$\gamma_k^m = (E,p,G_k(F^m))$ is locally trivial I'm currently studying bundles from Husemaller and I'm stuck with the proof that $\gamma_k^m = (E,p,G_k(F^m))$ at p.$25$.
The thing I don't get is why $U_H$ is open and $h_H$ is an isomorphism. It should be obvious since not many words have been spoken but I don't see clearly the reason since the topology on $G_k(F^m)$ is a little hard to visualize.
Trying to incorporate Hatcher notes p.$28$ Lemma $1.15$ it seeems that Hatcher defines $U_l$ which seems to have a different definition from the one given by Husemaller. Are the two definition the same?
Any help to disambiguate the problem or any solution would be appreciated.
 A: The openess of these sets is not obvious. But you can try to get more intuition by looking at the examples $G_1(\mathbb{R}^2)=\mathbb{RP}^1,\; G_2(\mathbb{R}^3) = \mathbb{RP}^2$, where you can draw a bit. Just visualize a point in $G_1(\mathbb{R}^2)$ as a line in $\mathbb{R}^2$ and now try to guess how an open neighbourhoud of this point looks like. The open sets given by Husemoller are generalisations of the "standard" trivialsiations for projective spaces.
Husemoller defines for each $H \subset \{ 1,\ldots ,m\} $ with $\#H = k$ a $k$ dimensional subspace of $F^m$ which can be written as $\{(x_1,\ldots,x_m) \in F^m \; |\; x_i = 0 \text{ for } i \notin H\}$. In the book its defined by the image of a monomorphism $u_H:F^k \to F^m$. Note that $u_H(F^k) \in G_k(F^m)$.
Also $\pi: G_k(F^m)\times F^m \to F^m$ is defined, such that $\pi(V,x)$ is the orthogonal (there seems to be a scalar product) projection of $x$ into $V$. So for $V \in G_k(F^m)$ we have a map (compare to Hatcher notes) $\pi(V, -) = \pi_V: F^m \to V$. Now the subset $U_H$ in Husemoller is defined to be (there is a typo in the definition in the book)
$$U_H := \left\{V \in G_k(F^m) \; |\; \pi_V\circ u_H:F^k \to V \text{ is bijective } \right\}.$$
In Husemoller's notation, Hatcher defines for each $W \in G_k(F^m)$ the set
$$ U_W := \left \{ V \in G_k(F^m)\; | \; \pi_W(V) \subset W \text{ has dimension }  k\right\}.$$
Since $W$ has dimension $k$, that is saying $\pi_W|_V : V \to W$ is bijective. So you see that $U_H = U_W$, if we take $W = u_H(F^k)$. So Hatcher is describing open neighbourhoods for any point in $G_k(F^m)$ while Husemoller is only doing it for the $\binom{m}{k}$ "standard" points. Now you can use Hatcher's proof for openess. The moral is, that if you have $V \in U_W$ and you "change $V$ a little" the dimension of $\pi_W(V)$ will not change. This can be very nicely drawn for $G_1(\mathbb{R}^2)$.
From these descriptions it shoud be clear that $$h_H: U_H \times F^k \to p^{-1}(U)\\ (V, x) \mapsto (V,\pi(V,u_H(x)))$$ is bijective and a linear isomorphism on the fibres. The proof for continuity is given in Hatcher's notes.
