In most references about Grassmann manifold, we usually introduce the following map: suppose that $S$ is a fixed subspace of codimension $k$, and let $\Omega_S \subset Gr(k,\mathbb{K})$ be all subspaces $W$ such that $W \cap S = \{0\}$. $\Omega_S$ is an open set.
Fix some $U \in \Omega_S$. One can show that $W \in \Omega_S$ if and only if there exists some linear map $\beta_W \in Hom(U,S)$ such that $W = \{x + \beta_W x : x \in U \}$. From there, one can easily check that there exists a bijection $\Omega_S \simeq Hom(U,S)$.
I'm interested in show that this bijection is actually an homeomorphism of topological spaces!
So far, I haven't been able to make a lot about this claim. What bugs me is that the topologies at stake are not necessarily intuitive to work with: $Gr(r, \mathbb{K})$ is endowed with the quotient topology, and $Hom(U,S)$ is really just an euclidean space, so might just take some random norm/distance.
Any help greatly appreciated !
EDIT : Look here Lee's page 22 for more informations.
