$ This has the evident projection $\pi: E \to X$ given by the first projection. We then have $ E_V = V.$ To see that this is a vector bundle, we have to check local triviality. We fix a $V \in Gr_k(\mathbb{R}^n)$, and let $$ U = \{W \in Gr_k(\mathbb{R}^n): W \cap V^\perp = \{0\}\}. $$ We now construct a map $\varphi: E|_U \to U \times V \cong U \times \mathbb{R}^k$ by mapping $(W, w)$ to $(W, pr_V(w))$, where $pr_V: \mathbb{R}^n \to V$ is the orthogonal projection.
Now if $W \in U$, then $pr_V(w) \not= 0$ since $W \cap V^\perp = \{0\}$. So $\varphi$ is a homeomorphism. We call this bundle $\gamma_{k, n}^\mathbb{R} \to Gr_k(\mathbb{R}^n)$.
I have some questions about that:
1) Could you please give me some intuition about how the topology we define "works"? I can understand the formal definition, but I cannot see what exactly we are doing.
2) Why is it $Gr_1(\mathbb{R}^{n + 1}) = \mathbb{RP}^n$? I assume that this might be a direct consequence of 1).
3)Why $U$ is an open neighborhood of V? I can see that $V\in U$ and I have an idea that goes like that: $W\in U$ means that for every $w\in W$ we have that $pr_V(w)\neq 0$. This holds for all the elements of $V$, and if we remain in a small neighborhood of $V$ then this will still hold for all elements of the spaces in this neighborhood, due to continuity. However, I cannot make that rigorous.
4) Why is $\phi$ a homeomorphism? I can see that it is bijection and continuous but not that its inverse is also continuous.
Thank you in advance
ps1: I used the source code of Dexter from here for the setting
ps2: I don't know if I should have split this post to several others. Please, let me know.