Proving that the Grassman manifold is indeed a manifold.
I'm trying to follow a proof that every point $X_o$ of a grassman manifold $G_n(\mathbb{R}^{n+k})$ has a neighborhood that is homemorphic to $\mathbb{R}^{nk}$ where $G_n(\mathbb{R}^{n+k})$ is the set of all n-planes through $n+k$ dimensional euclidean space.
I'm trying to follow the proof by Milnor in his classic text "Characteristic classes", page 58, it goes something like this: (italic font will be questions that I have regarding the proof):
Let $U$ be the open subset of $G_n(\mathbb{R}^{n+k})$ consisting of all n-planes $Y$ such that the orthogonal projection $p: X_0 \oplus X_0^t \rightarrow X_0$ maps $Y$ onto $X_0$. (i.e. all $Y$ such that $Y \cap X_0^t = 0$)
($X_0^t$ is the orthogonal complement of $X_0$)
Why is U an open subset of $G_n(\mathbb{R}^{n+k})?^1$
Then each $Y \in U$ can be represented by the graph of a linear transformation:
$T(Y): X_0 \rightarrow X_0^t$.
Can somebody help me understand why this is possible? I know what the graph of a function is. Maybe an explicit example would help.
Thus we have a one to one correspondence:
$T: U \rightarrow Hom(X_0,X_0^t)$.
Okay, so coming up is where I get really confused, it looks like there might be a typo, I'm typing it exactly as is written:
Let $x_1,....,x_n$ be a fixed orthonormal basis for $X_0$. Note that each n-plane $Y \in U$ has a unique basis $y_1,....,y_n$ such that:
$p(y_1)=x_1,.....,p(y_n)=x_n$ (this line is typed exactly as it appears in text, it is confusing me.)
It is easily verified that the n-frame (y_1,....,y_n) depends continuously on $Y$.
Now note the identity $y_i = x_i + T(Y)x_i$.
Since $y_i$ depends continuously on $Y$, it follows that the image $T(Y)x_i \in X_0^t$ depends continuously on $Y$. Why is this exactly?
On the other hand this identity shows that the n-frame $(y_1,...,y_n)$ depends contiously on $T(Y)$, and hence that $Y$ depends continuosuly on $T(Y)$. Thus the function $T^{-1}$ is also continuous. Thus $U$ is locally homeomorphic to $\mathbb{R}^{nk}$
So I start to get a bit confused on what is going on at the end there, with all the "this depends continuously on that" stuff, which seems to be arguements in showing that both $T$ and $T^{-1}$ are continuous. Any comments on this proof, whether they be helping me understand technical aspects or a bigger picture, is much appreciated. Thank you!